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Helping to close the gap by providing the evidence base to inform practice and policy in Aboriginal and Torres Strait Islander health
The Alcohol and Other Drugs Knowledge Centre acknowledges the Traditional Owners of the lands and waters of Australia and the Torres Strait.
We respect all Aboriginal and Torres Strait Islander people—their customs and their beliefs. We also pay our respects to Elders past and present, with particular acknowledgement to the Whadjuk people of the Nyoongar nation, the traditional owners of the lands where our offices are located.
This product, excluding the Australian Indigenous HealthInfoNet logo, artwork, and any material owned by a third party or protected by a trademark,
has been released under a Creative Commons BY-NC-ND 3.0 (CC BY-NC-ND 3.0) licence.
Excluded material owned by third parties may include, for example, design and layout, images obtained under licence from third parties and signatures.
Aboriginal and Torres Strait Islander people should be aware that this website may contain images, voices and names of people who have passed away. | null | minipile | NaturalLanguage | mit | null |
An EPA spokesman told the New York Times the agency had decided to abide by the White House’s edict (PDF) that the drop dead date for publishing new rules in the Federal Register was Nov. 1.
“We didn’t want to be faced with putting a midnight regulation in place,” the spokesman, Jonathan Shradar, told the Times. “It was better to leave those incomplete rather than force something through.”
It remains to be seen what the EPA will do about the other rules the agency has been pushing through—rules that, if they remain in place, will also be published well after the deadline.
We’d like to know why EPA chose to abandon these two rules and not the others still pending. We’re placing a call to the agency to find out.
Safeguard the public interest
Republish This Story for Free
Thank you for your interest in republishing the story. You are are free republish it so long as you do the following:
You can’t edit our material, except to reflect relative changes in time, location and editorial style. (For example, "yesterday" can be changed to "last week," and "Portland, Ore." to "Portland" or "here.")
If you’re republishing online, you have link to us and to include all of the links from our story, as well as our PixelPing tag.
You can’t sell our material separately.
It’s okay to put our stories on pages with ads, but not ads specifically sold against our stories.
You can’t republish our material wholesale, or automatically; you need to select stories to be republished individually. | null | minipile | NaturalLanguage | mit | null |
Alcoholic calories, red wine consumption and breast cancer among premenopausal women.
The role of alcohol consumption (alcoholic calories, alcoholic beverages) on breast cancer risk was investigated in a case-control study of 154 premenopausal female patients diagnosed with primary breast carcinoma. For each case, one control was matched for age (+/- 3 years) and socio-economic status. The survey was carried out in Northeastern France (Lorraine) between 1986 and 1989. While taking into account total caloric intakes and various breast cancer factors, breast cancer risk was shown to increase as consumption of alcohol increased (p value for trend = 0.007). A significant relative risk (RR = 2.69; 95% CI: 1.40-5.17) was shown above 60 kcal per day (approximately 9 g of alcohol per day). Breast cancer risk appeared to be restricted to red wine consumption among these premenopausal women, for monthly consumption (p value for trend = 0.003) as well as for duration of consumption (p value for trend = 0.01). A relative risk of 3.96 (95% CI: 1.59-9.84) was found for a monthly consumption higher than 4 liters per month. This reinforces the notion of a particular sensitivity of young women to breast cancer in relation to alcohol consumption. | null | minipile | NaturalLanguage | mit | null |
Viewpoint
MCH hopes to be a part of future state network
The Affordable Care Act, passed in March of 2010, aims to extend access to health insurance coverage to about 32 million uninsured Americans by expanding both private and public insurance. Legislators all agree it is a very complex law involving both federal and state agencies. With few exceptions, it requires individuals to have insurance. To help that occur, the law mandates the creation of state-based (or multi-state) insurance exchanges, to help individuals without insurance to purchase plans through the exchange. The State Health Benefit Exchange goes into effect Jan., 1, 2014. In addition, small businesses with fewer than 50 employees will be given the option to purchase similar plans through networks referred to as “Shop Exchange” administered through the federal government.
Here in New Hampshire, Anthem/Blue Cross, Blue Shield has applied and been identified as the insurance provider charged with creating and managing the limited provider network created for the ACA. Consequently, Anthem was given the responsibility and resources to control the network negotiations. Those organizations not selected (including MCH) were only notified after decisions had already been made. Unfortunately, we were not given the opportunity to negotiate our participation. We believe decisions were made primarily on the basis of geographic coverage, not other factors such as quality or patient satisfaction
Those not affected by the legislation in the upcoming year include the following groups: employees of companies with 50 or more individuals, those in small businesses who do not elect to purchase coverage for their employees through the Shop Exchange, those who have Medicaid or Medicare or other private coverage and those in the Children’s Health Insurance Program.
Again, this process only impacts individuals who purchase their insurance independently through Anthem Blue Cross Blue Shield’s new “Pathway Network.” The decision on Anthem’s part does not affect the majority of patients utilizing MCH’s services and participants in the exchange will still be able to utilize MCH for emergency care
What does this mean for those of us in the Monadnock region? One thing is certain, increasingly we all need to be better educated health consumers – and more conscientious stewards of our own health. Here at MCH, we have taken a very active stance in promoting health, improving quality and patient satisfaction, and reducing costs. These efforts have has earned us statewide and national attention with industry-leading results. In addition, we are grateful to have earned national recognition for quality and patient satisfaction measures. All of this would not be possible without the tremendous support of our dedicated employees, our physicians and you, our community. Looking forward, we are hopeful in 2015, the 2nd year of the contract that Anthem will allow us to be included in the discussion of participation options for the future. We also anticipate there will be other insurance payers in NH who will consider offering exchange coverage in 2015 as well. | null | minipile | NaturalLanguage | mit | null |
From massive registration fraud, to disenfranchising military votes, to backing spoiler candidates and more, Democrats are pulling every dirty trick they can think of to hold on to as many seats as possible. And these are just the stories we know about. Vote fraud is inherently difficult to catch, so if we are hearing about this many cases already, you can bet there is a great deal more fraud going undiscovered.
This is why we all need to volunteer to help get out the Republican vote. It’s very simple: we need to get SO many Republicans out to vote, that there simply aren’t any close elections. It’s the close elections that Democrats try to steal. Just ask George Bush, Dino Rossi or Norm Coleman. We can’t just win, we have to win by more than the margin of fraud.
You can volunteer to help your Republican candidate, or (if your local race isn’t competitive) OTHER vulnerable Republican candidates by going here, or by calling your local GOP. Volunteering is extremely easy. They give you a script, and tell you everything you need to do. Most campaigns are looking for people to call Republican voters and remind them to get their ballots in, go door to door at houses they have pegged as likely Republican leaners to ask if they plan to vote, and they are looking for people to poll watch on election day to prevent/record/and object to Democrat cheating.
A HUGE hat tip to Michelle Malkin, Gateway Pundit, and Hot Air for finding the vast majority of these stories. I just wanted to put them all together in one place. I will try to update this post as new stories come in. Be sure to send us tips if you see stories that aren’t reported here.
Illegal aliens canvassing across state for Patty Murray and Democrats . (ACORN was previously convicted in Washington for committing the worst case of registration fraud in state history. And heavily Democrat King County has been a regular source of corrupt shenanigans from disenfranchising military ballots, to disenfranchising voters in Republican leaning districts, to “finding” hundreds of previously uncounted and in some cases unsecured ballots to steal the gubernatorial race from Dino Rossi after he won the first two counts.)
Barney Frank, that most dishonest Democrat, may, the good Lord willing, lose his job this November. His district voted for Scott Brown in that special election, and it’s looking increasingly likely that they will give Barney the boot as well.
Upon exiting the most recent debate with Barney Frank, located at WGBH studios in Boston, MA, Republican Congressional candidate, Sean Bielat, gets heckled by a Barney Frank “supporter” while talking to the media. While watching this video, we realized that we recognized this “supporter”. We received confirmation from two eyewitnesses that the mysterious cameraman was none other than Barney Frank’s pot-growing boyfriend, James Ready.
I can’t think of a race that better exemplifies what this election is all about — defeating corrupt, ruling-class socialists and replacing them with principled, honest, good men and women. You can help Bielat defeat one of America’s most loathsome politicians here. Go Sean!
Electoral fraud? Let me count the ways. How about the “grubby little racket” of Obama’s online credit fraud. There are default security checks on the computers that accept donations that prevent basic fraud like fake addresses, names that don’t match. Apparently all those checks have been intentionally disabled. Donations from Mr. A Hitler of Berlin Germany, names such as Es Esh or Doodad Pro.
Or there are the scores of small online donations Mary T. Biskup discovered that had been made to the Obama campaign in her name even though she had not given him a cent. They added up to $174,800, obviously more than a little over the $2,300 legal limit.
Or the Obama campaign that claims “no ties” to ACORN; but a former staffertestified todaythat ACORN was provided a “donor list” from the presidential campaign. Obama contributors who had “maxed out” under federal limits could be targeted to give to ACORN’s Project Vote.
No home? There are ways around the rules. ACORN and the ACLU are anxious to be sure that the homeless can vote. Apparently all you need is a park bench and a cross street.
We are a little sensitive to this kind of thing here in Washington State. Our election four years ago put a Democrat governor into office through a highly questionable election that involved ACORN as well as improper actions by election officials.
American Elephant Adds: I happen to think the very reason we are seeing such widespread voter-registartion fraud by ACORN in all the battleground states this year is because it worked so well in Washington in 2004. Fake registrations is the first step, miraculously “finding” enough ballots on election day to overcome any lead the opponent may have is the second. Remember, ACORN was convicted here in WA, and the ballots that put Christine Gregoire in the governor’s office were unsecured, in violation of election law. Look for ballots to be “found” on election day if there is any crucial state where Obama is behind.
Argentine President Cristina Kirchner announced this week, the Wall Street Journal reports, that her government plans to nationalizethe country’s private pension system. If the Argentine Congress approves of this property confiscation, $30 billion in individually held retirement accounts(think 401(k)s) managed by private pension funds will become government property. This must strike Americans as appalling. That the state cold seize the private savings of their citizens is outrageous, but it is because of the “crisis” of course. Couldn’t happen here. We are Americans. If you have a 401(k), you have probably watched in dismay as your nest egg has grown smaller because of the current turmoil in the markets. Democrats are now talking about eliminating the tax deductabilityof 401(k)s. Workforce Management reported on a hearing of the House Education and Labor Committee earlier this month:
A plan by Teresa Ghilarducci, professor of economic-policy analysis at the New School for Social Research in New York, contains elements that are being considered… Under Ghilarducci’s plan, all workers would receive a $600 annual inflation-adjusted subsidy from the U.S. government but would be requiredto invest 5 percent of their pay into a guaranteed retirement account administered by the Social Security Administration. The money in turn would be invested in special government bonds that would pay 3 percent a year, adjusted for inflation. The current system of providing tax breaks on 401(k) contributions and earnings would be eliminated.
James Taranto goes on to explain: Ghilarducci outlined her plan last year in a paper for the left-liberal Economic Policy Institute, in which she acknowledges that her plan would amount to atax increaseon workers making more than $75,000 — considerably less than the $250,000 Barack Obama has said would be his tax-hike cutoff. In addition, workers would be able to pass on only half of their account balances to their heirs; presumably the government would seize the remaining half. (Under current law, 401(k) balances are fully heritable, although they are subject to the income tax.
Well, yes, apparently it could happen here. It is hardly certain that a President Obama or a Democrat Congress would be receptive to such a proposal, but I wouldn’t want to bet on it. They are very serious about “spreading the wealth”. Which only becomes troubling if it is your hard-earned wealth that they are spreading. (emphasis mine)
ADDENDUM: Democrats in Congress might want to take a hard look at what has happened in Argentina since the announcement. Its stock market lost 23% of its value in two days, for a 57% loss since January. The losses spread to other markets in Brazil, South Africa and Spain. Investors Business Daily goes on to explain:
Markets don’t like expropriation of private property — including savings. And this takes away a key source of private capital. Moreover, one quarter of private pension assets were by law invested in Argentine stocks, making up about a quarter of the bourse’s value. So the seizure of pensions amounts to government ownership across the entire private sector. “It’s a stealth nationalization of every single business in the country,” explained Diana Mondino, an Argentinian economist at Universidad del CEMA in Buenos Aires. “Will (the government) influence those companies? I would think so — anyone who owns 25% of a company will have a lot to say about how it’s run.” Growth will suffer, and Moody’s already warns it “undermines the government’s already weak policy credibility.” Nationalization may pay the bills now, but it poisons prospects for growth. For that reason, Argentina’s sovereign bonds now trade at 25 cents to the dollar and yield 30%.
If the next President and the next Congress see private corporations as the class enemy, and Congress continues to see private assets as a public piggy bank, we’re in for trouble.
The New York Times has a long article from the Obama campaign in today’s paper on Sarah Palin. They sent some 30 attorneys with pockets full of cash off to Alaska to dig up dirt. The Obama campaign must be really, really scared of her. Surely most Alaskans must have some nasty things to say.
A new poll from American Viewpoint (not commissioned by the McCain campaign) asks about the job approval for Governor Palin. Overall: 86%. Among Independents: 86%. Among Democrats: 75%.
Obama continues his thuggish campaign to keep Americans from learning about his relationship to unrepentant domestic terrorist, William Ayers:
DENVER — Sen. Barack Obama’s campaign organized its supporters Wednesday night to confront Tribune-owned WGN-AM in Chicago for having a critic of the Illinois Democrat on its air.
“WGN radio is giving right-wing hatchet man Stanley Kurtz a forum to air his baseless, fear-mongering terrorist smears,” Obama’s campaign wrote in an e-mail to supporters. “He’s currently scheduled to spend a solid two-hour block from 9:00 to 11:00 p.m. pushing lies, distortions, and manipulations about Barack and University of Illinois professor William Ayers.”
Kurtz, a conservative writer, recently wrote an article for the National Review that looked at Obama’s ties to Ayers, a former 1960s radical.
The magazine had been blocked in its initial attempts to obtain records from the University of Illinois at Chicago regarding the Chicago Annenberg Challenge, which Obama chaired and Ayers co-founded. The school later reserved [sic] its position and made the records available Tuesday.
…Christenson said the Obama campaign was asked to have someone appear on the show and declined the request. [emphasis mine. read more]
They can’t refute the charges because they are all true. Obama tries to obfuscate by saying that Ayers committed his crimes when Obama was only 11 years old. True, but that’s really beside the point because Ayers is unapologetic. He was unapologetic when he kicked off Obama’s campaign in his home: he is unrepentant even to this day.
The fact is, that in Barack Obama’s short political career he’s had a disturbing number of relationships with radicals, extremists and even terrorists.
And because he can’t refute the relationships, Obama is doing everything in his power, including some very disturbing tactics, to keep Americans from hearing about them before election day.
Now Democrats don’t want you to see them hob-knobbing with big-money donors, special interests and lobbyists:
DENVER–Police in Denver arrested an ABC News producer today as he and a camera crew were attempting to take pictures on a public sidewalk of Democratic Senators and VIP donors leaving a private meeting at the Brown Palace Hotel. [more]
It’s no wonder they dont want you to see, these are the same big-money donors, special interests, and lobbyists that Obama has promised he will have nothing to do with. | null | minipile | NaturalLanguage | mit | null |
Q:
Sitecore: Turning on HTML Caching blocks postback behavior
I have a sitecore page with an ASP dropdownlist, and the data on the form is populated from the selected value of the dropdown. When the selected item of the dropdownlist is changed, a postback is fired. In the postback, the new selected item is added to the querystring and the user is redirected (for linkability).
I recently enabled HTML caching (for all sublayouts, "Vary by querystring"), and now suddenly, this mechanism no longer works. What seems to happen is I select a new dropdown item, the page appears to post back (though if I'm debugging, none of my breakpoints get hit). After that, if I change the selected item again, I can see in Firebug the message "__doPostBack is not defined", which appears to mean the ASP-generated JavaScript is not being added to the page.
A:
Enabling caching for a sublayout means you are bypassing the code entirely and Sitecore is just serving up the same HTML it generated previously. So it's behaving as designed. In other words, this does not seem to be a scenario where you can take advantage of sublayout caching.
| null | minipile | NaturalLanguage | mit | null |
Molmen and his family have had a personal connection to Liberia since their ten month long mission trip to the country in 2006-07. While in Liberia, Molmen and his wife taught at Cuttington University, where they met and adopted Dolo, a nursing student at the time. Fast forward and Dolo is now leading a movement in Liberia that provides families with health information about the Ebola virus to help eliminate common myths and misconceptions.
The Community Action Against Ebola team was created to educate the public and spread awareness. In addition to the information campaign, the Molmens are collecting donations to support Dolo and his team as they provide food to Ebola victims and basic necessities for survivors whose possessions have been burned following their diagnoses. | null | minipile | NaturalLanguage | mit | null |
Your Best Shot
Truth Commission on Poverty
Published 12:09 am, Tuesday, July 18, 2017
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The Labor-Religion of Coalition of New York State convened a Truth Commission on Poverty in New York on July 13 at First United Methodist Church in Schenectady. At the public meeting, held in partnership with the national Poor People's Campaign, people told their stories about the effect of poverty and other injustices on their lives. This was the second meeting. A third will be on Long Island in September followed by a statewide gathering in Binghamton in October. | null | minipile | NaturalLanguage | mit | null |
The New Covenant
In everyday use a covenant describes an agreement between parties where there are benefits and responsibilities for the parties and penalties for non-compliance. It was the practice when Scripture was being written to sacrifice an animal and divide the carcase into pieces. The parties contracting the covenant would then pass between the pieces to confirm the covenant and in acknowledgement that their lives would be forfeit if they failed to keep their part of the agreement. In Genesis 15:18, where the promise of the land is confirmed to Abraham, it says: “In the same day the LORD made a covenant with Abram, saying, Unto thy seed have I given this land, from the river of Egypt unto the great river, the river Euphrates”. The word “made” should be translated ‘cut’. It is clear when this chapter is carefully read that Abraham passed between the pieces of several sacrifices (vv. 10,11) and God also did (v. 17; the burning lamp must have been the Spirit of God). Another example of this practice is found in Jeremiah 34:18,19, and in the New Testament the writer to the Hebrews also seems to refer to this procedure in 9:15-17, where he argues that Jesus is the covenant sacrifice, the mediator who died to confirm the new testament, that is, covenant.
Covenants of promise
Covenants necessarily involve the two parties making solemn promises to each other. Scrip-ture often links the ideas of covenants and promises, for example, Psalm 105:8,9 and Luke 1:72,73. Paul in Ephesians writes of ”the covenants of promise” to which Gentiles can now be related (2:12,13). Whilst God will never renege on his side of a covenant, men have always failed to live up to their side, Jesus being the one exception. It is a wonder that the great God of heaven will stoop to make covenants with erring men, and many men of faith have acknowledged this in their prayers and teaching, addressing the Almighty as “God, Which keepeth covenant and mercy” (Deut. 7:9; cf. Neh. 1:5; Dan. 9:4).
Examples of covenants
The earliest covenant was made in Eden, and sacrifice was associated with it (Gen. 3:15,21). It is not called a covenant here, but God called it so when he saved Noah from the Flood: “But with thee will I establish My covenant” (6:18). Later on God gave Noah (and us) the rainbow as a token of His everlasting covenant with the earth and with all flesh, never again to bring a flood to destroy all life (9:8-17).
When Abraham was given the promise of a seed he was told that this was an everlasting covenant between God and his seed. The sign of this covenant would be circumcision, and every living descendant of Abraham through his son Isaac is a reminder to us that God’s purpose with Israel will never fail (17:9-14,21).
Under Moses the nation of Israel entered into a covenant with God. This is called the old covenant, and it was confirmed, like the other covenants, by sacrifice. Moses sprinkled the blood of the offerings on the people, and they said: “All that the LORD hath said will we do” (Ex. 19:5; 24:4-8). But they soon broke the covenant.
The new covenant
When Jesus introduced the breaking of bread in the upper room in Jerusalem, shortly before his death, before he passed the cup of wine round he said: “this is my blood of the new testament [covenant], which is shed for many for the remission of sins” (Mt. 26:28). This Covenant is different from the old (Mosaic) covenant because under it God promises forgive-ness of sins. The old covenant was constantly broken by Israel, but, because of the excellence of Christ’s offering, under the new covenant God can forgive, and will be able to fulfil all His covenants of promise (Rom. 8:1-4). References such as 2 Corinthians 4:14,15 and Galatians 2:20 show that the love of Christ in offering himself acts as a powerful example in the outworking of the new covenant on the hearts of believers. It is this which is called to mind in the breaking of bread service.
The covenant promises to Abraham and David confirmed by the death and resurrection of Jesus
Jesus is introduced to us in the New Testament as “the son of David, the son of Abraham” (Mt. 1:1). Two of the great covenants of Scripture were given to these men (Gen. 17:7; 2 Sam. 23:5). Paul says that Jesus came “to confirm the promises made unto the fathers [of Israel]”, and that all the promises of God in him are yea and Amen (Rom. 15:8; 2 Cor. 1:20). So the offering of Jesus makes possible the fulfilment of the great and precious promises. This is seen in Genesis 15:4-21. Abraham was not lacking in faith, but wanted to know by what means he, a mortal and sinful man, might inherit the land of Canaan for an everlasting possession. God’s answer to him was a prophetic vision of the sacrifice of Christ (see above). The apostle calls Jesus “the mediator of the new testament [covenant]” (Heb. 9:15). Zacharias, when his tongue was loosed, saw the Lord Jesus as the one who would fulfil the covenants of promise to Abraham and David (Lk. 1:68-73).
The new covenant is to be fulfilled by the nation of Israel
When we read Jeremiah 31:31-34, and the apostle’s quotation of it in its entirety in Hebrews 8, we realise that the primary purpose of the new covenant is the cleansing and adoption of Israel as the people of God in the Kingdom. The apostle refers clearly to the house of Israel and the house of Judah, and the context is the people who rejected the old covenant now accepting the new. Other prophets teach the same. Isaiah 59:20,21 (quoted by the Apostle Paul in Romans 11:26,27) tells us that the redeemer (Christ) will come to Zion and turn away ungodliness’ from Jacob to fulfil the covenant. Ezekiel speaks of the everlasting covenant that God will make with His nation, and of their cleansing (37:23-26). Jeremiah tells us that the covenant of night and day is a guarantee to us that God will keep His promises to David (32:40; 33:20- 22). Zechariah 12 and 13 describes how Jews in the land will be cleansed, and Ezekiel 20:33- 44 tells how the Diaspora Jews will be cleansed by the work of the new covenant.
Some features of the new covenant
It is primarily made with the nation of Israel (Jer. 31:31)
It offers forgiveness of sins to many (Mt. 26:28)
Jesus is its mediator (Heb. 9:15)
Its work of forgiveness covers those under the old covenant and before (9:15)
It is everlasting (13:20)
It confirms the promises made in Eden, to Abraham and to David (Rom. 15:8)
It preceded the old covenant by 430 years, but was not ratified until Christ came (Gal. 3:17)
Today, believers enter the new covenant by baptism, which associates them with the sacrifice of Christ (Rom. 6:3-6; Gal. 3:27-29; Ps. 50:5)
Those in covenant relationship with God should have God’s laws written in their hearts and minds, and walk in faith to the Kingdom (Heb. 8:10; Jas. 2:20-22). | null | minipile | NaturalLanguage | mit | null |
Q:
kubectl auth can-i says I can, but I can’t
I am baffled. kubectl says I can (via the kubectl auth can-i subcommand), but then when I go to perform the action, I can't.
I have installed kubectl on a docker image which is running on a pod managed by a Deployment. When I kubectl exec -it into that pod (which only has a single container) I get this.
user@my-pod:~$ kubectl auth can-i get secrets -n myNamespace
yes
user@my-pod:~$ kubectl get secrets -n myNamespace
Error from server (Forbidden):
secrets is forbidden:
User "system:serviceaccount:myNamespace:myServiceAccount"
cannot list resource "secrets" in API group "" in the namespace "myNamespace"
Here is how my serviceaccount is configured
---
apiVersion: v1
kind: ServiceAccount
metadata:
name: myServiceAccount
namespace: myNamespace
---
kind: Role
apiVersion: rbac.authorization.k8s.io/v1
metadata:
name: myRole
namespace: myNamespace
rules:
- apiGroups: [""]
resources: ["secrets"]
verbs: ["get", "describe"]
---
apiVersion: rbac.authorization.k8s.io/v1
kind: RoleBinding
metadata:
name: myRoleBinding
namespace: myNamespace
roleRef:
apiGroup: rbac.authorization.k8s.io
kind: Role
name: myRole
subjects:
- kind: ServiceAccount
name: myServiceAccount
namespace: myNamespace
I am first of all curious to know if I am using kubectl auth can-i incorrectly.
Secondly, I would like to be able to authorize this serviceaccount to make this API call. Is there a misconfiguration in my yaml?
A:
What's going on here is the unfortunate collision between kubectl using get in two different ways, but can-i uses it only in one way. The list of supported verbs for can-i shows up on its reference page
Running:
kubectl auth can-i get secrets -n myNamespace
asks about the get verb specifically. That is the equivalent of kubectl get secret my-awesome-secret. If you want to know about kubectl get secret, that is using the list verb, and thus would be tested via:
kubectl auth can-i list secrets -n myNamespace
The distinction is called out in this table: https://kubernetes.io/docs/reference/access-authn-authz/authorization/#determine-the-request-verb
I believe the fix for your Role is to update verbs: to also include "list" if you want to enumerate the Secrets
rules:
- apiGroups: [""]
resources: ["secrets"]
verbs: ["get", "describe", "list"]
| null | minipile | NaturalLanguage | mit | null |
Hey guys - with the draft coming up and the floods of new members that start patroling these parts around this time of year, GRR and I thought that restarting this thread might get people to look at it again. It's a nice way to get a little bit more personal with the other members of the forum. Nobody should feel obligated to contribute to the thread, but if you'd like to let your co-posters a little bit about yourself this thread is a good outlet for that. Feel free to write as much or as little as you'd like.
I'll start - my real name's Joe. I'm 27 and I've lived in the west suburbs most of my life with a 7-year stint in AZ in the middle centered around my college years at ASU. I've been a Bears fan my whole life and am also a HUGE Blackhawks and White Sox fan (not a Cubs hater - just prefer the Sox). Ironically, Steve Young, Mario Lemieux and Tony Gwynn are my favorite all-time football/hockey/baseball players and none of them ever played for my teams. I've been married over 4 years now and we have a son who is 10 months old and already runs the house lol. In the little free time I get these days I like to play softball, Xbox (getting a ps3 soon too), poker and, of course, come on FF!_________________
My real name is Tim and I will be 18 in June. I have lived in Darien my whole life and next year I plan on attending the University of Iowa pursuing a degree in journalism in hopes of one day becoming a sports writer for Chicago (maybe one day you guys will get the chance to scrutinize every word in my articles and/or blogs). I currently work as a Jimmy Johns employee and have been there for almost two years, it is easy work. I am a die hard fan of the Bears (obviously), Blackhawks, and Cubs. Much like AZ, I only hate the Sox for two weekends in June. I would love to say I'm a die hard Bulls fan, but it is just hard to really get into the NBA nowadays.
I don't know what else to say so I will just wrap it up at that._________________👮
Adopt-a-Bear: Martellus Bennett, TE
Rec: 77 Yds: 821 TD: 6
AZBearsFan wrote:
He's a playmaker though, and we can use more of those in the Devin.Fart
My name is Byron. I'm From Sacramento, California I moved to Louisville, Kentucky to play baseball in college and i'm still here. I work as a registered nurse at the hospital. I'm 32 yrs old I've been a bears fan since 84. I like to watch old bears game like yesterday green bay and chicago 85. I don't remember alot of those games and as a matter of fact I only remember our losses in the playoffs to washington ( I think 87) and 49ers 88 nfc championship game. Alot of my memories of the bears are bad offenses and tough D. I liked players like head hunting Mark Carrier, Neal Anderson, Jim Harbaugh, Tom Waddle and Wilber Marshall. I don't know why I choose the bears as my team, I hate their colors, their philosophy towards offense Rashaan Salam, Anthoney Thomas and their choices in coaching. Growing up in Cali I seen alot of 49er games and raider games but for some reason the local channels showed alot of bears games. My favorite teams are the oakland A's, Sacramento Kings and of coarse the bears. I also love to gamble on 2k10 baseball if anybody is interested I'm not good yet but I plan on getting that million dollars for the perfect game.
My real name is Matt (stupid play on words for my SN, oh well) I am 24 and currently reside in Streator, IL. I spent the last few years in Dekalb,IL studying @ NIU and after receiving my bachelors in English I had to move back home and enjoy the struggle to find a career.
I've loved the Bears for as long as I remember, my Dad sort of instilled my love for the Bears at a young age and I've been hooked ever since. I consider myself a Chicago sports nut. I love the Bulls and watched them when they were absolutely terrible. The Blackhawks and hockey are becoming a new favorite of mine, and disappointingly I love the Chicago Cubs as much as I do the Bears and it's heartbreak over and over again, but I love them.
In my spare time I enjoy reading sports novels, playing Xbox 360, and basically reading any and everything about sports online, and I absolutely LOVE the NFL draft.
My name is Mark and I live in Austin, Tx. Grew up in small town Iowa, went to the University of Iowa and ended up in Texas because of work. Been married 21 years. Am a die hard Bears, Twins and Celtics fan. I know, teams all over the place. Glad to be a part of the discussion._________________The pioneers take the arrows!!!
My name is Byron. I'm From Sacramento, California I moved to Louisville, Kentucky to play baseball in college and i'm still here. I work as a registered nurse at the hospital. I'm 32 yrs old I've been a bears fan since 84. I like to watch old bears game like yesterday green bay and chicago 85. I don't remember alot of those games and as a matter of fact I only remember our losses in the playoffs to washington ( I think 87) and 49ers 88 nfc championship game. Alot of my memories of the bears are bad offenses and tough D. I liked players like head hunting Mark Carrier, Neal Anderson, Jim Harbaugh, Tom Waddle and Wilber Marshall. I don't know why I choose the bears as my team, I hate their colors, their philosophy towards offense Rashaan Salam, Anthoney Thomas and their choices in coaching. Growing up in Cali I seen alot of 49er games and raider games but for some reason the local channels showed alot of bears games. My favorite teams are the oakland A's, Sacramento Kings and of coarse the bears. I also love to gamble on 2k10 baseball if anybody is interested I'm not good yet but I plan on getting that million dollars for the perfect game.
Growing up in Northern CA in the 80's and you're NOT a 49ers fan? Wow. Nevertheless, Bears nation is glad to have you!_________________
My name is Mike, am 22 from just outside Glasgow in Scotland and have lived here all my life, been a Bears fan since 2002 when I was watching TV late one night and seen a then young Brian Urlacher smashing through an OG and stuffing a RB for a loss, I was just amazed at the size and speed of the guy, then learned about the history and tradition of the Bears and stuck with them as my team, have been trying to get into other American sports, I have always liked basketball and am a Bulls fan, starting to understand ice hockey and I like the Blackhawks...hate Baseball lol. Iím a human resource manager for my uncles IT firm...must admit I actually enjoy my job unlike most lol. In my spare time I play rugby and just chilli out with my girl.
Mike, if you can figure out how to type with an accent, please do so. While I have no Scottish blood in me, I Love the Scottish dialect, but I also love the bagpipes, so there ya have it.
My real name is Jeff. I've been a Bear fan since the late 1960's (Butkus era). I grew up in a small town in Central Illinois, Served 10 years in the US Army and NG as a Helicopter Crew Chief, spent 17 years living in Springfield, before my wife and I bought some timber near the small town of Shelbyville. I've spent the better part of the last two years selectively clearing timber and performing "part" of the construction of our house, so I haven't been on here much in the last 2 years until recently (we finally finished). My wife and I run our own business out of our house after working most of our lives for others. I'm 46 with 2 GREAT teenage kids (boy and Girl). My wife of 20 years and I both went to U of I, although mine was at the Springfield campus.
I also post on another board that will remain nameless, but I am recently growing more fond of this one. Not as flashy, but better discussions._________________Once the mind of man has expanded to the dimension of a new idea, it can never return to it's former size.
I'm Josh, 30 yo, from the Quad Cities originally, but have spent most of my life in Des Moines, IA. Just lost my job of 7 years thanks to the economy. I was born and bred a Cubs and Bears fan (for those who don't know the triple A Cubs team plays here in Des Moines) and of course as a kid I became a gigantic bulls fan and always rooted for the Hawks against the Wings. I'm also a converted Hawkeye fan (no affiliation to the school). I have an 11 yo daughter and my gf has twin 22 mo boys._________________2013 Bears Forum Mike Ditka Award Winner
2014 Adopt-A-Bear Alshon Jeffery
Mike, if you can figure out how to type with an accent, please do so. While I have no Scottish blood in me, I Love the Scottish dialect, but I also love the bagpipes, so there ya have it.
My real name is Jeff. I've been a Bear fan since the late 1960's (Butkus era). I grew up in a small town in Central Illinois, Served 10 years in the US Army and NG as a Helicopter Crew Chief, spent 17 years living in Springfield, before my wife and I bought some timber near the small town of Shelbyville. I've spent the better part of the last two years selectively clearing timber and performing "part" of the construction of our house, so I haven't been on here much in the last 2 years until recently (we finally finished). My wife and I run our own business out of our house after working most of our lives for others. I'm 46 with 2 GREAT teenage kids (boy and Girl). My wife of 20 years and I both went to U of I, although mine was at the Springfield campus.
I also post on another board that will remain nameless, but I am recently growing more fond of this one. Not as flashy, but better discussions.
Leave the flashy stuff to those mesmerized by the twinkling lights on some of those other sites. Content is what we're all about around here!*
*This message was brought to you by footballsfuture.com, the best source for all things football_________________
I started watching football about 3 years ago because my friend brought a ball into school and it was really fun to play. I used to play for a team a couple of years ago (London Blitz) and I was a Running Back in a two back system, My mate provided the speed, I provided the power
I decided to stop because I really want to coach instead of play, and this is why I began posting on these here forums (I used to have another sign in but it was deleted). You guys have provided me with the knowledge to understand the game of football on the field and off and that is why I want to thank you My dream is to coach somewhere in the USA, but the reality is that I can coach a team here in the UK as a hobby...
I started Supporting the Bears because they had a swagger to them and I want them to get it back. I think with Jay Cutler, Dr Peppers and a Ballhawk at the back *cough*Atogwe*cough* we can...
Just a reminder to all to not post personal info like your last name, address, anything like that...forum rules ya know.
My name is Mike...I found FF 5 years ago and haven't seen a better site yet to talk football on. I have been married for 23yrs to my wonderful and understanding wife that I have actually turned into a Bears fan over the years and who now understands why I can't leave the TV on Sundays in the Fall. We have two boys, 22 and 19.
Joined the Air Force out of HS and after spending some time in Texas, Japan and then Maryland I got out. Decided to stay here in MD and raise our family. Got a job with a company that deals with satellite communications and have been in that field in one manner or another ever since._________________
My name is Connor, I live in a small town called Richmond, which is very close to a town called Lake Geneva, Wisconsin, if any of you are familiar with it. I'm 17, and will be 18 in late July, and I'm a Senior in High School. I'm going to end up going to NIU next season, and right now I'm planning on having a double-major in Criminal Justice and Finance, but I've changed my mind on that so many times, I will probably have a different idea by the time I actually get there. Other than football, my biggest hobby would be to play video games, and I love games like Mass Effect, Fallout, and Fable. I joined in 2007, I believe, just before the Bears lost in the Superbowl._________________
My name is Connor, I live in a small town called Richmond, which is very close to a town called Lake Geneva, Wisconsin, if any of you are familiar with it. I'm 17, and will be 18 in late July, and I'm a Senior in High School. I'm going to end up going to NIU next season, and right now I'm planning on having a double-major in Criminal Justice and Finance, but I've changed my mind on that so many times, I will probably have a different idea by the time I actually get there. Other than football, my biggest hobby would be to play video games, and I love games like Mass Effect, Fallout, and Fable. I joined in 2007, I believe, just before the Bears lost in the Superbowl.
Enjoy yourself at NIU tank. I'm a recent alum
Name is Aaron, 26 years old, live in Downers Grove, grew up in Plainfield, went to NIU, then went to The John Marshall Law School. I'm currently a public defender.
I'm all kinds of nerd. I love the Bears (football in general), video games, and books _________________
My name is Connor, I live in a small town called Richmond, which is very close to a town called Lake Geneva, Wisconsin, if any of you are familiar with it. I'm 17, and will be 18 in late July, and I'm a Senior in High School. I'm going to end up going to NIU next season, and right now I'm planning on having a double-major in Criminal Justice and Finance, but I've changed my mind on that so many times, I will probably have a different idea by the time I actually get there. Other than football, my biggest hobby would be to play video games, and I love games like Mass Effect, Fallout, and Fable. I joined in 2007, I believe, just before the Bears lost in the Superbowl.
Enjoy yourself at NIU tank. I'm a recent alum
Name is Aaron, 26 years old, live in Downers Grove, grew up in Plainfield, went to NIU, then went to The John Marshall Law School. I'm currently a public defender.
I'm all kinds of nerd. I love the Bears (football in general), video games, and books
Hey neighbor!_________________👮
Adopt-a-Bear: Martellus Bennett, TE
Rec: 77 Yds: 821 TD: 6
AZBearsFan wrote:
He's a playmaker though, and we can use more of those in the Devin.Fart | null | minipile | NaturalLanguage | mit | null |
Five reasons why Indian equities could spice up your portfolio
Viktor Nossek, director of research at WisdomTree Europe, outlines a number of reasons why investors should capitalize on expected strong growth in India.
India is at an interesting crossroad where leadership is pro-actively taking tough reforms for long-term growth. Two pillars of the Indian economy, that is consumption and demographics, have encouraging projected growth numbers.
Furthermore, India has policymakers who are not shy of taking bold steps and opening the economy; a central bank that successfully fought inflation and is now supportive of growth through lenient monetary policy; financial planning that is accelerating consumption, infrastructure and digitisation; and a global macro environment that is not disruptive of growth in India.
We believe that investors interested in emerging markets who hold an overweight position in Indian equities (which are right at the junction of recovery and growth) will benefit from long-term growth. Here are five other reasons why we believe India presents a hot investment opportunity:
1. A fast-growing workforce: By 2050, India’s workforce[1] is expected to have grown from the current 674 million to a staggering 940 million. To put this into perspective, the US workforce will be a little over 200 million in 2050 at its current rate and China is likely to be facing a shrinking workforce. This will potentially drive up labour costs in China – which would be a dent to its competitiveness.
2. Strong projected growth compared to emerging market peers: India is the only large accessible economy that is projected to grow at 7 per cent or more for the foreseeable future.[2] Unlike China, which for many years economists emphasised should shift from an export-centric model to a more self-contained economy, India has been an economy sustained by robust consumption.
3. Low debt to accelerate growth: Due to its relatively low debt levels, India can increase debt issuance with ease, thereby accelerating growth without running into the risk of excess debt levels. In 2015 domestic credit to the private sector ( percentage of GDP) stood at 52.6 per cent for India compared to 153.3 per cent for China. This reflects China’s years of rising debt and state-driven investments which have created excess capacity in several sectors.
4. Consumption driven economy: Juxtaposing India’s favourable demographics with its consumption expenditure, it’s no surprise that India’s ~60 per cent consumption expenditure to GDP ratio is much higher than the ~39 per cent of China.[3] This indicates an economic model that is hugely influenced by local consumption with greater potential insulation from global headwinds.
5. Liberalisation of the economy and key reforms: Another key element to India’s growing economy is that the Modi government has been gradually opening various sectors of the economy to foreign investors. Areas like Construction Projects, Cable Networks, Agriculture and Plantation (Coffee, Rubber and Palm Oil, etc.), Air Transportation (non-scheduled and ground handling) are now allowed 100 per cent Foreign Direct Investment (FDI). This is compared to other sectors like Defence and Broadcast that have been allowed, for now, an increase to 49 per cent in FDI.
We believe this creates opportunities for investors to invest in sectors that were earlier not accessible. The Modi government is also in the process of implementing what could be the single biggest tax reform under the ‘Goods and Services Tax’ or GST bill. Under this taxation scheme, all states and central taxes would be combined to create a consistent tax structure across the entire country, converting the whole nation into one market place. Several other key reforms on boosting consumption, infrastructure spending, debt recovery for banks, etc., are also in the process of being implemented.
Subscribe to Money Observer magazine
Be the first to receive expert investment news and analysis of shares, funds, regions and strategies we expect to deliver top returns, plus free access to the digital issues on your desktop or via the Money Observer App. | null | minipile | NaturalLanguage | mit | null |
Q:
MySQL or Derby ? for online java chat server?
i am builds online chat application , using Applets and java chat server , the application will hold many connections and a lot of rooms ,
i used Derby , but i am worry about it's power of many connections and many queries , and solid of queries which it support ?
so i am thinking also in MySQL , but i don't know if it's ideal also for this type of application ? or Derby is the best ?
Thank you ,
A:
Well try the comparison first MySql vs Derby
Developing any application with embedded system we should think about Direct database connection, easy to update and alter the schema, easy to remove/handle data inconsistency, easy to backup and restore data easily. After finding these out, which RDBMS would be beneficial, is mostly up to you, as opinion will always vary.
MySql's innoDB can be configured to hyperthread read threads, hyperthread write threads with accessing multiple CPUs/Cores and efficient Buffer Pool; It's performance is quite good.
As the linked page is showing the differences: For for Derby's Full text, Hash, R-/R+ Tree for indexing capability and Range, Hash, Composite(Range + hash) for partitioning capability; My opinion is, if my JAVA application in server side, has also a client side which will need a database embedded, i would consider to go for Derby.
This is the article, i read prior to start Derby, it has a nice description and performance comparison with MySql using graph and charts.
| null | minipile | NaturalLanguage | mit | null |
Development of antibodies against the putative proteinase maturation protein A in relation to pneumococcal carriage and otitis media.
The putative pneumococcal proteinase maturation protein A is a potential pneumococcal vaccine candidate. We examined serum antipneumococcal proteinase maturation protein A antibodies at 6, 12, 18 and 24 months of age, and showed that the age-related development of antipneumococcal proteinase maturation protein A antibodies is associated with pneumococcal contacts. A higher antipneumococcal proteinase maturation protein A antibody concentration at 18 months of age tends to predict for a lower risk of pneumococcal acute otitis media in the following 6 months (relative risk: 0.84, 95% confidence interval: 0.62-1.13). | null | minipile | NaturalLanguage | mit | null |
Diane Wilson Archive: Democracy Now (Oct. 11, 2005)
DIANE WILSON: Dow’s responsibility, they claim all the profits, and we believe that they claim the liabilities also. I do know that they have taken on Union Carbide’s liabilities in the United States. There was a case where a child was contaminated with some of Union Carbide’s pesticide. And I believe the American child received up to $6 million. And the children over in India, a lot of them received nothing at all. And some of them just, you know, like $500.
AMY GOODMAN: You found Warren Andersen here in this country. Can you talk about what happened? DIANE WILSON: Well, it’s real interesting, because, you know, they had been trying to extradite him to India for a long time. And the FBI kept saying, well, they couldn’t find Warren Andersen. They just had no idea where that man was. Well, actually, it was Greenpeace who found him first. And once we heard that Warren Andersen was in South Hampton on Long Island, I was in New York one day. So I just decided just to go by his house and stand out front. And I had a big sign that said, “Warren, shouldn’t you be in India?” And I had actually had no idea that he was inside. You’d see — every once in a while you would see a curtain pull back. And I was really surprised when he and his wife walked out.
Note: In her preface to the interview, Goodman cites the Oct. 10 story by Corporate Crime Reporter. | null | minipile | NaturalLanguage | mit | null |
Comments on: Kings will unveil statue of Robitaille in 2014-15https://nhl.nbcsports.com/2014/08/07/kings-will-unveil-statue-of-robitaille-in-2014-15/
ProHockeyTalk on NBCSports.comThu, 24 May 2018 17:49:41 +0000hourly1http://wordpress.com/By: schusters3https://nhl.nbcsports.com/2014/08/07/kings-will-unveil-statue-of-robitaille-in-2014-15/comment-page-1/#comment-360391
Fri, 08 Aug 2014 03:39:49 +0000http://prohockeytalk.nbcsports.com/?p=2338391#comment-360391C’mon, the mullet energized the team and got them on an amazing hot streak while Gretzky had that piano on his back while he skated for half the season. They’re having a night for Granato as well. The honor paid by the organization and fans to former players is part of what has turned this team into a family that players league wide want to be a part of. Can’t hate on Barry…we would have had a Cup with him if it weren’t for that damned curved stick!
]]>By: pjblake2redwingshttps://nhl.nbcsports.com/2014/08/07/kings-will-unveil-statue-of-robitaille-in-2014-15/comment-page-1/#comment-360298
Thu, 07 Aug 2014 21:16:53 +0000http://prohockeytalk.nbcsports.com/?p=2338391#comment-360298why honor Barry Melrose. He is a Jack A** who doesnt know what he is talking about half the time. He picks a team like Boston or Pitt to win the cup in the start of the year then when LA wins he goes ” I knew all along that LA would win this year” lol guys a joke
]]>By: kingsfan93https://nhl.nbcsports.com/2014/08/07/kings-will-unveil-statue-of-robitaille-in-2014-15/comment-page-1/#comment-360297
Thu, 07 Aug 2014 21:15:45 +0000http://prohockeytalk.nbcsports.com/?p=2338391#comment-360297LLLUUUUUUCCCC! | null | minipile | NaturalLanguage | mit | null |
Joint determination of fluorine and chlorine in granitic rocks with ion-selective electrodes.
This enables both fluorine and chlorine in granitic rocks to be determined with a single sample. A rapid technique, using fusions in culture tubes, and a slower technique, employing fusions in platinum crucibles are described. The culture-tube technique is suitable for geochemical exploration and employs an oxidizing flux so that sulphide-bearing rocks can be analysed. The platinum-crucible technique yields fluorine and chlorine results for the standard rocks G-2 and GSP-1 that are comparable in value and precision to those obtained by other analytical methods. It is faster, easier to manipulate and yields higher fluorine values than the existing joint spectrophotometric method. | null | minipile | NaturalLanguage | mit | null |
Peptide dendrimers: applications and synthesis.
Peptide dendrimers are radial or wedge-like branched macromolecules consisting of a peptidyl branching core and/or covalently attached surface functional units. The multimeric nature of these constructs, the unambiguous composition and ease of production make this type of dendrimer well suited to various biotechnological and biochemical applications. Applications include use as biomedical diagnostic reagents, protein mimetics, anticancer and antiviral agents, vaccines and drug and gene delivery vehicles. This review focuses on the different types of peptide dendrimers currently in use and the synthetic methods commonly employed to generate peptide dendrimers ranging from stepwise solid-phase synthesis to chemoselective and orthogonal ligation. | null | minipile | NaturalLanguage | mit | null |
Informational Literacy Unit - Post Revised
Due to an overwhelming request by the devoted readers of this blog, I am revising this original post to include pdf's of the Guidelines for Critical Thinking rubric and the Code of Cooperation. Scroll down to find them in context.
If information literacy is central to success, survival, schooling, workplace and the community, then the teaching of information literacy and all the skills that it entails is critical, too. As many of you are quite aware, the new Common Core State Standards (CCSS) is building instructional practices and "shifts" into the roll-out...where teaching the standards alone is simply not enough to prepare our students for the 21st century.
Shift #1 for the CCSS in English/Language Arts (ELA) is this:
For my 3rd grade reading group, now up to 28 (high-achieving) students, we have been able to put some systems and structures into place to give students an opportunity to excercise their critical thinking skills....which of course, must be taught and relentlessly practiced, with informational content text. You will notice the wording in the CCSS Shift statement, does not use the term, a balance of "fiction and non-fiction" text...it states, "a true balance of narrative and informational text. In the elementary grades K-5, the percentage balance is truly pure, 50% narrative and 50% informational....but in the middle school and high school, the percentage for informational text reading rises while narrative text reading decreases, to the content based classes in middle and high school. As a class, we began brainstorming different types of non-fiction genres, then different types of informational text. Here is the final anchor chart we created:
From this chart, we created a few more. One about informational text features and their purpose and one about informational text structures using Charts found on Beth Newingham's Scholastic blog. The informational text features chart was created from my own original creations which can be found on my www.hellofirstgrade.com website under Literacy and Language.
After generating and exhausting the list of informational text types with the students, the students decided they would like to create an informational text museum using informational text from their home and world.
During guided reading groups, we used informational text to determine main idea, cause event, sequence of events, etc. We also used our Guide to Critical Thinking to guide our conversations, our justifications and interactions with each other. Here is the anchor chart we created.
I'm happy to share this Gudelines of Critical Thinking rubric I created with anyone in the best interest of "preparing all learners for their future"...this document is licensed under Creative Commons for non-profit, share-alike and proper attribution.
During guided reading, students take turns facilitating the group conversation asking questions of each other and justifying their opinions to one other...and I get to sit back and file my nails [just kidding]...seriously, we have talked like this for so long, they know how to play devil's advocate now and "play" being the teacher with each other. It is not uncommon to hear students say, "I disagree with xyz, because on page 10, the author says abc...so from that I infer 123." Students also hold each other accountable to this language by calling on students that rarely share by saying...."xyz, you haven't contributed to the conversation yet, what would you like to add?" To get students to justify their thinking and opinions, I made a big huge BECAUSE that hangs right above us on the wall...it's about 2' x 3'.
In order to help students facilitate this type of discussion, I created these sentence frames in order to help them frame their thoughts, ideas and opinions. They are free and you can download them HERE.
And my Revised Bloom's Taxonomy Posters to get all students using the maximum amount of signal strength during literacy....and in every subject all throughout the day.
During Independent Reading & Responding, students opted to use this form in their informational responding.
The last group is a cooperative learning literacy activity. We do so much modeling of how to speak and interact with each other about our literacy ideas, reading thinking and ideas, students are able to carry on without my assistance. Here was their task last week. {P.S. The green serving platter that the two book sets are resting in is from the Dollar Tree and serves as the tracing circles when the groups create their Venns.}
Here the product of two groups....
And this is the Cooperative Learning Anchor Chart that guided how students interact here:
Again, I'm happy to share this Code of Cooperation document I created with anyone in the best interest of "preparing all learners for their future"...this document is licensed under Creative Commons for non-profit, share-alike and proper attribution.
In addition, with Speaking and Listening having it's own domain in the Common Core for ELA, we created a sort of "code of cooperation" for the Rules for Discussion standard. Students respectfully discuss their ideas by following these rules and hold each other accountable to these rules as well. If you'd like to use these Rules for Discussion from the Common Core, here is the link to the Common Core Rules for Discussion.
Here are few more ideas for you from this week:
Anecdotal Record Sheet...I use one for my 2nd grade group and one for my 3rd grade group.
Teaching students to read and comprehend nonfiction and informational text is more difficult than reading and comprehending fiction for the sheer reason that there is no sense of story to hold the ideas together, no characters, setting or plot to contextualize the reading experience. As teachers, it is our responsibility to help students understand the various structures and features of informational text. I help my students do this by teaching each feature of nonfiction through mini-lessons where mentor texts are used to demonstrate the various features, what they show, how they are used and how they help us understand the topic better. Several years ago, I created a template for a nonfiction conventions notebook. This year, I have updated that notebook and aligned it to the common core standards related to nonfiction text features. If you' re interested in this, you can find it HERE.
To also my blog post about Sources for Informational Text online. And if you're interested in keeping anecdotal snapshots during guided reading or informal running records of informational texts, I created these "all standards at-a-glance" sheets to quickly documents the RL and RIT standards of the Common Core. The first grade one is shown below but I have created them for all grades K-5.
This is wonderful. I have been presenting along with a co-worker all the different shifts and this gives light to some with examples, used in a classroom. I really enjoyed your post and can't wait to guide others to come find your blog. Great job, thanks for informing all of us, and continue on! I am in Eastern, NC, and want to take a field trip to your classroom!
What a great wealth of knowledge and information. This is an excellent tool and I would like to have our English department create and or set up a web page for literacy in high school. It would be great to have some of our major literacy strategies and activities available at our finger tips. Thanks for sharing.
Thank you so much! You have absolutely amazing ideas/charts here. I teach high school and I can adapt much of it for my kids. Getting them to justify their responses with the big giant "because" - I want to make one this weekend! Thank you for sharing.
I love your blog! i teach 4th grade ... Also a high achieving bunch. I wish a 4th grade teacher would start this with common core. I would but between my own kids, finishing grad school, teaching I don't think i can take much more on solo. I would love to partner with someone though!
Dear Angie,Ideas like the CT rubric AND holding kids accountable for talking this way are "vital behaviors" (from the book The Influencer) where it only takes one teacher and class of students to start doing it and the rest is history, as that teacher shares it with her PLT, and those students go to the next grade talking like that, the principal asks you and your team to speak about it at a staff meeting...etc. Our school has determined that common CT language is so important to our mission "preparing all learners for their future" that we have wriiten it into our SIP. Good luck.
Thank you so much for sharing these educational resources, PDFs, and creative/clear/concise ideas! As a middle school language arts teacher, it is always helpful to gain a variety of perspectives and ideas on the literary genres, elements of reading, etc. that are taught in the elementary schools. When I have an understanding of my students’ potential prior knowledge, my “investigations” and pre-assessments are more informed and my teaching is more effective. Thanks again for letting us into your classroom, providing a springboard of ideas, and contributing to this tremendous virtual professional community.
Love your Blog! The critical thinking, making connections and comprehension ideas are very helpful as we plan for the new school year. Each year we strive to grow lifelong readers and writers in our classroom, and you have given us a multitude of tools with which to do just that! I job share, so our children have 2 "facilitators". We are excited to begin teaching our children and thank you for your ideas and strategies...all very beneficial to teaching our amazing third graders!Cynthia MitchenSpring, TX | null | minipile | NaturalLanguage | mit | null |
University of Canberra Village offers a choice of self-catered residential halls and stylish new apartments (opened in 2009). The new apartments feature environmentally sustainable design initiatives such as greywater recycling, solar powered hot water tanks and a wildlife corridor for native animals.
If you cannot find what you want then please try our smaller student accommodation options. | null | minipile | NaturalLanguage | mit | null |
87 Wn.2d 253 (1976)
551 P.2d 740
THE CITY OF SEATTLE, Appellant,
v.
OLLIE BIRT CROCKETT, Respondent.
No. 44061.
The Supreme Court of Washington, En Banc.
June 24, 1976.
John P. Harris, Corporation Counsel, and Richard S. Oettinger, Assistant, for appellant.
Jerome J. Doherty of Seattle-King County Public Defender, for respondent.
DOLLIVER, J.
This is an appeal from an order entered by the King County Superior Court dismissing defendant's conviction in Seattle Municipal Court for illegally discharging a firearm in violation of a city ordinance. The conviction was dismissed due to an alleged failure to comply with CrR 3.3(b): the defendant was not given a trial de novo within 90 days from the date his counsel requested and received a trial date from the clerk of the superior court.
The sole issue is whether the time limits of CrR 3.3(b) have been met in an appeal to the superior court for a trial de novo where a person charged with a crime is brought initially to trial in municipal court within 90 days following the preliminary appearance. We hold under these circumstances the requirements of CrR 3.3(b) have been met and reverse the trial court.
The following facts are relevant to the resolution of the issues presented:
February 25, 1975 A "Washington Uniform Citation and Notice
to Appear" was issued to defendant for
discharging a firearm within the city
limits.
March 14, 1975 Defendant made a preliminary appearance
before the Seattle Municipal Court pursuant
to JCrR 2.03 and entered a plea of not
guilty.
*255 April 22, 1975 Defendant was found guilty by the Seattle
Municipal Court 39 days after his
preliminary appearance.
June 4, 1975 Defendant was sentenced to 60 days in jail.
June 9, 1975 Defendant filed a notice of appeal in
municipal court.
July 25, 1975 A trial de novo in superior court was set
for December 8, 1975.
November 3, 1975 Defendant made a motion to dismiss.
November 24, 1975 The superior court granted the motion and
entered an order dismissing the criminal
conviction pursuant to CrR 3.3(f).
CrR 3.3(b) provides:
A criminal charge shall be brought to trial within 90 days following the preliminary appearance.
CrR 3.3 (f) provided:
A criminal charge not brought to trial as required by this rule shall be dismissed with prejudice.
The defendant contends that the requirements of CrR 3.3(b) have not been met because (1) a trial de novo is a new proceeding in superior court and the plain language of the rule gives him a right to a trial in superior court within 90 days; (2) the "preliminary appearance" occurred and the 90-day period commenced to run on the day the defendant's case was either set or noted for trial in the superior court; and (3) the reasons for the rule are not obviated by the fact that defendant has appealed to superior court for a trial de novo.
[1] A trial de novo in superior court as the result of a conviction and appeal taken in municipal court does not signal the initiation of an entirely new criminal process or procedure. The slate has not been wiped clean as, for example, where the defendant is to be tried again following a mistrial or order granting a new trial. See ABA Standards Relating to Speedy Trial § 2.2(c) (Approved Draft, 1968). *256 The judgment of conviction rendered by the municipal court still exists and the appeal merely suspends operation of the judgment. JCrR 6.02; Goulter v. Huse, 196 Wash. 652, 84 P.2d 126 (1938).
A trial de novo, such as in this case, represents the exercise of the appellate jurisdiction of the superior court and not its original jurisdiction. Camas v. Kiggins, 120 Wash. 40, 46, 206 P. 951 (1922); see Const. art. 4, § 6 (amendment 28). However, when such an appeal is taken, the superior court is vested with jurisdiction to proceed with the case as if it had been commenced originally in that court. State ex rel. Getman v. Webster, 193 Wash. 265, 267, 75 P.2d 124 (1938). The superior court may render such judgment as it deems is warranted.
The trial de novo involved in the present case is an unusual judicial procedure created by statute. RCW 35.20.070. It has the characteristics of both an appellate and a trial action. It cannot be viewed in isolation but must be considered as one step along a juridical continuum.
[2] Similarly, the procedural rules applicable to superior courts and courts of limited jurisdiction must be considered as a whole and cannot be sliced up, then construed and applied piece by piece to the resolution of issues that develop from or are related to the judicial process. The criminal rules must be viewed in relation to both the type of procedure involved and the totality of their purpose, which is to secure simple and fair as well as inexpensive and effective justice. CrR 1.1, 1.2; JCrR 1.02; JAR 2. The rules were designed to operate in conjunction with one another and not to require meaningless and useless duplication. When so considered and applied to this case there is no hiatus between the criminal rules for courts of limited jurisdiction and the criminal rules for superior courts.
CrR 3.3(a) provides that every person "charged with crime" is entitled to a speedy trial in accordance with the provisions of the rule. JCrR 2.03(a)(2) provides that a person arrested for any offense is entitled to a preliminary appearance at which time the judge informs the person (1) *257 of the charge for which he or she is arrested, and (2) of the rights of the person charged with the crime.
[3] In this case defendant was given a preliminary appearance pursuant to JCrR 2.03. This appearance triggered the time limits of CrR 3.3(b). State v. Parmele, 87 Wn.2d 139, 550 P.2d 536 (1976); State v. Elizondo, 85 Wn.2d 935, 540 P.2d 1370 (1975). Subsequently, the defendant was brought to trial within the 90 days after his preliminary appearance in municipal court. Defendant has been accorded a speedy trial pursuant to CrR 3.3(b) which initially fulfilled public policy and satisfied the purpose of requiring a speedy trial. See ABA Standards Relating to Speedy Trial § 1.1 (Approved Draft, 1968).
The language of the rule is clear. A trial on a criminal charge is required within 90 days following the preliminary appearance. Such a trial was held. The requirements of CrR 3.3 (b) have been met. Within the confines of this case and the language and purpose of JCrR 2.03 and CrR 3.3, there is no merit to defendant's attempt to transform the date on which a trial de novo is noted or set for trial in the superior court into the preliminary appearance referred to in CrR 3.3(b).
[4] Defendant further ignores the fact that the prosecuting attorney initially met the burden in providing an accused with a speedy trial within the time frame specified by our criminal rules. Thus, when the defendant appealed his conviction in municipal court, the responsibility for and the burden of compliance with the rules shifted away from the court and the prosecuting attorney. State v. Sodorff, 84 Wn.2d 888, 529 P.2d 1066 (1975). In fact, it is the defendant's burden to prosecute an appeal diligently and a failure to do so may result in dismissal on the clerk's motion. JCrR 6.03; State v. Twogood, 14 Wn. App. 447, 542 P.2d 793 (1975).
It should be noted that, even though the rigid requirements of CrR 3.3 have been met in the district court proceedings, the defendant continues at the trial de novo to have a constitutionally protected right to a speedy trial. *258 See Const. art. 1, § 22; cf. State v. Estes, 151 Wash. 51, 54, 274 P. 1053 (1929). Nevertheless, the burden is on him to pursue it.
The judgment of the superior court is reversed, the prosecution is reinstated and the cause is remanded for trial.
STAFFORD, C.J., and ROSELLINI, HUNTER, HAMILTON, WRIGHT, UTTER, BRACHTENBACH, and HOROWITZ, JJ., concur.
| null | minipile | NaturalLanguage | mit | null |
Item: Expect the press conference this week to announce the long talked-about indoor practice facility for the Orange football team.
He may have originally thought that the ground would be broken with Doug Marrone as his football coach, but Daryl Gross will be a happy man none-the-less with Scott Shafer running the ship when the ground is expected to be broken in early spring on a missing link for his football program about to embark in ACC play.
It’s the much-discussed indoor football piece to the puzzle that Marrone deemed mandatory during his four-year stay, which not only frees up having to make practice trips over to the Dome when inclement weather strikes, but turns the field at Manley Field House over to the other men’s and women’s teams to share with one less tenant.
» Related: Don’t expect final football schedule for a few weeks
While the specific details are to be announced at the university press conference, sources have told The Juice Online that besides a full 120-yard field with a small area for spectators, video camera crews, and minimal office space, it is also expected to have some sort of connectivity to the existing Iocolano-Petty football wing near the original Coyne Field.
In 2011, the Brooklyn-based Bernheimer Architecture released design blueprints and building renderings for a project titled “Syracuse University Football Training Facility,” indicating that the indoor practice field was part of a “larger strategic masterplan (sic) for the Lampe Athletic Complex.”
Bernheimer described the indoor practice facility building this way:
“For the football practice facility, we designed the building to create an environment that would enhance the practice session. Composed as a highly regulated and repetitive structure, all systems throughout the facility are deployed in the service of order and conceived to reinforce a sense of regularity and consistency.
Column bays and light queen post trusses occur every ten yards, in alignment with the field landmarks. Translucent wall panels provide a well-lit neutral background and daylighting (sic) to minimize energy use. The base of the building is formed from precast concrete panels that are cast with patterned dimples of embedded footballs.”
Translated from architectural-speak, it sounds like it’s going to look something like this rendering, but again, the building specifics are to be announced.
What’s clear is that as opposed to a “masterplan” for the Lampe Athletic Complex, the athletic department has been able to win over the university administration and board of trustees to be in unison on the essential need to at least go ahead with building the indoor practice field now, as the team under rookie head coach Shafer gets ready to compete on and off (recruiting) the field with the other ACC programs that have that indoor facility as part of their pitch to a highly coveted, impressionable prospect.
As to the financing of an estimated $15-20M structure, we’re told the athletic department is taking on the funding buoyed by the bump in annual ACC revenue sharing and backed-up by the administration and board, while simultaniously launching a vigorous fundraising campaign.
A final game bowl victory, a new coach and staff, a new league, and new facilities on the way. There’s a lot to like about the direction of ‘Cuse football.
» Related: ACC has early title contenders
Item: Unless Syracuse hosts Western Kentucky, the final football home game opponent is going to be a FCS member.
It’s simple math. According to FBSschedules.com there’s simply no other Bowl Subdivision team that needs a 2013 game on its schedule besides Western Kentucky. New Mexico State and Louisiana-Lafayette were the other two FBS teams that needed games, and they scheduled each other.
As we wrote two weeks ago, Syracuse was waiting on the release of the ACC schedule, expected anytime now, to see if it had an open date on Sept. 14 to fill with a Dome game after the season opener at MetLife Stadium against Penn State followed by a trip to Northwestern, and we speculated that the Orange would likely be left standing with a FCS opponent.
FBSchedules.com also reports that the Syracuse non-conference opening and the date of Duke’s non-league meeting with Troy are the only games holding up release of the entire ACC schedule.
For more Syracuse coverage, Like our Facebook page and follow us @TheJuiceOnline. | null | minipile | NaturalLanguage | mit | null |
WASHINGTON (AP) — Iowa Republican congressman Steve King says he’s not a racist, but he faced intensifying criticism Friday over his remarks about white supremacy, including from a black GOP senator who said such comments are a blight on the nation and the party.
For the second time in two days, King insisted that he is an advocate for “Western civilization,” not white supremacy or white nationalism. But he didn’t deny remarks published a day earlier in The New York Times in which he was quoted saying: “White nationalist, white supremacist, Western civilization — how did that language become offensive?”
Within hours Thursday, the House’s top three Republicans condemned his remarks, and on Friday, GOP Sen. Tim Scott of South Carolina published his disapproval in an op-ed column.
King, who has denied being racist, appeared on the House floor after most lawmakers had left town.
“One phrase in that long article has created an unnecessary controversy. That was my mistake,” King told his colleagues. King said terms describing bigotry, such as racism, are unfairly applied to “otherwise innocent” people.
King, in his ninth House term, spoke as key members of his party publicly took issue with his remarks and as a Republican from back home lined up to challenge him in a GOP primary.
Scott, who is black, cast King’s remarks and those like them as a blemish on the country and the Republican Party, which has long had a frosty relationship with black voters.
“When people with opinions similar to King’s open their mouths, they damage not only the Republican Party and the conservative brand but also our nation as a whole,” Scott wrote.
King’s views, Scott added, are separate from the conservative movement and “should be ridiculed at every turn possible.”
“Some in our party wonder why Republicans are constantly accused of racism — it is because of our silence when things like this are said,” Scott wrote.
In fact, House Republican leaders swiftly condemned King’s remarks as racist. And on Wednesday, King drew a 2020 primary challenger: Randy Feenstra, a GOP state senator.
Some Democrats have called for the House to condemn King’s remarks or somehow punish him. Speaker Nancy Pelosi acknowledged Friday there was “interest” in taking action — but no decision to do so, or how.
“We’ll see what we do about Steve King,” she told reporters. “Nothing is shocking anymore, right?”
King’s position in the GOP had been imperiled even before this week.
In 2017, he tweeted: “We can’t restore our civilization with somebody else’s babies.” Then he doubled down on CNN, telling the network, “I’d like to see an America that’s just so homogeneous that we look a lot the same.”
Shortly before the 2018 midterm elections, in which King was running, Rep. Steve Stivers, R-Ohio, then the head of the GOP campaign committee, issued an extraordinary public denunciation of him.
King on Friday suggested he’s been misunderstood. He said the foundation of the Times interview was partly a Sept. 12 tweet in which he wrote: ”‘Nazi’ is injected into Leftist talking points because the worn out & exhausted “racist” is over used & applied to everyone who lacks melanin & who fail to virtue signal at the requisite frequency & decibels. But...Nazis were socialists & Leftists are socialists.”
On Friday, King said on the House floor that the interview “also was discussion of other terms that have been used, almost always unjustly labeling otherwise innocent people. The word racist, the word Nazi, the word fascist, the phrase white nationalists, the phrase white supremacists.”
King said he was only wondering aloud: “How did that offensive language get injected into our political dialogue? Who does that, how does it get done, how do they get by with laying labels like this on people?” | null | minipile | NaturalLanguage | mit | null |
Ron Dellums, one of the most significant and radical members of Congress in U.S. history, died on Monday at age 82. Liberals, progressives, social democrats, or democratic socialists interested in fostering the left-wing political movement that emerged following Vermont Sen. Bernie Sanders’s 2016 primary loss should study how Dellums managed to get and hold the power of elected office for decades, and use it to make America better in numberless ways — most famously by overturning U.S. support for apartheid South Africa.
Dellums was first elected to Congress in 1970 as a 35-year-old Democrat from a California district that included Berkeley and Oakland.
He didn’t do this by challenging a Republican. Nor did he wait patiently for an open seat. Instead, he ran against a six-term Democrat named Jeffery Cohelan in that year’s primary.
Cohelan was a standard Cold War liberal. He was strong on labor and the environment. But on foreign policy, he’d faithfully supported President Lyndon Johnson’s massive escalation of the Vietnam War. When hundreds of constituents mailed him postcards opposing the war, he denounced them. When several peace activists held a sit-in at his office in 1966, they were, to their surprise, arrested.
The district was one of the most progressive in the U.S., and a hub for anti-Vietnam War resistance. So local activists decided they could do better and recruited Dellums, then a member of the Berkeley City Council, to run against Cohelan.
In a memoir written 30 years later, Dellums described seeing his political career as always more about commitment to a movement than himself. And in order to make his case honestly, he told supporters it was critical that he not “take any funny money; the campaign will have to raise all its financial support from the people.” He relied on the apparatus and experience built up over two previous attempts to challenge Cohelan, one in 1966 and another in 1968, and made sure to “carefully document and study the campaign” so that if he lost, candidates with the same values could avoid his mistakes and win in the future.
The goal of his campaign, Dellums explained, was to appeal “to people across the divides of race, class, gender, sexual orientation, and disability status … to speak to people on the basis of their own mutual self-interest. … Equality and justice were themes of relevance to women, to trade unionists, to those with disabilities, and to seniors, just as they were to blacks, Latinos, Asian Americans, and Native Americans.”
As it turns out, it worked: Dellums knocked off Cohelan with room to spare, getting 42,619 votes to Cohelan’s 35,137. A New York Times article, headlined “When Radicals Are Elected to the Hated System,” reported that Dellums “brought out thousands of new student voters.”
Dellums’s primary victory immediately led to a gigantic Republican freakout. Spiro Agnew, then vice president, assailed Dellums as a “radical extremist.” Dellums’s GOP opponent in the general election called him the candidate of “the crazies” and said that he was part of “the lunatic left wing.”
But Dellums didn’t tack to the center by even a millimeter — on Vietnam or anything else. He continued to identify as a socialist, just as he had throughout the primary. And on Election Day that fall, he became both the first avowed socialist to win a seat in Congress since World War II, and the first black candidate in American history ever elected from a majority-white district.
After winning, Dellums faced some of his strongest opposition from other Democrats. When he first walked onto the House floor to attend a meeting of the Democratic Caucus, he said he overheard one of his colleagues ask about him: “Where is that radical son of bitch?” But he was undeterred.
Dellums immediately held unofficial hearings about Vietnam — unofficial because his fellow Democrats would not permit official ones. He started pushing for the U.S. to sanction apartheid South Africa that same year, and after 15 years, finally won in 1986 when Congress passed the Comprehensive Anti-Apartheid Act over President Ronald Reagan’s veto. He co-founded both the Congressional Black Caucus and the Congressional Progressive Caucus. He helped popularize the tradition, born in these caucuses, of creating alternative federal budgets — visions for where the U.S. should be trying to go. He worked with then-Rep. John Kasich to kill the preposterously wasteful B-2 bomber, and rose to become chair of the House Armed Services Committee. | null | minipile | NaturalLanguage | mit | null |
Q:
How to convert time of the date and count the days in python?
I have a dataset about users taking online courses. It has features like, 'id', 'event', 'time'. I groupby them and want to know the frequency of a user doing every event on specific days. I want to count them in days.
lt = log_train.groupby(['enrollment_id','event','time']).size()
print(lt)
enrollment_id event time
1 access 2014-06-14T09:38:39 2
2014-06-14T09:38:48 1
2014-06-19T06:21:16 2
2014-06-19T06:21:32 1
2014-06-19T06:21:45 1
..
200887 navigate 2014-07-24T03:27:16 1
200887 navigate 2014-07-24T03:27:16 1
page_close 2014-07-24T04:19:55 1
video 2014-07-24T04:19:57 1
200888 access 2014-07-24T03:48:14 2
discussion 2014-07-24T03:47:57 1
navigate 2014-07-24T03:47:17 1
2014-07-24T03:47:28 1
2014-07-24T03:48:01 1
From the information I have seen in another dataset there are userIDs, courseIDs and course range time.
usercourse = pd.merge(enroll,date,how="left", on= 'course_id' )
enrollment_id username \
0 1 9Uee7oEuuMmgPx2IzPfFkWgkHZyPbWr0
1 3 1qXC7Fjbwp66GPQc6pHLfEuO8WKozxG4
2 4 FIHlppZyoq8muPbdVxS44gfvceX9zvU7
course_id from to
0 DPnLzkJJqOOPRJfBxIHbQEERiYHu5ila 2014-06-12 2014-07-11
1 7GRhBDsirIGkRZBtSMEzNTyDr2JQm4xx 2014-06-19 2014-07-18
2 DPnLzkJJqOOPRJfBxIHbQEERiYHu5ila 2014-06-12 2014-07-11
Every single user has only 1 course and all the courses have the same range with 30 days. So what I want to have should be similar like this,
enrollment_id event #ofDays #ofActionTimes
1 access 2 2
10 6
30 2
..
200887 navigate 23 1
page_close 30 1
video 1 1
200888 access 12 2
discussion 2 1
navigate 5 3
29 4
**#ofDays means at the Nth day of a course.
#ofActionTimes means how often an event happens on the Nth day.**
Since every course started from different dates I have no idea how to generate this data form on python.
Hope someone could help me to solve the problem!
A:
IIUC, you can use merge, groupby, and count to get what you want.
First, some example data. This is based on the data you provided, but I've modified it so that the output can be clearly traced from the starting data.
data1 = {"enrollment_id":[1,1,1,1,2,2,3,3,3],
"event":["access","access","access","navigate","access",
"page_close","navigate","navigate","video"],
"time":["2014-06-14T09:38:39", "2014-06-14T09:38:48",
"2014-06-19T06:21:16", "2014-06-19T06:21:32",
"2014-06-21T06:21:45", "2014-06-22T06:21:16",
"2014-06-19T06:21:32", "2014-06-20T06:21:16",
"2014-06-20T06:21:16"]}
data2 = {"enrollment_id":[1,2,3],
"username":["user1", "user2", "user3"],
"course_id":["course1", "course2", "course3"],
"course_from":["2014-06-12", "2014-06-19", "2014-06-12"],
"course_to":["2014-07-11", "2014-07-18", "2014-07-11"]}
df1 = pd.DataFrame(data1)
df1
enrollment_id event time
0 1 access 2014-06-14T09:38:39
1 1 access 2014-06-14T09:38:48
2 1 access 2014-06-19T06:21:16
3 1 navigate 2014-06-19T06:21:32
4 2 access 2014-06-21T06:21:45
5 2 page_close 2014-06-22T06:21:16
6 3 navigate 2014-06-19T06:21:32
7 3 navigate 2014-06-20T06:21:16
8 3 video 2014-06-20T06:21:16
df2 = pd.DataFrame(data2)
df2
course_id enrollment_id course_from course_to username
0 course1 1 2014-06-12 2014-07-11 user1
1 course2 2 2014-06-19 2014-07-18 user2
2 course3 3 2014-06-12 2014-07-11 user3
We want to know how many times a specific event happened for a specific enrollment_id, with a separate count for each day of the course.
Derive the course day number course_day_num by subtracting course_from (the course start date) from event_date.
df = (df1.merge(df2[["enrollment_id", "course_from"]],
on="enrollment_id", how="left")
)
df["event_date"] = pd.to_datetime(pd.to_datetime(df1.time).dt.date)
df["course_from"] = pd.to_datetime(df["course_from"])
df["course_day_num"] = (df.event_date - df["course_from"]).dt.days
Then groupby each course_day_num to get the event count, per person, per course day:
groupby_cols = ["enrollment_id", "event", "event_date", "course_day_num"]
df.groupby(groupby_cols).event_date.count()
enrollment_id event event_date course_day_num
1 access 2014-06-14 2 2
2014-06-19 7 1
navigate 2014-06-19 7 1
2 access 2014-06-21 2 1
page_close 2014-06-22 3 1
3 navigate 2014-06-19 7 1
2014-06-20 8 1
video 2014-06-20 8 1
Name: event_date, dtype: int64
| null | minipile | NaturalLanguage | mit | null |
The implementation of a restrictive worksite smoking policy in a large decentralized organization.
This study investigates the implementation of a restrictive smoking policy in decentralized worksites. A model which includes four elements--concept, context, process, and outcomes--is used as a framework for identifying characteristics that influence implementation. The organization studied was a state human services agency with approximately 400 worksites spread across 12 geographic regions. Quantitative data collection included three cross-sectional surveys of employees and supervisors administered before and after the date the policy became effective. Qualitative data were collected from three sources, including written comments on surveys, focus groups, and structured interviews with supervisors and top administrators. Tabular analyses and one-way analyses of variance were used to analyze quantitative data. Qualitative data were examined for key themes and have been used to elucidate findings. Those characteristics related to concept, context, and process which appeared to have the strongest influence on expected and unexpected outcomes of the restrictive smoking policy were degree of policy restrictiveness, job characteristics, perceived level of participation in formulation and implementation, and support of supervisors responsible for day to day enforcement. In particular, in this decentralized organization, lack of participation was found to underlie many of the problems experienced in implementation. The practical implications for developing and implementing a worksite restrictive smoking policy are discussed. | null | minipile | NaturalLanguage | mit | null |
Determines how well an adhesive "wets-out" or flows uniformly on the surface being bonded to. High surface energy materials, e.g. stainless steel and glass, offer excellent bonding characteristics. Low surface energy materials, e.g. molded polypropylene, resists "wetting-out" and require the use of more aggressive adhesive or a special surface treatment for proper bonding.
Surface energy quantifies the disruption of chemical bonds that occurs when a surface is created. In the physics of solids, surfaces must be intrinsically less energetically favourable than the bulk of a material; otherwise there would be a driving force for surfaces to be created, and surface is all there would be (see sublimation (physics)). Cutting a solid body into pieces disrupts its bonds, and therefore consumes energy. | null | minipile | NaturalLanguage | mit | null |
[Cardiac surgery in the elderly: perioperative care and operative strategies].
Caused by the age-dependent prevalence of cardiac diseases, the number of cardiac surgical interventions to geriatric patients is increasing. High life quality and life expectancy can be reached by cardiac operations. The advantage of cardiac surgical interventions is the decade's long positive effect. Accordingly also elderly benefit from complete revascularisation and from aortic valve replacement with biological prosthesis, which rarely degenerate in old age. A weak point is the surgical trauma, which can be reduced by less-invasive methods, such as OPCAB with aortic non-touch-technique, resulting in less than 1 % stroke. The indications for heart operations will be based on age-independent evidence-based guidelines. The decision for surgery is influenced by the expectation of the risk. This is defined by the co-morbidities and to lesser extent by the age per se. The operation risk can be calculated by risk-scores and hospital-specific data. The patient's expectations from the operation and his ability to overcome the accompanying stress must be thoroughly assessed. The operation must take place electively and at the right time. A good nutritional status and preoperative optimization of the organ functions are decisive for the prognosis. The blood-sugar-level must be optimized; thyroid function, (hidden) infections, anaemia and depression must be excluded or treated. The required screening tests should have been done already by the family doctor. The elderly are postoperatively susceptible to complications; especially low cardiac output, renal failure, respiratory insufficiency and stroke. Subsequently they need more intensive care. | null | minipile | NaturalLanguage | mit | null |
I bought this on day one. I had been wanting Crayon Physics Deluxe for a long time, and I had seen VVVVVV before but did not really know what it was. They are both awesome and the other games are good too. I wish I had gotten the first two bundles too, but I did not hear about them in time. In fact I just stumbled on the Humble Indie Bundle #3 when I was googling for some action script tutorials. Remember the more you pay the more likely they will continue having these awesome bundles. Also if you tell all your friends, this could be the most successful Humble Bundle yet.
They just announced every one who buys Humble Indie Bundle #3 can play Minecraft for free until August 14th! | null | minipile | NaturalLanguage | mit | null |
News in Brief
<bt>Reston founder Robert Simon, Jr., will be honored as a National Planning Pioneer and Reston will be named a National Historic Planning Landmark, according to the American Institute of Certified Planners (AICP). A bronze plaque will be cast and presented at the Lake Anne Village Center, Washington Plaza at 6 p.m. on Thursday, Sept. 5.
Simon will be honored for the his work in creating the nation's first planned unit community zone and is credited with introducing urban living to suburban sprawl when he purchased 6,700 acres in northern Virginia and founded Reston. Earlier this year, Reston was honored by the AICP for being "one of the finest examples of American 20th century conceptual new town planning."
<sh>Murder Suspect Nabbed
<bt>Authorities arrested Brian Orlando Sanchez-Maradiga, 21, on Aug. 4 in Raleigh, N.C., for the murder of a man in Brown's Chapel Park in Reston last year, police said.
The suspect, who does not have a known address, will be extradited to Virginia from North Carolina, Fairfax County Police said.
Sanchez-Maradiga is the fourth of four suspects arrested and accused of killing Fredy Del Carmen Reyes-Castillo last June 17. On that day, a couple walking in the park found Reyes-Castillo lying the parking lot with severe upper body trauma, police said. Authorities said all four suspects are members of the MS-13 gang.
<sh>Gang Meeting at Langston Hughes
<bt>Responding to concerns about escalating gang activity and youth violence in Hunter Mill-area schools, the Community Focus Group, a coalition of nearly 70 educators, youth-service, law enforcement and community leaders will hold a public meeting Wednesday, Sept. 4, at Langston Hughes Middle School from 7:30 p.m. until 9:30 p.m.
"As recent media reports have made clear, gang involvement is on the rise in Fairfax County," stated Supervisor Cathy Hudgins, in a release. "As responsible adults, we owe it to the children in our schools to make the community free of gangs and to thus create a safe environment for their development.
<sh>LHMS Students Plant Trees for Dogwood
<bt>In a tribute to Sept. 11, students from Langston Hughes Middle School in coordination with the Reston Association and the Reston Character Counts! Coalition will be leading Dogwood's student body in a replanting of the landscaping lost during the Reston elementary school's 2000 fire.
For four hours, beginning at 9 a.m., students will be planting six different species of trees, representing the six pillars of good character: trustworthiness, respect, responsibility, fairness, caring and citizenship.
The event will kick off with the dedication of a special tree representing the "Spirit of American Character." With the planting and dedication, the coalition and RA hopes to establish a precedent for sponsoring community service projects as a "fitting annual remembrance of the events of Sept. 11."
<sh>Pro-BRT Group Sees DEIS Bias
<bt>The Rapid Transit Action Committee, also known as MoveIt.org, which represent a group of civic and business interests in Fairfax and Loudoun counties, announced last week that the Draft Environmental Impact Study was "substantially flawed and must be redone." After a private review of the document, officials with the group have concluded there were serious problems with it, ranging from insufficient financial analysis to excessively optimistic assumptions on transit ridership. "The people promoting rail have interests in the outcome," said Tom Hirst, the RTRAC/MoveIt chairman, in a press conference last week to tout the benefits of BRT, an express bus rapid transit system. "A lot of what rail advocates have bought into is pure fantasy." The Reston Association and the Herndon Town Council, among others, have endorsed the rail option. | null | minipile | NaturalLanguage | mit | null |
Localized cystic disease of the kidney.
Localized cystic disease of the kidney is a benign nonsurgical condition. Its imaging and clinical features are characterized and differentiated from autosomal dominant polycystic kidney disease, multilocular cystic nephroma, and cystic neoplasm. Localized cystic disease was diagnosed in 18 patients on the basis of a review of imaging studies, clinical histories, and pathologic proof in four of the 18 patients. Average age at diagnosis was 54 years (age range, 24-83 years). Fifteen of the patients (83%) were men. CT was performed on 18 patients, sonography on nine, excretory urography on six, arteriography on four, and MR imaging on two. Localized cystic disease was unilateral in all patients and characterized by multiple cysts of various sizes separated by normal (or atrophic) renal tissue in a conglomerate mass suggestive of cystic neoplasm. In some patients, involvement of the entire kidney, which was suggestive of unilateral autosomal dominant polycystic kidney disease, was seen. No cysts were seen in the contralateral kidney in 14 patients, and only one or two scattered small cysts were present in four patients. Clinical presentations included hematuria, flank pain, palpable abdominal mass, and localized cystic disease as an incidental finding. None of the patients had a family history of autosomal dominant polycystic kidney disease. Ten patients underwent follow-up (follow-up range, 1-12 years); nine patients underwent imaging follow-up and one patient underwent clinical follow-up, which showed stability of disease. Four patients underwent nephrectomy for suspected renal neoplasm. Familiarity with localized cystic disease of the kidney and its imaging findings is important to avoid unnecessary surgery and to differentiate the disease from autosomal dominant polycystic kidney disease. | null | minipile | NaturalLanguage | mit | null |
---
abstract: |
This paper completes and extends some earlier studies by the author to show that Morris-Thorne wormholes are compatible with quantum field theory. The strategy is to strike a balance between reducing the size of the unavoidable exotic region and the degree of fine-tuning of the metric coefficients required to achieve this reduction, while simultaneously satisfying the constraints from quantum field theory. The fine-tuning also serves to satisfy various traversability criteria such as tidal constraints and proper distances through the wormhole. The degree of fine-tuning turns out to be a generic feature of the type of wormhole discussed.
PAC numbers: 04.20.Jb, 04.20.Gz
author:
- |
Peter K.F. Kuhfittig\
Department of Mathematics\
Milwaukee School of Engineering\
Milwaukee, WI 53202-3109 USA\
[email protected]\
title: 'Theoretical construction of Morris-Thorne wormholes compatible with quantum field theory'
---
Introduction
============
Wormholes are handles or tunnels in the spacetime topology linking two separate and distinct regions of spacetime. These regions may be part of our Universe or of different universes altogether. The pioneer work of Morris and Thorne [@MT88] has shown that macroscopic wormholes may be actual physical objects. Furthermore, such wormholes require the use of exotic matter to prevent self-collapse. Such matter is confined to a small region around the throat, a region in which the weak energy condition is violated. Since exotic matter is rather problematical, it is desirable to keep this region as small as possible. However, the use of arbitrarily small amounts of exotic matter leads to severe problems, as discussed by Fewster and Roman [@FR05a; @FR05b]. The discovery by Ford and Roman [@FR95; @FR96] that quantum field theory places severe constraints on the wormhole geometries has shown that most of the “classical" wormholes could not exist on a macroscopic scale. The wormholes described by Kuhfittig [@pK06; @pK08] are earlier attempts to strike a balance between two conflicting requirements, reducing the amount of exotic matter and fine-tuning the values of certain parameters. The purpose of this paper is to refine and solidify the earlier ideas, particularly the use of the quantum inequalities of Ford and Roman, here slightly extended, all with the aim of demonstating that wormholes, which are based on Einstein’s theory, are compatible with quantum field theory. The models discussed will therefore (1) satisfy all the constraints imposed by quantum field theory, (2) strike a reasonable balance between a small proper thickness of the exotic region and the degree of fine-tuning of the metric coefficients, Eq. (\[E:line\]) below, (3) minimize the assumptions on these metric coefficients, and (4) satisfy certain traversabilty criteria.
Problems with arbitrarily small amounts of exotic matter are also discussed in Ref. [@oZ07], but the author states explicitly that the issues discussed here and in Ref. [@pK06] are beyond the scope of his paper.
The problem
===========
Consider the general line element $$\label{E:line}
ds^2 =-e^{2\beta(r)}dt^2+e^{2\alpha(r)}dr^2+r^2(d\theta^2+
\text{sin}^2\theta\, d\phi^2),$$ where $\beta(r)\rightarrow 0$ and $\alpha(r)\rightarrow 0$ as $r\rightarrow \infty$. (We are using units in which $G=c=1$.) The function $\beta$ is called the *redshift function*; this function must be everywhere finite to avoid an event horizon at the throat. The function $\alpha$ is related to the *shape function* $b=b(r)$: $$\label{E:line2}
e^{2\alpha(r)}=\frac{1}{1-b(r)/r}.$$ (The shape function determines the spatial shape of the wormhole when viewed, for example, in an embedding diagram.) It now follows that $$\label{E:shape}
b(r)=r(1-e^{-2\alpha(r)}).$$ The minimum radius $r=r_0$ is the *throat* of the wormhole, where $b(r_0)=r_0$. As a result, $\alpha$ has a vertical asymptote at the throat $r=r_0$: $$\label{E:asymptote}
\lim_{r \to r_0+}\alpha(r)=+\infty.$$ So $\alpha(r)$ is assumed to be monotone decreasing near the throat. The qualitative features (again near the throat) of $\alpha(r)$, $\beta(r)$, and $-\beta(r)$, the reflection of $\beta(r)$ in the horizontal axis, are shown in Fig. 1. It is assumed that $\beta$ and $\alpha$ are twice differentiable with $\beta'(r)\ge 0$ and $\alpha'(r)<0$.
The next step is to list the components of the Einstein tensor in the orthonormal frame. From Ref. [@pK08], $$\label{E:E1}
G_{\hat{t}\hat{t}}=\frac{2}{r}e^{-2\alpha(r)}\alpha'(r)
+\frac{1}{r^2}(1-e^{-2\alpha(r)}),$$ $$\label{E:E2}
G_{\hat{r}\hat{r}}=\frac{2}{r}e^{-2\alpha(r)}\beta'(r)
-\frac{1}{r^2}(1-e^{-2\alpha(r)}),$$ and $$\label{E:E3}
G_{\hat{\theta}\hat{\theta}}=G_{\hat{\phi}\hat{\phi}}=
e^{-2\alpha(r)}\left[\beta''(r)
+\alpha'(r)\beta'(r)
+[\beta'(r)]^2+\frac{1}{r}\beta'(r)
-\frac{1}{r}\alpha'(r)\right].$$ Now recall that since the Einstein field equations $G_{\hat{\alpha}\hat{\beta}}=8\pi T_{\hat{\alpha}\hat{\beta}}$ in the orthonormal frame imply that the stress-energy tensor is proportional to the Einstein tensor, the only nonzero components are $T_{\hat{t}\hat{t}}=\rho,$ $T_{\hat{r}\hat{r}}=-\tau,$ and $T_{\hat{\theta}\hat{\theta}}=T_{\hat{\phi}\hat{\phi}}=p,$ where $\rho$ is the energy density, $\tau$ the radial tension, and $p$ the lateral pressure. The weak energy condition (WEC) may now be stated as follows: the stress-energy tensor $T_{\hat{\alpha}\hat{\beta}}$ must obey $$T_{\hat{\alpha}\hat{\beta}}\mu^{\hat{\alpha}}\mu^{\hat{\beta}}\ge0$$ for all time-like vectors and, by continuity, all null vectors. Using the radial outgoing null vector $\mu^{\hat{\alpha}}=(1,1,0,0)$, the condition becomes $T_{\hat{t}\hat{t}}+T_{\hat{r}\hat{r}}=
\rho-\tau\ge 0.$ So if the WEC is violated, then $\rho-\tau<0$. The field equations $G_{\hat{\alpha}\hat{\beta}}=
8\pi T_{\hat{\alpha}\hat{\beta}}$ now imply that $$\label{E:WEC}
\rho-\tau=\frac{1}{8\pi}\left(\frac{2}{r}e^{-2\alpha(r)}
\left[\alpha'(r)+\beta'(r)\right]\right).$$ Sufficiently close to the asymptote, $\alpha'(r)+\beta'(r)$ is clearly negative. (Recall that $\alpha'<0$ and $\beta'\ge 0$.) According to Ford and Roman [@FR95; @FR96], the exotic matter must be confined to a thin band around the throat. To satisfy these constraints, we would like the WEC to be satisfied outside of some small interval $[r_0,r_1]$. In other words, $$\label{E:FR1}
|\alpha'(r_1)|=\beta'(r_1),$$ $$\label{E:FR2}
\alpha'(r)+\beta'(r)<0\quad \text{for}\quad r_0<r<r_1,$$ and $$\label{E:FR3}
\alpha'(r)+\beta'(r)\ge 0 \quad \text{for} \quad r\ge r_1.$$ (See Fig. 1.) Condition (\[E:FR3\]) implies that $|\alpha'(r)|\le \beta'(r)$ for $r\ge r_1$. So if $\beta(r)\equiv$ constant, then $\alpha'(r)\equiv0$ for $r\ge
r_1$. In the neighborhood of $r=r_1$, we also require that $\alpha''(r)>0$, $\beta''(r)<0$, and $\alpha''(r)>|\beta''(r)|$. We now have the minimum requirements for constructing the type of wormhole that we are interested in.
Using the components of the stress-energy tensor allows us to restate Eqs. (\[E:E1\]) and (\[E:E2\]) in terms of $b=b(r)$: $$\label{E:stress1}
T_{\hat{t}\hat{t}}=\rho=\frac{b'(r)}{8\pi r^2}$$ and $$\label{E:stress2}
T_{\hat{r}\hat{r}}=-\tau
=-\frac{1}{8\pi}\left[\frac{b(r)}{r^3}-\frac{2\beta'(r)}{r}
\left(1-\frac{b(r)}{r}\right)\right].$$ Because of Eq. (\[E:stress1\]), we require that $b'(r)>0$. Eq. (\[E:line2\]) implies that $\alpha(r)=-\frac{1}{2}
\text{ln}(1-b(r)/r)$. From $$\label{E:alphaprime}
\alpha'(r)=\frac{1}{2}\frac{1}{1-b(r)/r}\frac{b'(r)-b(r)/r}{r},$$ we conclude that $b'(r_0)\le 1$ to keep $\alpha'(r)$ negative near the throat. In fact, $\lim_{r \to r_0+}\alpha'(r)=-\infty$. (The condition $b'(r_0)\le 1$ is called the *flare-out* condition in Ref. [@MT88].)
The extended quantum inequality (first version)
-----------------------------------------------
The sought-after compatibility with quantum field theory is based on the so-called quantum inequality in Ref. [@FR96], applied to different situations. (A modified version, based on Ref. [@FR05a], is given in Sec. 6.) This inequality deals with an inertial Minkowski spacetime without boundaries. If $u^{\mu}$ is the observer’s four-velocity (i.e., the tangent vector to a timelike geodesic), then $\langle T_{\mu\nu}u^{\mu}u^{\nu}\rangle$ is the expectation value of the local energy density in the observer’s frame of reference. It is shown that $$\label{E:FR}
\frac{\tau_0}{\pi}\int^{\infty}_{-\infty}
\frac{\langle T_{\mu\nu}u^{\mu}u^{\nu}\rangle d\tau}
{\tau^2+\tau_0^2}\ge -\frac{3}{32\pi^2\tau_0^4},$$ where $\tau$ is the observer’s proper time and $\tau_0$ the duration of the sampling time. (See Ref. [@FR96] for details.) Put another way, the energy density is sampled in a time interval of duration $\tau_0$ which is centered around an arbitrary point on the observer’s worldline so chosen that $\tau=0$ at this point. It is shown in Ref. [@FR96] that the inequality can be applied in a curved spacetime as long as $\tau_0$ is small compared to the local proper radii of curvature, as illustrated in Ref. [@FR96] by several examples. To obtain an estimate of the local curvature, we need to list the nonzero components of the Riemann curvature tensor in the orthonormal frame. From Ref. [@pK08] $$\label{E:Riemann1}
R_{\hat{r}\hat{t}\hat{r}\hat{t}}=e^{-2\alpha(r)}
\left(\beta''(r)-\alpha'(r)\beta'(r)
+\left[\beta'(r)\right]^2\right),$$ $$\label{E:Riemann2}
R_{\hat{\theta}\hat{t}\hat{\theta}\hat{t}}
=R_{\hat{\phi}\hat{t}\hat{\phi}\hat{t}}
=\frac{1}{r}
e^{-2\alpha(r)}\beta'(r),$$ $$\label{E:Riemann3}
R_{\hat{\theta}\hat{r}\hat{\theta}\hat{r}}
= R_{\hat{\phi}\hat{r}\hat{\phi}\hat{r}}
=\frac{1}{r}
e^{-2\alpha(r)}\alpha'(r),$$ and $$\label{E:Riemann4}
R_{\hat{\theta}\hat{\phi}\hat{\theta}\hat{\phi}}
=\frac{1}{r^2}\left(1-e^{-2\alpha(r)}\right).$$ Still following Ref. [@FR96], we need to introduce the following length scales over which various quantities change: $$\label{E:rsubm}
r_m \equiv\text{min}\left[r,
\left|\frac{b(r)}{b'(r)}\right|,
\frac{1}{|\beta'(r)|},
\left|\frac{\beta'(r)}{\beta''(r)}\right|\right].$$ The reason is that the above components of the Riemann curvature tensor can be reformulated as follows: $$\begin{gathered}
\label{E:newRiemann1}
R_{\hat{r}\hat{t}\hat{r}\hat{t}}=
\left(1-\frac{b(r)}{r}\right)\frac{1}
{\frac{\beta'(r)}{\beta''(r)}\frac{1}{\beta'(r)}}
-\frac{b(r)}{2r}\left(\frac{1}
{\frac{1}{\beta'(r)}\frac{b(r)}{b'(r)}}
-\frac{1}{r\frac{1}{\beta'(r)}}\right)\\
+\left(1-\frac{b(r)}{r}\right)
\frac{1}{\left(\frac{1}{\beta'(r)}\right)^2},\end{gathered}$$ $$\label{E:newRiemann2}
R_{\hat{\theta}\hat{t}\hat{\theta}\hat{t}}
=R_{\hat{\phi}\hat{t}\hat{\phi}\hat{t}}
=\left(1-\frac{b(r)}{r}\right)
\frac{1}{r\frac{1}{\beta'(r)}},$$ $$\label{E:newRiemann3}
R_{\hat{\theta}\hat{r}\hat{\theta}\hat{r}}
= R_{\hat{\phi}\hat{r}\hat{\phi}\hat{r}}
=\frac{b(r)}{2r}\left(
\frac{1}{r\frac{b(r)}{b'(r)}}
-\frac{1}{r^2}\right),$$ and $$\label{E:newRiemann4}
R_{\hat{\theta}\hat{\phi}\hat{\theta}\hat{\phi}}
=\frac{1}{r^2}\frac{b(r)}{r}.$$ When it comes to curvature, we are going to be primarily interested in magnitudes. So we let $R_{\text{max}}$ denote the magnitude of the maximum curvature. We know that the largest value of $(1-b(r)/r)$ and of $b(r)/r$ is unity; it follows from Eqs. (\[E:rsubm\])-(\[E:newRiemann4\]) that $R_{\text{max}}\le 1/r^2_m$ (disregarding the coefficient $\frac{1}{2}$). So the smallest radius of curvature $r_c$ is $$r_c\approx \frac{1}{\sqrt{R_{\text{max}}}}\ge r_m.$$ The point is that working on this scale, the spacetime is Minkowskian (at least approximately), so that inequality (\[E:FR\]) can be applied with an appropriate $\tau_0$.
As noted earlier, we assume that $b'(r)$ and hence $\rho$ are positive. Being nonnegative, it is suggested in Ref. [@FR96] that a bound can be obtained by Lorentz transforming to the frame of a radially moving *geodesic* observer who is moving with velocity $v$ relative to the static frame. In this “boosted frame" $$r'_c\approx \frac{1}{\sqrt{R'_{\text{max}}}}
\ge \frac{r_m}{\gamma},$$ where $\gamma=(1-v^2)^{-1/2}$, so that the spacetime should be approximately flat. The suggested sampling time is $$\tau_0=\frac{fr_m}{\gamma}\ll r'_c,$$ where $f$ is a scale factor such that $f\ll 1$. The energy density in the boosted frame is $$T_{\hat{0}'\hat{0}'}=\rho'=\gamma^2(\rho+v^2p_r),$$ where $v$ is the velocity of the boosted observer. It is stated in Ref. [@FR96] that in this frame the energy density does not change very much over the short sampling time and is therefore approximately constant: $$\begin{gathered}
\frac{\tau_0}{\pi}\int^{\infty}_{-\infty}
\frac{\langle T_{\mu\nu}u^{\mu}u^{\nu}\rangle d\tau}
{\tau^2+\tau_0^2}\approx
\langle T_{\mu\nu}u^{\mu}u^{\nu}\rangle
\frac{\tau_0}{\pi}\int^{\infty}_{-\infty}
\frac{d\tau}{\tau^2+\tau_0^2}\\
=\langle T_{\mu\nu}u^{\mu}u^{\nu}\rangle=\rho'
\ge -\frac{3}{32\pi^2\tau_0^4}.\end{gathered}$$ From Eqs. (\[E:stress1\]) and (\[E:stress2\]), $$\rho'=\frac{\gamma^2}{8\pi r^2}\left[b'(r)
-v^2\frac{b(r)}{r}+v^2r(2\beta'(r))
\left(1-\frac{b(r)}{r}\right)\right].$$ In order for $\rho'$ to be negative, $v$ has to be sufficiently large: $$\label{E:velocity}
v^2>\frac{b'(r)}{\frac{b(r)}{r}-2r\beta'(r)
\left(1-\frac{b(r)}{r}\right)};$$ (observe that $v^2$ is dimensionless.) In particular, at the throat, $v^2>b'(r_0)$. Given $b(r)$, inequality (\[E:velocity\]) places a restriction on $\beta'(r)$. We will return to this point in Sec. \[S:solution\].
Next, from $$\frac{3}{32\pi^2\tau_0^4}\ge -\rho'$$ we have $$\frac{32\pi^2\tau_0^4}{3}\le\\\frac{8\pi r^2}{\gamma^2}
\left[v^2\frac{b(r)}{r}-b'(r)-v^2r(2\beta'(r))
\left(1-\frac{b(r)}{r}\right)\right]^{-1}.$$ Using $\tau_0=fr_m/\gamma$ and dividing both sides by $r^4$, we have (disregarding a small coefficient) $$\frac{f^4r_m^4}{r^4\gamma^4} \le\\\frac{1}{r^2\gamma^2}
\left[v^2\frac{b(r)}{r}-b'(r)-2v^2r\beta'(r)
\left(1-\frac{b(r)}{r}\right)\right]^{-1}$$ and, after inserting $l_p$ to produce a dimensionless quantity, $$\label{E:genQI}
\frac{r_m}{r}\le
\left(\frac{1}{v^2\frac{b(r)}{r}-b'(r)-2v^2r\beta'(r)
\left(1-\frac{b(r)}{r}\right)}\right)^{1/4}\\
\frac{\sqrt{\gamma}}{f}
\left(\frac{l_p}{r}\right)^{1/2}.$$ This is the first version of the extended quantum inequality. At the throat, where $b(r_0)=r_0$, inequality (\[E:genQI\]) reduces to Eq. (95) in Ref. [@FR96]: $$\label{E:QI}
\frac{r_m}{r_0}\le\left(\frac{1}{v^2-b'(r_0)}\right)^{1/4}
\frac{\sqrt{\gamma}}{f}\left(\frac{l_p}{r_0}\right)^{1/2}.$$ Observe that inequality (\[E:QI\]) is trivially satisfied if $b'(r_0)=1$ but not if $b'(r_0)<1$. In view of inequality (\[E:genQI\]) and the tidal constraints in the next subsection, we would like $b'(r)$ to be close to unity in the exotic region. (The need for $b'(r_0)$ to be close to 1 is also pointed out in Ref. [@FR96].)
The tidal constraints
---------------------
Much of what follows is based on the discussion in Ref. [@MT88]. In particular, we have for the radial tidal constraint $$\begin{gathered}
\label{E:radial}
\left|R_{\hat{1}'\hat{0}'\hat{1}'\hat{0}'}\right|=
\left|R_{\hat{r}\hat{t}\hat{r}\hat{t}}\right|\\
=e^{-2\alpha(r)}
\left|\beta''(r)-\alpha'(r)\beta'(r)
+\left[\beta'(r)\right]^2\right|
\le \frac{g_{\oplus}}{c^2\times 2\,\text{m}}
\approx (10^8\,\text{m})^{-2},\end{gathered}$$ that is, assuming a traveler with a height of 2 m. This constraint is trivially satisfied if $\beta(r)\equiv$ constant, referred to as the zero-tidal-force solution in Ref. [@MT88]. The lateral tidal constraints are (reinserting $c$) $$\begin{gathered}
\label{E:lateral}
\left|R_{\hat{2}'\hat{0}'\hat{2}'\hat{0}'}\right|
=\left|R_{\hat{3}'\hat{0}'\hat{3}'\hat{0}'}\right|
=\gamma^2\left|R_{\hat{\theta}\hat{t}\hat{\theta}\hat{t}}\right|
+\gamma^2\left(\frac{v}{c}\right)^2\left|
R_{\hat{\theta}\hat{r}\hat{\theta}\hat{r}}\right|\\
=\gamma^2\left(\frac{1}{r}e^{-2\alpha(r)}\beta'(r)\right)
+\gamma^2\left(\frac{v}{c}\right)^2
\left(\frac{1}{r}e^{-2\alpha(r)}\alpha'(r)\right)
\le (10^8\,\text{m})^{-2};\end{gathered}$$ here $\gamma^2=1/\left[1-(v/c)^2\right]$.
Returning to Eq. (\[E:shape\]), we have for the shape function, $$\label{E:bprime}
b'(r_0)=\frac{d}{dr}\left[r(1-e^{-2\alpha(r)})\right]_{r=r_0}\\
=2r_0e^{-2\alpha(r_0)}\alpha'(r_0)+1-e^{-2\alpha(r_0)}.$$ In order for $b'(r_0)\approx 1,$ we require that $$\lim_{r \to r_0+}e^{-2\alpha(r)}\alpha'(r)=0.$$ As a consequence, the radial tidal constraint (\[E:radial\]) is satisfied at the throat, while the lateral tidal constraints (\[E:lateral\]) merely constrain the velocity of the traveler in the vicinity of the throat.
One of the consequences of the condition $b'(r_0)\approx 1$ is that the wormhole will flare out very slowly, so that the coordinate distance from $r=r_0$ to $r=r_1$ will be much less than the proper distance.
The exotic region
-----------------
We saw in the last section that $\alpha$ has to go to infinity fast enough so that $\lim_{r \to r_0+}e^{-2\alpha(r)}\alpha'(r)=0.$ At the same time, $\alpha$ has to go to infinity slowly enough so that the proper distance $$\ell(r)=\int\nolimits_{r_0}^{r}e^{\alpha(r')}dr'$$ is finite. Then by the mean-value theorem, there exists a value $r=r_2$ such that $$\ell(r)=e^{\alpha(r_2)}(r-r_0),\quad r_0<r_2<r.$$ In particular, $\ell(r_0)=0$ and $$\label{E:meanvalue}
\ell(r_1)=e^{\alpha(r_2)}(r_1-r_0).$$ With this information we can examine the radial tidal constraint at $r=r_1.$ From Eq. (\[E:Riemann1\]) $$\begin{gathered}
\label{E:radial1}
|R_{\hat{r}\hat{t}\hat{r}\hat{t}}|=e^{-2\alpha(r_1)}
\left|\beta''(r_1)-\alpha'(r_1)\beta'(r_1)
+\left[\beta'(r_1)\right]^2\right|\\
=e^{-2\alpha(r_1)}\left|\beta''(r_1)
-\alpha'(r_1)[-\alpha'(r_1)]
+\left[\alpha'(r_1)\right]^2\right|\end{gathered}$$ by Eq. (\[E:FR1\]). So by inequality (\[E:radial\]), $$|R_{\hat{r}\hat{t}\hat{r}\hat{t}}|
=e^{-2\alpha(r_1)}\left|\beta''(r_1)+
\alpha'(r_1)\alpha'(r_1)
+\left[\alpha'(r_1)\right]^2\right|\\
\le (10^8\text{m})^{-2}.$$ Since $e^{-2\alpha(r)}$ is strictly increasing, it follows that $$e^{-2\alpha(r_2)}\left|\beta''(r_1)
+2\left[\alpha'(r_1)\right]^2\right|
<10^{-16} \text{m}^{-2}.$$ From Eq. (\[E:meanvalue\]), we now get the following: $$\frac{(r_1-r_0)^2}{[\ell(r_1)]^2}
\left|\beta''(r_1)+2\left[\alpha'(r_1)\right]^2\right|
<10^{-16}\text{m}^{-2}$$ and $$\label{E:abs1}
\left|\beta''(r_1)+2\left[\alpha'(r_1)\right]^2\right|
<\frac{[\ell(r_1)]^2}{10^{16}(r_1-r_0)^2}.$$ As a consequence, $$\label{E:abs2}
\beta''(r_1)+2\left[\alpha'(r_1)\right]^2
<\frac{[\ell(r_1)]^2}{10^{16}(r_1-r_0)^2}$$ or $$\label{E:abs3}
\beta''(r_1)+2\left[\alpha'(r_1)\right]^2
>-\frac{[\ell(r_1)]^2}{10^{16}(r_1-r_0)^2}.$$ So if either condition (\[E:abs2\]) or condition (\[E:abs3\]) is satisfied, then so is condition (\[E:abs1\]).
A class of models; fine-tuning
==============================
To estimate the size of the exotic region, we need some idea of the magnitude of $\alpha(r)$, which depends on the specific model chosen. The only information available is that $\alpha(r)$ increases slowly enough as $r\rightarrow
r_0+$ to keep $\int\nolimits_{r_0}^{r}e^{\alpha(r')}dr'$ finite. (We have not made any assumptions regarding $\beta(r)$, except for some of the basic requirements.)
First we need to recall that Morris-Thorne wormholes are not just concerned with traversability in general but more specifically with humanoid travelers. According to Ref. [@MT88], the space station should be far enough away from the throat so that $$\label{E:flat}
1-\frac{b(r)}{r}=e^{-2\alpha(r)}\approx 1,$$ making the space nearly flat. Another condition involves the redshift function: at the station we must also have $$\label{E:station1}
|\beta'(r)|\le g_{\oplus}/\left(c^2\sqrt{1-b(r)/r}\right).$$ It will be seen below that for our wormhole, the first condition, Eq. (\[E:flat\]), is easily satisfied. By condition (\[E:FR3\]), as well as Fig. 1, $|\alpha'(r)|\le \beta'(r)$ for $r\ge r_1$. So if $1-b(r)/r\approx 1,$ then we have $$\label{E:station2}
|\alpha'(r)|<10^{-16}\,\text{m}^{-1}$$ at the station. This inequality should give us at least a rough estimate of the distance to the station: for large $r$, $|\alpha(r)|\sim|\beta(r)|$, since both $\alpha$ and $\beta$ go to zero. It must be kept in mind, however, that the inequality $|\alpha'(r)|\le \beta'(r)$ implies that this procedure does underestimate the distance. The main reason for using $\alpha$ in the first place is to avoid making additional assumptions involving $\beta$. Instead, $\beta$ can be left to its more obvious role, helping to meet the tidal constraints and the quantum inequality, Eq. (\[E:genQI\]). We will return to this point after discussing $\alpha.$
Consider next a class of models based on the following set of functions: $$\label{E:hyper}
\alpha(r)=a\,\text{ln}\left(\frac{1}{(r-r_0)^b}
+\sqrt{\frac{1}{(r-r_0)^{nb}}+1}\right).$$ (Other models are discussed in Ref. ([@pK08]).) For convenience let us concentrate for now on the special case $n=2$ and return to Eq. (\[E:hyper\]) later. For $n=2$, the equation becomes $$\label{E:sinh}
\alpha(r)=a\,\text{sinh}^{-1}\frac{1}{(r-r_0)^b},
\quad b>\frac{1}{2a}.$$ The need for the assumption $b>1/(2a)$ comes from the shape function $$b(r)=r\left(1-e^{-2a\,\text{sinh}^{-1}
[1/(r-r_0)^b]}\right):$$ $$\begin{gathered}
b'(r)=1-e^{-2a\,\text{sinh}^{-1}[1/(r-r_0)^b]}\\
+r\left(-e^{-2a\,\text{sinh}^{-1}[1/(r-r_0)^b]}\right)
\frac{2ab}{(r-r_0)\sqrt{(r-r_0)^{2b}+1}};\end{gathered}$$ $b'(r)\rightarrow 1$ as $r\rightarrow r_0$, as long as $b>1/(2a).$ To see this, it is sufficient to examine $$e^{-2a\,\text{sinh}^{-1}[1/(r-r_0)^b]}\frac{1}{r-r_0}$$ as $r\rightarrow r_0$: $$\begin{gathered}
\frac{1}{\left[\frac{1}{(r-r_0)^b}+\sqrt{\frac{1}{(r-r_0)^{2b}}+1}
\right]^{2a}}\frac{1}{r-r_0}\\
=\frac{1}{\frac{1}{(r-r_0)^{2ab}}\left[1+(r-r_0)^b
\sqrt{\frac{1}{(r-r_0)^{2b}}+1}\right]^{2a}}\frac{1}{r-r_0}\\
=\frac{1}{\frac{1}{(r-r_0)^{2ab-1}}
\left[1+\sqrt{1+(r-r_0)^{2b}}\right]^{2a}}. \end{gathered}$$ So if $2ab-1>0$, then the second factor in the denominator becomes negligible for $r\approx r_0$. The result is $$e^{-2a\,\text{sinh}^{-1}[1/(r-r_0)^b]}\frac{1}{r-r_0}
\sim(r-r_0)^{2ab-1}\rightarrow 0.$$ For computational purposes, however, we will simply let $b=1/(2a)$. Consider next, $$\label{E:D1alpha}
\alpha'(r)=-\frac{ab}{(r-r_0)\sqrt{(r-r_0)^{2b}+1}},\quad r>r_0,$$ and $$\label{E:D2alpha}
\alpha''(r)=\frac{ab\left[(1+b)(r-r_0)^{2b}+1\right]}
{(r-r_0)^2\left[(r-r_0)^{2b}+1\right]^{3/2}}.$$
We know that the wormhole flares out very slowly at the throat, which suggests assigning a small coordinate distance to the exotic region, at least initially. A good choice is $r-r_0=0.000001$ m, as in Ref. [@pK08]. Then from Eqs. (\[E:D1alpha\]) and (\[E:D2alpha\]), we get $$\alpha'(r_1)\approx -\frac{ab}{r_1-r_0}\quad \text{and}
\quad \alpha''(r_1)\approx\frac{ab}{(r_1-r_0)^2}.$$ For future reference, let us replace $ab$ by $A$: $$\label{E:alphageneral}
\alpha'(r_1)\approx -\frac{A}{r_1-r_0}\quad \text{and}
\quad \alpha''(r_1)\approx \frac{A}{(r_1-r_0)^2}.$$ Since we also want $\alpha''(r_1)>|\beta''(r_1)|$ \[or $\alpha''(r_1)>-\beta''(r_1)$\], we have in view of inequality (\[E:abs2\]), $$\label{E:main}
\frac{A}{(r_1-r_0)^2}>-\beta''(r_1)>\frac{2A^2}{(r_1-r_0)^2}-
\frac{[\ell(r_1)]^2}{10^{16}(r_1-r_0)^2}.$$ Conversely, the inequality $$\frac{2A^2}{(r_1-r_0)^2}-\frac{[\ell(r_1)]^2}
{10^{16}(r_1-r_0)^2}\\
=2[\alpha'(r_1)]^2-\frac{[\ell(r_1)]^2}{10^{16}(r_1-r_0)^2}
<-\beta''(r_1)$$ implies condition (\[E:abs2\]). Since $-\beta''(r_1)<
\alpha''(r_1)$, we conclude that inequality (\[E:main\]) is valid if, and only if, condition (\[E:abs2\]) is met.
Inequality (\[E:main\]) now implies that $$2A^2-A-\frac{[\ell(r_1)]^2}{10^{16}}<0.$$ The critical values are $$A=\frac{1\pm\sqrt{1+\frac{8[\ell(r_1)]^2}{10^{16}}}}{4}.$$ Hence $$\frac{1-\sqrt{1+\frac{8[\ell(r_1)]^2}{10^{16}}}}{4}<A<
\frac{1+\sqrt{1+\frac{8[\ell(r_1)]^2}{10^{16}}}}{4}.$$ Returning to the condition $b'(r_0)\le 1$ for a moment, note that $ab$ and hence $A$ must exceed 1/2. It follows that $$\label{E:Afinal}
\frac{1}{2}<A<\frac{1+\sqrt{1+\frac{8[\ell(r_1)]^2}{10^{16}}}}{4}$$ and, replacing $A$, $$\label{E:abfinal}
\frac{1}{2}<ab<\frac{1+\sqrt{1+\frac{8[\ell(r_1)]^2}{10^{16}}}}{4}.$$ The left inequality confirms that $b>1/(2a)$. This solution shows that considerable fine-tuning is required. We will return to this point in Sec. \[S:finetune\].
Finally, observe that with the extra condition $|\beta''(r_1)|
<\alpha''(r_1)$, the qualitative features in Fig. 1 are retained, so that no additional assumptions are needed.
Letting $b=1/(2a)$ once again for computational purposes, we now have $$\ell(r_1)=\int_{r_0}^{r_0+0.000001}
e^{a\,\text{sinh}^{-1}[1/(r-r_0)^{1/(2a)}]}dr.$$ These values change very little with $a$. For example, if $a$ ranges from 0.1 to 0.5, then $\ell(r_1)$ ranges from 0.0021 m to 0.0028 m; $\ell(r_1)$ is much larger than $r_1-r_0$, a consequence of the slow flaring out. From inequality (\[E:station2\]) we can estimate the distance $r_s$ to the space station: if $|\alpha'(r_s)|=10^{-16}\,\text{m}^{-1}$, then $r_s=70\,000$ km. Of course, we can always reduce the coordinate distance. Thus for $r_1-r_0=0.000000001\,\text{m}$ and $a=0.5$, we get $\ell(r_1)=0.000089\,\text{m}<0.1\,\text{mm}$.
A good alternative is to use Eq. (\[E:hyper\]), subject to the condition $$nab-b+\frac{1}{2}nb>1.$$ (As before, this condition comes from the requirement that $b'(r_0)\le 1$; in fact, if $n=2,$ we are back to $2ab>1$.) For example, retaining $r_1-r_0=0.000001\,\text{m}$, if $a=0.2$ and $b=1$, then $nb=2.857$. These values yield $\ell(r_1)\approx 0.0000725\,\text{m}<0.1\,\text{mm}.$ The corresponding distance $r_s$, obtained from $\alpha'(r)$ \[now referring to Eq. (\[E:hyper\])\], is about $45\,000$ km. Both $\ell(r_1)$ and $r_s$ are relatively small.
Using the equation $nab-b+\frac{1}{2}nb=1$ to eliminate $n$ in Eq. (\[E:hyper\]) shows that further reductions in $\ell(r_1)$ are only significant if $a$ and $b$ get unrealistically close to zero. So practically speaking, a further reduction in the proper distance $\ell(r_1)$ requires a reduction in the coordinate distance $r_1-r_0$.
Returning to the radial tidal constraint, based on experience with specific functions (as in Ref. [@pK06]), $\left|R_{\hat{r}\hat{t}\hat{r}\hat{t}}\right|$ is likely to reach its peak just to the right of $r=r_1$. The simplest way to handle this problem is to tighten the constraint in Eq. (\[E:radial\]) at $r=r_1$ by reducing the right side. This change increases the degree of fine-tuning in condition (\[E:Afinal\]).
A final consideration is the time dilation near the throat. Denoting the proper distance by $\ell$ and the proper time by $\tau$, as usual, we let $\gamma v=d\ell/d\tau$, so that $d\tau=d\ell/(\gamma v)$. Assume that $\gamma\approx 1$. Since $d\ell=e^{\alpha(r)}dr$ and $d\tau=e^{\beta(r)}dt$, we have for any coordinate time interval $\Delta t$: $$\Delta t=\int\nolimits_{t_a}^{t_b}dt=
\int\nolimits_{\ell_a}^{\ell_b}e^{-\beta(r)}\frac{d\ell}{v}=
\int\nolimits_{r_a}^{r_b}\frac{1}{v}e^{-\beta(r)}e^{\alpha(r)}
dr.$$ From Eq. (\[E:hyper\]), we have on the interval $[r_0, r_1]$ $$\Delta t= \int_{r_0}^{r_1}\frac{1}{v}
e^{-\beta(r)}\left(\frac{1}{(r-r_0)^b}
+\sqrt{\frac{1}{(r-r_0)^{nb}}+1}\right)^adr.$$ Since $\beta(r)$ is finite, the small size of the interval $[r_0,r_1]$ implies that $\Delta t$ is going to be relatively small for a wide variety of choices for $a$ and $b$.
The fine-tuning problem in general {#S:finetune}
==================================
The forms of inequalities (\[E:Afinal\]) and (\[E:abfinal\]) suggest that the degree of fine-tuning encountered is a general property of the type of wormhole being considered, namely wormholes for which $b'(r_0)\le 1$ and $\alpha(r)=A\,\text{ln}f(r-r_0),$ where (generalizing from earlier cases) $f(r-r_0)|_{r=r_0}$ is undefined $(+\infty)$ and $f(\frac{1}{r-r_0})|_{r=r_0}$ is a constant (possible zero). If we also assume that $g(r-r_0)=f(\frac{1}{r-r_0})$ can be expanded in a Maclaurin series, then we have for $r\approx r_0$, $$\begin{gathered}
f\left(\frac{1}{r-r_0}\right)=g(r-r_0)=a_0+a_1(r-r_0)\\
+a_2(r-r_0)^2+a_3(r-r_0)^3+\cdot\cdot\cdot
\approx a_0+a_1(r-r_0).\end{gathered}$$ It follows that $$f(r-r_0)=a_0+\frac{a_1}{r-r_0}$$ near the throat. So $$\alpha(r)=A\,\text{ln}\left(a_0+\frac{a_1}{r-r_0}\right),$$ $$\label{E:der1}
\alpha'(r_1)=\frac{-Aa_1}{a_0+\frac{a_1}{r_1-r_0}}
\frac{1}{(r_1-r_0)^2}\sim -\frac{A}{r_1-r_0},$$ and $$\label{E:der2}
\alpha''(r_1)\sim\frac{A}{(r_1-r_0)^2}.$$
To show that $b'(r_0)\le 1,$ we need to show that $e^{-2\alpha(r)}\alpha'(r)\rightarrow 0$ as $r\rightarrow
r_0$: $$\begin{gathered}
e^{-2A\,\text{ln}[a_0+a_1/(r-r_0)]}\frac{-Aa_1}
{a_0+\frac{a_1}{r-r_0}}\frac{1}{(r-r_0)^2}\\
=\frac{1}{\left(a_0+\frac{a_1}{r-r_0}\right)^{2A}}
\frac{-Aa_1}{a_0+\frac{a_1}{r-r_0}}
\frac{1}{(r-r_0)^2}
=\frac{-Aa_1}{\left(a_0+\frac{a_1}{r-r_0}\right)^{2A+1}
(r-r_0)^2}\\
=\frac{-Aa_1}{\left[\left(a_0+\frac{a_1}{r-r_0}\right)
(r-r_0)\right]^{2A+1}\frac{(r-r_0)^2}{(r-r_0)^{2A+1}}}
=\frac{-Aa_1}{[a_0(r-r_0)+a_1]^{2A+1}
\frac{1}{(r-r_0)^{2A-1}}}. \end{gathered}$$ The first factor in the denominator becomes negligle for $r\approx r_0$ as long as $2A-1>0$ and $A>\frac{1}{2}.$ We obtain $$e^{-2\alpha(r)}\alpha'(r)\sim(r-r_0)^{2A-1}
\rightarrow 0.$$ Comparing Eqs. (\[E:der1\]) and (\[E:der2\]) to Eq. (\[E:alphageneral\]), we conclude that $$\label{E:finetuning}
\frac{1}{2}<A<\frac{1+\sqrt{1+\frac{8[\ell(r_1)]^2}
{10^{16}}}}{4}.$$ So the amount of fine-tuning required appears to be a general property of wormholes of the present type. (Exactly which parameter needs fine-tuning depends on the precise form of $f(r-r_0)$.) While the degree of fine-tuning considered so far is quite severe, it is considerably milder than most of the cases discussed in Ref. [@FR05a].
The solution {#S:solution}
============
The discussion of Morris-Thorne wormholes in Ref. [@MT88] is concerned not just with traversability but, more specifically, with traversability by humanoid travelers. So the length of the trip, possible time dilations, and the tidal constraints are important considerations.
The first part of this paper deals with the size of the unavoidable exotic region around the throat. It was found that the size can be reduced almost indefinitely by carefully fine-tuning $\alpha=\alpha(r)$ or, equivalently, the shape function $b=b(r)$. The degree of fine-tuning required of some parameter turns out to be a general property of the type of wormhole considered. To achieve this fine-tuning, it is necessary to assume that $b'(r)$ is close to unity near the throat. This assumption proved to be sufficient to satisfy the tidal constraints.
Concerning the quantum inequalities, if $b'(r_0)=1$, then inequality (\[E:QI\]) is trivially satisfied at or near the throat. Away from the throat that may not be the case. Fortunately, we have made no assumptions on $\beta=\beta(r)$ beyond the basic requirements, no event horizon and $\beta'(r)\ge |\alpha'(r)|$ for $r\ge r_1$. For convenience, we restate inequalities (\[E:velocity\]) and (\[E:genQI\]), $$\label{E:velocity2}
v^2>\frac{b'(r)}{\frac{b(r)}{r}-2r\beta'(r)
\left(1-\frac{b(r)}{r}\right)},$$
$$\label{E:last}
\frac{r_m}{r}\le
\left(\frac{1}{v^2\frac{b(r)}{r}-b'(r)-2v^2r\beta'(r)
\left(1-\frac{b(r)}{r}\right)}\right)^{1/4}\\
\frac{\sqrt{\gamma}}{f}
\left(\frac{l_p}{r}\right)^{1/2},$$
where $v$ is the velocity of the radially moving geodesic observer. Since $\beta'(r)>0$, it now becomes evident that $\beta'(r)$ can be adjusted (or constructed “by hand") to become part of the fine-tuning strategy: according to Fig. 2, for a typical shape function
$b=b(r)$, the slope of the tangent line at $(r, b(r))$ is less than the slope $b(r)/r$ of the chord extending from the origin to $(r, b(r))$. This allows us to construct (or adjust) $\beta'(r)$ so that $$\label{E:construct}
\frac{b(r)}{r}-b'(r)-2r\beta'(r)\left(1-\frac{b(r)}
{r}\right)$$ is 0 or very nearly 0. (According to Eq. (\[E:alphaprime\]), $\beta'(r)$ is large enough for $r>r_1$.) Hence the right-hand side of inequality (\[E:velocity2\]) is 1 or very nearly 1, thereby forcing $v$ to be 1 or very nearly 1. As a consequence, the denominator on the right-hand side of inequality (\[E:last\]) is 0 or very nearly 0; so the inequality is satisfied for any $r_m$.
*Remark:* As noted earlier, at $r=r_0$, inequality (\[E:last\]) is trivially satisfied. Similarly, at $r=r_1$, expression (\[E:construct\]) is zero since $\beta'(r_1)=|\alpha'(r_1)|$. To the right of $r_1$, $\beta'(r)$ is large enough to overtake $b(r)/r-b'(r)$ and can therefore be adjusted to produce 0 or very nearly 0. Inside the small interval $[r_0,r_1]$, however, it may be necessary to fine-tune $b(r)$ to keep $b'(r)$ close to 1 inside the interval, or, which amounts to the same thing, $\alpha(r)$ must turn sharply upward after crossing $r=r_1$ from the right. (Recall the qualitative features in Fig. 1.)
Observe that, given any particular $b=b(r)$, the choice $\beta \equiv$ constant is not likely to work, basically in agreement with the analysis in Ref. [@FR96], since the original wormhole models in Ref. [@MT88] all assumed a constant $\beta$, at least near the throat.
Since inequality (\[E:last\]) is satisfied, the radius of the throat, $r=r_0$, is macroscopic since $r_m$ includes $r_0$. The wormholes satisfy the various traversability criteria for humanoid travelers. All the while the exotic region is made as small as possible while keeping the degree of fine-tuning within reasonable bounds. The models discussed have led to the following promising results: approximately 0.1 mm for the proper thickness of the exotic region, corresponding to a distance much less than $100\,000$ km to the space station. By decreasing the coordinate distance, it is theoretically possible to decrease the proper thickness of the exotic region indefinitely. While the decrease may be thought of as an engineering challenge, the fact remains that the concomitant increase in the degree of fine-tuning would eventually exceed any practical limit.
Additional remarks: the extended quantum inequality (second version)
====================================================================
The extended quantum inequality discussed in Subsection 2.1 is not the most general form available: a version of the original quantum inequality based on the violation of the null energy condition is obtained in Ref. [@FR05a]. This inequality, about to be extended, eliminates the need for a boosted frame since it features a static observer.
For present purposes it is sufficient to note that for the null vector $\textbf{k}=\textbf{e}_{\hat{t}}+\textbf{e}_{\hat{r}}$, $$T_{\hat{\alpha}\hat{\beta}}k^{\hat{\alpha}}k^{\hat{\beta}}
=\rho-\tau=-\frac{e^{2\beta(r)}}{8\pi r}\frac{d}{dr}
\left[e^{-2\beta(r)}\left(1-\frac{b(r)}{r}\right)\right],$$ which is readily obtained from Eq. (\[E:WEC\]). It follows from the subsequent discussion in Ref. [@FR05a] that $$\frac{e^{2\beta(r)}}{8\pi rl^2_p}\frac{d}{dr}
\left[e^{-2\beta(r)}\left(1-\frac{b(r)}{r}\right)\right]
\le \frac{C}{\tau^4_0}$$ after inserting the Planck length $l_p$. The constant $C$ is given in Eq. (7) of Ref. [@FR05a]. It is assumed that $\tau_0=f\ell_{min}$, where $\ell_{min}$ is the proper minimum length scale. Taking the derivative and solving for $\ell_{min}/r$, we obtain $$\frac{\ell_{min}}{r}\le
\left(\frac{1}{\frac{b(r)}{r}-b'(r)-2r\beta'(r)
\left(1-\frac{b(r)}{r}\right)}\right)^{1/4}\\
\frac{1}{f}\left(\frac{l_p}{r}\right)^{1/2}
(8\pi C)^{1/4}.$$ According to Ref. [@FR05a], $(8\pi C)^{1/4}\approx 3.2$. Since we are primarily interested in estimating orders of magnitude, we now have inequality (\[E:last\]) in Sec. 5 with $v$ and $\gamma$ omitted, while $\ell_{min}$ replaces $r_m$: $$\frac{\ell_{min}}{r}\le
\left(\frac{1}{\frac{b(r)}{r}-b'(r)-2r\beta'(r)
\left(1-\frac{b(r)}{r}\right)}\right)^{1/4}\\
\frac{1}{f}\left(\frac{l_p}{r}\right)^{1/2}.$$ As a result, our conclusions are unaltered.
[10]{} M.S. Morris and K.S. Thorne, Wormholes in spacetime and their use for interstellar travel: A tool for teaching general relativity, *Am. J. Phys.* **56**, pp. 395-412, 1988. C.J. Fewster and T.A. Roman, On wormholes with arbitrarily small quantities of exotic matter, *Phys. Rev. D* **72**, Artivle ID 044023 (15 pages), 2005. C.J. Fewster and T.A. Roman, Problems with wormholes which involve arbitrarily small amounts of exotic matter, arXiv: gr-qc/0510079. L.H. Ford and T.A. Roman, Averaged energy conditions and quantum inequalities, *Phys. Rev. D* **51**, pp. 4277-4286, 1995. L.H. Ford and T.A. Roman, Quantum field theory constrains traversable wormhole geometries, *Phys. Rev. D* **53**, pp. 5496-5507, 1996. P.K.F. Kuhfittig, More on wormholes supported by small amounts of exotic matter, *Phys. Rev. D* **73**, Article ID 084014 (5 pages), 2006. P.K.F. Kuhfittig, Viable models of traversable wormholes supported by small amounts of exotic matter, *Int. J. Pure Appl. Math.* **44**, pp. 467-482, 2008. O.B. Zaslavskii, Traversable wormholes: Minimum violations of the null energy conditions revisited, *Phys. Rev. D* **76**, Article ID 044017 (6 pages), 2007.
| null | minipile | NaturalLanguage | mit | null |
---
abstract: 'The eigen mode of spin oscillations with $\omega\simeq \sqrt{58/35}\Delta$ is predicted to exist besides already known spin waves with $\omega \simeq\Delta /\sqrt{5}$ in the triplet superfluid neutron condensate in the inner core of neutron stars. The new mode is kinematically able to decay into neutrino pairs through neutral weak currents. The problem is considered in BCS approximation for the case of $^{3}P_{2}-$$^{3}F_{2}$ pairing with a projection of the total angular momentum $m_{j}=0$ which is conventionally considered as preferable one at supernuclear densities.'
address: 'Institute of Terrestrial Magnetism, Ionosphere and Radio Wave Propagation RAS, 142190 Troitsk, Moscow Region, Russia'
author:
- 'L. B. Leinson'
title: 'New eigen-mode of spin oscillations in the triplet superfluid condensate in neutron stars'
---
Neutron star, Superfluidity, Spin waves
A superfluidity of the inner core of neutron stars plays a crucial role in theirs cooling scenario. The energy gap $\Delta $ arising in the quasiparticle spectrum below the critical condensation temperature $T_{c}$ suppresses the most of neutrino emission mechanisms [@YL]. According to the minimal cooling paradigm [@Page04; @Page09; @SY; @PPLS], under these conditions, the most efficient energy losses from the star volume can take place at a recombination of thermal excitations in the form of broken Cooper pairs. The neutrino emission at the pair-recombination processes occurs through neutral weak currents in the axial channel of weak interactions[^1] and can be very efficient, in the triplet superfluid neutron liquid, a few below the critical temperature [@YKL]. However, the corresponding neutrino emissivity falls rapidly with lowering of the temperature because the number of broken pairs, having the excitation energy larger than $2\Delta $, decreases exponentially. In this case the *collective* excitations of the condensate can dominate in the neutrino production.
Since we assume that the condensate consist of neutron pairs in the triplet state it is natural to expect the collective modes associated with spin fluctuations of the condensate[^2]. Such collective excitations with the energy lower than $%
2\Delta $ might undergo the weak decay into neutrino pairs. Recently spin waves with the excitation energy $\omega =\Delta /\sqrt{5}$ was predicted to exist in the superfluid spin-triplet condensate of neutrons [L09a,L10a,L10b]{}. Because of a rather small excitation energy, the weak decay of such waves leads to a substantial neutrino emission at the lowest temperatures $T\ll T_{c}$, when all other mechanisms of the neutrino energy losses are killed by the superfluidity.
In Refs. [@L09a; @L10a; @L10b], the eigen-mode of spin oscillations in the $%
^{3}P_{2}$ superfluid neutron liquid was studied in a simple model restricted to excitations of the condensate with $l=1$. In this paper we demonstrate that extending of the decomposition up to $l=1,3$ leads to a very small frequency shift of the known mode, $\omega =\Delta /\sqrt{5}$, but opens the new additional mode of spin oscillations with the finite energy gap $\omega \left( \mathbf{q}=0\right) <2\Delta $. The problem is considered for the case of $^{3}P_{2}-$$^{3}F_{2}$ pairing with a projection of the total angular momentum $m_{j}=0$ which is conventionally considered as preferable one at supernuclear densities.
We will examine the spin modes within the BCS approximation[^3]. Let us remind briefly the theory of spin density excitations in the condensate. The order parameter, $\hat{D}\equiv
D_{\alpha \beta }$, arising due to triplet pairing of quasiparticles, represents a $2\times 2$ symmetric matrix in spin space, $\left( \alpha
,\beta =\uparrow ,\downarrow \right) $. The spin-orbit interaction among quasiparticles is known to dominate in the nucleon matter of a high density. Therefore it is conventional to represent the triplet order parameter of the system as a superposition of standard spin-angle functions of the total angular momentum $\left( j,m_{j}\right) $, $$\Phi _{\alpha \beta }^{\left( j,l,m_{j}\right) }\left( \mathbf{n}\right)
\equiv \sum_{m_{s}+m_{l}=m_{j}}\left( \frac{1}{2}\frac{1}{2}\alpha \beta
|sm_{s}\right) \left( slm_{s}m_{l}|jm_{j}\right) Y_{l,m_{l}}\left( \mathbf{n}%
\right) . \label{sa}$$Assuming that the pair condensation occurs into the state with a total angular momentum $j=2$ we use the vector notation which involves a set of mutually orthogonal complex vectors $\mathbf{b}_{l,m_{j}}\left( \mathbf{n}%
\right) $ defined as$$\mathbf{b}_{l,m_{j}}\left( \mathbf{n}\right) =-\frac{1}{2}\mathrm{Tr}\left(
\hat{g}\bm{\hat{\sigma}}\hat{\Phi}^{2,l,m_{j}}\right) ~,~\mathbf{b}%
_{l,-m_{j}}=\left( -\right) ^{m_{j}}\mathbf{b}_{l,m_{j}}^{\ast },
\label{blm}$$where $\bm{\hat{\sigma}}=\left( \hat{\sigma}_{1},\hat{\sigma}_{2},\hat{\sigma%
}_{3}\right) $ are Pauli spin matrices, $\hat{g}=i\hat{\sigma}_{2}$, and the angular dependence of the order parameter is represented by the unit vector $%
\mathbf{n=p}/p$ which defines the polar angles $\left( \theta ,\varphi
\right) $ on the Fermi surface. The vectors $\mathbf{b}_{l,m_{j}}$ are mutually orthogonal and are normalized by the condition$$\left\langle \mathbf{b}_{l^{\prime },m_{j}^{\prime }}^{\ast }\mathbf{b}%
_{l,m_{j}}\right\rangle =\delta _{ll^{\prime }}\delta _{m_{j}m_{j}^{\prime
}}. \label{lmnorm}$$Hereafter the angle brackets denote angle averages, $\left\langle
...\right\rangle \equiv \left( 4\pi \right) ^{-1}\int d\mathbf{n}...$.
The block of interaction diagrams irreducible in the channel of two quasiparticles, $\Gamma _{\alpha \beta ,\gamma \delta }$, is usually generated by expansion over spin-angle functions. The spin-orbit interaction among quasiparticles is known to dominate at high densities. This implies that the spin $\mathbf{s}$ and orbital momentum $\mathbf{l}$ of the pair cease to be conserved separately, and the complete list of channels includes the pair states with $j=0,1,2$, and $\left\vert m_{j}\right\vert \leq j$. These nine complex states exhaust the number of independent components in the matrix order parameter arising at the $P$-wave pairing caused by the strong spin-orbit forces. The pairing in the $j=2$ channel dominates, and due to relatively small tensor components of the neutron-neutron interaction the condensation of pairs occurs in the $^{3}P_{2}+$$^{3}F_{2}$ state. In this pairing model, contributions from $^{3}P_{2}\rightarrow $$^{3}P_{0}$ or $^{3}P_{2}\rightarrow $$^{3}P_{1}$ transitions are deemed to be unimportant. Such assumption is somewhat vulnerable especially when considering excited state of the condensate. Unfortunately the detailed information on the in-medium effective interaction between neutrons in the channels $j=0,1$ is currently unavailable and requires a special investigation. Hence we take the approximation to neglect the $j=0,1$ coupling throughout this paper. From now on we omit the suffix j everywhere by assuming that the interaction occurs in the state with $j=2$. Thus we assume $l=j\pm 1$, and $$\varrho \Gamma _{\alpha \beta ,\gamma \delta }\left( \mathbf{p,p}^{\prime
}\right) =\sum_{l^{\prime }lm_{j}}\left( -1\right) ^{\frac{l-l^{\prime }}{2}%
}V_{ll^{\prime }}\left( p,p^{\prime }\right) \left( \mathbf{b}_{lm_{j}}(%
\mathbf{n})\bm{\hat{\sigma}}\hat{g}\right) _{\alpha \beta }\left( \hat{g}%
\bm{\hat{\sigma}}\mathbf{b}_{l^{\prime }m_{j}}^{\ast }(\mathbf{n}^{\prime
})\right) _{\gamma \delta }, \label{ppint}$$where $V_{ll^{\prime }}\left( p,p^{\prime }\right) $ are the interaction amplitudes, and $l,l^{\prime }=1,3$, in the case of tensor forces; $\varrho
=p_{F}M^{\ast }/\pi ^{2}$ is the density of states near the Fermi surface in the normal state. The effective mass of a neutron quasiparticle is defined as $M^{\ast }=p_{F}/\upsilon _{F}$, where $\upsilon _{F}\ll 1$ is the Fermi velocity of the non-relativistic neutrons.
The order parameter is of the following general form $$\hat{D}\left( \mathbf{n}\right) =\sum_{lm_{j}}\Delta _{l,m_{j}}\left( %
\bm{\hat{\sigma}}\mathbf{b}_{l,m_{j}}\right) \hat{g}. \label{Dnlm}$$
The ground state occurring in neutron matter has a relatively simple structure (unitary triplet) [@Tamagaki; @Takatsuka], where $$\sum_{lm_{j}}\Delta _{l,m_{j}}\mathbf{b}_{l,m_{j}}\left( \mathbf{n}\right)
=\Delta ~\mathbf{\bar{b}}\left( \mathbf{n}\right) . \label{bbar}$$On the Fermi surface, $\Delta $ is a complex constant, and $\mathbf{\bar{b}}%
\left( \mathbf{n}\right) $ is a real vector which we normalize by the condition $$\left\langle \bar{b}^{2}\left( \mathbf{n}\right) \right\rangle =1.
\label{Norm}$$The following orthogonality relations are also valid:$$\int \frac{d\varphi }{2\pi }\mathbf{b}_{l,m_{j}}^{\ast }\mathbf{b}%
_{l^{\prime },m_{j}^{\prime }}=\delta _{m_{j}m_{j}^{\prime }}\mathbf{b}%
_{l,m_{j}}^{\ast }\mathbf{b}_{l^{\prime },m_{j}}, \label{bbdfi}$$$$\int \frac{d\varphi }{2\pi }\left( \mathbf{\bar{b}b}_{l,m_{j}}^{\ast
}\right) \left( \mathbf{\bar{b}b}_{l^{\prime },m_{j}^{\prime }}\right)
=\delta _{m_{j}m_{j}^{\prime }}\left( \mathbf{\bar{b}b}_{l,m_{j}}^{\ast
}\right) \left( \mathbf{\bar{b}b}_{l^{\prime },m_{j}}\right) .
\label{bbbbdfi}$$Thus the triplet order parameter can be written as $$\hat{D}\left( \mathbf{n}\right) =\Delta \mathbf{\bar{b}}\bm{\hat{\sigma}}%
\hat{g}. \label{Dn}$$
Making use of the adopted graphical notation for the ordinary and anomalous propagators, $\hat{G}=\parbox{1cm}{\includegraphics[width=1cm]{Gn.eps}}$, $%
\hat{G}^{-}(p)=\parbox{1cm}{\includegraphics[width=1cm,angle=180]{Gn.eps}}$, $\hat{F}^{(1)}=\parbox{1cm}{\includegraphics[width=1cm]{F1.eps}}$, and $\hat{F}^{(2)}=\parbox{1cm}{\includegraphics[width=1cm]{F2.eps}}$, it is convenient to employ the Matsubara calculation technique for the system in thermal equilibrium. Then the analytic form of the propagators is as follows [@AGD; @Migdal]$$\begin{aligned}
\hat{G}\left( p_{\eta },\mathbf{p}\right) & =G\left( p_{\eta },\mathbf{p}%
\right) \delta _{\alpha \beta }~,\ \ \ \ \ \ \ \hat{G}^{-}\left( p_{\eta },%
\mathbf{p}\right) =G^{-}\left( p_{\eta },\mathbf{p}\right) \delta _{\alpha
\beta }, \notag \\
\hat{F}^{\left( 1\right) }\left( p_{\eta },\mathbf{p}\right) & =F\left(
p_{\eta },\mathbf{p}\right) \mathbf{\bar{b}}\bm{\hat{\sigma}}\hat{g}~,\ \ \
\hat{F}^{\left( 2\right) }\left( p_{\eta },\mathbf{p}\right) =F\left(
p_{\eta },\mathbf{p}\right) \hat{g}\bm{\hat{\sigma}}\mathbf{\bar{b}},
\label{GF}\end{aligned}$$where the scalar Green’s functions are of the form $G^{-}\left( p_{\eta },%
\mathbf{p}\right) =G\left( -p_{\eta },-\mathbf{p}\right) $ and$$G\left( p_{\eta },\mathbf{p}\right) =\frac{-ip_{\eta }-\varepsilon _{\mathbf{%
p}}}{p_{\eta }^{2}+E_{\mathbf{p}}^{2}}~,\ F\left( p_{\eta },\mathbf{p}%
\right) =\frac{-\Delta }{p_{\eta }^{2}+E_{\mathbf{p}}^{2}}. \label{GFc}$$Here $p_{\eta }\equiv i\pi \left( 2\eta +1\right) T$ with $\eta =0,\pm 1,\pm
2...$ is the Matsubara’s fermion frequency, and $\varepsilon _{\mathbf{p}%
}=p^{2}/\left( 2M^{\ast }\right) -p_{F}^{2}/\left( 2M^{\ast }\right) $. $%
\allowbreak \allowbreak $The quasiparticle energy is given by $E_{\mathbf{p}%
}^{2}=\varepsilon _{\mathbf{p}}^{2}+\Delta ^{2}\bar{b}^{2}\left( \mathbf{n}%
\right) $, where the (temperature-dependent) energy gap, $\Delta \bar{b}%
\left( \mathbf{n}\right) $, is anisotropic. In the absence of external fields, the gap amplitude $\Delta $ is real.
Finally we introduce the following notation used below. We designate as $%
\mathcal{I}_{XX^{\prime }}\left( \omega ,\mathbf{n,q};T\right) $ the analytical continuations onto the upper-half plane of complex variable $%
\omega $ of the following Matsubara sums:$$\mathcal{I}_{XX^{\prime }}\left( \omega _{\kappa },\mathbf{n,q};T\right)
\equiv T\sum_{\eta }\frac{1}{2}\int_{-\infty }^{\infty }d\varepsilon _{%
\mathbf{p}}X\left( p_{\eta }+\omega _{\kappa },\mathbf{p+}\frac{\mathbf{q}}{2%
}\right) X^{\prime }\left( p_{\eta },\mathbf{p-}\frac{\mathbf{q}}{2}\right) ,
\label{Ixx}$$where $X,X^{\prime }\in G,F,G^{-}$, and $\omega _{\kappa }=2i\pi T\kappa $ with $\kappa =0,\pm 1,\pm 2...$.These are functions of $\omega $, $\mathbf{q}
$ and the direction of the quasiparticle momentum $\mathbf{p}=p\mathbf{n}$.
We will focus on the processes with $\omega ^{2}<2\Delta ^{2}\bar{b}^{2}$ and with a time-like momentum transfer, $q^{2}<\omega ^{2}$. In this case the key role in the response theory belongs to the loop integral $\mathcal{I}%
_{FF}$. A straightforward calculation yields $\mathcal{I}_{FF}\left( \mathbf{%
n},\omega ,\mathbf{qn};T\right) =\mathcal{I}_{0}\left( \mathbf{n,}\omega
;T\right) +O\left( q^{2}\upsilon _{F}^{2}/\omega ^{2}\right) $, where $$\mathcal{I}_{0}\left( \mathbf{n,}\omega ;T\right) =\int_{0}^{\infty }\frac{%
d\varepsilon }{E}\frac{\Delta ^{2}}{4E^{2}-\left( \omega +i0\right) ^{2}}%
\tanh \frac{E}{2T}. \label{FFq0}$$Insofar as $q^{2}\upsilon _{F}^{2}/\omega ^{2}\ll 1$ and $q^{2}\upsilon
_{F}^{2}/\Delta ^{2}\ll 1$ we will neglect everywhere small corrections caused by a finite value of space momentum $\mathbf{q}$.
The gap equations are of the form [Tamagaki,Takatsuka,Khod,Baldo,Elg,Khodel,Schwenk]{}: $$\Delta _{l,m_{j}}\left( p\right) =-\sum_{l^{\prime }}\frac{1}{2\varrho }\int
dp^{\prime }p^{\prime 2}i^{l-l^{\prime }}V_{ll^{\prime }}\left( p,p^{\prime
}\right) \Delta \left( p^{\prime }\right) \left\langle T\sum_{\eta }\frac{%
\mathbf{b}_{l^{\prime },m_{j}}^{\ast }(\mathbf{n}^{\prime })\mathbf{\bar{b}}(%
\mathbf{n}^{\prime })}{p_{\eta }^{2}+E_{\mathbf{p}^{\prime }}^{2}}%
\right\rangle . \label{gap}$$We are interested in the processes occurring in a vicinity of the Fermi surface. To get rid of the integration over the regions far from the Fermi surface we renormalize the interaction as suggested in Refs. [Leggett,Leggett1]{}: we define $$V_{ll^{\prime }}^{\left( r\right) }\left( p,p^{\prime };T\right)
=V_{ll^{\prime }}\left( p,p^{\prime }\right) -\sum_{l^{\prime \prime }}\int
\frac{dp^{\prime \prime }p^{\prime \prime 2}}{\pi ^{2}}V_{ll^{\prime \prime
}}\left( p,p^{\prime \prime }\right) \left( GG^{-}\right) _{N}^{\prime
\prime }V_{l^{\prime \prime }l^{\prime }}^{\left( r\right) }\left( p^{\prime
\prime },p^{\prime };T\right) \ , \label{Vr}$$where the loop $\left( GG^{-}\right) _{n}$ is evaluated in the normal (non-superfluid) state. In terms of $V_{ll^{\prime }}^{\left( r\right) }$ the renormalized gap equations can be written in the following matrix form $$\left(
\begin{array}{c}
\Delta _{1,m_{j}} \\
\Delta _{3,m_{j}}%
\end{array}%
\right) =-\Delta \left(
\begin{array}{cc}
V_{11}^{\left( r\right) } & -V_{13}^{\left( r\right) } \\
-V_{13}^{\left( r\right) } & V_{33}^{\left( r\right) }%
\end{array}%
\right) \left(
\begin{array}{c}
\left\langle \mathbf{\bar{b}b}_{1,m_{j}}^{\ast }A\right\rangle \\
\left\langle \mathbf{\bar{b}b}_{3,m_{j}}^{\ast }A\right\rangle
\end{array}%
\right) , \label{gapeq}$$assuming that in the narrow vicinity of the Fermi surface the smooth functions $V_{ll^{\prime }}^{\left( r\right) }\left( p,p^{\prime }\right) $ and $\Delta \left( p^{\prime }\right) $ may be replaced with constants. In obtaining Eq. (\[gapeq\]) the fact is used that the interaction matrix is symmetric on the Fermi surface, $V_{31}=V_{13}$. The function $A\left(
\mathbf{n}\right) $ arises due to the renormalization procedure. It is given by $$A\left( \mathbf{n}\right) =\frac{1}{2}\int_{0}^{\infty }d\varepsilon \left(
\frac{1}{\sqrt{\varepsilon ^{2}+\Delta ^{2}\bar{b}^{2}}}\tanh \frac{\sqrt{%
\varepsilon ^{2}+\Delta ^{2}\bar{b}^{2}}}{2T}-\frac{1}{\varepsilon }\tanh
\frac{\varepsilon }{2T}\right) . \label{An}$$
The interaction matrix can be diagonalized by unitary transformations $%
V^{^{\prime }}=UVU^{\dagger }$with $U$ being an unitary matrix $$U=\left( U^{-1}\right) ^{\dagger }=\frac{1}{\left( V_{+}+V_{-}\right) ^{%
\frac{1}{2}}\allowbreak }\left(
\begin{array}{cc}
\sqrt{V_{+}} & \sqrt{V_{-}} \\
-\sqrt{V_{-}} & \sqrt{V_{+}}%
\end{array}%
\right) , \label{Umatr}$$where$\allowbreak $ $V_{\pm }=\sqrt{\left( V_{33}^{\left( r\right)
}-V_{11}^{\left( r\right) }\right) ^{2}+4V_{13}^{\left( r\right) 2}}\pm
\left( V_{33}^{\left( r\right) }-V_{11}^{\left( r\right) }\right) $.
One has $UVU^{\dagger }=\mathrm{diag}\left( W_{-},W_{+}\right) $ with $$W_{\pm }=\frac{1}{2}\left( V_{11}^{\left( r\right) }+V_{33}^{\left( r\right)
}\pm \sqrt{\left( V_{33}^{\left( r\right) }-V_{11}^{\left( r\right) }\right)
^{2}+4V_{13}^{\left( r\right) 2}}\right) . \label{W}$$Applying the unitary transformation $U$ to the gap equations (\[gapeq\]) yields two coupled equations:$$\begin{aligned}
&&\sqrt{V_{+}}\Delta _{1,m_{j}}+\sqrt{V_{-}}\Delta _{3,m_{j}} \notag \\
&=&-W_{-}\sum_{l}\Delta _{l,m_{j}}\left\langle \left( \sqrt{V_{+}}\mathbf{b}%
_{1,m_{j}}^{\ast }\mathbf{b}_{l,m_{j}}+\sqrt{V_{-}}\mathbf{b}%
_{3,m_{j}}^{\ast }\mathbf{b}_{l,m_{j}}\right) A\right\rangle , \label{Ge1}\end{aligned}$$$$\begin{aligned}
&&\sqrt{V_{-}}\Delta _{1,m_{j}}-\sqrt{V_{+}}\Delta _{3,m_{j}} \notag \\
&=&-W_{+}\sum_{l}\Delta _{l,m_{j}}\left\langle \left( \sqrt{V_{-}}\mathbf{b}%
_{1,m_{j}}^{\ast }\mathbf{b}_{l,m_{j}}-\sqrt{V_{+}}\mathbf{b}%
_{3,m_{j}}^{\ast }\mathbf{b}_{l,m_{j}}\right) A\right\rangle . \label{Ge2}\end{aligned}$$In obtaining these equations we made use of Eq. (\[bbar\]) and orthogonality relations (\[bbdfi\]), assuming that the energy gap $\bar{b}%
\left( \mathbf{n}\right) \Delta $ is azimuth-symmetric [Tamagaki,Takatsuka,Khod,Baldo,Elg,Khodel,Schwenk]{}.
We are interested in the linear medium response onto the external axial-vector field. The field interaction with a superfluid should be described with the aid of two ordinary and two anomalous three-point effective vertices. In the BCS approximation, the ordinary axial-vector vertices of a particle and a hole are to be taken as $\bm{\hat{\sigma}}$ and $\bm{\hat{\sigma}}^{T}$, respectively. The anomalous effective vertices, $%
\mathbf{\hat{T}}^{\left( 1\right) }\left( \mathbf{n;}\omega ,\mathbf{q}%
\right) $ and $\mathbf{\hat{T}}^{\left( 2\right) }\left( \mathbf{n;}\omega ,%
\mathbf{q}\right) $ are given by the infinite sums of the diagrams taking account of the pairing interaction in the ladder approximation [@Larkin]. These $2\times 2$ vector matrices are to satisfy the Dyson’s equations symbolically depicted by graphs in Fig. \[fig1\]. Analytic form of the above diagrams is derived in Refs. [@L09a]. After some algebraic manipulations the BCS equations for anomalous vertices can be found in the following form (for brevity we omit the dependence of functions on $\omega $ and $\mathbf{q}$**)**:
![Dyson’s equations for the anomalous vertices. The shaded rectangle represents the pairing interaction.[]{data-label="fig1"}](bcseq.eps)
$$\begin{gathered}
\mathbf{\hat{T}}^{\left( 1\right) }\left( \mathbf{n}\right) =\sum_{lm_{j}}%
\bm{\hat{\sigma}}\mathbf{b}_{lm_{j}}(\mathbf{n})\hat{g}\sum_{l^{\prime
}}V_{ll^{\prime }}\frac{1}{2}\left\langle \mathcal{I}_{GG^{-}}\mathrm{Tr}%
\left[ \hat{g}\left( \bm{\hat{\sigma}}\mathbf{b}_{l^{\prime }m_{j}}^{\ast
}\right) \mathbf{\hat{T}}^{\left( 1\right) }\right] \right. \notag \\
\left. -\mathcal{I}_{FF}\mathrm{Tr}\left[ \left( \bm{\hat{\sigma}}\mathbf{b}%
_{l^{\prime }m_{j}}^{\ast }\right) \left( \bm{\hat{\sigma}}\mathbf{\bar{b}}%
\right) \hat{g}\mathbf{\hat{T}}^{\left( 2\right) }\left( \bm{\hat{\sigma}}%
\mathbf{\bar{b}}\right) \right] -\frac{\omega }{\Delta }\mathcal{I}%
_{FF}2i\left( \mathbf{b}_{l^{\prime }m_{j}}^{\ast }\mathbf{\times \bar{b}}%
\right) \right\rangle , \label{EqT1}\end{gathered}$$
$$\begin{gathered}
\mathbf{\hat{T}}^{\left( 2\right) }\left( \mathbf{n}\right) =\sum_{lm_{j}}%
\hat{g}\bm{\hat{\sigma}}\mathbf{b}_{lm_{j}}^{\ast }(\mathbf{n}%
)\sum_{l^{\prime }}V_{ll^{\prime }}\frac{1}{2}\left\langle \mathcal{I}%
_{G^{-}G}\mathrm{Tr}\left[ \left( \bm{\hat{\sigma}}\mathbf{b}_{l^{\prime
}m_{j}}\right) \hat{g}\mathbf{\hat{T}}^{\left( 2\right) }\right] \right.
\notag \\
\left. -\mathcal{I}_{FF}\mathrm{Tr}\left[ \left( \bm{\hat{\sigma}}\mathbf{b}%
_{l^{\prime }m_{j}}\right) \left( \bm{\hat{\sigma}}\mathbf{\bar{b}}\right)
\mathbf{\hat{T}}^{\left( 1\right) }\hat{g}\left( \bm{\hat{\sigma}}\mathbf{%
\bar{b}}\right) \right] -\frac{\omega }{\Delta }\mathcal{I}_{FF}2i\left(
\mathbf{b}_{l^{\prime }m_{j}}\mathbf{\times \bar{b}}\right) \right\rangle .
\label{EqT2}\end{gathered}$$
Inspection of the equations reveals that the anomalous axial-vector vertices can be found in the following form $$\mathbf{\hat{T}}^{\left( 1\right) }\left( \mathbf{n},\omega \right)
=\sum_{lm_{j}}\mathbf{B}_{l,m_{j}}\left( \omega \right) \left( %
\bm{\hat{\sigma}}\mathbf{b}_{l,m_{j}}\right) \hat{g}, \label{T1A}$$$$\mathbf{\hat{T}}^{\left( 2\right) }\left( \mathbf{n},\omega \right)
=\sum_{lm_{j}}\mathbf{B}_{l,m_{j}}\left( \omega \right) \hat{g}\left( %
\bm{\hat{\sigma}}\mathbf{b}_{l,m_{j}}^{\ast }\right) . \label{T2A}$$As explained above we are interested in solutions with $\mathbf{q}=0$. Then inserting of these forms into Eqs. (\[EqT1\]), (\[EqT2\]) allows to obtain the equations for $\mathbf{B}_{l,m_{j}}\left( \omega \right) $. We write the result in the matrix form (For brevity we omit the dependence on $%
\mathbf{n}$ and $\omega $)$$\begin{gathered}
\left(
\begin{array}{c}
\mathbf{B}_{1,m_{j}} \\
\mathbf{B}_{3,m_{j}}%
\end{array}%
\right) =-\left(
\begin{array}{cc}
V_{11} & -V_{13} \\
-V_{13} & V_{33}%
\end{array}%
\right) \left\{ \left(
\begin{array}{c}
\sum_{l}\left\langle \left( A+\frac{\omega ^{2}}{2\Delta ^{2}}\mathcal{I}%
_{0}\right) \left( \mathbf{b}_{1,m_{j}}^{\ast }\mathbf{b}_{l,m_{j}}\right)
\right\rangle \mathbf{B}_{l,m_{j}} \\
\sum_{l}\left\langle \left( A+\frac{\omega ^{2}}{2\Delta ^{2}}\mathcal{I}%
_{0}\right) \left( \mathbf{b}_{3,m_{j}}^{\ast }\mathbf{b}_{l,m_{j}}\right)
\right\rangle \mathbf{B}_{l,m_{j}}%
\end{array}%
\right) \right. \notag \\
\left. -2\left(
\begin{array}{c}
\sum_{l}\left\langle \mathcal{I}_{0}\left( \mathbf{\bar{b}b}_{1,m_{j}}^{\ast
}\right) \left( \mathbf{\bar{b}b}_{l,m_{j}}\right) \right\rangle \mathbf{B}%
_{l,m_{j}} \\
\sum_{l}\left\langle \mathcal{I}_{0}\left( \mathbf{\bar{b}b}_{3,m_{j}}^{\ast
}\right) \left( \mathbf{\bar{b}b}_{l,m_{j}}\right) \right\rangle \mathbf{B}%
_{l,m_{j}}%
\end{array}%
\right) +\frac{\omega }{\Delta }i\left(
\begin{array}{c}
\left\langle \mathcal{I}_{0}\left( \mathbf{b}_{1,m_{j}}^{\ast }\mathbf{%
\times \bar{b}}\right) \right\rangle \\
\left\langle \mathcal{I}_{0}\left( \mathbf{b}_{3,m_{j}}^{\ast }\mathbf{%
\times \bar{b}}\right) \right\rangle%
\end{array}%
\right) \right\} . \label{B}\end{gathered}$$
In this equation, the interaction matrix can be diagonalized by the unitary transformation (\[Umatr\]). Further simplification is possible due to the fact that by virtue of Eqs. (\[Ge1\]), (\[Ge2\]) the coupling constants $%
W_{\pm }$ can be removed out of the equations. Explicit evaluation of equations obtained in this way for arbitrary values of $\omega $ and $T$ requires numerical computation. However, we can get a clear idea of the behavior of the vertex functions using the angle-averaged energy gap $\Delta
^{2}\bar{b}^{2}\rightarrow \left\langle \Delta ^{2}\bar{b}^{2}\right\rangle
=\Delta ^{2}$ in the quasiparticle energy $E_{\mathbf{p}}$. In this approximation, the functions $\mathcal{I}\left( \omega ,T\right) $ and $%
A\left( T\right) $ can be moved beyond the angle integrals. Performing trivial integrations we then get a set of linear equations (two equations for each value of $m_{j})$. It is convenient to denote$$\beta _{l,l^{\prime }}^{\left( m_{j}\right) }\equiv \left\langle \left(
\mathbf{b}_{l,m_{j}}\mathbf{\bar{b}}\right) \left( \mathbf{b}_{l^{\prime
},m_{j}}^{\ast }\mathbf{\bar{b}}\right) \right\rangle , \label{beta}$$and $$\Omega =\frac{\omega }{2\Delta }. \label{OMEGA}$$Then the set of equations can be written in the form$~$$$\begin{gathered}
\mathbf{B}_{1,m_{j}}\left[ \sqrt{V_{+}}\left( \Omega ^{2}-\beta
_{1,1}^{\left( m_{j}\right) }\right) -\sqrt{V_{-}}\beta _{1,3}^{\left(
m_{j}\right) }\right] \notag \\
+\mathbf{B}_{3,m_{j}}\left[ \sqrt{V_{-}}\left( \Omega ^{2}-\beta
_{3,3}^{\left( m_{j}\right) }\right) -\sqrt{V_{+}}\beta _{3,1}^{\left(
m_{j}\right) }\right] \notag \\
=-i\Omega \left( \sqrt{V_{+}}\left\langle \mathbf{b}_{1,m_{j}}^{\ast }\times
\mathbf{\bar{b}}\right\rangle +\sqrt{V_{-}}\left\langle \mathbf{b}%
_{3,m_{j}}^{\ast }\times \mathbf{\bar{b}}\right\rangle \right) ,
\label{eqB1}\end{gathered}$$$$\begin{gathered}
\mathbf{B}_{1,m_{j}}\left[ -\sqrt{V_{-}}\left( \Omega ^{2}-\beta
_{1,1}^{\left( m_{j}\right) }\right) -\sqrt{V_{+}}\beta _{1,3}^{\left(
m_{j}\right) }\right] \notag \\
+\mathbf{B}_{3,m_{j}}\left[ \sqrt{V_{+}}\left( \Omega ^{2}-\beta
_{3,3}^{\left( m_{j}\right) }\right) +\sqrt{V_{-}}\beta _{3,1}^{\left(
m_{j}\right) }\right] \notag \\
=-i\Omega \left( -\sqrt{V_{-}}\left\langle \mathbf{b}_{1,m_{j}}^{\ast
}\times \mathbf{\bar{b}}\right\rangle +\sqrt{V_{+}}\left\langle \mathbf{b}%
_{3,m_{j}}^{\ast }\times \mathbf{\bar{b}}\right\rangle \right) ,
\label{eqB2}\end{gathered}$$which can be solved to give $\allowbreak $$$\mathbf{B}_{1,m_{j}}=\frac{-i\Omega }{\chi }\left[ \left( \Omega ^{2}-\beta
_{3,3}^{\left( m_{j}\right) }\right) \left\langle \mathbf{b}_{1,m_{j}}^{\ast
}\times \mathbf{\bar{b}}\right\rangle +\beta _{3,1}^{\left( m_{j}\right)
}\left\langle \mathbf{b}_{3,m_{j}}^{\ast }\times \mathbf{\bar{b}}%
\right\rangle \right] , \label{B1m}$$$$\mathbf{B}_{3,m_{j}}=\frac{-i\Omega }{\chi }\left[ \left( \Omega ^{2}-\beta
_{1,1}^{\left( m_{j}\right) }\right) \left\langle \mathbf{b}_{3,m_{j}}^{\ast
}\times \mathbf{\bar{b}}\right\rangle +\beta _{1,3}^{\left( m_{j}\right)
}\left\langle \mathbf{b}_{1,m_{j}}^{\ast }\times \mathbf{\bar{b}}%
\right\rangle \right] ~ \label{B3m}$$with $$\chi \left( \Omega \right) \equiv \Omega ^{4}-\Omega ^{2}\left( \beta
_{1,1}^{\left( m_{j}\right) }+\beta _{3,3}^{\left( m_{j}\right) }\right)
+\beta _{1,1}^{\left( m_{j}\right) }\beta _{3,3}^{\left( m_{j}\right)
}-\beta _{1,3}^{\left( m_{j}\right) }\beta _{3,1}^{\left( m_{j}\right) }.
\label{hiw}$$
As is well known, poles of the vertex function correspond to collective eigen-modes of the system. Eigen- frequencies, $\Omega =\Omega ^{\left(
m_{j}\right) }$, of such oscillations satisfy the equation $\chi \left(
\Omega ^{\left( m_{j}\right) }\right) =0$. This equation gives$$\left( \Omega _{\pm }^{\left( m_{j}\right) }\right) ^{2}=\frac{1}{2}\left(
\beta _{1,1}^{\left( m_{j}\right) }+\beta _{3,3}^{\left( m_{j}\right) }\pm
\sqrt{\left( \beta _{1,1}^{\left( m_{j}\right) }-\beta _{3,3}^{\left(
m_{j}\right) }\right) ^{2}+4\beta _{1,3}^{\left( m_{j}\right) }\beta
_{3,1}^{\left( m_{j}\right) }}\right) .\allowbreak \label{Omega}$$ Notice that the interaction parameters,$V_{\pm }$, drop out of the above solutions, which depend explicitly only on the partial gap amplitudes. This means that the contribution of excited bound pairs with $l=3$ into the spin oscillations is caused basically by spin-orbit interactions but not by the tensor forces.
Indeed, in Eqs. (\[B1m\]) - (\[Omega\]), the equilibrium order parameter is specified solely by means of the real vector $\mathbf{\bar{b}}$. If we switch off the interaction in the $^{3}F_{2}$ and $^{3}P_{2}-$$%
^{3}F_{2}$ channels and consider pure $^{3}P_{2}$ pairing with $m_{j}=0$ we are then left with $\mathbf{\bar{b}=b}_{1,0}$ and $\Delta =\Delta _{1,0}$. In this case, in Eqs. (\[B1m\]), (\[B3m\]), one has: $$\int \frac{d\varphi }{2\pi }\left( \mathbf{b}_{1,m_{j}}^{\ast }\times
\mathbf{\bar{b}}\right) =\int \frac{d\varphi }{2\pi }\left( \mathbf{b}%
_{3,m_{j}}^{\ast }\times \mathbf{\bar{b}}\right) =0~~\mathsf{for}%
~~m_{j}=0,\pm 2, \label{cross}$$and the non-trivial solutions exist only for $m_{j}=\pm 1$. The explicit form of $\mathbf{b}_{l,m_{j}}$ can be obtained from Eq. (\[blm\]): $$\mathbf{b}_{1,0}=\sqrt{\frac{1}{2}}\left(
\begin{array}{c}
-n_{1} \\
-n_{2} \\
2n_{3}%
\end{array}%
\right) \,,~\mathbf{b}_{1,1}=-\sqrt{\frac{3}{4}}\left(
\begin{array}{c}
n_{3} \\
in_{3} \\
n_{1}+in_{2}%
\end{array}%
\right) , \label{b1mj}$$$$\mathbf{b}_{3,0}=\sqrt{\frac{3}{4}}\left(
\begin{array}{c}
n_{1}\left( 1-5n_{3}^{2}\right) \\
n_{2}\left( 1-5n_{3}^{2}\right) \\
n_{3}\left( 3-5n_{3}^{2}\right)
\end{array}%
\right) \,,~\mathbf{b}_{3,1}=\sqrt{\frac{1}{2}}\left(
\begin{array}{c}
n_{3}\left( 1-5n_{1}\left( n_{1}+in_{2}\right) \right) \\
in_{3}\left( 1+5in_{2}\left( n_{1}+in_{2}\right) \right) \\
\left( n_{1}+in_{2}\right) \left( 1-5n_{3}^{2}\right)
\end{array}%
\right) . \label{b3mj}$$Making use of these expressions in Eq. (\[beta\]) we find $$\beta _{1,1}^{\left( \pm 1\right) }=\frac{1}{20}~,~\beta _{3,3}^{\left( \pm
1\right) }=\frac{29}{70}~,~\beta _{1,3}^{\left( \pm 1\right) }=\beta
_{3,1}^{\left( \pm 1\right) }=-\frac{1}{70}\sqrt{\frac{3}{2}}.
\label{betamj}$$Inserting these values into Eq. (\[Omega\]) we find $4\beta _{1,3}^{\left(
\pm 1\right) }\beta _{3,1}^{\left( \pm 1\right) }\ll \left( \beta
_{3,3}^{\left( \pm 1\right) }-\beta _{1,1}^{\left( \pm 1\right) }\right) ^{2}
$. By neglecting the small term $4\beta _{1,3}^{\left( \pm 1\right) }\beta
_{3,1}^{\left( \pm 1\right) }$ under the root in Eq. (\[Omega\]) we obtain two (twofold) eigen-frequencies of spin oscillations in the condensate with $%
m_{j}=\pm 1$: $$\omega _{-}^{\left( m_{j}\right) }~\simeq 2\Delta \sqrt{\beta _{1,1}^{\left(
\pm 1\right) }}=\frac{1}{\sqrt{5}}\Delta , \label{w1p}$$$$\omega _{+}^{\left( m_{j}\right) }\simeq 2\Delta \sqrt{\beta _{3,3}^{\left(
\pm 1\right) }}=\sqrt{\allowbreak \frac{58}{35}}\Delta . \label{w2p}$$
In Refs. [@L10a; @L10b], eigen-modes of spin oscillations in the $^{3}P_{2}
$ superfluid neutron liquid was studied in a simple model restricted to excitations of the condensate with $l=1$. The spin wave energy (at $\mathbf{q%
}=0$) was found to be $\omega _{m_{j}}=\Delta /\sqrt{5}$. Equations ([Omega]{}), (\[w1p\]), (\[w2p\]) show that extending of the decomposition up to $l=1,3$ in Eqs. (\[T1A\]), (\[T2A\]) leads to a very small frequency shift of the known mode, $\omega =\omega _{-}^{\left( m_{j}\right)
}\simeq \omega _{m_{j}}$, but opens the new additional mode of spin oscillations with $\omega =\omega _{+}^{\left( m_{j}\right) }$.
Neutrino decays of spin waves can play an important role in the cooling scenario of neutron stars. A simple estimate made in Ref. [@L10b] has shown that the decays of spin waves with $\omega _{m_{j}}=\Delta /\sqrt{5}~$can become the dominant cooling mechanism in a wide range of low temperatures and modify the cooling trajectory of neutron stars. As well as the first mode, the second mode of spin oscillations is kinematically able to decay into neutrino pair. Therefore let us examine the wave excitation energies more accurately with taking into account the tensor forces. We will again focus on the condensation with $m_{j}=0~$by assuming $\Delta
^{2}=\Delta _{1,0}^{2}+\Delta _{3,0}^{2}$, and $$\mathbf{\bar{b}}\left( \mathbf{n}\right) =\frac{\Delta _{1,0}}{\Delta }%
\mathbf{b}_{1,0}\left( \mathbf{n}\right) +\frac{\Delta _{3,0}}{\Delta }%
\mathbf{b}_{3,0}\left( \mathbf{n}\right) . \label{bbarPF}$$In this case Eqs. (\[cross\]) are still valid and the non-trivial solutions to Eqs. (\[B1m\]), (\[B3m\]) exist only for $m_{j}=\pm 1$. Insertion of the expression (\[bbarPF\]) into Eq. (\[beta\]) results in $$\begin{aligned}
\beta _{1,1}^{\left( \pm 1\right) } &=&\frac{1}{20}\frac{\Delta _{1}^{2}}{%
\Delta ^{2}}\left( 1-\frac{2}{7}\sqrt{6}\frac{\Delta _{3}}{\Delta _{1}}+%
\frac{58\Delta _{3}^{2}}{7\Delta _{1}^{2}}\right) , \notag \\
\beta _{3,3}^{\left( \pm 1\right) } &=&\frac{29}{70}\frac{\Delta _{1}^{2}}{%
\Delta ^{2}}\left( 1-\frac{32}{87}\sqrt{6}\frac{\Delta _{3}}{\Delta _{1}}+%
\frac{28}{87}\frac{\Delta _{3}^{2}}{\Delta _{1}^{2}}\right) , \notag \\
\beta _{1,3}^{\left( \pm 1\right) } &=&\beta _{3,1}^{\left( \pm 1\right) }=-%
\frac{1}{140}\sqrt{6}\frac{\Delta _{1}^{2}}{\Delta ^{2}}\left( 1-11\sqrt{6}%
\frac{\Delta _{3}}{\Delta _{1}}+\frac{32}{3}\frac{\Delta _{3}^{2}}{\Delta
_{1}^{2}}\right) . \label{beta31}\end{aligned}$$Because $\beta _{l,l^{\prime }}^{\left( 1\right) }=\beta _{l,l^{\prime
}}^{\left( -1\right) }\equiv \beta _{l,l^{\prime }}$ we further omit the superscript $m_{j}=\pm 1$ by assuming that all the frequencies are twofold. Making use of Eqs. (\[beta31\]) we find $$\begin{aligned}
\omega _{-}^{2}& =\Delta _{1,0}^{2}\left( \frac{13}{14}-\frac{1}{3}\sqrt{6}%
\frac{\Delta _{3}}{\Delta _{1}}+\frac{23}{21}\frac{\Delta _{3}^{2}}{\Delta
_{1}^{2}}\right. \notag \\
& \left. -\sqrt{\frac{15}{28}-\frac{25}{49}\sqrt{6}\frac{\Delta _{3}}{\Delta
_{1}}+\frac{485}{147}\frac{\Delta _{3}^{2}}{\Delta _{1}^{2}}-\allowbreak
\frac{370}{441}\sqrt{6}\frac{\Delta _{3}^{3}}{\Delta _{1}^{3}}+\frac{55}{63}%
\frac{\Delta _{3}^{4}}{\Delta _{1}^{4}}}\right) , \label{w3}\end{aligned}$$$$\begin{aligned}
\omega _{+}^{2}& =\Delta _{1,0}^{2}\left( \frac{13}{14}-\frac{1}{3}\sqrt{6}%
\frac{\Delta _{3}}{\Delta _{1}}+\frac{23}{21}\frac{\Delta _{3}^{2}}{\Delta
_{1}^{2}}\right. \notag \\
& \left. +\sqrt{\frac{15}{28}-\frac{25}{49}\sqrt{6}\frac{\Delta _{3}}{\Delta
_{1}}+\frac{485}{147}\frac{\Delta _{3}^{2}}{\Delta _{1}^{2}}-\allowbreak
\frac{370}{441}\sqrt{6}\frac{\Delta _{3}^{3}}{\Delta _{1}^{3}}+\frac{55}{63}%
\frac{\Delta _{3}^{4}}{\Delta _{1}^{4}}}\right) . \label{w4}\end{aligned}$$In Fig. \[fig2\], the energy of the collective spin excitations (at $%
\mathbf{q}=0$) is shown vs the ratio of the partial gaps $x=\Delta
_{3,0}/\Delta _{1,0}$.
![The energy gaps for the collective spin excitations $\protect\omega %
_{-}^{\left( m_{j}\right) }$ and $\protect\omega _{+}^{\left( m_{j}\right) }$ vs the ratio of partial gap amplitudes in the $^{3}F_{2}$ and $^{3}P_{2}$ channels. The energy gap of a neutron quasiparticle is given by $\Delta
^{2}=\Delta _{1,0}^{2}+\Delta _{3,0}^{2}$.[]{data-label="fig2"}](PltW.eps)
According calculations of different authors, at the Fermi surface one has $%
\Delta _{3}\simeq 0.17\Delta _{1}$ (see, *e.g.*, Ref. [@Khod]). In this case our theoretical analysis predicts two degenerate modes with $%
\omega =\omega _{-}=0.42\Delta $, and two degenerate modes with $\omega
=\omega _{+}=1.\,\allowbreak 19\Delta $.
Because of a rather small excitation energy the decay of the corresponding collective spin excitations into neutrino pairs should lead to an extension of the low-temperature domain where the volume neutrino emission dominates the surface gamma radiation in the star cooling. This effect was already demonstrated in Ref. [@L10b], where only the lowest branch of the collective spin excitations $\omega =\Delta /\sqrt{5}\simeq \omega _{-}$ has been taken into account. The neutrino emissivity caused by the decay of the new spin modes predicted in this paper will be considered elsewhere.
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D. Page, M. Prakash, J. M. Lattimer, A. W. Steiner, Phys. Rev. Lett. 106 (2011) 081101
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[^1]: The vector channel of the neutrino radiation through neutral weak currents is strongly suppressed in the non-relativistic case [@LP06; @L09a].
[^2]: Previously spin modes have been thoroughly studied in the $p$-wave superfluid liquid $^{3}He$ with a central interaction between quasiparticles [@Maki; @C1; @C2; @W; @Wolfle] . These results cannot be applied directly to the triplet-spin neutron superfluid condensate, where the pairing is caused mostly by the spin-orbit interaction between quasiparticles (see details in Ref. [@L10a])
[^3]: Throughout this paper, we use the system of units $\hbar =c=1$ and the Boltzmann constant $k_{B}=1$.
| null | minipile | NaturalLanguage | mit | null |
Ava Gardner: Movie star was almost killed by a Hollywood legend at the Savoy Hotel
BRUISED and bloodied, screen siren Ava Gardner thought she would die in her suite at The Savoy
Battered black and blue by a crazed alcoholic wielding a broken bottle and threatening to kill her she had locked herself in the bathroom as her assailant raged at the door. But her attacker was no random madman.
Astonishingly it was movie legend George C Scott, star of films including Dr Strangelove and The Hustler, who went on to win an Oscar for Patton. Ava Gardner is famed for her boozing and volatile love life, with failed marriages to actor Mickey Rooney, bandleader Artie Shaw and crooner Frank Sinatra.
But a new biography reveals shocking details of her abusive relationship with Scott. “He repeatedly beat Ava brutally but she kept going back to him,” says Kendra Bean, co-author with Anthony Uzarowski of Ava: A Life In Movies.
Related articles
“She was almost addicted to him but he was relentlessly violent. In one beating he even broke her collar bone yet she kept pursuing him. She knew he was damaged but thought she could fix him even if it put her life in danger.”
Gardner called it “a dark and ugly” chapter in her life that began when the duo met in Italy filming John Huston’s 1966 movie The Bible and began a torrid affair. But Scott’s mood unexpectedly turned ugly one night in his hotel room.
“His eyes narrowed and reddened,” Gardner recalled. “His jaw clenched. When I got up to leave he leapt up and threw a punch. He hit me in the face. I was trapped in that room and he beat me for God knows how long until I managed to escape.”
GETTY
Ava Gardner is famed for her boozing and volatile love life
Ava Gardner: 1940s Hollywood icon
Tue, January 26, 2016
We remember Hollywood icon Ava Gardner on the anniversary of her death on 25th January 1990.
But as their volatile fling continued so did the beatings. “No one understood why Ava kept going back to him,” says Bean. Scott pleaded for forgiveness and begged Ava to marry him, even though he was already married to actress Colleen Dewhurst.
When filming finished Gardner ended the tormented romance, fleeing to London. But Scott stalked her, tracking her down to The Savoy. “Ava was a naturally kind person and found it hard to reject him,” says Uzarowski. “I think she felt sorry for him. But he beat Ava up savagely and threatened to kill her assistant René with a broken bottle.”
GETTY
George C Scott mood unexpectedly turned ugly one night in his hotel room
Ava and René locked themselves in the hotel bathroom, finally escaping through a small window, fleeing into the night down The Strand. “Scott trashed the hotel room but he was never arrested and Ava was banned for life from The Savoy.” Gardner fl ed to Los Angeles but Scott followed her again, finding her in a bungalow at the Beverly Hills Hotel. “Scott beat her again and in desperation she called her ex-husband Sinatra asking for help,” says Uzarowski. “Frank sent some of his shady mobster friends to set Scott straight and she never saw him again.”
Though Gardner and Sinatra divorced in 1957 it’s hardly surprising she turned to him for help a decade later: their romance was the love of their lives and they remained close after their split. “She loved him dearly but they could not stand to be together any more than they could stand to be apart,” say the authors.
“They’d been drinking, got their hands on a couple of pistols and began driving around shooting out shop windows and mailboxes. MGM publicist Jack Keller was called to bribe the police with $20,000 to keep mum about the incident. The studio couldn’t afford to have their leading lady busted for a drunken joy ride.”
But Gardner’s career was soaring while Sinatra’s was fading when they wed in 1951 and his fragile ego couldn’t cope. “Frank’s self-esteem was plummeting and Ava felt helpless, unable to cope with his dark moods and self-pity,” say the authors. But when Sinatra’s career revived with his Oscar-winning performance in From Here To Eternity in 1953 he became even harder to live with.
“Frank had returned to his arrogant ways, seducing women everywhere he went just to spite her,” say the authors. “She was tired of his jealousy, his possessiveness and over-sensitive ego. She didn’t want to continue to suppress her personality for any man, not even for Frank.”
GETTY
Ava Gardner and her first husband Mickey Rooney
AS THEIR marriage foundered Ava fell pregnant twice and had two abortions, much to the horror of Catholic-raised Sinatra. He con- fided: “I shoulda beaten her brains out for what she did to me and the baby but I loved her too much.”
They divorced after six turbulent years but their passion never died. Gardner kept Sinatra’s love letters in a shoebox and in her later years friends often found her sitting alone listening to Sinatra records, drowning her sorrows in booze.
As wild as she was with Sinatra, Gardner was innocent when she married Hollywood superstar Mickey Rooney in 1942. “She was only 18 and still a virgin,” says Bean.
“He swept her off her feet and though she wasn’t initially interested he wore her down.” She experienced a “sexual awakening” but Rooney chased every girl on the MGM lot and after nine months Gardner filed for divorce.
GETTY
Gardner kept Sinatra’s love letters in a shoebox
Her second marriage to bandleader Artie Shaw in 1946 was equally fraught as he made her feel intellectually inadequate. They divorced a year later and she turned to drink, which became a lifetime addiction. Gardner’s lovers included Errol Flynn, David Niven, Peter Lawford, Robert Taylor, millionaire Howard Hughes and even a Spanish bullfighter but after Sinatra she never married again and remained haunted by insecurities.
“I was never really an actress,” Gardner confessed. “None of us who came from MGM was. We were just good to look at. I don’t enjoy making films. I just enjoy making money.” Gardner spent her final 18 years in Knightsbridge, where she died alone in 1990 with a drink in her hand, having lived with one terrible regret. “I am sorry I spent 25 years making films,” she confessed. “I am very conscious that as a woman I have not entirely fulfilled myself. After all I have no husband and no children and those are really the two reasons a woman has for being.”
● To pre-order Ava: A Life In Movies, by Kendra Bean and Anthony Uzarowski (published by Running Press, £20, on July 27) please call the Express Bookshop with your card details on 01872 562310. Alternatively send a cheque or postal order made payable to The Express Bookshop to: Ava Offer, PO Box 200, Falmouth, Cornwall, TR11 4WJ or visit expressbookshop.com UK delivery is free. | null | minipile | NaturalLanguage | mit | null |
A lesbian basketball coach is suing her former employer, after claiming she was fired for being gay.
Courtney Graham was a women’s basketball assistant at Drake University, until she was asked to resign without reason in May 2015, according to the New York Post.
She is now suing the university as well as Bulldogs coach Jennie Baranczyk, claiming they forcer her to resign after it became public she was lesbian.
In November 2014, Graham brought her wife, then still her girlfriend, to a home game.
In her federal lawsuit, the former coach alledges Baranczyk started stripping her of her duties as assistant head coach.
She also alledges she was left out of team meetings and scouting trips, and faced ‘hostile interactions’ at work.
‘After discovery of plaintiff’s sexual orientation, she was subjected to hostile interactions on the job,’ the lawsuit says.
‘And the subject of inter-office slander, speculation and gossip related to her sexual orientation.’
Baranczyk allegedly also told her colleague she ‘was not acting like herself’, the New York Post quotes, and forced Graham to take time off work shortly after her sexual orientation was made public.
When she was asked to resign, Graham first refused, but subsequently says she resigned three weeks later, under duress.
Her lawsuit alleges discrimination based on sexual orientation, as well as two counts of retaliation, negligence, intentional infliction of emotional distress and harassment.
On Wednesday (28 December), Drake released a statement denying the allegations.
‘Drake University and head women’s basketball coach Jennie Baranczyk have a strong commitment to diversity, tolerance and non-discrimination,’ it reads.
‘Consistent with university policy regarding personnel matters and out of respect to the parties involved, we will not provide further comment.’
According to the university, Graham made similar claims in a complaint to the Iowa Civil Rights Commission.
The commission dismissed her complaint at the time and, according to CBS Philly, said it ‘didn’t believe that further investigation would reveal that Graham was discriminated against.’ | null | minipile | NaturalLanguage | mit | null |
Therapeutic drug monitoring and pharmacogenetic tests as tools in pharmacovigilance.
Therapeutic drug monitoring (TDM) and pharmacogenetic tests play a major role in minimising adverse drug reactions and enhancing optimal therapeutic response. The response to medication varies greatly between individuals, according to genetic constitution, age, sex, co-morbidities, environmental factors including diet and lifestyle (e.g. smoking and alcohol intake), and drug-related factors such as pharmacokinetic or pharmacodynamic drug-drug interactions. Most adverse drug reactions are type A reactions, i.e. plasma-level dependent, and represent one of the major causes of hospitalisation, in some cases leading to death. However, they may be avoidable to some extent if pharmacokinetic and pharmacogenetic factors are taken into consideration. This article provides a review of the literature and describes how to apply and interpret TDM and certain pharmacogenetic tests and is illustrated by case reports. An algorithm on the use of TDM and pharmacogenetic tests to help characterise adverse drug reactions is also presented. Although, in the scientific community, differences in drug response are increasingly recognised, there is an urgent need to translate this knowledge into clinical recommendations. Databases on drug-drug interactions and the impact of pharmacogenetic polymorphisms and adverse drug reaction information systems will be helpful to guide clinicians in individualised treatment choices. | null | minipile | NaturalLanguage | mit | null |
ITER Live: Start of Machine Assembly - RainforestCx
https://youtu.be/2-7GyVLKE6A
======
mrtksn
I recall multiple conflicting stories about how this is the future and how
this is waste of money on a ill conceived giga-project.
I also recall a skunk works reactor that takes radically different approach
(smaller reactors, instead of big) and was supposed to have results if few
years.
[2014]:
[https://news.ycombinator.com/item?id=8458339](https://news.ycombinator.com/item?id=8458339)
Then there are new stories about skunk works being on track.
[https://aviationweek.com/defense-space/lockheeds-skunk-
works...](https://aviationweek.com/defense-space/lockheeds-skunk-works-
building-bigger-fusion-reactor)
~~~
maccam94
IIRC, when ITER was conceived, it wasn't a waste of money. They just chose a
conservative development path that was expensive but very likely achievable.
Technology and theories have since advanced that might leapfrog it, but the
project has still contributed a great deal to our understanding of fusion and
the engineering required to produce it.
Relevant talk by one of the MIT professors working with Commonwealth Fusion
Systems:
Breakthrough in Nuclear Fusion? - Prof. Dennis Whyte (2016) -
[https://www.youtube.com/watch?v=KkpqA8yG9T4](https://www.youtube.com/watch?v=KkpqA8yG9T4)
Timeline (in case you want to skip over some parts):
00:01:00 - introducing Dennis Whyte, MIT department head for nuclear science
00:04:24 - presentation starts
00:06:00 - identifies breakthrough with REBCO magnets
00:07:25 - explains deuterium-tritium fusion
00:12:30 - basic metrics for reactor performance
00:17:15 - energy output of other previous fusion experiments
00:19:00 - examines ITER and the problems of its approach
00:22:00 - problems solved by high energy magnetic fields
00:28:15 - full scale reactor concept, teardown of REBCO magnets
00:37:00 - design limits and margins
00:39:00 - fixes plasma instabilities found in weaker magnetic chambers
00:40:00 - maintainability, lifespan, component replacement
00:45:00 - solution to neutron damage and energy capture
00:50:30 - cost and profitability
00:54:00 - full graph of field strength vs reactor scale (and thus funding requirements)
01:01:50 - Q&A
01:30:00 - question about the biggest risks
A more recent (2019) talk with more numbers and even more confidence:
[https://www.youtube.com/watch?v=rY6U4wB-
oYM](https://www.youtube.com/watch?v=rY6U4wB-oYM)
~~~
sgift
> Technology and theories have since advanced that might leapfrog it
With "might" being the operative word. As long as none of these new approaches
has achieved viable fusion (so, more power out than in) I think it's not a bad
idea to just continue with the less radical plan that will probably work, even
if it is slower.
~~~
xorcist
ITER is research. There are numerous engineering problems yet to be solved,
plasma physics, materials science, you name it. Research projects are always a
"waste of money" in some way so that particular criticism is neither
surprising nor very helpful.
------
inputmice
Does this mean we are only 10 years away from fusion?
~~~
swebs
No, ITER will not be used as a power plant. Think of it as an experimental
device like the LHC. The findings from ITER will be used to build DEMO which
will be able to output power continuously. And DEMO's findings will be used to
create PROTO, a commercial power plant. Sometime after 2050. Maybe longer.
And if you want to be really technical, we've had fusion reactors for decades,
but ITER should be the first one that produces more energy than it consumes.
[https://en.wikipedia.org/wiki/DEMOnstration_Power_Station](https://en.wikipedia.org/wiki/DEMOnstration_Power_Station)
[https://en.wikipedia.org/wiki/PROTO_(fusion_reactor)](https://en.wikipedia.org/wiki/PROTO_\(fusion_reactor\))
------
waynenilsen
How far behind? How much over budget?
------
DiabloD3
All the while, we have the new company formed out of the work of the SAFIRE
project that uses self-organizing plasma fields that makes traditional
tokamaks look entirely stoneage.
Not only are they doing fusion at a tiny fraction of the power, they're doing
it with a much much much smaller system that is producing heavier elements out
of thin air _while_ having started as just a "sun in a bottle" research system
to understand how the Sun interacts with the electromagnetic field of the
galaxy (and why the 1960s nuclear fusion model of the Sun explains _nothing_
of the past 50-60 years of observations published since).
They did it on basically a shoestring budget, too.
~~~
Zealotux
I really wanted to believe in the SAFIRE project, but it looks like an
embarrassing scam, unfortunately:
[https://www.youtube.com/watch?v=vmVdPgkudC8](https://www.youtube.com/watch?v=vmVdPgkudC8)
Really hope I get proved wrong in the next years.
~~~
DiabloD3
Yeah, I don't get the point of those kinds of Youtube videos. If they think
its a "scam", they should publish a paper on the inability to replicate of the
results.
Instead its just some weird reaction video by some rando on Youtube that has
not cited his sources. The irony of this is, he accuses them of doing the same
thing EU proponents have accused modern Cosmology of doing: filtering out
anything that doesn't fit their narrative, even when its established science
in another field for the past few decades.
The SAFIRE guys aren't cosmologists, they're approaching the problem of the
Sun the same way a plasma physicist would. Literally, they are following in
the footsteps of Hannes Alfvén (the guy that got the Nobel for magneto-
hydrodynamics), and Kristian Birkeland (nominated for a Nobel seven times due
to his work in this field), and continuing their work.
The people who claim that plasma physics has no place in space, in
solar/planet interactions, in galactic/star interactions, etc are, frankly,
insulting both the Nobel committee _and_ the work of NASA for the past 30+
years. They're on the same delusion spectrum that flat earthers are on, just
not quite to that extreme.
I'm not saying SAFIRE is right, but SAFIRE isn't even saying they're right.
They've literally asked for other labs to replicate their results because
they've found a novel way of replicating the Sun's activity in their lab, and
they need to rule out measurement error in ways they haven't already. They are
_seeking_ replication and validation, something scammers don't do.
| null | minipile | NaturalLanguage | mit | null |
A Wednesday report by Keith Laing at the Hill failed to point out a quite obvious contradiction during departing Transportation Secretary LaHood’s appearance on NPR’s Diane Rehm show.
From all appearances, based on the video available at her site, Rehm, once LaHood launched into a predictable rant about how our transportation infrastructure is in serious disrepair, didn’t ask — and should have asked — why the hundreds of billions of dollars spent on the stimulus plan accompanied by those ubiquitous Recovery Act promotional signs seen at road construction projects didn’t stabilize things two or three years ago. Excerpts from Laing’s lackluster effort follow the jump (bolds are mine):
LaHood: ‘America is one big pothole’
Outgoing Transportation Secretary Ray LaHood lamented the amount of infrastructure spending that was approved by Congress during his tenure at the Department of Transportation (DOT) on Wednesday.
“America is one big pothole right now,” LaHood said in an interview on “The Diane Rehm Show” on National Public Radio.
“At one time … we were the leader in infrastructure,” LaHood continued. “We built the interstate system. It’s the best road system in the world, and we’re proud of it. But we’re falling way behind other countries, because we have not made the investments.”
… “The next decisions that will be made by this Congress, by this administration will have to be bold if we’re going to continue our efforts to fix up our roads, keep our highways in a state of good repair, to fix up unsafe bridges,” he said. “We need a bold plan, and a bold way to fund it.”
You had a “bold plan” in 2009, Ray. It was called the “stimulus.” The administration spent 2009 and 2010 bragging about how infrastructure improvements were bringing the nation back from the eeeeevil Bush recession.
The problem was, first, that much of the funding went to projects already on the books; in other words, federal money replaced state and local money. Second, much of the stimulus plan money was spent on nonsense like failed “green energy” projects which had nothing to do with improving infrastructure.
Now the administration acts as if little has ever been done (largely true, and their fault), and wants us to believe that another round of supposed infrastructure spending will be properly managed. Sure it will.
More fundamentally, money from the Highway Trust Fund which should have been sent on highway improvements and enhancements has been flushed down the black hold of mass transit, light rail, and other non-automotive means of transportation Americans have historically shunned and will continue to shun unless statists in government financially force them into it.
what’s really driving LaHood? He’s pursued an anti-car ideological zeal from Day One — from entertaining proposals to impose mileage taxes on drivers and to track drivers’ routes, to redistributing tax dollars to pie-in-the-sky high-rail projects that no private business will touch, to peddling a “livability initiative” that would discourage suburban growth and corral residents in high-density areas dependent on public transportation.
Of course, Rehm never challenged LaHood on this legacy from his tenure, and Laing never pointed to it.
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Mughal-e-Azam (musical)
Mughal-e-Azam () is a Broadway-style musical based on the 1960 Bollywood film of the same name, directed by K. Asif and produced by Shapoorji Pallonji. The musical was directed by Feroz Abbas Khan and produced by Shapoorji Pallonji Group.
Based on the love story between Mughal Prince Salim and courtesan Anarkali, Mughal-e-Azam portrays the conflict faced by Mughal Emperor Akbar: his responsibility towards the future of his empire and his duty as the father of a beloved son.
It was the first large-scale Indian Broadway-style musical, and was in production for ten months. The show was well received by critics in the media and entertainment industry. In 2017, it won seven out of the fourteen BroadwayWorld India Awards including Best Play, Best Director, and Best Costume Design.
Plot
In the era of the Mughals, Emperor Akbar's desire of a son is fulfilled when his wife, Jodhabai, gives birth to Salim, who grows up to be a spoiled brat filled with disrespect and self-indulgence. Consequently, he's sent off to war in Akbar's army and returns as a reformed person with perseverance and discipline fourteen years later. Salim, now a young man, falls in love with Anarkali, a lowly courtesan. Bahar, a dancer who covets the love of the Prince, is filled with jealousy upon discovering the secret affair and exposes it to Akbar. The emperor, full of royal pride, vehemently disapproves of the relationship and imprisons Anarkali, which leads Salim to declare an open rebellion against him. The war between father and son changes the lives of everyone in the empire threatening the foundations of the Mughal rule in India.
Background
The 1960 K. Asif film was inspired by a play called Anarkali, written in Lahore in 1922 by dramatist Imtiaz Ali Taj. Feroz Abbas Khan had wanted to recreate the film onstage since he saw the black-and-white film re-released in colour in 2004. In a 2017 interview, Khan said he hadn't wanted to do an imitation of Asif’s Mughal-e-Azam, but rather "pay a tribute to K. Asif sahab". For him, the play was “an interpretation; carrying the legacy forward in a different medium.” Prior to Mughal-e-Azam, Khan was known for minimalistic productions like Tumhari Amrita, and as a director, he wanted to do something entirely different. Inspired by Mughal-e-Azam, he approached Shapoorji Pallonji Group to get the stage rights. The current CEO and director of Shapoorji Pallonji Group, Deepesh Salgia, agreed on the condition that the play kept up the tradition of the film and provided creative consultancy for the production.
According to Salgia, after the colourisation of the original black-and-white movie, the company was looking for new ways to promote the film. The musical retained Naushad’s soundtrack and Shakeel Badayuni’s lyrics from the original film, along with two newly composed songs. The songs are sung live by the singers on stage to a pre-recorded orchestral and choral score. The play is in both the Urdu and Hindi languages, with LED screens displaying English subtitles of the dialogue.
Production
The technical team included Drama Desk Award-winning light designer David Lander; Emmy-nominated John Narun for projection design, who has previously worked on Madonna’s concerts and productions at Radio City Music Hall in New York; and production designer Neil Patel, a recipient of the Obie Award and Helen Hayes Award, who recreated the design of Emperor Akbar’s Sheesh Mahal (mirror hall) on set. Bollywood designer Manish Malhotra designed over 550 costumes and choreographer Mayuri Upadhya recreated the dance sequences from the film for the play.
Cast
Mughal-e-Azam recruited a cast and crew of over 350 people, including 30 classically trained Kathak dancers who had been living and training in Mumbai for three months. Since the lead actresses had to both sing and dance simultaneously, two actresses — Neha Sargam and Priyanka Barve — played the part of Anarkali alternatively. Theatre actors Nissar Khan, Syed Shahab Ali, and Dhanveer Singh played the parts of Akbar and Salim respectively. The part of Jodha Bai was played by Sonal Jha Rajesh jais & Tareeq Ahmed Khan played part of Sangtarash/Narrator,and Palvi Jaswal essayed the role of Suraiya. Durjan Singh was portrayed by Chirag Garg and Amit Pathak alternatively.
Reception
The premiere public shows of the musical were held at the Jamshed Bhabha Theatre. The musical had 57 shows in Mumbai, and one successful season in Delhi. The premiere of the musical in Delhi was attended by Union Finance Minister Arun Jaitley, Information and Broadcasting and Textiles Minister Smriti Irani, Nitin Gadkari, and Jyotiraditya Scindia. Bollywood actresses Rekha and Hema Malini, and directors Karan Johar, Ashutosh Gowariker, and Gurinder Chadha attended the last show of the musical at the Jamshed Bhabha Theatre in Mumbai.
In her review of the Broadway-style musical, Eshita Bhargava of the Indian Express called it “an audio-visual extravaganza that will be a joy to behold”. Rishi Kapoor, a Bollywood actor and the grandson of actor Prithviraj Kapoor who played the role of Mughal emperor Akbar in the original film, tweeted “I don't have enough of adjectives to describe it. This will play forever.” Karan Johar, Bollywood director and actor, tweeted “Mughal-e-Azam, the musical play, was a spectacular throwback to the legendary film; the team and director Feroz Khan have excelled themselves.” The Quint called it “Indian theatre’s coming of age”, “succeeding in giving the audience its own, new set of timeless memories”. In her theatre review of The Times of India, Purvaja Sawant stated “the live performances of 'Mughal-e-Azam's hit songs transport you back to a nostalgic era, one you don't want to come out of.”
Awards
Mughal-e-Azam: The Musical won seven trophies, including the Best Indian Play, at the Broadway World India Awards 2017.
References
Category:2016 musicals
Category:Indian musicals
Category:Musicals based on films | null | minipile | NaturalLanguage | mit | null |
WASHINGTON – Democrats have an "army of lawyers" working on the likely recount that will decide the winner of the Florida Senate race, the head of the Democratic National Committee said Thursday.
Florida Sen. Marco Rubio, a Republican, responded that the other party isn't playing fair.
"Now democrat lawyers are descending on #Florida," Rubio tweeted Thursday. "They have been very clear they aren’t here to make sure every vote is counted. They are here to change the results of election."
Fewer than 17,400 votes out of more than 8.1 million cast separated Republican challenger Gov. Rick Scott over Democratic Sen. Bill Nelson as of Thursday afternoon. That's inside the 0.25 point margin that dictates a hand recount of ballots that showed either no vote or more than one vote cast in the Senate race.
But the totals could change before noon Saturday when counties submit preliminary unofficial results to the state.
"It's a jump ball," said Marc Elias, the lawyer overseeing the recount effort for Nelson.
Recounts are also possible in the Florida races for governor and agriculture commissioner.
Elias, who has been called by The Washington Post a "go-to lawyer for Democrats in recount fights and redistricting battles," represents the national Democratic Party organizations.
Democratic National Committee Chairman Tom Perez was asked by reporters at a Christian Science Monitor breakfast what help the party is giving to Nelson.
"There's an army of lawyers down there now that are working on the recount," Perez responded. "And and I'm glad they're doing that."
He also said the party has "people on our voter protection team right now chasing provisional ballots" and "whatever help anyone needs" in both Florida and Georgia where Democratic gubernatorial candidate Stacey Abrams has not conceded to Republican Brian Kemp.
"Voter suppression is a permanent staple in their playbook," Perez said of Republicans, "and that's a lesson that we have to learn."
For his part, Rubio pointed the finger at election officials in heavily Democratic Broward County. He accused them of not following the legal requirements on timely reporting of results.
"It has opened the door for lawyers to come here & try to steal a seat in the U.S. Senate & Florida Cabinet," Rubio tweeted of Nelson's seat and the race for state agriculture commissioner.
A spokeswoman for the Republican National Committee didn't immediately respond to a request for comment about how the organization is involved in Florida's Senate contest.
Ron Klain, the lawyer who headed Al Gore's recount team in the 2000 presidential race, said Nelson has two main advantages that Gore lacked.
After that contentious battle, Florida allowed voters to cast a provisional ballot if their eligibility is in dispute at a polling place. While the number of provisional ballots outstanding in Florida is unknown, if the total ends up being similar to the 25,000 cast in 2016, that's a "huge potential gain" for Nelson, Klain said.
Nelson also doesn't face an expiring clock as Gore did. The recount in the presidential race was stopped by the Supreme Court because of the December 12 deadline for electoral college votes to be submitted. There's no similar time limit for a congressional recount, Klain notes.
In fact, the 2008 battle over Minnesota Sen. Al Franken's close election – which was handled by Elias – lasted eight months.
Contributing: John McCarthy, USA TODAY Network. | null | minipile | NaturalLanguage | mit | null |
I just spent 99p for nothing from McDonald's pic.twitter.com/1OHgYTEZ3Q
Aug. 23 (UPI) -- A teen in England spent 99 cents to order "nothing" from McDonald's after removing all of the ingredients from the order.
Twitter user Ari shared photos of the order and the resulting receipt that featured a charge of 99 cents for the list of ingredients that added up to nothing.
"I just spent 99p for nothing from McDonald's," Ari wrote.
Ari originally planned to order a cheeseburger without pickles, but soon realized the order station offered the option to remove every single ingredient.
After removing the onion, ketchup, mustard, cheese, bun and beef patty from the order the teen was left with nothing but an empty bag and a 99 cent bill.
"This tweet actually cost me money to make, I'm gonna be pissed off if it doesn't blow up," Ari tweeted.
The post eventually went viral, receiving more than 150,000 likes and 60,000 retweets.
It also inspired other Twitter users to imagine other unusual orders including removing all of the ingredients except for ketchup. | null | minipile | NaturalLanguage | mit | null |
Mum slain minutes after drop off
A young mother found dead in a pool of blood by her stepbrother near the back door to her home had been dropped off only minutes earlier.
Police investigating the death of 22-year-old Sina Nerisa Solomona in Ashburton early on Saturday say they are keeping an open mind but believe is likely to be a homicide.
Ms Solomona, a mother of a small daughter, was found inside her Cass St house house after 2.30am.
She had suffered severe head trauma.
Only minutes earlier the meat packer had been dropped home by her sister's partner after a shift at CMP Canterbury.
Sina lived at the house with her mother Anuella, her mother's partner, her 3-year-old daughter Kaira, her twin sister Loretta, her sister's partner and her two brothers.
None of the family were believed to have been home because they were at another family property in the town.
Anuella Solomona said her daughter was a loving and beautiful person and the family was in deep shock.
"They just don't believe someone could do something like this, if there was any people hate us, any people had trouble, problem with us ... I don't think so," she said.
The family had lived in Ashburton since 2002, with many members working at CMP Canterbury.
Kaira believed her mother was at work and Anuella explained that the only time Sina was not with her daughter was when she was at work.
"She loved her daughter, she loves her friends, she loves anyone. Until now I'm still questioning who would do this," Anuella said.
Her stepson had discovered Sina fatally injured at the house.
He could not initially open the back door and thought Sina was playing around and not letting him in, but then he saw she was injured and ran in a panic down the street to get help.
He bumped into a man on a bike who cycled to a service station to raise the alarm.
Police said an emergency call was made at 2.43am. Sina was pronounced dead by paramedics at the scene.
It was believed Sina was dropped home sometime between 2.20am and 2.30am and had left work at CMP after finishing her shift at 1am.
Detective Senior Sergeant John Rae said police were preparing a time line and looking at CCTV footage from locations in Ashburton.
An unsuccessful search for a weapon on surrounding streets is believed to have been undertaken, and Mr Rae confirmed some items, one of particular interest, was missing from the house and could have been discarded by someone leaving the scene.
Sina did not have a partner, and there was no suggestion she had been drinking after being at work for the evening. She was with friends when she needed a ride home, and got a family member to pick her up.
Police were examining the scene in conjunction with ESR staff and making enquiries to establish her movements in the hours leading up to when her body was found.
"Enquiries have are also being made with her associates and others who may have had relevant information to pass to the investigation. The enquiries are ongoing," Mr Rae said.
He said family members were being helpful.
"They are deeply shocked, they are trying to resolve she's no longer with them."
Sina's body was moved from the house around midday yesterday, and the property was likely to remain cordoned off today as scene examinations continue.
A post-mortem examination would be carried out in Christchurch today.
One neighbour said her husband heard "a bit of a noise" which sounded like yelling or "young ones going home from the pub" around 2.30am.
"Her husband had not thought much of it and had gone back to sleep."
Nikayla Thompson, who lives in a house at the rear of the victim's home said she heard a scream sometime between 1am and 3am.
Her flatmate Marc Ellery was asleep at the time, but he said he had heard the noise of drunk people on the street through the night.
Another neighbour knew the victim and described her as a "real bubbly and happy go lucky chick".
"She was a lovely girl, she was so full of life."
A tribute page on Facebook called Rest In Paradise Sina Solomona has been set up. | null | minipile | NaturalLanguage | mit | null |
About Tingre Nagar, Pune
New Projects in Tingre Nagar
Tingre Nagar is one of the popular localities in Pune having
1 apartment projects .
Some of the popular residential builders in the Tingre Nagar are Mantra Properties. The average price per square feet of apartments in Tingre Nagar is Rs. 4190. | null | minipile | NaturalLanguage | mit | null |
The present invention relates to a technique for detecting a specific image area and performing a removal process.
There is known a technique such as a volume rendering for generating a two-dimensional projection drawing from a three-dimensional image (volume data) which is made up of tomographic images (angio data) of a test object, the tomographic images being obtained by a tomographic apparatus such as an X-ray CT (Computed Tomography) apparatus and an MRI (Magnetic Resonance Imaging) apparatus.
The MIP (Maximum Intensity Projection) method is one of the volume rendering techniques. The MIP method is a method for applying a projection process to the volume data in an optional direction, and displaying on a projection plane, a maximum luminance value in the projection path. With this method, if there exists an area where bones and the like are displayed, which may hamper the observation of blood vessels or the like being a diagnostic object on the projection plane, a user has been allowed to select to a high degree of detail, the area which is desired to be removed. Accordingly, it has been possible to create a display where such area is removed (see, “Oh, I see!! Medical three-dimensional image; Bible of Way of thinking and Processing method”, supervised by Yutaka Imai, p. 141 (2003)(non-patent document 1), for instance). | null | minipile | NaturalLanguage | mit | null |
We think of interactions between predators as always antagonistic. Meat is hard to come by, and if one comes by meat on the hoof, it is unlikely that the owner-operator of said flesh will give it up willingly. Meat is a prized food source, and it is little wonder that most predators spend quite a bit of energy driving out competitors from hunting grounds.
Because of this antagonism, the domestication of wolves by ancient hunter-gatherers is difficult to explain. Indeed, the general way of getting wolves associated with people is see them as scavengers that gradually evolved to fear our species less.
This idea is pretty heavily promoted in the dog domestication literature, for it is difficult for experts to see how wolves could have been brought into the human fold any other way.
But there are still writers out there who posit a somewhat different course for dog domestication. Their main contentions are that scavengers don’t typically endear themselves to those from which they are robbing, and further, the hunter-gatherers of the Pleistocene did not produce enough waste to maintain a scavenging population of wolves.
It is virtually impossible to recreate the conditions in which some wolves hooked up with people. With the exception of those living on the some the Queen Elizabeth Islands, every extant wolf population has been persecuted heavily by man. Wolves generally avoid people, and there has been a selection pressure through our centuries of heavy hunting for wolves to have extreme fear and reactivity. It is unlikely that the wolves that were first encountered on the Mammoth Steppe were shy and retiring creatures. They would have been like the unpersecuted wolves of Ellesmere, often approaching humans with bold curiosity.
As I have noted in an earlier post, those Ellesmere wolves are an important population that have important clues to how dog domestication might have happened, but the truth of the matter is that no analogous population of wolves or other wild canids exists in which cooperation with humans is a major part of the survival strategy. The wolves on Ellesmere are not fed by anyone, but they don’t rely upon people for anything.
But they are still curious about our species, and their behavior is so tantalizing. Yet it is missing that cooperative analogy that might help us understand more.
I’ve searched the literature for this analogy. I’ve come up short every time. The much-celebrated cooperation between American badgers and coyotes is still quite controversial, and most experts now don’t believe the two species cooperate. Instead, they think the badger goes digging for ground squirrels, and the coyote stand outside the burrow entrance waiting for the prey to bolt out as the badger’s digging approaches its innermost hiding place in the den. The coyote gets the squirrel, and the badger wastes energy on its digging.
But there is a story that is hard to dispute. It has only been recorded once, but it is so tantalizing that I cannot ignore it.
Both of these species do engage in cooperative hunting behavior. Black-backed jackals often work together to hunt gazelles and other small antelope, and they are well-known to work together to kill Cape fur seal pups on Namibia’s Skeleton Coast. Male cheetahs form coalitions that work together to defend territory and to hunt cooperatively.
However, the two species generally have a hostile relationship. Cheetahs do occasionally prey upon black-backed jackals, and black-backed jackals will often mob a cheetah after it has made a kill, in hopes of forcing the cat to abandon all that meat.
So these animals usually cannot stand each other, and their interactions are not roseate in the least. Eaton described the “normal interaction” as follows:
The normal interaction between these two predators occurs when the jackals hunt in the late afternoon and come into a group of cheetahs. The jackals, often four or five, are normally spread out over several hundred yards and maintain contact by barking as they move. When cheetahs are encountered by one of the jackals, it barks to the others and they all come to the cheetahs, sniffing the air as they approach apparently looking for a kill. If the cheetahs are not on a kill, the jackals search the immediate area looking for a carcass that might have just been left by the cheetahs. If nothing is found, they remain near the cheetahs for some time, following them as they move ; and when a kill is made the jackals feed on the leftover carcass. If the cheetahs have already fed and are inactive and if a carcass is not found nearby, the jackals move on.
However, Eaton discovered that one particular group of jackals and one female cheetah had developed a different strategy:
At the time I was there in November, 1966, one area of the park was often frequented by a female cheetah with four cubs and was also the territory of a pair of jackals with three pups. The jackal young remained at the den while the adults hunted either singly or together. Upon encountering the cheetah family, the jackals approached to about 20 yards and barked but were ignored except for an occasional chase by the cubs. The jackals ran back and forth barking between the cheetahs and a herd of Grant’s gazelles (Gazella granti) feeding nearby. The two jackals had gone on to hunt and were almost out of sight by the time the adult cheetah attacked two male Grant’s gazelles that had grazed away from the herd. The hunt was not successful. The jackals took notice of the chase and returned to look for a kill ; it appeared that they associated food with the presence of the cheetahs and perhaps with the chase.
One month later, while observing the same cheetah family, I noticed that the entire jackal family was hunting as a group. The cheetah and her cubs were about 300 yards from a herd of mixed species. This same herd had earlier spotted the cheetahs and given alarm calls. The adult cheetah was too far away for an attack,there was little or no stalking cover and the herd was aware of her presence. The cheetahs had been lying in the shade for about one-half an hour since the herd spotted them when the jackals arrived. Upon discovering the cheetahs lying under an Acacia tree, one of the adult jackals barked until the others were congregated around the cheetah family. The jackal that had found the cheetahs crawled to within ten feet of the adult cheetah which did not respond. The jackal then stood up and made a very pneumatic sound by forcing air out of the lungs in short staccato bursts. This same jackal turned towards the game herd, ran to it and, upon reaching it, ran back and forth barking. The individuals of the herd watched the jackal intently. The cheetah sat up and watched the herd as soon as it became preoccupied with the activity of the jackal. Then the cheetah quickly got up and ran at half-speed toward the herd, getting to within 100 yards before being seen by the herd. The prey animals then took flight while the cheetah pursued an impala at full speed.
Upon catching the impala and making the kill, the cheetah called to its cubs to come and eat. After the cheetahs had eaten their fill and moved away from the carcass, the waiting jackals then fed on the remains.
Eaton made several observations of this jackal family working with this female cheetah, and by his calculations, the cheetah was twice as successful when the jackals harassed the herds to aid her stalk.
Eaton made note of this behavior and speculated that this sort of cooperative hunting could have been what facilitated dog domestication:
If cheetah and jackal can learn to hunt mutually then it is to be expected that man’s presence for hundreds, of thousands of years in areas with scavenging canines would have led to cooperative hunting between the two. In fact, it is hard to believe otherwise. It is equally possible that it was man who scavenged the canid and thereby established a symbiosis. Perhaps this symbiosis facilitated the learning of effective social hunting by hominids. Selection may have favored just such an inter-specific cooperation.
Agriculture probably ended the importance of hunting as the binding force between man and dog and sponsored the more intensive artificial selection of breeds for various uses. It is possible that until this period men lived closely with canids that in fossil form are indistinguishable from wild stock (Zeuner, 1954).
Domestication may have occurred through both hunting symbiosis and agricultural life; however, a hunting relationship probably led to the first domestication. Fossil evidence may eventually reconstruct behavioral associations between early man and canids.
Wolves are much more social and much more skilled as cooperative hunters than black-backed jackals are. Humans have a complex language and a culture through which techniques and technology can be passed from generation to generation.
So it is possible that a hunting relationship between man and wolf in the Paleolithic could have been maintained over many generations.
The cheetah had no way of teaching her cubs to let the jackals aid their stalks, and one family of jackals is just not enough to create a population of cheetah assistants.
But humans and these unpersecuted Eurasian wolves of the Pleistocene certainly could create these conditions.
I imagine that the earliest wolf-assisted hunts went much like these jackal-cheetah hunts. Wolves are always testing prey to assess weakness. If a large deer species or wild horse is not weak, it will stand and confront the wolves, and in doing so, it would be exposing itself to a spear being thrown in its direction.
If you’ve ever tried a low-carbohydrate diet, you will know that your body will crave fat. Our brains require quite a bit of caloric intake from fat to keep us going, which is one of those very real costs of having such a large brain. Killing ungulates that stood to fight off wolves meant that would target healthy animals in the herds, and healthy animals have more fat for our big brains.
Thus, working together with wolves would give those humans an advantage, and the wolves would be able to get meat with less effort.
So maybe working together with these Ellesmere-like wolves that lived in Eurasia during the Paleolithic made us both more effective predators, and unlike with the cheetah and the black-backed jackals, human intelligence, language, and cultural transmission allowed this cooperation to go on over generations.
Eaton may have stumbled onto the secret of dog domestication. It takes more than the odd population of scavenging canids to lay the foundations for this unusual domestication. Human agency and foresight joined with the simple cooperative nature of the beasts to make it happen. | null | minipile | NaturalLanguage | mit | null |
The only suspect ever charged, Minneapolis activist and graduate student Scott Ryan DeMuth, has always maintained his innocence. Under a plea bargain in September, federal prosecutors dropped allegations that he played a role in the attack, and he instead pleaded guilty to a misdemeanor for helping in a 2006 break-in at a ferret farm in Minnesota.
Barring a surprise development, that means nobody will be held criminally responsible for the raid in which masked men broke into Spence Laboratories, freed 400 rats and mice, dumped chemicals on data, damaged computers and equipment and publicized researchers' home addresses. The university estimated the damage at $500,000, not including the years of research lost.
"While active investigation of the attack on the University of Iowa Laboratory has been completed, there are still ongoing investigations into other potential criminal activities that came to light during the course of the investigation," Weysan Dun, special agent in charge of the FBI's Omaha office, said in a statement Monday.
DeMuth's lawyer, Michael Deutsch of Chicago, said the five-year statute of limitations on the attack has run out and the FBI "hasn't come up with anybody." The government could still theoretically bring charges against suspects by arguing a criminal conspiracy existed for years afterward, he said.
UI psychology researchers whose work was destroyed still call the Nov. 14, 2004, attack an act of domestic terrorism. They said they are disappointed nobody will be held responsible, but praised the FBI's aggressive pursuit of the case.
"While this wasn't the outcome we had hoped for ... the message was clearly sent that felony acts committed in the name of liberating animals are not going to be tolerated," UI Psychology Department Chairman Alan Christensen said.
But Deutsch said the investigation into DeMuth, who was originally indicted on one count of conspiring to commit animal enterprise terrorism, was an overreach.
DeMuth came under scrutiny after authorities investigating alleged criminal acts by protesters during the 2008 Republican National Convention in St. Paul, Minn., raided his home and seized his computer and a journal. In a 2005 entry, he wrote he was worried about federal scrutiny and that "it's been almost a year since Iowa" but did not elaborate.
Authorities said DeMuth's height was similar to one of the masked intruders seen in a video released by the Animal Liberation Front, an extremist group that took responsibility for the attack.
Deutsch said the journal entry was referring to a 2004 meeting of protesters in Des Moines, not the university attack, and he noted the identities of those in the video could not be ascertained because of their disguises.
DeMuth contends his indictment was vindictive, coming after he refused to answer questions before a grand jury, and that he was targeted because his political views are anarchist. The charge also came days before the statute of limitations ran out.
His former girlfriend, Carrie Feldman, was detained in connection with the investigation but released without being charged.
"They made a desperate effort to haul Mr. DeMuth into it, but he wasn't involved in it in any way," Deutsch said. "It just seemed like they were desperate to hold somebody accountable. The statute was about to run out, and they falsely accused this young man of being involved. The FBI has put in a lot of resources to figure it out and hasn't come up with anybody."
But UI psychology professor Amy Poremba, whose research was among that destroyed, scoffed at the notion that DeMuth was innocent. The 23-year-old faces prison time when he is sentenced in January for the April 2006 release of ferrets from Lakeside Ferrets Inc., a ferret breeder in Minnesota.
"By pleading guilty to the ferret break-in, Demuth admits he is clearly connected to groups willing to break the law," Poremba said. "The FBI investigation was very useful in gaining information about the population involved in these terror activities."
Dun said the case was a priority for the FBI, which collaborated with federal prosecutors, UI police and the Joint Terrorism Task Force. Hundreds of interviews were conducted, and investigators collected and analyzed extensive amounts of evidence, he said.
An FBI affidavit last year spelled out the painstaking investigation, which concluded the overalls worn by suspects were bought in Cedar Rapids, and the Animal Liberation Front may have sent its message taking responsibility from a computer in UI's law library.
Investigators raided the home of an activist in Salt Lake City in connection with the case, but that person, Peter Daniel Young, was never charged.
Young, who has been convicted in a string of incidents in which he freed mink from fur farms, called the case "among the most egregious examples of prosecutorial overzealousness in the animal liberation movement's history" on his blog. He denied involvement in the Iowa attack, which he called one of the largest and most successful on a university research lab.
"They were able to get in deep inside a laboratory that had some fairly sophisticated security," he said. "They were able to get animals out, smash the labs up and not be apprehended. That was very empowering for lots of activists." | null | minipile | NaturalLanguage | mit | null |
The aim of this project is to define the immunopathogenetic mechanisms responsible for human immunodeficiency syndromes. During the past year we have focused on the functional capacities of T and B cells from patients with common variable immunodeficiency (CVI). In studies of CVI patient T cells, we showed that while purified CD4 T cells proliferate normally in response to proliferation by PHA, staphylococcal enterotoxin B (SEB) and anti-CD2 antibodies, these stimuli induce significantly less IL-2 than normal CD4+ T cells. On the other hand, immobilized anti-CD3 antibody and this stimulus in combination with anti-CD28 antibody induced CVI T cells to produce normal amounts of IL-2. These studies thus demonstrate that the IL-2 production defect in CD4+ T cells is due to a specific signalling pathway abnormality. In a second series of studies, in this case focused on B cell function in CVI patients, we showed first that circulating B cells express normal amounts of sIgM+, but greatly reduced amounts of sIgG+ and sIgA+; this result implies that patients have an in vivo defect of isotype switch differentiation. In other studies, we showed that upon stimulation of purified patient B cells with anti-CD40 antibody plus IL-10, patient B cells expressed Cmu, Cgamma and Calpha mRNA; furthermore, stimulation of cells with anti-CD40 plus IL-10, followed by stimulation with activated T cells (which express the CD40 ligand), led to partial restoration of IgG and IgA synthesis. These studies thus indicate that CVI B cells can be partially restored to normal function in vitro, suggesting that the cells are not irreversibly blocked from isotype and terminal differentiation. | null | minipile | NaturalLanguage | mit | null |
Effect of calcitonin on bone lesions in chronic dialysis patients.
The effects of synthetic salmon CT, administered subcutaneously and intermittently (1 MRC U/kg/day for 15 days/month over 6 months) were investigated in 15 uremic patients on regular dialysis treatment (RDT), all presenting various degrees of osteodystrophy. Clinically, osteoarticular pain disappeared in 8 out of 10 cases; 1 patient with rib fractures had a rapid calcification of the bone fracture repair tissue. No significant changes were found in serum calcium and PTH levels. Phosphotemia showed a significant decrease within the first 20 days. The varying individual hypophosphatemic response proved to be related to the initial level of phosphatemia. The alkaline phosphatase, when increased, showed a decrease to the normal range. A significant decrease in osteoclastic hyperactivity (active resorption surface, osteoclast index) and a slight increase in osteoblastic pool (active osteoid surface) were documented. No change was noted when osteomalacia predominated. Side effects included: anorexia, nausea, vomiting, face flushing. Our data suggest that salmon CT may be usefully employed in chronic uremic patients on RDT, when secondary hyperparathyroidism predominates. | null | minipile | NaturalLanguage | mit | null |
---
abstract: 'The present paper deals with some results of almsot semi-invariant submanifolds of generalized Sasakian-space-forms in [@ALEGRE3] with respect to semisymmetric metric connection, semisymmetric non-metric connection, Schouten-van Kampen connection and Tanaka-Webster connection.'
author:
- 'Pradip Mandal and Shyamal Kumar Hui$^{*}$'
title: 'A note on submanifolds of $\bar{M}^{2n+1}(f_1,f_2,f_3)$ with respect to certain connections'
---
Introduction
============
The notion of generalized Sasakian-space-form was introduced by Alegre et al. [@ALEGRE1]. An almost contact metric manifold $\bar{M}(\phi,\xi,\eta,g)$ whose curvature tensor $\bar{R}$ satisfies $$\begin{aligned}
\label{eqn1.1}
\bar{R}(X,Y)Z &=f_1\big\{g(Y,Z)X-g(X,Z)Y\big\}+f_2\big\{g(X,\phi Z)\phi Y\\
\nonumber& - g(Y,\phi Z)\phi X + 2g(X,\phi Y)\phi Z\big\}+f_3\big\{\eta(X)\eta(Z)Y\\
\nonumber& - \eta(Y)\eta(Z)X+g(X,Z)\eta(Y)\xi - g(Y,Z)\eta(X)\xi\big\}\end{aligned}$$ for all vector fields $X$, $Y$, $Z$ on $\bar{M}$ and $f_1,f_2,f_3$ are certain smooth functions on $\bar{M}$ is said to be generalized Sasakian-space-form [@ALEGRE1]. Such a manifold of dimension $(2n+1)$, $n>1$ (the condition $n>1$ is assumed throughout the paper), is denoted by $\bar{M}^{2n+1}(f_1,f_2,f_3)$ [@ALEGRE1]. Many authors studied this space form with different aspects. For this, we may refer, ([@PM1], [@HUI1], [@HUI2], [@HUI3], [@HUI4], [@HUI5], [@HUI6], [@KISH] and [@HUI8]). It reduces to Sasakian-space-form if $f_1 = \frac{c+3}{4}$, $f_2 = f_3 = \frac{c-1}{4}$ [@ALEGRE1]. We denote Sasakian-space-form of dimension $(2n+1)$ by $M^{2n+1}(c)$.
After introduced the semisymmetric linear connection by Friedman and Schouten [@FRID], Hayden [@HAYD] gave the idea of metric connection with torsion on a Riemannian manifold. Later, Yano [@YANO] and many others (see, [@HUI7], [@SHAIKH1], [@SULAR] and references therein) studied semisymmetric metric connection in different context. The idea of semisymmetric non-metric connection was introduced by Agashe and Chafle [@AGAS].
The Schouten-van Kampen connection introduced for the study of non-holomorphic manifolds ([@SCHO], [@VRAN]). In $2006$, Bejancu [@BEJA3] studied Schouten-van Kampen connection on foliated manifolds. Recently Olszak [@OLSZ] studied Schouten-van Kampen connection on almost(para) contact metric structure.
The Tanaka-Webster connection ([@TANA], [@WEBS]) is the canonical affine connection defined on a non-degenerate pseudo-Hermitian CR-manifold. Tanno [@TANN] defined the Tanaka-Webster connection for contact metric manifolds.
In [@ALEGRE3], Alegre and Carriazo studied almost semi-invariant submanifolds of generalized Sasakian-space-form with respect to Levi-Civita connection. In this paper, we have studied the results of [@ALEGRE3] with respect to certain connections, namely semisymmetric metric connection, semisymmetric non-metric connection, Schouten-van Kampen connection, Tanaka-Webster connection.
preliminaries
=============
In an almost contact metric manifold $\bar{M}(\phi,\xi,\eta,g)$, we have [@BLAIR] $$\begin{aligned}
\label{eqn2.1}
\phi^2(X) = -X+\eta(X)\xi,\ \phi \xi=0,\end{aligned}$$ $$\begin{aligned}
\label{eqn2.2}
\eta(\xi) = 1,\ g(X,\xi) = \eta(X),\ \eta(\phi X) = 0,\end{aligned}$$ $$\begin{aligned}
\label{eqn2.3}
g(\phi X,\phi Y) = g(X,Y)-\eta(X)\eta(Y),\end{aligned}$$ $$\begin{aligned}
\label{eqn2.4} g(\phi X,Y) = -g(X,\phi Y).\end{aligned}$$ In $\bar{M}^{2n+1}(f_1,f_2,f_3)$, we have [@ALEGRE1] $$\begin{aligned}
\label{eqn2.5}
(\bar{\nabla}_X\phi)(Y) = (f_1-f_3)[g(X,Y)\xi - \eta(Y)X],\end{aligned}$$ $$\begin{aligned}
\label{eqn2.6}
\bar{\nabla}_X\xi = -(f_1-f_3) \phi X,\end{aligned}$$ where $\bar{\nabla}$ is the Levi-Civita connection of $\bar{M}^{2n+1}(f_1,f_2,f_3)$.
The semisymmetric metric connection $\tilde{\bar{\nabla}}$ and the Riemannian connection $\bar{\nabla}$ on ${\bar{M}}^{2n+1}(f_{1},f_{2},f_{3})$ are related by [@YANO] $$\begin{aligned}
\label{eqn2.10}
\tilde{\bar{\nabla}}_{X}Y= \bar{\nabla}_X Y+\eta(Y)X-g(X,Y)\xi.\end{aligned}$$ The Riemannian curvature tensor $\tilde{\bar{R}}$ of $\bar{M}^{2n+1}(f_{1},f_{2},f_{3})$ with respect to $\tilde{\bar{\nabla}}$ is $$\begin{aligned}
\label{eqn2.11}
\tilde{\bar{R}}(X,Y)Z &=&(f_1-1)\big\{g(Y,Z)X-g(X,Z)Y\big\}+f_2\big\{g(X,\phi Z)\phi Y\\
\nonumber&-&g(Y,\phi Z)\phi X + 2g(X,\phi Y)\phi Z\big\}+(f_3-1)\big\{\eta(X)\eta(Z)Y \\
\nonumber&-& \eta(Y)\eta(Z)X+g(X,Z)\eta(Y)\xi - g(Y,Z)\eta(X)\xi\big\}\\
\nonumber&+&(f_1-f_3)\{g( X, \phi Z)Y-g( Y,\phi Z)X\\
\nonumber&+&g(Y,Z)\phi X-g(X,Z)\phi Y\}.\end{aligned}$$ The semisymmetric non-metric connection ${\bar{\nabla}}^{'}$ and the Riemannian connection $\bar{\nabla}$ on ${\bar{M}}^{2n+1}(f_{1},f_{2},f_{3})$ are related by [@AGAS] $$\begin{aligned}
\label{eqn2.12}
\bar{\nabla}^{'}_{X}Y= \bar{\nabla}_X Y+\eta(Y)X.\end{aligned}$$ The Riemannian curvature tensor ${\bar{R}}^{'}$ of $\bar{M}^{2n+1}(f_{1},f_{2},f_{3})$ with respect to ${\bar{\nabla}}^{'}$ is $$\begin{aligned}
\label{eqn2.13}
{\bar{R}}^{'}(X,Y)Z &=&f_1\big\{g(Y,Z)X-g(X,Z)Y\big\}+f_2\big\{g(X,\phi Z)\phi Y\\
\nonumber&-&g(Y,\phi Z)\phi X + 2g(X,\phi Y)\phi Z\big\}+f_3\big\{\eta(X)\eta(Z)Y\\
\nonumber& -&\eta(Y)\eta(Z)X+g(X,Z)\eta(Y)\xi - g(Y,Z)\eta(X)\xi\big\}\\
\nonumber&+&(f_1-f_3)[g(X,\phi Z)Y-g( Y,\phi Z) X]\\
\nonumber&+&\eta(Y)\eta(Z)X-\eta(X)\eta(Z)Y.\end{aligned}$$ The Schouten-van Kampen connection $\hat{\bar{\nabla}}$ and the Riemannian connection $\bar{\nabla}$ of ${\bar{M}}^{2n+1}(f_{1},f_{2},f_{3})$ are related by [@OLSZ] $$\begin{aligned}
\label{eqn2.14}
\hat{\bar{\nabla}}_{X}Y=\bar{\nabla}_X Y+(f_1-f_3)\eta(Y)\phi X-(f_1-f_3)g(\phi X,Y)\xi.\end{aligned}$$ The Riemannian curvature tensor $\hat{\bar{R}}$ of $\bar{M}^{2n+1}(f_{1},f_{2},f_{3})$ with respect to $\hat{\bar{\nabla}}$ is $$\begin{aligned}
\label{eqn2.15}
\hat{\bar{R}}(X,Y)Z&=&f_1\big\{g(Y,Z)X-g(X,Z)Y\big\}+f_2\big\{g(X,\phi Z)\phi Y\\
\nonumber&-&g(Y,\phi Z)\phi X +2g(X,\phi Y)\phi Z\big\}\\
\nonumber&+&\{f_3+(f_1-f_3)^2\}\big\{\eta(X)\eta(Z)Y - \eta(Y)\eta(Z)X\\
\nonumber&+&g(X,Z)\eta(Y)\xi - g(Y,Z)\eta(X)\xi\big\}\\
\nonumber&+&(f_1-f_3)^2\{g(X,\phi Z)\phi Y-g(Y, \phi Z)\phi X\}.\end{aligned}$$ The Tanaka-Webster connection $\stackrel{\ast}{\bar{\nabla}}$ and the Riemannian connection $\bar{\nabla}$ of ${\bar{M}}^{2n+1}(f_{1},f_{2},f_{3})$ are related by [@CHO] $$\begin{aligned}
\label{eqn2.16}
\stackrel{\ast}{\bar{\nabla}}_{X}Y= \bar{\nabla}_X Y+\eta(X)\phi Y+(f_1-f_3)\eta(Y)\phi X-(f_1-f_3)g(\phi X,Y)\xi.\end{aligned}$$ The Riemannian curvature tensor $\stackrel{*}{\bar{R}}$ of $\bar{M}^{2n+1}(f_{1},f_{2},f_{3})$ with respect to $\stackrel{*}{\bar{\nabla}}$ is $$\begin{aligned}
\label{eqn2.17}
\stackrel{*}{\bar{R}}(X,Y)Z &=&f_1\big\{g(Y,Z)X-g(X,Z)Y\big\}+f_2\big\{g(X,\phi Z)\phi Y\\
\nonumber&-& g(Y,\phi Z)\phi X+ 2g(X,\phi Y)\phi Z\big\}\\
\nonumber&+&\{f_3+(f_1-f_3)^2\}\big\{\eta(X)\eta(Z)Y - \eta(Y)\eta(Z)X\\
\nonumber&+&g(X,Z)\eta(Y)\xi - g(Y,Z)\eta(X)\xi\big\}\\
\nonumber&+&{(f_1-f_3)^2}\{g(X,\phi Z)\phi Y-g(Y,\phi Z)\phi X\}\\
\nonumber&+&2(f_1-f_3)g(X,\phi Y)\phi Z.\end{aligned}$$
Let $M$ be a $(m+1)$-dimensional submanifold of $\bar{M}^{2n+1}(f_1,f_2,f_3)$. If $\nabla$ and $\nabla^\perp$ are the induced connections on the tangent bundle $TM$ and the normal bundle $T^\perp{M}$ of $M$, respectively then the Gauss and Weingarten formulae are given by [@YANOKON1] $$\begin{aligned}
\label{eqn2.7}
\bar{\nabla}_XY = \nabla_XY +h(X,Y),\ \bar{\nabla}_XV = -A_VX + \nabla_X^{\perp}V\end{aligned}$$ for all $X,Y\in\Gamma(TM)$ and $V\in\Gamma(T^{\perp}M)$, where $h$ and $A_V$ are second fundamental form and shape operator (corresponding to the normal vector field V), respectively and they are related by $ g(h(X,Y),V) = g(A_VX,Y)$ [@YANOKON1].
Moreover, if $h(X,Y)=0$ for all $X,Y \in \Gamma(TM)$ then $M$ is said to be [*[totally geodesic]{}*]{} and if $H=0$ then $M$ is [*[minimal]{}*]{} in $\bar{M}$, where $H$ is the mean curvature tensor.
From (\[eqn2.7\]), we have the Gauss equations as $$\begin{aligned}
\label{eqn2.9}
\bar{R}(X,Y,Z,W)&=R(X,Y,Z,W)-g\big(h(X,W),h(Y,Z)\big)\\
\nonumber&+g\big(h(X,Z),h(Y,W)\big),\end{aligned}$$ where $R$ is the curvature tensor of $M$. Let $\tilde{\nabla}$, $\nabla^{'}$, $\hat{\nabla}$ and $\stackrel{*}{\nabla}$ are the induced connection of $M$ from the connection $\tilde{\bar{\nabla}}$, $\bar{\nabla}^{'}$, $\hat{\bar{\nabla}}$ and $\stackrel{*}{\bar{\nabla}}$ of $\bar{M}^{2n+1}(f_1,f_2,f_3)$ respectively. Then Gauss equation with respect to $\tilde{\bar{\nabla}}$, $\bar{\nabla}^{'}$, $\hat{\bar{\nabla}}$ and $\stackrel{*}{\bar{\nabla}}$ are $$\begin{aligned}
\label{eqn2.9a}
\tilde{\bar{R}}(X,Y,Z,W)&=\tilde{R}(X,Y,Z,W)-g\big(\tilde{h}(X,W),\tilde{h}(Y,Z)\big)\\
\nonumber&+g\big(\tilde{h}(X,Z),\tilde{h}(Y,W)\big),\end{aligned}$$ $$\begin{aligned}
\label{eqn2.9b}
\bar{R}^{'}(X,Y,Z,W)&=R^{'}(X,Y,Z,W)-g\big(h^{'}(X,W),h^{'}(Y,Z)\big)\\
\nonumber&+g\big(h^{'}(X,Z),h^{'}(Y,W)\big),\end{aligned}$$ $$\begin{aligned}
\label{eqn2.9c}
\hat{\bar{R}}(X,Y,Z,W)&=\hat{R}(X,Y,Z,W)-g\big(\hat{h}(X,W),\hat{h}(Y,Z)\big)\\
\nonumber&+g\big(\hat{h}(X,Z),\hat{h}(Y,W)\big),\end{aligned}$$ $$\begin{aligned}
\label{eqn2.9d}
\stackrel{*}{\bar{R}}(X,Y,Z,W)&=\stackrel{*}{R}(X,Y,Z,W)-g\big(\stackrel{*}{h}(X,W),\stackrel{*}{h}(Y,Z)\big)\\
\nonumber&+g\big(\stackrel{*}{h}(X,Z),\stackrel{*}{h}(Y,W)\big),\end{aligned}$$ where $\tilde{h}$, $h^{'}$, $\hat{h}$, $\stackrel{*}{h}$ are the second fundamental forms with respect to $\tilde{\nabla}$, ${\nabla}^{'}$, $\hat{\nabla}$ and $\stackrel{*}{\nabla}$ respectively. Also $\tilde{H}$, $H^{'}$, $\hat{H}$, $\stackrel{*}{H}$ be the mean curvature of $M$ with respect to $\tilde{\nabla}$, ${\nabla}^{'}$, $\hat{\nabla}$ and $\stackrel{*}{\nabla}$ respectively.
For any $X\in\Gamma(TM)$, we may write $$\label{eqn2.10a} \phi X=TX+FX,$$ where $TX$ is the tangential component and $FX$ is the normal component of $\phi X$.
([@ALEGRE3], [@TRIPs]) A submanifold $M$ of an almost contact metric manifold $\bar{M}$, $\xi$ tangent to $M$, is said to be an almost semi-invariant submanifold if their exist $l$ functions $\lambda_1,\cdots,\lambda_l$, defined on $M$ with values in $(0,1)$, such that
1. $-\lambda_1^2(p),\cdots,-\lambda_l^2(p)$ are distinct eigenvalues of $T^2|_D$ at $p\in M$, with $$T_pM=D^1_p\oplus D^0_p\oplus D^{\lambda_1}_p\oplus\cdots\oplus D^{\lambda_l}_p\oplus span\{\xi_p\},$$ where $D^\lambda_p$, $\lambda\in \{1,0,\lambda_1(p),\cdots,\lambda_l(p)\}$, denotes the eigenspace associated to the eigenvalue $-\lambda^2$.
2. the dimension of $D^1_p,D^0_p, D^{\lambda_1}_p,\cdots, D^{\lambda_l}_p$ are independent of $p\in M$.
Let the orthogonal projection from $TM$ on $D^\lambda$ be $U^\lambda$. Then we have $$\label{eqn2.10b}
g(TX,TY)=\sum_{\lambda}^{}\lambda^2g(U^\lambda X,U^\lambda Y).$$
Let us consider $\{E_1,\cdots,E_m,E_{m+1}=\xi\}$ and $\{F_1,\cdots,F_{2n-m}\}$ local orthonormal basis of $TM$ and $T^\bot M$ respectively, and denote $A_{F_k}=A_k$.
Ricci tensor on $ M$ of $\bar{M}^{2n+1}(f_1,f_2,f_3)$ with $\tilde{\bar{\nabla}}$
=================================================================================
The Ricci tensor $\tilde{S}$ of submanifold $M$ of $\bar{M}^{2n+1}(f_1,f_2,f_3)$ with respect to $\tilde{\bar{\nabla}}$ is $$\begin{aligned}
\label{eqn3.1}
\tilde{S}(X,Y) &=& mf_1g(X,Y)+3f_2g(TX,TY)-(f_3-1)\{g(X,Y)\\
\nonumber&+&(m-1)\eta(X)\eta(Y)\}+(f_1-f_3)(m-1)g(TX,Y) \\
\nonumber&+&\sum_{k=1}^{2n-m}\{(m+1)(trace \tilde{A}_k)g(\tilde{A}_kX,Y)-g(\tilde{A}_kX,\tilde{A}_kY)\}\end{aligned}$$ for any vector fields $X,Y$ on $M$.
Using (\[eqn2.11\]) and (\[eqn2.9a\]) we have the above Lemma.
The Ricci tensor $\tilde{S}$ of almost semi-invariant submanifolds $M$ of $\bar{M}^{2n+1}(f_1,f_2,f_3)$ with respect to $\tilde{\bar{\nabla}}$ is $$\begin{aligned}
\label{eqn3.2}
\tilde{S}(X,Y)&=&\sum_{\lambda}^{}(mf_1+3f_2\lambda^2-f_3+1)g(U^\lambda X,U^\lambda Y)\\
\nonumber&&+m(f_1-f_3+1)\eta(X)\eta(Y)+(f_1-f_3)(m-1)g(T X,Y)\\
\nonumber&+&\sum_{k=1}^{2n-m}\{(m+1)(trace \tilde{A}_k)g(\tilde{A}_kX,Y)-g(\tilde{A}_kX,\tilde{A}_kY)\}\end{aligned}$$ for any vector fields $X,Y$ on $M$.
Using (\[eqn2.10b\]) and (\[eqn3.1\]) we have the above Lemma.
For an almost semi-invariant submanifolds $M$ of Sasakian-space-form $\bar{M}^{2n+1}(c)$ with respect to $\tilde{\bar{\nabla}}$ is $$\begin{aligned}
\label{eqn3.3}
\tilde{S}(X,Y) &=& \frac{(m-1+3\lambda^2)c+3(m-\lambda^2)+5}{4}g(U^\lambda X,U^\lambda Y) \\
\nonumber&+&(m-1)g(T X,Y)+\sum_{k=1}^{2n-m}\{(m+1)(trace \tilde{A}_k)g(\tilde{A}_kX,Y)\\
\nonumber&-&g(\tilde{A}_kX,\tilde{A}_kY)\}\end{aligned}$$ for any vector fields $X,Y$ on $M$.
Putting $f_1=\frac{c+3}{4},\ f_2=f_3=\frac{c-1}{4}$ in (\[eqn3.2\]) we obtain the result.
The scalar curvature $\tilde{\tau}$ of an almost semi-invariant submanifold $M$ of $\bar{M}^{2n+1}(f_1,f_2,f_3)$ with respect to $\tilde{\bar{\nabla}}$ is $$\begin{aligned}
\label{eqn3.4}
\tilde{\tau} &=& f_1+\frac{1}{m(m+1)}\{3f_2\sum_{\lambda}^{}n(\lambda)\lambda^2-2mf_3+2m\} \\
\nonumber&+&(m+1)^2||\tilde{H}||^2-||\tilde{h}||^2.\end{aligned}$$
Let us consider an orthonormal frame $\{E_1,\cdots,E_{n(\lambda)}\}$ in $D^\lambda$. Then we have $$\label{eqn3.7}
\tilde{\tau}=\frac{1}{m(m+1)}\sum\limits_{i,j=1}^{m+1}\tilde{R}(E_i,E_j,E_j,E_i).$$ Using (\[eqn2.11\]), (\[eqn2.9a\]) in (\[eqn3.7\]) we get (\[eqn3.4\]).
If $M$ is an almost semi-invariant minimal submanifold of $\bar{M}^{2n+1}(f_1,f_2,f_3)$ with respect to $\tilde{\bar{\nabla}}$, then the following relation holds:
- $\tilde{S}(X,X)\leq\sum\limits_{\lambda}^{}(mf_1+3f_2\lambda^2-f_3+1)g(U^\lambda X,U^\lambda X)$ $+m(f_1-f_3+1)\eta(X)\eta(X)+(f_1-f_3)(m-1)g(TX,X)$,\
- $\tilde{\tau}\leq f_1+\frac{1}{m(m+1)}\{3f_2\sum\limits_{\lambda}^{}n(\lambda)\lambda^2-2m(f_3-1)\}$.
Since $M$ is minimal submanifold with respect to $\tilde{\bar{\nabla}}$, then we have $$\begin{aligned}
\label{eqn3.5}
\sum_{k=1}^{2n-m}(trace\ \tilde{A}_k)g(\tilde{A}_kX,X)&=&\sum_{i=1}^{m+1}g(\tilde{h}(X,X),\tilde{h}(E_i,E_i))\\
\nonumber&=&(m+1)g(\tilde{h}(X,X),\tilde{H})=0.\end{aligned}$$ Using (\[eqn3.5\]) in (\[eqn3.2\]) we have $$\begin{aligned}
\label{eqn3.6}
&&\tilde{S}(X,X)-\sum_{\lambda}^{}(mf_1+3f_2\lambda^2-f_3+1)g(U^\lambda X,U^\lambda X)\\
\nonumber&&-m(f_1-f_3+1)\eta(X)\eta(X)-(f_1-f_3)(m-1)g(T X,X)\\
\nonumber&=&-\sum_{k=1}^{2n-m}g(\tilde{A}_kX,\tilde{A}_kX)\leq 0,\end{aligned}$$ which proves (i).\
The second part (ii) comes directly from Lemma $3.3$.
The equality of (i) and (ii) in Theorem $3.1$ holds if $M$ is almost semi-invariant totally geodesic submanifolds of $\bar{M}^{2n+1}(f_1,f_2,f_3)$ with respect to $\tilde{\bar{\nabla}}$.
If $M$ is totally geodesic submanifold with respect to $\tilde{\bar{\nabla}}$, then $M$ is minimal submanifold with respect to $\tilde{\bar{\nabla}}$. Then by virtue of Lemma $3.2$ we have the equality case (i) and by virtue of Lemma $3.3$ we have equality case of (ii).
Submanifolds of $\bar{M}^{2n+1}(f_1,f_2,f_3)$ with ${\bar{\nabla}}^{'}$
=======================================================================
The Ricci tensor ${S}^{'}$ of submanifold $M$ of $\bar{M}^{2n+1}(f_1,f_2,f_3)$ with respect to ${\bar{\nabla}}^{'}$ is $$\begin{aligned}
\label{eqn4.1}
{S}^{'}(X,Y) &=& mf_1g(X,Y)+3f_2g(TX,TY)-(f_3-1)\{g(X,Y)\\
\nonumber&+&(m-1)\eta(X)\eta(Y)\}-m\eta(X)\eta(Y)+(f_1-f_3)g(TX,Y) \\
\nonumber&+&\sum_{k=1}^{2n-m}\{(m+1)(trace A^{'}_k)g(A^{'}_kX,Y)-g(A^{'}_kX,A^{'}_kY)\}\end{aligned}$$ for any vector fields $X,Y$ on $M$.
Using (\[eqn2.13\]) and (\[eqn2.9b\]) we have the above Lemma.
The Ricci tensor $S^{'}$ of almost semi-invariant submanifolds $M$ of $\bar{M}^{2n+1}(f_1,f_2,f_3)$ with respect to $\bar{\nabla}^{'}$ is $$\begin{aligned}
\label{eqn4.2}
S^{'}(X,Y)&=&\sum_{\lambda}^{}(mf_1+3f_2\lambda^2-f_3+1)g(U^\lambda X,U^\lambda Y)\\
\nonumber&+&m(f_1-f_3)\eta(X)\eta(Y)+(f_1-f_3)g(T X,Y)\\
\nonumber&+&\sum_{k=1}^{2n-m}\{(m+1)(traceA^{'}_k)g(A^{'}_kX,Y)-g(A^{'}_kX,A^{'}_kY)\}\end{aligned}$$ for any vector fields $X,Y$ on $M$.
Using (\[eqn2.10b\]) and (\[eqn4.1\]) we have the above Lemma.
The Ricci tensor $S^{'}$ of almost semi-invariant submanifolds $M$ of $\bar{M}^{2n+1}(c)$ with respect to $\bar{\nabla}^{'}$ is $$\begin{aligned}
\label{eqn4.3}
\ \ \ {S}^{'}(X,Y) &=& \frac{(m-1+3\lambda^2)c+3(m-\lambda^2)+5}{4}g(U^\lambda X,U^\lambda Y)+g(T X,Y) \\
\nonumber&+&\sum_{k=1}^{2n-m}\{(m+1)(trace A^{'}_k)g(A^{'}_kX,Y)-g(A^{'}_kX,A^{'}_kY)\}\end{aligned}$$ for any vector fields $X,Y$ on $M$.
Putting $f_1=\frac{c+3}{4},\ f_2=f_3=\frac{c-1}{4}$ in (\[eqn4.2\]) we obtain the result.
The scalar curvature $\tau^{'}$ of an almost semi-invariant submanifold $M$ of $\bar{M}^{2n+1}(f_1,f_2,f_3)$ with respect to $\bar{\nabla}^{'}$ is $$\begin{aligned}
\label{eqn4.4}
\tau^{'}&=& f_1+\frac{1}{m(m+1)}\{3f_2\sum_{\lambda}^{}n(\lambda)\lambda^2-2mf_3+m\}\\
\nonumber&+&(m+1)^2||H^{'}||^2-||h^{'}||^2.\end{aligned}$$
It is known that $$\label{eqn4.7}
\tau^{'}=\frac{1}{m(m+1)}\sum\limits_{i,j=1}^{m+1}{R}^{'}(E_i,E_j,E_j,E_i).$$ Using (\[eqn2.13\]), (\[eqn2.9b\]) in (\[eqn4.7\]) we get (\[eqn4.4\]).
If $M$ is an almost semi-invariant minimal submanifolds of $\bar{M}^{2n+1}(f_1,f_2,f_3)$ with respect to $\bar{\nabla}^{'}$, then the following condition holds:
- $S^{'}(X,X)\leq\sum\limits_{\lambda}^{}(mf_1+3f_2\lambda^2-f_3+1)g(U^\lambda X,U^\lambda X)+m(f_1-f_3)\eta(X)\eta(X)+(f_1-f_3) g(T X,X)$,\
- $\tau^{'}\leq f_1+\frac{1}{m(m+1)}\{3f_2\sum\limits_{\lambda}^{}n(\lambda)\lambda^2-2mf_3+m\}$.
Since $M$ is minimal submanifold with respect to $\bar{\nabla}^{'}$, then we have $$\begin{aligned}
\label{eqn4.5}
\sum_{k=1}^{2n-m}(trace A^{'}_k)g(A^{'}_kX,X)&=&\sum_{i=1}^{m+1}g(h^{'}(X,X),h^{'}(E_i,E_i))\\
\nonumber&=&(m+1)g(h^{'}(X,X),H^{'})=0.\end{aligned}$$ Using (\[eqn4.2\]) and (\[eqn4.5\]) we have $$\begin{aligned}
\label{eqn4.6}
&&S^{'}(X,X)-\sum_{\lambda}^{}(mf_1+3f_2\lambda^2-f_3+1)g(U^\lambda X,U^\lambda X) \\
\nonumber&&-m(f_1-f_3)\eta(X)\eta(X)-(f_1-f_3) g(T X,X)\\
\nonumber&=&-\sum_{k=1}^{2n-m}g(A^{'}_kX,A^{'}_kX)\leq 0.\end{aligned}$$ This proves (i).\
The second part (ii) is comes directly from Lemma $4.3$.
The equality of (i) and (ii) in Theorem $4.1$ holds if $M$ is almost semi-invariant totally geodesic submanifolds of $\bar{M}^{2n+1}(f_1,f_2,f_3)$ with respect to $\bar{\nabla}^{'}$.
If $M$ is totally geodesic submanifold with respect to $\bar{\nabla}^{'}$, then $M$ is minimal submanifold with respect to $\bar{\nabla}^{'}$. Then by virtue of Lemma $4.2$ we have the equality case of (i) and by virtue of Lemma $4.3$ we have the equality case (ii).
Submanifolds of $\bar{M}^{2n+1}(f_1,f_2,f_3)$ with $\hat{\bar{\nabla}}$
=======================================================================
The Ricci tensor $\hat{S}$ of submanifold $M$ of $\bar{M}^{2n+1}(f_1,f_2,f_3)$ with respect to $\hat{\bar{\nabla}}$ is $$\begin{aligned}
\label{eqn5.1}
\hat{S}(X,Y) &=& mf_1g(X,Y)+\{3f_2+(f_1-f_3)^2\}g(TX,TY)\\
\nonumber&-&\{f_3+(f_1-f_3)^2\}\{g(X,Y)+(m-1)\eta(X)\eta(Y)\} \\
\nonumber&+&\sum_{k=1}^{2n-m}\{(m+1)(trace \hat{A}_k)g(\hat{A}_kX,Y)-g(\hat{A}_kX,\hat{A}_kY)\}\end{aligned}$$ for any vector fields $X,Y$ on $M$.
Using (\[eqn2.15\]) and (\[eqn2.9c\]) we have the above Lemma.
The Ricci tensor $\hat{S}$ of almost semi-invariant submanifolds $M$ of $\bar{M}^{2n+1}(f_1,f_2,f_3)$ with respect to $\hat{\bar{\nabla}}$ is $$\begin{aligned}
\label{eqn5.2}
\hat{S}&=&\sum_{\lambda}^{}\big[mf_1+3f_2\lambda^2-f_3+(f_1-f_3)^2(\lambda^2-1)\big]g(U^\lambda X,U^\lambda Y)\\
\nonumber&+&m\big[f_1-\{f_3+(f_1-f_3)^2\}\big]\eta(X)\eta(Y)\\
\nonumber&+&\sum_{k=1}^{2n-m}\{(m+1)(trace \hat{A}_k)g(\hat{A}_kX,Y)-g(\hat{A}_kX,\hat{A}_kY)\}\end{aligned}$$ for any vector fields $X,Y$ on $M$.
Using (\[eqn2.10b\]) and (\[eqn5.1\]) we have the above Lemma.
The Ricci tensor $\hat{S}$ of almost semi-invariant submanifolds $M$ of $\bar{M}^{2n+1}(c)$ with respect to $\hat{\bar{\nabla}}$ is $$\begin{aligned}
\label{eqn5.3}
\hat{S}(X,Y) &=& \frac{(m-1+3\lambda^2)c+3(m-1)+\lambda^2}{4}g(U^\lambda X,U^\lambda Y) \\
\nonumber&+&2m\eta(X)\eta(Y)+\sum_{k=1}^{2n-m}\{(m+1)(trace \hat{A}_k)g(\hat{A}_kX,Y)\\
\nonumber&-&g(\hat{A}_kX,\hat{A}_kY)\}.\end{aligned}$$
Putting $f_1=\frac{c+3}{4},\ f_2=f_3=\frac{c-1}{4}$ in (\[eqn5.2\]) we obtain the result.
The scalar curvature $\hat{\tau}$ of an almost semi-invariant submanifold $M$ of $\bar{M}^{2n+1}(f_1,f_2,f_3)$ with respect to $\hat{\bar{\nabla}}$ is $$\begin{aligned}
\label{eqn5.4}
\hat{\tau} &=& f_1+\frac{1}{m(m+1)}\big[\{3f_2+(f_1-f_3)^2\}\sum_{\lambda}^{}n(\lambda)\lambda^2\\
\nonumber&-&2m\{f_3+(f_1-f_3)^2\}+(m+1)^2||\hat{H}||^2-||\hat{h}||^2\big].\end{aligned}$$
It is known that $$\label{eqn5.7}
\hat{\tau}=\frac{1}{m(m+1)}\sum\limits_{i,j=1}^{m+1}\hat{R}(E_i,E_j,E_j,E_i).$$ Using (\[eqn2.11\]), (\[eqn2.9c\]) in (\[eqn5.7\]) we get (\[eqn5.4\]).
If $M$ is an almost semi-invariant minimal submanifolds of $\bar{M}^{2n+1}(f_1,f_2,f_3)$ with respect to $\hat{\bar{\nabla}}$, then the following condition holds:
- $\hat{S} \leq\sum\limits_{\lambda}^{}\big[mf_1+3f_2\lambda^2-f_3+(f_1-f_3)^2(\lambda^2-1)\big]g(U^\lambda X,U^\lambda Y)+m\big[f_1-\{f_3+(f_1-f_3)^2\}\big]\eta(X)\eta(Y)$,\
- $\hat{\tau}\leq f_1+\frac{1}{m(m+1)}\big[\{3f_2+(f_1-f_3)^2\}\sum\limits_{\lambda}^{}n(\lambda)\lambda^2-2m\{f_3+(f_1-f_3)^2\}\big]$.
Since $M$ is minimal submanifold with respect to with respect to $\hat{\bar{\nabla}}$, then we have $$\begin{aligned}
\label{eqn5.5}
\sum_{k=1}^{2n-m}(trace \hat{A}_k)g(\hat{A}_kX,X)&=&\sum_{i=1}^{m+1}g(\hat{h}(X,X),\hat{h}(E_i,E_i))\\
\nonumber&=&(m+1)g(\hat{h}(X,X),\hat{H})=0.\end{aligned}$$ Using (\[eqn5.2\]) and (\[eqn5.5\]) we have $$\begin{aligned}
\label{eqn5.6}
&&\hat{S}-\sum_{\lambda}^{}\big[mf_1+3f_2\lambda^2-f_3+(f_1-f_3)^2(\lambda^2-1)\big]g(U^\lambda X,U^\lambda Y)\\
\nonumber&&-m\big[f_1-\{f_3+(f_1-f_3)^2\}\big]\eta(X)\eta(Y)\\
\nonumber&=&-\sum_{k=1}^{2n-m}g(\hat{A}_kX,\hat{A}_kX)\leq 0.\end{aligned}$$ This complete the proves (i).\
The second part (ii) is comes directly from Lemma $5.3$.
The equality of (i) and (ii) in Theorem $5.1$ holds if $M$ is almost semi-invariant totally geodesic submanifolds of $\bar{M}^{2n+1}(f_1,f_2,f_3)$ with respect to $\hat{\bar{\nabla}}$.
If $M$ is totally geodesic submanifold with respect to $\hat{\bar{\nabla}}$, then $M$ is minimal submanifold with respect to $\hat{\bar{\nabla}}$. Then by virtue of Lemma $5.2$ we have the equality case (i) and by virtue of Lemma $5.3$ we have the equality case of (ii).
Submanifolds of $\bar{M}^{2n+1}(f_1,f_2,f_3)$ with $\stackrel{*}{\bar{\nabla}}$
===============================================================================
The Ricci tensor $\stackrel{*}{S}$ of submanifold $M$ of $\bar{M}^{2n+1}(f_1,f_2,f_3)$ with respect to $\stackrel{*}{\bar{\nabla}}$ is $$\begin{aligned}
\label{eqn6.1}
\stackrel{*}{S}(X,Y) &=& mf_1g(X,Y)+\{3f_2+2(f_1-f_3)+(f_1-f_3)^2\}g(TX,TY)\\
\nonumber&-&\{f_3+(f_1-f_3)^2\}\{g(X,Y)+(m-1)\eta(X)\eta(Y)\} \\
\nonumber&+&\sum_{k=1}^{2n-m}\{(m+1)(trace \stackrel{*}{A}_k)g(\stackrel{*}{A}_kX,Y)-g(\stackrel{*}{A}_kX,\stackrel{*}{A}_kY)\}\end{aligned}$$ for any vector fields $X,Y$ on $M$.
Using (\[eqn2.17\]) and (\[eqn2.9d\]) we have the above Lemma.
The Ricci tensor $\stackrel{*}{S}$ of almost semi-invariant submanifolds $M$ of $\bar{M}^{2n+1}(f_1,f_2,f_3)$ with respect to $\stackrel{*}{\bar{\nabla}}$ is $$\begin{aligned}
\label{eqn6.2}
\stackrel{*}{S}(X,Y)&=&\sum_{\lambda}^{}\big[mf_1+\{3f_2+2(f_1-f_3)\}\lambda^2-f_3\\
\nonumber&+&(f_1-f_3)^2(\lambda^2-1)\big]g(U^\lambda X,U^\lambda Y)\\
\nonumber&+&m\big[f_1-\{f_3+(f_1-f_3)^2\}\big]\eta(X)\eta(Y)\\
\nonumber&+&\sum_{k=1}^{2n-m}\{(m+1)(trace \stackrel{*}{A}_k)g(\stackrel{*}{A}_kX,Y)-g(\stackrel{*}{A}_kX,\stackrel{*}{A}_kY)\}\end{aligned}$$ for any vector fields $X,Y$ on $M$.
Using (\[eqn2.10b\]) and (\[eqn6.1\]) we have the above Lemma.
The Ricci tensor $\stackrel{*}{S}$ of almost semi-invariant submanifolds $M$ of $\bar{M}^{2n+1}(c)$ with respect to $\stackrel{*}{\bar{\nabla}}$ is $$\begin{aligned}
\label{eqn6.3}
\stackrel{*}{S}(X,Y) &=& \frac{(m-1+3\lambda^2)c+3(m-1+3\lambda^2)}{4}g(U^\lambda X,U^\lambda Y) \\
\nonumber&+&2m\eta(X)\eta(Y)+\sum_{k=1}^{2n-m}\{(m+1)(trace \stackrel{*}{A}_k)g(\stackrel{*}{A}_kX,Y)\\
\nonumber&-&g(\stackrel{*}{A}_kX,\stackrel{*}{A}_kY)\}\end{aligned}$$ for any vector fields $X,Y$ on $M$.
Putting $f_1=\frac{c+3}{4},\ f_2=f_3=\frac{c-1}{4}$ in (\[eqn6.2\]) we obtain the result.
The scalar curvature $\stackrel{*}{\tau}$ of an almost semi-invariant submanifold $M$ of $\bar{M}^{2n+1}(f_1,f_2,f_3)$ with respect to $\stackrel{*}{\bar{\nabla}}$ is $$\begin{aligned}
\label{eqn6.4}
\stackrel{*}{\tau} &=& f_1+\frac{1}{m(m+1)}\big[\{3f_2+2(f_1-f_3)+(f_1-f_3)^2\}\sum_{\lambda}^{}n(\lambda)\lambda^2\\
\nonumber&-&2m\{f_3+(f_1-f_3)^2\}+(m+1)^2||\stackrel{*}{H}||^2-||\stackrel{*}{h}||^2\big].\end{aligned}$$
It is known that $$\label{eqn6.7}
\stackrel{*}{\tau}=\frac{1}{m(m+1)}\sum\limits_{i,j=1}^{m+1}\stackrel{*}{R}(E_i,E_j,E_j,E_i).$$ Using (\[eqn2.11\]), (\[eqn2.9d\]) in (\[eqn6.7\]) we get (\[eqn6.4\]).
Let $M$ be an almost semi-invariant minimal submanifolds of $\bar{M}^{2n+1}(f_1,f_2,f_3)$ with respect to $\stackrel{*}{\bar{\nabla}}$, then the following condition holds:
- $\stackrel{*}{S}(X,Y)\leq\sum\limits_{\lambda}^{}\big[mf_1+\{3f_2+2(f_1-f_3)\}\lambda^2-f_3+(f_1-f_3)^2(\lambda^2-1)\big]g(U^\lambda X,U^\lambda X)+m\big[f_1-\{f_3+(f_1-f_3)^2\}\big]\eta(X)\eta(X)$,\
- $\stackrel{*}{\tau}\leq f_1+\frac{1}{m(m+1)}\big[\{3f_2+2(f_1-f_3)+(f_1-f_3)^2\}\sum_{\lambda}^{}n(\lambda)\lambda^2-2m\{f_3+(f_1-f_3)^2\}\big]$.
Since $M$ is minimal submanifold with respect to with respect to $\stackrel{*}{\bar{\nabla}}$, then we have $$\begin{aligned}
\label{eqn6.5}
\sum_{k=1}^{2n-m}(trace \stackrel{*}{A}_k)g(\stackrel{*}{A}_kX,X)&=&\sum_{i=1}^{m+1}g(\stackrel{*}{h}(X,X),\stackrel{*}{h}(E_i,E_i))\\
\nonumber&=&(m+1)g(\stackrel{*}{h}(X,X),\stackrel{*}{H})=0.\end{aligned}$$ Using (\[eqn6.2\]) and (\[eqn6.5\]) we have $$\begin{aligned}
\label{eqn6.6}
&&\stackrel{*}{S}-\sum_{\lambda}^{}\big[mf_1+\{3f_2+2(f_1-f_3)\}\lambda^2-f_3+(f_1-f_3)^2(\lambda^2-1)\big]\\
\nonumber&&g(U^\lambda X,U^\lambda Y)-m\big[f_1-\{f_3+(f_1-f_3)^2\}\big]\eta(X)\eta(Y)\\
\nonumber&=&-\sum_{k=1}^{2n-m}g(\stackrel{*}{A}_kX,\stackrel{*}{A}_kX)\leq 0,\end{aligned}$$ which proves (i).\
The proof of (ii) comes directly from Lemma $6.3$.
The equality of (i) and (ii) in Theorem $6.1$ holds if $M$ is almost semi-invariant totally geodesic submanifolds of $\bar{M}^{2n+1}(f_1,f_2,f_3)$ with respect to $\stackrel{*}{\bar{\nabla}}$.
If $M$ is totally geodesic submanifold with respect to $\stackrel{*}{\bar{\nabla}}$, then $M$ is minimal submanifold with respect to $\stackrel{*}{\bar{\nabla}}$. Then by virtue of Lemma $6.2$ we have the equality case (i) and by virtue of Lemma $6.3$ we have the equality case of (ii).
[**Acknowledgement:**]{} The first author (P. Mandal) gratefully acknowledges to the CSIR(File No.:09/025(0221)/2017-EMR-I), Govt. of India for financial assistance. The second author (S. K. Hui) are thankful to University of Burdwan for providing administrative and technical support.
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Pradip Mandal and Shyamal Kumar Hui\
Department of Mathematics\
The University of Burdwan\
Burdwan – 713104\
West Bengal, India\
E-mail: [email protected] and [email protected]
| null | minipile | NaturalLanguage | mit | null |
---
abstract: 'We study in a strip of ${{\mathbb R}}^2$ a combustion model of flame propagation with stepwise temperature kinetics and zero-order reaction, characterized by two free interfaces, respectively the ignition and the trailing fronts. The latter interface presents an additional difficulty because the non-degeneracy condition is not met. We turn the system to a fully nonlinear problem which is thoroughly investigated. When the width $\ell$ of the strip is sufficiently large, we prove the existence of a critical value ${{\rm{{Le}}}}_c$ of the Lewis number ${{\rm{{Le}}}}$, such that the one-dimensional, planar, solution is unstable for $0<{{\rm{{Le}}}}<{{\rm{{Le}}}}_c$. Some numerical simulations confirm the analysis.'
address:
- 'Dipartimento di Matematica e Applicazioni, Università degli Studi di Milano Bicocca, via R. Cozzi 55, I-20125 Milano (Italy)'
- 'School of Mathematical Sciences, University of Science and Technology of China, Hefei 230026 (China), and Institut de Mathématiques de Bordeaux, Université de Bordeaux, 33405 Talence Cedex (France).'
- 'Plesso di Matematica e Informatica, Dipartimento di Scienze Matematiche, Fisiche e Informatiche, Università di Parma, Parco Area delle Scienze 53/A, I-43124 Parma (Italy)'
- 'School of Science, East China University of Technology, Nanchang 330013 (China). Former affiliation: School of Mathematical Sciences, Xiamen University, Xiamen 361005 (China).'
author:
- Davide Addona
- 'Claude-Michel Brauner'
- Luca Lorenzi
- 'Wen Zhang$^\dag$'
title: |
Instabilities in a combustion model\
with two free interfaces
---
[^1]
Introduction
============
This paper is devoted to the analysis of cellular instabilities of planar traveling fronts for a thermo-diffusive model of flame propagation with stepwise temperature kinetics and zero-order reaction. In non-dimensional form, the model reads: $$\begin{aligned}
\label{eq:i1}
\left\{
\begin{array}{l}
\Theta_t=\Delta \Theta+W(\Phi,\Theta), \\[2mm]
\Phi_t=\frac{1}{{{{\rm{{Le}}}}}} \Delta \Phi-W(\Phi,\Theta),
\end{array}
\right.\end{aligned}$$ where $\Theta$ and $\Phi$ are appropriately normalized temperature and concentration of deficient reactant, ${{{\rm{{Le}}}}}$ is the Lewis number and $W(\Phi, \Theta)$ is a reaction rate given by $$\begin{aligned}
\label{eq:i2}
W(\Theta,\Phi)=
\left\{
\begin{array}{lllll}
A, & \mbox{if} & \Theta\ge \Theta_i & \mbox{and}& \Phi>0, \\[2mm]
0, & \mbox{if} &\Theta<\Theta_i & \mbox{and/or} & \Phi=0.
\end{array}
\right.\end{aligned}$$ Here, $0<\Theta_i<1$ is the ignition temperature and $A>0$ is a normalizing factor.
Combustion models involving discontinuous reaction terms, including the system -, have been used by physicists and engineers since the very early stage of the development of the combustion science (see Mallard and Le Châtelier [@i4]), primarily due to their relative simplicity and mathematical tractability (see, e.g., [@i7; @i5; @i6], and more recently [@i8; @BGKS15]). These models have drawn several mathematical studies on systems with discontinuous nonlinearities and related Free Boundary Problems which include, besides the pioneering work of K.-C. Chang [@C78], the references [@BRS95; @G95; @GH92; @GM93a; @GM93b; @NS87; @NS89], to mention a few of them. In particular, models with ignition temperature were introduced in the mathematical description of the propagation of premixed flames to solve the so-called “cold-boundary difficulty” (see, e.g., [@BL83 Section 2.2], [@BSN85]).
More specifically, in this paper we consider the free interface problem associated to the model -. The domain is the strip $\mathbb{R} \times (-{\ell}/{2},{\ell}/{2})$, the spatial coordinates are denoted by $(x,y)$, $t>0$ is the time. The free interfaces are respectively the [*ignition interface*]{} $x =F(t,y)$ and the [*trailing interface*]{} $x = G(t,y)$, $G(t,y)<F(t,y)$, defined by $\Theta(t, F(t,y),y) = \theta_i$, $\Phi(t, G(t,y),y) = 0$. The system reads as follows, for $ t>0$ and $y\in (-\ell/2,\ell/2)$: $$\begin{aligned}
\left\{
\begin{array}{ll}
\displaystyle\frac{\partial\Theta}{\partial t}(t,x,y) = \Delta \Theta(t,x,y),\quad &x<G(t,y),\\[3mm]
\Phi(t,x,y)=0, \quad &x<G(t,y),\\[2mm]
\displaystyle\frac{\partial\Theta}{\partial t}(t,x,y) = \Delta \Theta(t,x,y) + A, \quad &G(t,y)<x<F(t,y),\\[3mm]
\displaystyle\frac{\partial\Phi}{\partial t}(t,x,y) = ({{\rm{{Le}}}})^{-1}\Delta \Phi(t,x,y) - A, \quad &G(t,y)<x<F(t,y), \\[3mm]
\displaystyle\frac{\partial\Theta}{\partial t}= \Delta \Theta(t,x,y), \quad &x>F(t,y),\\[3mm]
\displaystyle\frac{\partial\Phi}{\partial t}= ({{\rm{{Le}}}})^{-1}\Delta \Phi(t,x,y), \quad &x>F(t,y),
\end{array}
\right.
\label{system-1}\end{aligned}$$ where the normalizing factor $A$ will be fixed below. The functions $\Theta$ and $\Phi$ are continuous across the interfaces for $t>0$, as well as their normal derivatives. As $x\to \pm \infty$, it holds $$\label{infty}
\Theta(t,-\infty,y) = \Phi(t,+\infty,y) =1, \qquad\;\, \Theta(t,+\infty,y) = 0.$$ Finally, periodic boundary conditions are assumed at $y=\pm \ell/2$.
As was noted in earlier studies (see [@BGKS15; @BGZ]), this system is very different from models arising in conventional thermo-diffusive combustion. Two are the principal differences. (i) The first one is that in the model considered here, the reaction zone is of order unity, whereas in the case of Arrhenius kinetics the reaction zone is infinitely thin. This fact suggests to refer to flame fronts for stepwise temperature kinetics as thick flames, in contrast to thin flames arising in Arrhenius kinetics. (ii) The second, even more important difference, is that, in the case of Arrhenius kinetics, there is a single interface separating burned and unburned gases. In contrast to that, in case of the stepwise temperature kinetics given by , there are two interfaces, namely the [*ignition interface*]{} where $\Theta=\Theta_i$ located at $x=F(t,y)$, and [*trailing interface*]{} at $x=G(t,y)$ being defined as a largest value of $x$ where the concentration is equal to zero. As a consequence of (i), the normal derivatives are continuous across both interfaces, in contrast to classical models with Arrhenius kinetics where jumps occur at the flame front (see e.g., [@BL83 Section 11.8] and [@MS79; @S80] for the related Kuramoto-Sivashinsky equation). There have been a number of mathematical works in the latter case based on the method of [@BHL00] that we are going to extend below, see in particular [@BHL13; @BHLS10; @BLSX10; @BL00; @lorenzi-1; @lorenzi-2; @lorenzi-3] for the flame front, and the references therein. Finally, note that Free Boundary Problems with two interfaces have already been considered in the literature, especially in Stefan problems, see e.g., [@DL10; @DL10-bis; @WZ17] (one-dimensional problem) and [@DG11] (radial solutions).
The above system admits a one-dimensional traveling wave (planar) solution $(\Theta^{(0)},\Phi^{(0)})$ which propagates with constant positive velocity $V$ (see [@BGKS15 Section 4]). It is convenient to choose the normalizing factor $A= 1/R$ in such a way that $V=1$, where the positive number $R = R(\theta_i)$ is given by: $$\theta_i R = 1 - e^{-R}, \quad 0<\theta_i<1.
\label{theta-i-R}$$ Thus, in the moving frame coordinate $x'=x-t$, the system for the travelling wave solution reads as follows: $$\begin{aligned}
\left\{
\begin{array}{ll}
D_{x'}\Theta^{(0)} + D_{x'x'}\Theta^{(0)}= 0, & {\rm in}~(-\infty,0],\\[1mm]
D_{x'}\Theta^{(0)}+ D_{x'x'}\Theta^{(0)}=- R^{-1}, & {\rm in}~(0,R),\\[1mm]
{{\rm{{Le}}}}\, D_{x'}\Phi^{(0)}+D_{x'x'}\Phi^{(0)}={{\rm{{Le}}}}R^{-1}, & {\rm in}~(0,R),\\[1mm]
D_{x'}\Theta^{(0)}+ D_{x'x'}\Theta^{(0)}= 0, &{\rm in}~[R,+\infty),\\[1mm]
{{\rm{{Le}}}}\, D_{x'}\Phi^{(0)}_{x'}+D_{x'x'}\Phi^{(0)}=0, &{\rm in}~[R,+\infty),
\end{array}
\right.\end{aligned}$$ whose solution is $$\Theta^{(0)}(x')=
\left\{
\begin{array}{ll}
1, & x'\le 0,\\[2mm]
\displaystyle 1 + \frac{1-x' - e^{-x'}}{R}, &x'\in (0,R),\\[2mm]
\displaystyle \theta_i e^{R-x'}, & x'\ge R,
\end{array}
\right.
\qquad\;\,
\Phi^{(0)}(x')=
\left\{
\begin{array}{ll}
0, & x'\le 0,\\[2mm]
\displaystyle\frac{e^{-{{\rm{{Le}}}}x'} -1}{{{\rm{{Le}}}}\, R} + \frac{x'}{R}, &x'\in (0,R),\\[2mm]
\displaystyle 1 + \frac{1- e^{ {{\rm{{Le}}}}R}}{{{\rm{{Le}}}}\, R\, e^{{{\rm{{Le}}}}x'}}, & x'\ge R.
\end{array}
\right.
\label{planar-front}$$
![$\Theta^{(0)}$ (solid curve) and $\Phi^{(0)}$ (dashed curve) with $\theta_i = 0.75 $, $ {{\rm{{Le}}}}= 0.75$ ($R = 0.60586$). []{data-label="Theta&Phi"}](ThetaPhi.pdf){height="5cm" width="8cm"}
The existence of traveling fronts poses a natural question of one and multidimensional stability, or especially instabilities of such fronts. It is known (see [@MS79; @S80]) that diffusional-thermal instabilities of planar flame fronts, when the Lewis number is less than unity, generates cellular flames and pattern formation. In this paper, we focus our attention on instabilities of the traveling wave $(\Theta^{(0)},\Phi^{(0)})$, and thus for the ignition and the trailing interfaces. Earlier studies have shown (see [@BGZ]) that instabilities depend on the Lewis number and occur only when the width of the strip $\ell$ is large enough (in [@BGKS15], $\ell$ is taken to infinity), which motivates the present study.
The main result of the paper is the following:
\[thm-main\] Let $0<\theta_i<1$ be fixed. There exist $\ell_0(\theta_i)$ sufficiently large, such that, whenever $\ell>\ell_0(\theta_i)$, there exists a critical value of the Lewis number, say ${{\rm{{Le}}}}_c\in (0,1)$ $($see $)$. If ${{\rm{{Le}}}}\in (0,{{\rm{{Le}}}}_c)$, then the traveling wave solution to problem - is unstable with respect to smooth and sufficiently small two dimensional perturbation. Further, also the ignition and the trailing interfaces are pointwise unstable.
The paper is organized as follows. In Section \[sect-2\], we introduce the main notation and the functional spaces. Section \[sect-3\] is devoted to transforming problem - in a [*fully nonlinear*]{} for problem for the perturbation of the traveling wave solution $(\Theta^{(0)},\Phi^{(0)})$ in , set in a fixed domain (see ). We determine that the ignition interface meets the transversality (or non-degeneracy) condition of [@BHL00]. Unfortunately, this is not the case of the trailing interface which is of different nature. In short, the idea is to differentiate the mass fraction equation, taking advantage of the structure of the problems.
Then, in Section \[sect-4\] we collect some tools that are needed to prove the main result. The theory of analytic semigroups plays a crucial role in all our analysis. For this reason, one of the main tools of this section is a generation result: we will show that a suitable realization of the linearized (at zero) elliptic operator associated with the fully nonlinear problem generates an analytic semigroup and we characterize the interpolation spaces. This will allow us to prove an optimal regularity result for classical solutions to problem . Section \[sect-6\] contains the proof of Theorem \[thm-main\]. Finally, Section \[sect-7\] is devoted to a numerical method and computational results which show two-cell patterns (see [@BGZ] for further results).
Notation, functional spaces and preliminaries {#sect-2}
=============================================
In this section, we collect all the notation, the functional spaces and the preliminary results that we use throughout the paper
Notation {#sub-notation}
--------
We find it convenient to set, for each $\tau>0$, $$\begin{aligned}
\begin{array}{lll}
S={{\mathbb R}}\times (-\ell/2,\ell/2),\quad\;\, &S_{\tau}^+=(\tau,+\infty)\times (-\ell/2,\ell/2),\quad\;\, &S_{\tau}^-=(-\infty,\tau)\times (-\ell/2,\ell/2),\\[1mm]
H_{\tau}^{-}=(-\infty,\tau)\times{{\mathbb R}},\quad\;\, &H_{\tau}^{+}=(\tau,+\infty)\times{{\mathbb R}}, \quad\;\, &R_T=[0,T]\times [-\ell/2,\ell/2].
\end{array}\end{aligned}$$
#### *[**Functions.**]{}*
Given a function $f:(a,b)\to{{\mathbb R}}$ and a point $x_0\in (a,b)$, we denote by $[f]_{x_0}$ the jump of $f$ at $x_0$, i.e., the difference $f(x_0^+)-f(x_0^-)$ whenever defined. For each function $f:[-\ell/2,\ell/2)\to{{\mathbb C}}$ we denote by $f^{\sharp}$ its $\ell$-periodic extension to ${{\mathbb R}}$. If $f$ depends also on $x$ running in some interval $I$, we still denote by $f^{\sharp}$ its periodic (with respect to $y$) extension to $I\times{{\mathbb R}}$. For every $f\in L^2((-\ell/2,\ell/2))$ and $k\in{{\mathbb Z}}$, we denote by $\hat f_k$ the $k$-th Fourier coefficient of $f$, i.e., $$\begin{aligned}
\hat f_k=\frac{1}{\ell}\int_{-\frac{\ell}{2}}^{\frac{\ell}{2}}f\overline{e_k}dy,\end{aligned}$$ where $e_h(y)=e^{\frac{2h\pi i}{\ell}y}$ for each $h\in{{\mathbb Z}}$ and $y\in{{\mathbb R}}$. When $f$ depends also on the variable $x$ running in some interval $I$, $\hat f_k(x)$ stands for the $k$-th Fourier coefficient of the function $f(x,\cdot)$.
The time and the spatial derivatives of a given function $f$ are denoted by $D_tf$ ($=f_t$), $D_xf$ ($=f_x$), $D_yf$ ($=f_y$) $D_{xx}f$ ($=f_{xx}$) $D_{xy}f$ ($=f_{xy}$) and $D_{yy}f$ ($=f_{yy}$), respectively. If $\beta=(\beta_1,\beta_2)$ with $\beta_1,\beta_2\in{{\mathbb N}}\cup\{0\}$, then we set $D^{\gamma}=D^{\gamma_1}_xD^{\gamma_2}_y$.
Finally, we denote by $\chi_A$ the characteristic function of the set $A\subset{\mathbb R^d}$ ($d\ge 1$).
#### **Miscellanea**
Throughout the paper, we denote by $c_{\lambda}$ a positive constant, possibly depending on $\lambda$ but being independent of $k$, $n$, $x$ and the functions that we will consider, which may vary from line to line. We simply write $c$ when the constant is independent also of $\lambda$.
The subscript “$b$” stands for bounded. For instance $C_b(\Omega;{{\mathbb C}})$ denotes the set of bounded and continuous function from $\Omega$ to ${{\mathbb C}}$. When we deal with spaces of real-valued functions we omit to write “${{\mathbb C}}$”.
Vector-valued functions are displayed in bold.
Main function spaces
--------------------
Here, we collect the main function spaces used in the paper pointing out the (sub)section where they are used for the first time.
#### **The spaces $\boldsymbol{\mathcal X}$ and $\boldsymbol{\mathcal X}_{k+\alpha}$ (Section \[sect-4\])**
By $\boldsymbol{\mathcal X}$ we denote the set of all pairs ${\bf f}=(f_1,f_2)$, where $f_1:\overline{S}\to{{\mathbb C}}$ and $f_2:\overline{S_0^{+}}\to{{\mathbb C}}$ are bounded functions, $f_1\in C(\overline{S_0^-};{{\mathbb C}})\cap C([0,R]\times [-\ell/2,\ell/2];{{\mathbb C}})\cap C(\overline{S_R^+};{{\mathbb C}})$, $f_2\in C([0,R]\times [-\ell/2,\ell/2];{{\mathbb C}})\cap C(\overline{S_R^+};{{\mathbb C}})$ and $\lim_{x\to \pm\infty}f_1(x,y)=\lim_{x\to +\infty}f_2(x,y)=0$ for each $y\in [-\ell/2,\ell/2]$. It is endowed with the sup-norm, i.e., $\|{{\bf f}}\|_{\infty}=\|f_1\|_{L^{\infty}(S;{{\mathbb C}})}+
\|f_2\|_{L^{\infty}(S_0^+;{{\mathbb C}})}$.
For each $\alpha\in (0,1]$, $\boldsymbol{\mathcal X}_{\alpha}$ denotes the subset of $\boldsymbol{\mathcal X}$ of all ${{\bf f}}$ such that (i) $f_1\in C^{\alpha}_b(\overline{S_0^-};{{\mathbb C}})\cap C^{\alpha}_b([0,R]\times[-\ell/2,\ell/2];{{\mathbb C}})\cap C^{\alpha}_b(\overline{S_R^+};{{\mathbb C}})$, (ii) $f_2\in C^{\alpha}_b([0,R]\times[-\ell/2,\ell/2];{{\mathbb C}})\cap C^{\alpha}_b(\overline{S_R^+};{{\mathbb C}})$, (iii) $f_j(\cdot,-\ell/2)=f_j(\cdot,\ell/2)$ (and $\nabla f_j(\cdot,-\ell/2)=
\nabla f_j(\cdot,\ell/2)$ if $\alpha=1$) for $j=1,2$. It is endowed with the norm $\|{\bf f}\|_{\alpha}
=\|f_1\|_{C^{\alpha}_b(\overline{S_0^-};{{\mathbb C}})}+\sum_{j=1}^2(\|f_j\|_{C^{\alpha}_b([0,R]\times[-\ell/2,\ell/2];{{\mathbb C}})}
+\|f_j\|_{C^{\alpha}_b(\overline{S_R^+};{{\mathbb C}})})$.
For $k\in{{\mathbb N}}$ and $\alpha\in (0,1)$, $\boldsymbol{\mathcal X}_{k+\alpha}$ ($k\in{{\mathbb N}}$, $\alpha\in (0,1)$) denotes the set of all ${\bf f}\in \boldsymbol{\mathcal X}$ such that $D^\beta{\bf f}=(D^{\gamma}f_1,D^{\gamma}f_2)\in\boldsymbol{\mathcal X}$, $D^{\gamma}f_j(\cdot,-\ell/2)=D^{\gamma}f_j(\cdot,\ell/2)$ for each $|\gamma|\leq k$, $j=1,2$, and $D^{\gamma}{\bf f}\in \boldsymbol{\mathcal X}_{\alpha}$ for $|\gamma|=k$. It is endowed with the norm $\|{\bf f}\|_{k+\alpha}:=\sum_{|\gamma|< k}\|{D^{\gamma}{\bf f}}\|_{\infty}+\sum_{|\gamma|=k}\|D^{\gamma}{\bf f}\|_{\alpha}$.
#### **The spaces $\boldsymbol{\mathcal Y}_{\alpha}(a,b)$ and $\boldsymbol{\mathcal Y}_{2+\alpha}(a,b)$ (Section \[sect-5\])**
For $\alpha\in (0,1)$ and $0\le a<b$, we define by $\boldsymbol{\mathcal Y}_{\alpha}(a,b)$ the space of all pairs ${\bf f}=(f_1,f_2)$ such that $f_1:[a,b]\times\overline{S}\to{{\mathbb R}}$, $f_2:[a,b]\times \overline{S_0^+}\to{{\mathbb R}}$ and $$\begin{aligned}
\|{\bf f}\|_{\boldsymbol{\mathcal Y}_{\alpha}(a,b)}=\sup_{a<t<b}\|{\bf f}(t,\cdot,\cdot)\|_{\alpha}+\sup_{(x,y)\in S}\|f_1(\cdot,x,y)\|_{C^{\alpha/2}((a,b))}+\sup_{(x,y)\in S^+_0}\|f_2(\cdot,x,y)\|_{C^{\alpha/2}((a,b))}<+\infty.\end{aligned}$$
Similarly, $\boldsymbol{\mathcal Y}_{2+\alpha}(a,b)$ denotes the space of all the pairs ${\bf u}$ such that $D^{\gamma_1}_tD^{\gamma_2}_xD^{\gamma_3}_y{\bf u}$ belongs to $\boldsymbol{\mathcal Y}_{\alpha}(a,b)$ for every $\gamma_1, \gamma_2, \gamma_3\ge 0$ such that $2\gamma_1+\gamma_2+\gamma_3\leq 2$. These are Banach spaces with the norms $\|\cdot\|_{\boldsymbol{\mathcal Y}_{\alpha}(a,b)}$ and $\|{\bf u}\|_{\boldsymbol{\mathcal Y}_{2+\alpha}(a,b)}
= \sum_{2\gamma_1+\gamma_2+\gamma_3\leq 2}\|D^{\gamma_1}_tD^{\gamma_2}_xD^{\gamma_3}_y{\bf u}\|_{\boldsymbol{\mathcal Y}_{\alpha}(a,b)}$.
If $a=0$ and $b=T$ then we simply write $\boldsymbol{\mathcal Y}_{\alpha}$ and $\boldsymbol{\mathcal Y}_{2+\alpha}$ instead of $\boldsymbol{\mathcal Y}_{\alpha}(0,T)$ and $\boldsymbol{\mathcal Y}_{2+\alpha}(0,T)$, respectively.
Derivation of the fully nonlinear problem {#FNL1}
=========================================
\[sect-3\]
The system on a fixed domain {#fix-domain}
----------------------------
To begin with, we rewrite System in the coordinates $t'=t$, $x'=x-t$, $y'=y$, $D_t=D_{t'}-D_{x'}$. Next, we look for the free interfaces respectively as: $$G(t',y') = g(t',y'), \qquad\;\, F(t',y') = R + f(t',y'),$$ where $f$ and $g$ are small perturbations. In other words, the space variable $x'$ varies from $-\infty$ to $g(t',y')$, from $g(t',y')$ to $R + f(t',y')$, and eventually from $R + f(t',y')$ to $+\infty$. As usual, it is convenient to transform the problem on a variable domain to a problem on a fixed domain. To this end, we define a coordinate transformation in the spirit of [@BHL00 Section 2.1]: $$t'=\tau, \qquad\;\, x'=\xi+\beta(\xi)g(\tau,\eta)+\beta(\xi-R)f(\tau,\eta),\qquad\;\, y'=\eta,$$ where $\beta$ is a smooth mollifier, equal to unity in a small neighborhood of $\xi=0$, say $[-\delta,\delta]$, and has compact support contained in $(-2\delta,2\delta)\subset (-R,R)$. When $x'=g$, $\xi=0$, and $\xi=R$ when $x'=R+f$. Then, the trailing front and the ignition front are fixed at $\xi=0$ and $\xi=R$, respectively. Thanks to the translation invariance, holds with the variable $\xi$. For convenience, we introduce the notation $${\varrho}(\tau,\xi,\eta) = \beta(\xi)g(\tau,\eta)+\beta(\xi-R)f(\tau,\eta)
\label{notation}$$ and we expand $(1+{\varrho}_\xi)^{-1}= 1-{\varrho}_\xi + ({\varrho}_{\xi})^2(1+{\varrho}_\xi)^{-1}$. It turns out that $$\begin{aligned}
&D_{x'}=D_{\xi} - {\varrho}_\xi D_{\xi}+({\varrho}_{\xi})^2(1+{\varrho}_\xi)^{-1}D_{\xi},\\
&D_{t'}=D_{\tau}-{\varrho}_\tau D_{\xi}+ {\varrho}_\tau{\varrho}_\xi D_{\xi}-{\varrho}_\tau({\varrho}_{\xi})^2(1+{\varrho}_\xi)^{-1}D_{\xi},\\
&D_{y'}=D_{\eta}-{\varrho}_\eta D_{\xi}+ {\varrho}_\eta{\varrho}_\xi D_{\xi}-{\varrho}_\eta({\varrho}_{\xi})^2(1+{\varrho}_\xi)^{-1}D_{\xi}\end{aligned}$$ and System reads: $$\begin{gathered}
\label{Model2-i}
\begin{split}
\Theta_\tau = & \Theta_{\xi}+ \Delta\Theta+ ({\varrho}_\eta^2 - {\varrho}_\xi^2 -2 {\varrho}_\xi)(1+{\varrho}_\xi)^{-2}\Theta_{\xi\xi}-2 {\varrho}_\eta (1+{\varrho}_\xi)^{-1}\Theta_{\xi\eta} \\
&+ \big[({\varrho}_\tau - {\varrho}_\xi-{\varrho}_{\eta\eta})(1+{\varrho}_\xi)^{-1} + 2 {\varrho}_\eta {\varrho}_{\xi\eta} (1+{\varrho}_\xi)^{-2} - {\varrho}_{\xi\xi} (1+ {\varrho}_\eta^2) (1+{\varrho}_\xi)^{-3} \big ] \Theta_\xi,\\[2mm]
\Phi = & 0
\end{split}\end{gathered}$$ in $(0,+\infty)\times (-\infty,0)\times (-\ell/2,\ell/2)$, $$\begin{gathered}
\label{Model2-ii}
\begin{split}
\Theta_\tau =& \Theta_{\xi}+ \Delta\Theta+({\varrho}_\eta^2 - {\varrho}_\xi^2 -2 {\varrho}_\xi)(1+{\varrho}_\xi)^{-2}\Theta_{\xi\xi} -2 {\varrho}_\eta (1+{\varrho}_\xi)^{-1}\Theta_{\xi\eta} + R^{-1} \\
&+ \big [({\varrho}_\tau - {\varrho}_\xi-{\varrho}_{\eta\eta})(1+{\varrho}_\xi)^{-1} + 2 {\varrho}_\eta {\varrho}_{\xi\eta} (1+{\varrho}_\xi)^{-2} - {\varrho}_{\xi\xi} (1+ {\varrho}_\eta^2) (1+{\varrho}_\xi)^{-3} \big ] \Theta_\xi,\\[2mm]
\Phi_\tau =& \Phi_{\xi}+ {{\rm{{Le}}}}^{-1}\Delta\Phi+{{\rm{{Le}}}}^{-1}({\varrho}_\eta^2 - {\varrho}_\xi^2 -2 {\varrho}_\xi)(1+{\varrho}_\xi)^{-2}\Phi_{\xi\xi}-2{{\rm{{Le}}}}^{-1} {\varrho}_\eta (1+{\varrho}_\xi)^{-1}\Phi_{\xi\eta} - R^{-1}\\
&+[({\varrho}_\tau\! -\! {\varrho}_\xi\!-\!{{\rm{{Le}}}}^{-1}{\varrho}_{\eta\eta})(1+{\varrho}_\xi)^{-1}\! +\! 2{{\rm{{Le}}}}^{-1} {\varrho}_\eta {\varrho}_{\xi\eta} (1+{\varrho}_\xi)^{-2}-{{\rm{{Le}}}}^{-1} {\varrho}_{\xi\xi} (1+ {\varrho}_\eta^2) (1+{\varrho}_\xi)^{-3}] \Phi_\xi
\end{split}\end{gathered}$$ in $(0,+\infty)\times (0,R)\times (-\ell/2,\ell/2)$ and $$\begin{gathered}
\label{Model2-iii}
\begin{split}
\Theta_\tau =& \Theta_{\xi}+\Delta\Theta+ ({\varrho}_\eta^2 - {\varrho}_\xi^2 -2 {\varrho}_\xi)(1+{\varrho}_\xi)^{-2} \Theta_{\xi\xi} -2 {\varrho}_\eta (1+{\varrho}_\xi)^{-1} \Theta_{\xi\eta} \\
&+ \big [({\varrho}_\tau - {\varrho}_\xi-{\varrho}_{\eta\eta})(1+{\varrho}_\xi)^{-1} + 2 {\varrho}_\eta {\varrho}_{\xi\eta} (1+{\varrho}_\xi)^{-2} - {\varrho}_{\xi\xi} (1+ {\varrho}_\eta^2) (1+{\varrho}_\xi)^{-3} \big ] \Theta_\xi ,\\[2mm]
\Phi_\tau =& \Phi_{\xi}+ {{\rm{{Le}}}}^{-1}\Delta\Phi+{{\rm{{Le}}}}^{-1}({\varrho}_\eta^2 - {\varrho}_\xi^2 -2 {\varrho}_\xi)(1+{\varrho}_\xi)^{-2}\Phi_{\xi\xi}-2{{\rm{{Le}}}}^{-1} {\varrho}_\eta (1+{\varrho}_\xi)^{-1}\Phi_{\xi\eta}\\
&+ \big [({\varrho}_\tau\! -\! {\varrho}_\xi\!-\!{{\rm{{Le}}}}^{-1}{\varrho}_{\eta\eta})(1+{\varrho}_\xi)^{-1}\! +\! 2{{\rm{{Le}}}}^{-1} {\varrho}_\eta {\varrho}_{\xi\eta} (1+{\varrho}_\xi)^{-2}-{{\rm{{Le}}}}^{-1} {\varrho}_{\xi\xi} (1+ {\varrho}_\eta^2) (1+{\varrho}_\xi)^{-3} \big ] \Phi_\xi
\end{split}\end{gathered}$$ in $(0,+\infty)\times (R,+\infty)\times (-\ell/2,\ell/2)$. Moreover, $\Theta$ and $\Phi$ are continuous at the (fixed) interfaces $\xi=0$ and $\xi=R$, and so are their first-order derivatives. Thus, $$[\Theta(\tau,\cdot,\eta)]_0= [\Theta_\xi(\tau,\cdot,\eta)]_0=\Phi(\tau,0,\eta)=\Phi_\xi(\tau,0,\eta)=0$$ and $$\Theta (\tau,R,\eta)=\theta_i, \quad [\Theta(\tau,\cdot,\eta)]_R=[\Theta_\xi(\tau,\cdot,\eta)]_R= [\Phi(\tau,\cdot,\eta)]=[\Phi_\xi(\tau,\cdot,\eta)]=0.$$ Conditions hold at $\xi=\pm\infty$ and periodic boundary conditions are assumed at $\eta=\pm \ell/2$.
Elimination of the interfaces {#sect-3.2}
-----------------------------
From now on, with a slight abuse of notation, which does not cause confusion, we write $t$ instead of $\tau$ and $x$, $y$ instead of $\xi$ and $\eta$.
In the spirit of [@BHL00; @lorenzi-2], we introduce the splitting: $$\begin{gathered}
\Theta(t,x,y) = \Theta^{(0)}(x) + {\varrho}(t,x,y)\Theta^{(0)}_x(x) + u(t, x, y),\label{splitTheta}\\
\Phi( t,x,y) = \Phi^{(0)}(x) + {\varrho}(t,x,y) \Phi^{(0)}_x(x) + v(t,x, y),\label{splitPhi}\end{gathered}$$ which is a sort of Taylor expansion of $(\Theta,\Phi)$ around the travelling wave solution $(\Theta^{(0)},\Phi^{(0)})$. Thus, the pair $(u,v)$ plays the role of a remainder and, since we are interested in stability issues, we can assume that $u$ and $v$ are “sufficiently small” in a sense which will be made precise later on.
A long but straightforward computation reveals that the pair $(u,v)$ satisfies the differential equations $$\begin{gathered}
\label{PertTheta+}
\begin{split}
u_t = & u_x + \Delta u+ {\varrho}_t (1+{\varrho}_x)^{-1}({\varrho}\Theta^{(0)}_{xx} + u_x)-(1+{\varrho}_x)^{-3}{\varrho}_{xx} (1+{\varrho}_y^2) ({\varrho}\Theta^{(0)}_{xx} + u_x) \\
& -(1+{\varrho}_x)^{-1}\big [({\varrho}_{x} + {\varrho}_{yy}) ({\varrho}\Theta^{(0)}_{xx}+u_x) +2 {\varrho}_y ({\varrho}_y \Theta^{(0)}_{xx}+u_{xy})\big ]\\
& + (1+{\varrho}_x)^{-2} \big [2 {\varrho}_{y} {\varrho}_{xy} ({\varrho}\Theta^{(0)}_{xx}+u_x) + ({\varrho}_{y}^2 -{\varrho}_{x}^2) ({\varrho}\Theta^{(0)}_{xxx}+\Theta^{(0)}_{xx}+u_{xx})\\
&\phantom{+ (1+{\varrho}_x)^{-2} \big [\;\,}- 2 {\varrho}_x ({\varrho}\Theta^{(0)}_{xxx} -{\varrho}_y^2 \Theta^{(0)}_{xx}+u_{xx})\big ],
\end{split}\end{gathered}$$ in $(0,+\infty)\times ({{\mathbb R}}\setminus\{0,R\})\times (-\ell/2,\ell/2)$ and $$\begin{gathered}
\label{PertPhi}
\begin{split}
v_t = & v_x + {{\rm{{Le}}}}^{-1}\Delta v+ {\varrho}_t (1+{\varrho}_x)^{-1}
({\varrho}\Phi^{(0)}_{xx} + v_x)-{{\rm{{Le}}}}^{-1}(1+{\varrho}_x)^{-3}{\varrho}_{xx} (1+{\varrho}_y^2) ({\varrho}\Phi^{(0)}_{xx} + v_x)\\
& - {{\rm{{Le}}}}^{-1}(1+{\varrho}_x)^{-1}\big [({{\rm{{Le}}}}\, {\varrho}_{x} + {\varrho}_{yy}) ({\varrho}\Phi^{(0)}_{xx}+v_x) + 2 {\varrho}_y ({\varrho}_y \Phi^{(0)}_{xx}+v_{xy})\big ] \\
& + {{\rm{{Le}}}}^{-1}(1+{\varrho}_x)^{-2} \big [2 {\varrho}_{y} {\varrho}_{xy} ({\varrho}\Phi^{(0)}_{xx}+v_x) + ({\varrho}_{y}^2 -{\varrho}_{x}^2) ({\varrho}\Phi^{(0)}_{xxx}+\Phi^{(0)}_{xx}+v_{xx})\\
&\phantom{+ {{\rm{{Le}}}}^{-1}(1+{\varrho}_x)^{-2}\big [\;\,}- 2 {\varrho}_x ({\varrho}\Phi^{(0)}_{xxx} -{\varrho}_y^2 \Phi^{(0)}_{xx}+v_{xx})\big ]
\end{split}\end{gathered}$$ in $(0,+\infty)\times [(0,R)\cup(R,+\infty)]\times (-\ell/2,\ell/2)$ and $v=0$ in $(0,+\infty)\times (-\infty,0)\times (-\ell/2,\ell/2)$.
Two steps are still needed:
\(a) we have to determine the jump conditions satisfied by $u$ and $v$;
\(b) again in the spirit of [@BHL00; @lorenzi-2], we have to get rid of the function $\varphi$ from the right-hand sides of and . As we will see, some difficulties appear and, to overcome them, we will differentiate the differential equation .
### **The ignition interface $\boldsymbol{x=R}$.**
Note that $\Theta^{(0)}$, $\Phi^{(0)}$ belong to $C^1({{\mathbb R}})$. Thus, $[u]_R=[v]_R= 0$. Moreover, $\Theta^{(0)}_{x}(R)= -\theta_i$ and $\Phi^{(0)}_{x}(R)=(1-\exp(-{{\rm{{Le}}}}R))/R$, so that they do not vanish at the interface $x=R$. The latter is a kind of transversality or non-degeneracy condition (see [@BHL00]). Evaluating at $x=R$, we get $u(R) = \theta_i f$.
Next, differentiating and for $x \neq R$, and taking the jumps across $x=R$, it is not difficult to show that $[u_x]_R = -R^{-1}f$ and $[v_x]_R= R^{-1}{{\rm{{Le}}}}\, f$. This is a key point since we are able to express $f$ in terms of $u$ and write $$\label{elim-f}
f(t,y)={\theta_i}^{-1} u(t,R,y).$$ Summing up, the interface conditions at $x=R$ are the following: $$\label{IC1}
[u]_R=[v]_R= 0, \quad u(R) +\theta_i R [u_x]_R =0, \qquad\;\, {{\rm{{Le}}}}[u_x]_R + [v_x]_R =0.$$
### **The trailing interface $\boldsymbol{x=0}$.**
Taking the jump at $x=0$ of both sides of and , we get the conditions $[u]_0=v(0)=0$ for $u$ and $v$. The trailing interface has a different nature with respect to the ignition interface. Indeed, since $\Theta^{(0)}_{x}(0)=\Phi^{(0)}_{x}(0)=0$, the non-degeneracy condition of [@BHL00] is not verified and we are able to express $g$ in terms neither of $u$ or $v$. On the other hand, $\Theta^{(0)}_{xx}(0^+)= -R^{-1}$ and $\Phi^{(0)}_{xx}(0^+)=R^{-1}{{\rm{{Le}}}}$, so that they do not vanish. Differentiating and for $x \neq 0$, and taking the jumps yields: $[u_x]_0 = R^{-1}g$, $v_x(\cdot,0^+,\cdot) = - R^{-1}g{{\rm{{Le}}}}$. Hence, we get the additional interface condition ${{\rm{{Le}}}}\,[u_x]_0 + v_x(\cdot,0^+,\cdot) =0$, so that the interface conditions at $x=0$ are $$[u]_0=v(0)=0,\quad {{\rm{{Le}}}}\,[u_x]_0 + v_x(\cdot,0^+,\cdot) =0.
\label{IC2}$$ We can also write $$\label{elim-g}
g(t,y)= - R\,{{{\rm{{Le}}}}}^{-1} v_x(t, 0^+, y).$$ Although the front $g$ could be eliminated, the method of [@BHL00] is not applicable since, in contrast to , $g$ is related to the derivative of $v$ in the equation .
Differentiation and new interface conditions
--------------------------------------------
To overcome the difficulty pointed out above, the trick is to differentiate with respect to $x$, taking advantage of the structure of the system and consider the problem satisfied by the pair $(u,v_{x})$. From and we get the following interface conditions for $u$ and $w=v_x$: $$\begin{gathered}
\label{IC-diff}
\begin{split}
&[u]_0=0, \qquad\;\, {{\rm{{Le}}}}\, [u_x]_0 + w(0^+) =0,\\
&[u]_R= 0, \qquad\;\, u(R) +\theta_i R [u_x]_R =0, \qquad\;\, {{\rm{{Le}}}}[u_x]_R + [w]_R =0.
\end{split}\end{gathered}$$
We missed two jump conditions: one at the trailing interface and the other one at the ignition interface. To obtain the additional condition at the trailing interface $x=0$, we differentiate twice in a neighborhood of $x=0$ and take the trace at $x=0^+$. Using , we get $$\label{new-interface1}
\Phi_{xx}(\cdot,0^+,\cdot) = {{\rm{{Le}}}}\, R^{-1} +{{\rm{{Le}}}}\, w(\cdot,0^+,\cdot) + w_x(\cdot,0^+,\cdot).$$ To get rid of $\Phi_{xx}(\cdot,0^+,\cdot)$ from the left-hand side of , we observe that, for $x>0$ sufficiently small, the second equation in reduces to $$\begin{gathered}
\label{neighbor-0}
\begin{split}
\Phi_t =& \Phi_{x}+ {{\rm{{Le}}}}^{-1}\Delta\Phi+ {{\rm{{Le}}}}^{-1}g_y^2 \Phi_{xx}
- 2{{\rm{{Le}}}}^{-1} g_y \Phi_{xy} - R^{-1}+(g_t -{{\rm{{Le}}}}^{-1}g_{yy}) \Phi_x.
\end{split}\end{gathered}$$ Taking the trace of at $x=0^+$ it is easy to check that $\Phi_{xx}(\cdot,0^+,\cdot)(1+g_y^2) = {{\rm{{Le}}}}\, R^{-1}$. Hence, using and we get the additional interface condition $$\label{new-interface2}
{{\rm{{Le}}}}\, w(\cdot,0^+,\cdot) + w_x(\cdot,0^+,\cdot)= {{\rm{{Le}}}}\,R^{-1}\{[1+R^2{{\rm{{Le}}}}^{-2}(w_y(\cdot,0^+,\cdot))^2]^{-1}-1\}.$$
We likewise identify the additional interface condition at the ignition interface $x=R$. Differentiating twice in a neighborhood of $x=R$, taking the jump at $x=R$ and using , gives $$\label{new-interface3}
[\Phi_{xx}]_R = -R^{-1}{{\rm{{Le}}}}+{{\rm{{Le}}}}\, [w]_R + [w_x]_R.$$ We need to compute $[\Phi_{xx}]_R$: in a neighborhood of $R^-$, the second equation in yields $$\begin{aligned}
\Phi_t =\Phi_{x}+ {{\rm{{Le}}}}^{-1}\Delta\Phi
+ {{\rm{{Le}}}}^{-1}(f_y)^2 \Phi_{xx}- 2{{\rm{{Le}}}}^{-1} f_y \Phi_{xy} - R^{-1}+(f_t -{{\rm{{Le}}}}^{-1}f_{yy}) \Phi_x,\end{aligned}$$ while in a neighborhood of $R^+$ from the second equation in we get $$\begin{aligned}
\Phi_t =& \Phi_{x}+ {{\rm{{Le}}}}^{-1}\Delta\Phi+ {{\rm{{Le}}}}^{-1}(f_y)^2 \Phi_{xx}
- 2{{\rm{{Le}}}}^{-1} f_y \Phi_{xy}+(f_t -{{\rm{{Le}}}}^{-1}f_{yy}) \Phi_x.\end{aligned}$$ Using the previous two equations it can be easily shown that $[\Phi_{xx}]_R (1+ (f_y)^2) = -R^{-1}{{\rm{{Le}}}}$, which, together with and , gives $$\label{new-interface4}
{{\rm{{Le}}}}[w]_R + [w_x]_R=- {{\rm{{Le}}}}\, R^{-1}\{[1+\theta_i^{-2}(u_y(\cdot,R,\cdot))^2]^{-1}-1\}.$$ This is the additional condition we were looking for.
Elimination of ${\varrho}$ and its time and spatial derivatives
---------------------------------------------------------------
Formulae enable the elimination of the fronts $f$ and $g$ from the differential equations satisfied by $u$ and $w$. First, they allow to write the following formula for ${\varrho}$ (see ): $$\label{vp1}
{\varrho}(t,x,y) = {\theta_i}^{-1}\beta(x-R)u(t,R,y) - R\,{{{\rm{{Le}}}}}^{-1}\beta(x) w(t, 0^+, y).$$ Differentiation of with respect to $x$ and $y$ is benign. The right-hand sides of and depend also on $\varrho_t$. Hence, we need to compute such a derivative and express it in terms of (traces of) spatial derivatives of $u$ and $w$. Since $$\begin{aligned}
\label{stefan1}
{\varrho}_{t}(t,x,y) = {\theta_i}^{-1}\beta(x-R)u_{t}(t,R,y) - R\,{{{\rm{{Le}}}}}^{-1}\beta(x) w_{t}(t, 0^+, y),\end{aligned}$$ we need to get rid of $u_{t}(t,R,y)$ and $w_{t}(t,0^+,y)$. For simplicity, we forget the arguments $t$ and $y$. We evaluate at $x=R^+$ (it would be equivalent at $x=R^-$). Recalling that all the derivatives of ${\varrho}$ with respect of $x$ vanish and taking into account, we get $$\begin{aligned}
u_t(R) = & u_x(R^+) +\Delta u(R^+)+\theta_i^{-1}u_t(R) (u(R)+ u_x(R^+))-\theta_i^{-1}u_{yy}(R)u_x(R^+)\\
& -2\theta_i^{-1}u_y(R)u_{xy}(R^+)+\theta_i^{-2}(u_{y}(R))^2(u_{xx}(R^+)-u(R)-\theta_i)-\theta_i^{-1}u(R)u_{yy}(R).\end{aligned}$$ Since $\theta_i$ is fixed, assuming that the perturbations are small we may invert and write $$\begin{aligned}
\label{stefan2}
u_t(R)=[1\!-\!{\theta_i}^{-1}(u(R)\!+\!u_x(R^+))]^{-1}
[&u_x(R^+)+ \Delta u(R^+)-\theta_i^{-1}u_{yy}(R)u_x(R^+)-\theta_i^{-1}u(R)u_{yy}(R)\notag\\
&-2\theta_i^{-1}u_y(R)u_{xy}(R^+)+\theta_i^{-2}(u_{y}(R))^2(u_{xx}(R^+)-u(R)-\theta_i)].\end{aligned}$$
Similarly, differentiating and evaluating at $x=0^+$ we get $$\begin{aligned}
w_t(0^+) = & w_x(0^+) + {{\rm{{Le}}}}^{-1}\Delta w(0^+)-R{{\rm{{Le}}}}^{-1}w_t(0^+)({{\rm{{Le}}}}\, w(0^+)+w_{x}(0^+))\\
& +R{{\rm{{Le}}}}^{-2}\big [w_{yy}(0^+)({{\rm{{Le}}}}\, w(0^+)+w_{x}(0^+))+2w_y(0^+) w_{xy}(0^+)\big ] \\
& + R^2{{\rm{{Le}}}}^{-3}(w_{y}(0^+))^2(-{{\rm{{Le}}}}^2w(0^+)+R^{-1}{{\rm{{Le}}}}^2+w_{xx}(0^+)),\end{aligned}$$ so that $$\begin{aligned}
w_t(0^+)=\{&{{\rm{{Le}}}}\, w_x(0^+) +\Delta w(0^+)+R{{\rm{{Le}}}}^{-1}\big [w_{yy}(0^+)({{\rm{{Le}}}}\, w(0^+)+w_{x}(0^+))+2w_y(0^+) w_{xy}(0^+)\big ]\notag\\
&+R^2{{\rm{{Le}}}}^{-2}(w_{y}(0^+))^2(-{{\rm{{Le}}}}^2w(0^+)+R^{-1}{{\rm{{Le}}}}^2+w_{xx}(0^+))\}\notag\\
&\qquad\quad\times[{{\rm{{Le}}}}+ R({{\rm{{Le}}}}\, w(0^+) +w_{x}(0^+))]^{-1}.
\label{core-2}\end{aligned}$$ A related remark is that Equation for ${\varrho}_{t}$, together with formulas -, may be viewed as a *second-order Stefan condition*, see [@BL18].
The final system
----------------
Using , , , , -, we can write the final problem for ${\bf u}=(u,w)$, which is [*fully nonlinear*]{} since the nonlinear part of the differential equations contains traces at $\xi=0^+$ and $R$ of (first- and) second-order derivatives of the unknown ${{\bf u}}$ itself.
Summing up, the pair ${\bf u}=(u,w)$ solves the nonlinear system $$\begin{aligned}
\left\{
\begin{array}{lll}
D_t{\bf u}(t,\cdot,\cdot)={\mathscr L}{\bf u}(t,\cdot,\cdot)+\mathscr F({\bf u}(t,\cdot,\cdot)),& t\geq0,\\[1mm]
\mathscr B({\bf u}(t,\cdot))=\boldsymbol{\mathscr H}({{\bf u}}(t,\cdot)), & t\geq0,
\end{array}
\right.
\label{asta}\end{aligned}$$ and satisfies periodic boundary conditions at $y=\pm\ell/2$, where $${\mathscr L}{\bf v}=(\Delta\zeta+\zeta_x,{{\rm{{Le}}}}^{-1}\Delta \upsilon+\upsilon_x),
\label{operatore-L}$$ $${\mathscr B}{\bf v}=\left (
\begin{array}{l}
\zeta(0^+,\cdot)-\zeta(0^-,\cdot)\\[1mm]
\zeta(R^+,\cdot)-\zeta(R^-,\cdot)\\[1mm]
{\rm Le}[\zeta_x(0^+,\cdot)-\zeta_x(0^-,\cdot)]+\upsilon(0^+,\cdot)\\[1mm]
{\rm Le}\,\upsilon(0^+,\cdot)+\upsilon_x(0^+,\cdot)\\[1mm]
\zeta(R^+,\cdot)+\theta_iR[\zeta_x(R^+,\cdot)-\zeta_x(R^-,\cdot)]\\[1mm]
{\rm Le}[\zeta_x(R^+,\cdot)-\zeta_x(R^-,\cdot)]+\upsilon(R^+,\cdot)-\upsilon(R^-,\cdot)\\[1mm]
{\rm Le}[\upsilon(R^+,\cdot)-\upsilon(R^-,\cdot)]+\upsilon_x(R^+,\cdot)-\upsilon_x(R^-,\cdot)
\end{array}
\right ),
\label{boundary-B}$$ $$\begin{aligned}
\mathscr F({\bf v})
= & (\Lambda({\bf v})\mathscr F_1({\bf v})-\mathscr F_2({\bf v}),
\Lambda({\bf v})D_x\mathscr G_1({\bf v})+D_x\Lambda({\bf v})\mathscr G_1({\bf v})-{\rm Le}^{-1}D_x\mathscr G_2({\bf v})), \\[1mm]
{\mathscr F}_1({\bf v})=& \frac{\theta_i^{-1}\beta_R\Theta^{(0)}_{xx}\zeta(R^+,\cdot)-R{\rm Le}^{-1}\beta\Theta^{(0)}_{xx}\upsilon(0,\cdot)+\zeta_x}{1+\theta_i^{-1}\beta'(\cdot-R)\zeta(R^+,\cdot)-R{\rm Le}^{-1}\beta'\upsilon(0,\cdot)},\\[2mm]
{\mathscr F}_2({\bf v})=&\{\big (\theta_i^{-1}\beta'(\cdot-R)\zeta(R^+,\cdot)-R{\rm Le}^{-1}\beta'\upsilon(0,\cdot)+\theta_i^{-1}\beta_R \zeta_{yy}(R^+,\cdot)-R{\rm Le}^{-1}\beta \upsilon_{yy}(0,\cdot)\big )\\
&\qquad\quad\times(\theta_i^{-1}\beta_R\Theta^{(0)}_{xx}\zeta(R^+,\cdot)-R{\rm Le}^{-1}\beta\Theta^{(0)}_{xx}\upsilon(0,\cdot)+\zeta_x)\\
&\;\,+2(\theta_i^{-1}\beta_R \zeta_y(R^+,\cdot)-R{\rm Le}^{-1}\beta \upsilon_y(0,\cdot))^2\Theta^{(0)}_{xx}\\
&\;\,+2(\theta_i^{-1}\beta_R \zeta_y(R^+,\cdot)-R{\rm Le}^{-1}\beta \upsilon_y(0,\cdot))\zeta_{xy}\}(1+\theta_i^{-1}\beta'(\cdot-R)\zeta(R^+,\cdot)-R{\rm Le}^{-1}\beta'\upsilon(0,\cdot))^{-1}\\
&-\big\{2\big (\theta_i^{-1}\beta_R \zeta_y(R^+,\cdot)-R{\rm Le}^{-1}\beta \upsilon_y(0,\cdot)\big )\big (\theta_i^{-1}\beta'(\cdot-R)\zeta_y(R^+,\cdot)-R{\rm Le}^{-1}\beta'\upsilon_y(0,\cdot)\big )\\
&\qquad\quad\times
\big (\theta_i^{-1}\beta_R\Theta^{(0)}_{xx}\zeta(R^+,\cdot)-R{\rm Le}^{-1}\beta\Theta^{(0)}_{xx}\upsilon(0,\cdot)+\zeta_x\big )\\
&\;\;\;\;\;\;+\big [\big (\theta_i^{-1}\beta_R \zeta_y(R^+,\cdot)-R{\rm Le}^{-1}\beta \upsilon_y(0,\cdot)\big )^2-\big (\theta_i^{-1}\beta'(\cdot-R)\zeta(R^+,\cdot)-R{\rm Le}^{-1}\beta'\upsilon(0,\cdot)\big )^2\big ]\\
&\qquad\quad\;\;\;\;\times\big (\theta_i^{-1}\beta_R \Theta^{(0)}_{xxx}\zeta(R^+,\cdot)-R{\rm Le}^{-1}\beta \Theta^{(0)}_{xxx}\upsilon(0,\cdot)+\Theta^{(0)}_{xx}+\zeta_{xx}\big )\\
&\;\;\;\;\;\;-2\big (\theta_i^{-1}\beta'(\cdot-R)\zeta(R^+,\cdot)-R{\rm Le}^{-1}\beta'\upsilon(0,\cdot)\big )\\
&\qquad\quad\;\;\;\;\times\big [\theta_i^{-1}\beta_R\Theta^{(0)}_{xxx}\zeta(R^+,\cdot)-R{\rm Le}^{-1}\beta\Theta^{(0)}_{xxx} \upsilon(0,\cdot)\\
&\qquad\qquad\quad\;\;\;\;-\big (\theta_i^{-1}\beta_R \zeta_y(R^+,\cdot)-R{\rm Le}^{-1}\beta \upsilon_y(0,\cdot)\big )^2\Theta^{(0)}_{xx}+\zeta_{xx}\big ]\big\}\\
&\qquad\qquad\;\;\;\;\;\;\times(1+\theta_i^{-1}\beta'(\cdot-R)\zeta(R^+,\cdot)-R{\rm Le}^{-1}\beta'\upsilon(0,\cdot))^{-2}\\
&+\big\{\big (\theta_i^{-1}\beta''(\cdot-R)\zeta(R^+,\cdot)-R{\rm Le}^{-1}\beta''\upsilon(0,\cdot)\big )\big [1+\big (\theta_i^{-1}\beta_R \zeta_y(R^+,\cdot)-R{\rm Le}^{-1}\beta \upsilon_y(0,\cdot)\big )^2\big ]\\
&\qquad\quad\times (\theta_i^{-1}\beta_R\Theta^{(0)}_{xx}\zeta(R^+,\cdot)-R{\rm Le}^{-1}\beta\Theta^{(0)}_{xx}\upsilon(0,\cdot)+\zeta_x)\big\}\\
&\qquad\qquad\qquad\times(1+\theta_i^{-1}\beta'(\cdot-R)\zeta(R^+,\cdot)-R{\rm Le}^{-1}\beta'\upsilon(0,\cdot))^{-3},\\[2mm]
\mathscr G_1({\bf v})= &\frac{\theta_i^{-1}\beta_R\Phi^{(0)}_{xx}\zeta(R^+,\cdot)-R{\rm Le}^{-1}\beta\Phi^{(0)}_{xx}\upsilon(0,\cdot)+\upsilon}{1+\theta_i^{-1}\beta'(\cdot-R)\zeta(R^+,\cdot)-R{\rm Le}^{-1}\beta'\upsilon(0,\cdot)},\\[2mm]
\mathscr G_2({\bf v})= &
\{[{\rm Le}\,\theta_i^{-1}\beta'(\cdot-R)\zeta(R^+,\cdot)-R\beta'\upsilon(0,\cdot)+\theta_i^{-1}\beta_R \zeta_{yy}(R^+,\cdot)-R{\rm Le}^{-1}\beta \upsilon_{yy}(0,\cdot)]\\
&\qquad\quad\times[\theta_i^{-1}\beta_R\Phi^{(0)}_{xx}\zeta(R^+,\cdot)-R{\rm Le}^{-1}\beta\Phi^{(0)}_{xx}\upsilon(0,\cdot)+\upsilon]\\
&\;\,+2(\theta_i^{-1}\beta_R \zeta_y(R^+,\cdot)-R{\rm Le}^{-1}\beta \upsilon_y(0,\cdot))^2\Phi^{(0)}_{xx}\\
&\;\,+2(\theta_i^{-1}\beta_R \zeta_y(R^+,\cdot)-R{\rm Le}^{-1}\beta \upsilon_y(0,\cdot))\upsilon_y\}
(1+\theta_i^{-1}\beta'(\cdot-R)\zeta(R^+,\cdot)-R{\rm Le}^{-1}\beta'\upsilon(0,\cdot))^{-1}\\
&-\big\{2\big (\theta_i^{-1}\beta_R \zeta_y(R^+,\cdot)-R{\rm Le}^{-1}\beta \upsilon_y(0,\cdot)\big )\big (\theta_i^{-1}\beta'(\cdot-R)\zeta_y(R^+,\cdot)-R{\rm Le}^{-1}\beta'\upsilon_y(0,\cdot)\big )\\
&\qquad\quad\times
\big (\theta_i^{-1}\beta_R\Phi^{(0)}_{xx}\zeta(R^+,\cdot)-R{\rm Le}^{-1}\beta\Phi^{(0)}_{xx}\upsilon(0,\cdot)+\upsilon\big )\\
&\;\;\;\;\;\;+\big [\big (\theta_i^{-1}\beta_R \zeta_y(R^+,\cdot)-R{\rm Le}^{-1}\beta \upsilon_y(0,\cdot)\big )^2-\big (\theta_i^{-1}\beta'(\cdot-R)\zeta(R^+,\cdot)-R{\rm Le}^{-1}\beta'\upsilon(0,\cdot)\big )^2\big ]\\
&\qquad\quad\;\;\;\;\times\big (\theta_i^{-1}\beta_R\Phi^{(0)}_{xxx}\zeta(R^+,\cdot)-R{\rm Le}^{-1}\beta \Phi^{(0)}_{xxx}\upsilon(0,\cdot)+\Phi^{(0)}_{xx}+\upsilon_x\big )\\
&\;\;\;\;\;\;-2\big (\theta_i^{-1}\beta'(\cdot-R)\zeta(R^+,\cdot)-R{\rm Le}^{-1}\beta'\upsilon(0,\cdot)\big )\\
&\qquad\quad\;\;\;\;\times\big [\theta_i^{-1}\beta_R\Phi^{(0)}_{xxx}\zeta(R^+,\cdot)-R{\rm Le}^{-1}\beta\Phi^{(0)}_{xxx} \upsilon(0,\cdot)\\
&\qquad\qquad\quad\;\;\;\;-\big (\theta_i^{-1}\beta_R \zeta_y(R^+,\cdot)-R{\rm Le}^{-1}\beta \upsilon_y(0,\cdot)\big )^2\Phi^{(0)}_{xx}+\upsilon_x\big ]\big\}\\
&\qquad\qquad\;\;\;\;\;\;\times(1+\theta_i^{-1}\beta'(\cdot-R)\zeta(R^+,\cdot)-R{\rm Le}^{-1}\beta'\upsilon(0,\cdot))^{-2}\\
&+\big\{\big (\theta_i^{-1}\beta''(\cdot-R)\zeta(R^+,\cdot)-R{\rm Le}^{-1}\beta''\upsilon(0,\cdot)\big )\big [1+\big (\theta_i^{-1}\beta_R\zeta_y(R^+,\cdot)-R{\rm Le}^{-1}\beta \upsilon_y(0,\cdot)\big )^2\big ]\\
&\qquad\quad\times (\theta_i^{-1}\beta_R\Phi^{(0)}_{xx}\zeta(R^+,\cdot)-R{\rm Le}^{-1}\beta\Phi^{(0)}_{xx}\upsilon(0,\cdot)+\upsilon)\big\}\\
&\qquad\qquad\qquad\times(1+\theta_i^{-1}\beta'(\cdot-R)\zeta(R^+,\cdot)-R{\rm Le}^{-1}\beta'\upsilon(0,\cdot))^{-3},\\[1mm]
\Lambda({\bf v})=& (\theta_i\!-\!\zeta(R^+,\cdot)\!-\!\zeta_x(R^+,\cdot))^{-1}\\
&\qquad\quad\times \big [\zeta_x(R^+,\cdot)+ \Delta \zeta(R^+,\cdot)-\theta_i^{-1}\zeta_{yy}(R,\cdot)\zeta_x(R^+,\cdot)-2\theta_i^{-1}\zeta_y(R,\cdot)\zeta_{xy}(R^+,\cdot)\\
&\qquad\qquad\;\,+\theta_i^{-2}(\zeta_{y}(R,\cdot))^2(\zeta_{xx}(R^+,\cdot)-\zeta(R,\cdot)-\theta_i)-\theta_i^{-1}\zeta(R,\cdot)\zeta_{yy}(R,\cdot)\big ]\\
&-[{\rm Le}^2+R{{\rm{{Le}}}}({\rm Le}\, \upsilon(0^+,\cdot)+\upsilon_x(0^+,\cdot))]^{-1}\\
&\qquad\quad\times \big\{{{\rm{{Le}}}}\, \upsilon_x(0^+,\cdot) +\Delta \upsilon (0^+,\cdot)\\
&\qquad\qquad\;\,+R{{\rm{{Le}}}}^{-1}\big [\upsilon_{yy}(0^+,\cdot)({{\rm{{Le}}}}\, \upsilon(0^+,\cdot)+\upsilon_{x}(0^+,\cdot))+2\upsilon_y(0^+,\cdot) \upsilon_{xy}(0^+,\cdot)\big ]\notag\\
&\qquad\qquad\;\,+R^2{{\rm{{Le}}}}^{-2}(\upsilon_{y}(0^+,\cdot))^2(-{{\rm{{Le}}}}^2\upsilon(0^+,\cdot)+R^{-1}{{\rm{{Le}}}}^2+\upsilon_{xx}(0^+,\cdot))\big\},\\[2mm]
\mathscr H_j({\bf v})=&0~{\rm if}~j\neq 4,7,\qquad
\mathscr H_4({\bf v})=-\frac{R{\rm Le}(\upsilon_y(0,\cdot))^2}{{\rm Le}^2+R^2(\upsilon_y(0,\cdot))^2},\qquad
\mathscr H_7({\bf v})= \frac{{\rm Le}(\zeta_y(R^+,\cdot))^2}{R(\theta_i^2+(\zeta_y(R^+,\cdot))^2)},\end{aligned}$$ on smooth enough functions ${\bf v}=(\zeta,\upsilon)$, where $\beta_R=\beta(\cdot-R)$.
\[rmk-opB\] [Note that each smooth enough function ${{\bf u}}$, which solves problem , has its first component $u_1$ which is continuous on $\{R\}\times [-\ell/2,\ell/2]$. Therefore, the operator ${\mathscr B}$ can be replaced with the operator $\widetilde {\mathscr B}$ which is defined as ${\mathscr B}$ with the fifth equation being replaced by the condition $\frac{1}{2}(v_1(R^+,\cdot)+v_1(R^-,\cdot))+\theta_iR[D_xv_1(R^+,\cdot)-D_xv_1(R^+,\cdot)]=0$.]{}
We will use the above remark in Subsection \[subsection-lifting\].
Tools {#sect-4}
=====
In this section we collect some technical results which are used in the next (sub)sections.
Preliminary results needed to prove Theorems \[banca\] and Proposition \[perdita\]
----------------------------------------------------------------------------------
We find it convenient to set $$\begin{aligned}
({\mathcal F}_{k,\rho}f)(x):=\frac{1}{Z_k}\int_{{{\mathbb R}}}e^{-\frac{\rho}{2}s}e^{-\frac{1}{2}Z_k|s|}\hat f_k(x-s)ds,\qquad\;\,x\in{{\mathbb R}},\;\,f\in C_b(\overline{S};{{\mathbb C}}),\;\,k\in{{\mathbb Z}},\end{aligned}$$ where $Z_k=Z_k(\lambda,\rho)=\sqrt{\rho^2+4\lambda\rho+4\lambda_{k}}$ for every $k\in{{\mathbb Z}}$ and $\lambda_k=(4\ell^2)^{-1}k^2\pi^2$. Moreover, we denote by $\Sigma_0$ the set of $\lambda\in{{\mathbb C}}$ with positive real part.
\[finito\] For every $\rho\in (0,+\infty)$, $\lambda\in{{\mathbb C}}$ with ${\rm Re}\lambda>-\rho^{-1}({\rm Im}\lambda)^2$ and $f\in C_b(\overline{S};{{\mathbb C}})$, the series $\ell^{-1}\sum_{k\in{{\mathbb Z}}}(\mathcal F_{k,\rho}f)e_k$ defines a bounded and continuous function ${\mathscr R}_{\lambda,\rho}f$ in ${{\mathbb R}}^2$ which, clearly, is periodic with respect to $y$. Moreover,
1. ${\mathscr R}_{\lambda,\rho}f\in \bigcap_{p<+\infty} W^{2,p}_{\rm loc}({{\mathbb R}}^2;{{\mathbb C}})$ and $\lambda {\mathscr R}_{\lambda,\rho}f-\rho^{-1}\Delta {\mathscr R}_{\lambda,\rho}f-D_x{\mathscr R}_{\lambda,\rho}f=f$ in $S$;
2. $\nabla {\mathscr R}_{\lambda,\rho}f\in C_b({{\mathbb R}}^2;{{\mathbb C}})\times C_b({{\mathbb R}}^2;{{\mathbb C}})$ and $$|\lambda|\|{\mathscr R}_{\lambda,\rho}f\|_{\infty}+\sqrt{|\lambda|}\|\nabla{\mathscr R}_{\lambda,\rho}f\|_{\infty}\le c\|f\|_{\infty},\qquad\;\,\lambda\in\Sigma_0;
\label{braccio}$$
3. if further $\lim_{x\to -\infty}f(x,y)=0$ [(resp.]{} $\lim_{x\to +\infty}f(x,y)=0)$ for every $y\in [-\ell/2,\ell/2]$, then $({\mathscr R}_{\lambda,\rho}f)(\cdot,y)$ vanishes as $x\to -\infty$ [(resp.]{} $x\to +\infty)$ for every $y\in{{\mathbb R}}$;
4. for every $f\in C^{\alpha}_b(S;{{\mathbb C}})$, such that $f(\cdot,-\ell/2)=f(\cdot,\ell/2)$, and $\lambda\in{{\mathbb C}}$ with ${\rm Re}\lambda>-\rho^{-1}({\rm Im}\lambda)^2$, the function $\mathscr R_{\lambda,\rho}f$ admits classical derivatives up to the second-order which belong to $C^{\alpha}_b({{\mathbb R}}^2;{{\mathbb C}})$. Moreover, $$\begin{aligned}
\label{cotto}
\|\mathscr R_{\lambda,\rho}f\|_{C^{2+\alpha}_b({{\mathbb R}}^2;{{\mathbb C}})}\leq c_{\lambda}\|f\|_{C^{\alpha}_b(S;{{\mathbb C}})}.\end{aligned}$$
To begin with, we claim that, for each $f\in C_b(\overline{S};{{\mathbb C}})$ and $\lambda\in{{\mathbb C}}$, such that $\rho{\rm Re}\lambda+({\rm Im}\lambda)^2>0$, the series in the statement converges uniformly in ${{\mathbb R}}^2$. To prove the claim, we observe that ${\rm Re}(Z_k)>\rho$ and $|Z_k|\ge {\rm Re}(Z_k)\ge c_{\lambda}(k+1)$ for every $k\in{{\mathbb Z}}$. Thus, we can estimate $$\begin{aligned}
|(\mathcal F_{k,\rho}f)(x)|\le \sup_{x\in{{\mathbb R}}}|\hat f_k(x)|
\frac{1}{|Z_k|}\int_{{{\mathbb R}}}e^{\frac{\rho-{\rm Re}(Z_k)}{2}|s|}ds
\le \frac{c_{\lambda}}{k^2+1}\|f\|_{C_b(\overline{S};{{\mathbb C}})},\qquad\;\,x\in{{\mathbb R}},\;\,k\in{{\mathbb Z}},
\label{spugna}\end{aligned}$$ and this is enough to infer that the series converges locally uniformly on ${{\mathbb R}}^2$ and, as a byproduct, that the operator ${\mathscr R}_{\lambda,\rho}$ is bounded from $C_b(\overline{S};{{\mathbb C}})$ into $C_b({{\mathbb R}}^2;{{\mathbb C}})$. Moreover, if $f(\cdot,y)$ vanishes at $-\infty$ (resp. $+\infty$) for each $y\in [-\ell/2,\ell/2]$, then by dominated convergence the function ${\mathcal F}_kf$ vanishes at $-\infty$ (resp. $+\infty$) and, in view of the uniform convergence of the series which defines the function ${\mathscr R}_{\lambda,\rho}$, this is enough to conclude that this latter function tends to $0$ as $x\to-\infty$ (resp. $x\to+\infty$) for each $y\in{{\mathbb R}}$.
Now, we prove properties (i), (ii) and (iv).
\(i) Let us prove that the function ${\mathscr R}_{\lambda,\rho} f$ is the unique solution to the equation $\lambda u-\rho^{-1}\Delta u-D_xu=f^{\sharp}$ in ${\mathcal D}=\{u\in C^1_b({{\mathbb R}}^2;{{\mathbb C}})\cap \bigcap_{p<+\infty}W^{2,p}_{\rm loc}({{\mathbb R}}^2;{{\mathbb C}}): \Delta u\in L^{\infty}({{\mathbb R}}^2;{{\mathbb C}})\cap C_b(\overline{S};{{\mathbb C}}), u(\cdot,\cdot+\ell)=u\}$. For this purpose, for every $n\in{{\mathbb N}}$ we introduce the functions $u_n=\ell^{-1}\sum_{k=-n}^n(\mathcal F_{k,\rho}f)e_k$ and $f_n=\ell^{-1}\sum_{k=-n}^n\hat f(\cdot,k)e_k$. Note that $\lambda u_n-\rho^{-1}\Delta u_n-D_xu_n=f_n$ in ${{\mathbb R}}^2$, for every $n\in{{\mathbb N}}$, since the function ${\mathcal F}_{k,\rho}f$ $(k\in{{\mathbb Z}})$ solves the differential equation $(\lambda+\rho^{-1}\lambda_k)w-\rho^{-1}w''-w'=f_k$ in ${{\mathbb R}}$. Thus, $$\begin{aligned}
\langle u_n,\rho^{-1}\Delta\varphi-\varphi_x\rangle
=\int_{{{\mathbb R}}^2}u_n(\rho^{-1}\Delta\varphi-\varphi_x)dxdy=\int_{{{\mathbb R}}^2}(\lambda u_n-f_n)\varphi dxdy
=:\langle \lambda u_n-f_n,\varphi\rangle\end{aligned}$$ for every $\varphi\in C^{\infty}_c({{\mathbb R}}^2;{{\mathbb C}})$. Letting $n$ tend to $+\infty$ and applying the dominated convergence theorem, it follows that ${\mathscr R}_{\lambda,\rho}f$ is a distributional solution to the equation $\lambda {\mathscr R}_{\lambda,\rho}f-\rho^{-1}\Delta {\mathscr R}_{\lambda,\rho}f-D_x {\mathscr R}_{\lambda,\rho} f=f^{\sharp}$. By elliptic regularity (see e.g., [@gilbarg]), we can infer that ${\mathscr R}_{\lambda,\rho}f\in\bigcap_{p<+\infty}W^{2,p}_{\rm loc}({{\mathbb R}}^2;{{\mathbb C}})$. Since ${\mathscr R}_{\lambda,\rho}f$ is bounded and continuous in ${{\mathbb R}}^2$ and $\Delta {\mathscr R}_{\lambda,\rho}f+D_x{\mathscr R}_{\lambda,\rho}f=\lambda {\mathscr R}_{\lambda,\rho}f-f^{\sharp}$ belongs to $L^{\infty}({{\mathbb R}}^2;{{\mathbb C}})$, again by classical results we can infer that ${\mathscr R}_{\lambda,\rho}f\in C^{1+\gamma}_b({{\mathbb R}}^2;{{\mathbb C}})$ for each $\gamma\in (0,1)$ and, as a byproduct, that $\Delta {\mathscr R}_{\lambda,\rho} f\in L^{\infty}({{\mathbb R}}^2;{{\mathbb C}})\cap C_b(\overline{S};{{\mathbb C}})$. We can thus conclude that ${\mathscr R}_{\lambda,\rho} f$ belongs to ${\mathcal D}$.
To prove uniqueness, we assume that $v$ is another solution in ${\mathcal D}$ of the equation $\lambda u-\rho^{-1}\Delta u-u_x=f^{\sharp}$. The smoothness of $v$ implies that, for each $k\in{{\mathbb Z}}$, the function $\hat v_k$ belongs to $C^1_b({{\mathbb R}};{{\mathbb C}})$. Moreover, integrating by parts we obtain that $$\begin{aligned}
\int_{{{\mathbb R}}}\hat v_k\varphi''dx=&\frac{1}{\ell}\int_{S}\varphi''v\overline{e_k}dx dy
=\frac{1}{\ell}\int_{S}\varphi v_{xx}\overline{e_k}dx dy\end{aligned}$$ for each $\varphi\in C^{\infty}_c({{\mathbb R}})$. By Fubini theorem, $\hat v_k$ belongs to $W^{2,p}_{\rm loc}({{\mathbb R}};{{\mathbb C}})$. Since $\lambda v-\rho^{-1}\Delta v-v_x=f$ in $S$, we can write $$\begin{aligned}
\int_{S}\varphi(x)v_{xx}\overline{e_k}dx dy
=&\lim_{n\to +\infty}\int_{S}\varphi(x)v_{xx}(x,y)\psi_n(y)\overline{e_k(y)}dx dy\notag\\
=&\lambda\rho\int_{{{\mathbb R}}}\varphi v_kdx
-\lim_{n\to +\infty}\int_{S}\varphi v_{yy}\psi_n\overline{e_k}dx dy-\rho\int_{{{\mathbb R}}}\varphi\hat v'_kdx
-\rho\int_{{{\mathbb R}}}\varphi f_kdx,
\label{clonata-1}\end{aligned}$$ where $\psi_n(y)=\psi(n|y|/\ell+1-n/2)$ for each $y\in [-\ell/2,\ell/2)$, $n\in{{\mathbb N}}$, and $\psi$ is a smooth function such that $\psi=1$ in $(-\infty,1/2]$ and $\psi=0$ outside $(-\infty,3/4]$. Clearly, $\psi_n$ converges to $1$ in $L^1((-\ell/2,\ell/2))$ as $n$ tends to $+\infty$. An integration by parts shows that $$\begin{aligned}
&\lim_{n\to+\infty}\int_{S}\varphi(x)v_{yy}(x,y)\psi_n(y)\overline{e_k(y)}dx dy\notag\\
=&-\lim_{n\to+\infty}\frac{n}{\ell}\int_{{{\mathbb R}}^2}\varphi(x)\chi_{A}(x){\rm signum}(y)v_y(x,y)\psi'\bigg (\frac{n}{\ell}|y|+1-\frac{n}{2}\bigg )
\chi_{B_n}(y)\overline{e_k(y)}dx dy\notag\\
&-\lambda_k\int_{{{\mathbb R}}}\varphi(x)v_k(x)dx,
\label{clonata-2}\end{aligned}$$ where $A={\rm supp}(\varphi)$ and $B_n=\left\{y\in{{\mathbb R}}: \frac{\ell}{2}-\frac{\ell}{2n}\le |y|\le \frac{\ell}{2}-\frac{\ell}{4n}\right\}$ for every $n\in{{\mathbb N}}$. We claim that the first term in the last side of is zero. For this purpose, we split $$\begin{aligned}
&\frac{n}{\ell}\int_{{{\mathbb R}}^2}\varphi(x)\chi_A(x){\rm signum}(y)v_y(x,y)\psi'\bigg (\frac{n}{\ell}|y|+1-\frac{n}{2}\bigg )
\chi_{B_n}(y)\overline{e_k(y)}dx dy\\
=&\frac{n}{\ell}\int_{{{\mathbb R}}^2}\varphi(x)\chi_{A}(x){\rm signum}(y)(v_y(x,y)-v_y(x,\ell/2))\psi'\bigg (\frac{n}{\ell}|y|+1-\frac{n}{2}\bigg )
\chi_{B_n}(y)\overline{e_k(y)}dx dy\\
&+\frac{n}{\ell}\int_A\varphi(x)v_y(x,\ell/2)dx\int_{{{\mathbb R}}}{\rm signum}(y)\psi'\bigg (\frac{n}{\ell}|y|+1-\frac{n}{2}\bigg )
\chi_{B_n}(y)\overline{e_k(y)}dy\\
=&\!:{\mathcal K}_{1,n}(x,y)+{\mathcal K}_{2,n}(x,y)\end{aligned}$$ for every $(x,y)\in{{\mathbb R}}^2$. Since $v$ is $\ell$-periodic with respect to $y$, $D_yv$ is $\ell$-periodic with respect to $y$ as well. Moreover, this latter function is $1/2$-Hölder continuous in ${{\mathbb R}}^2$ since it belongs to ${\mathcal D}$. Hence, we can estimate $$\begin{aligned}
|v_y(x,y)-v_y(x,\ell/2)|=|v_y(x,y)-v_y(x,-\ell/2)|
\le [v_y]_{C^{1/2}(A\times [-\ell/2,\ell/2];{{\mathbb C}})}\min\{|y-\ell/2|,|y+\ell/2|\}^{\frac{1}{2}}\end{aligned}$$ for every $x\in A$ and $y\in [-\ell/2,\ell/2]$. Thus, $$\begin{aligned}
|{\mathcal K}_{1,n}|\le c\|\varphi\|_{\infty}\|\psi'\|_{\infty}n^{\frac{1}{2}}m(A\times B_n)
\le c\|\varphi\|_{\infty}\|\psi'\|_{\infty}n^{-\frac{1}{2}},\qquad\;\,n\in{{\mathbb N}},\end{aligned}$$ where $m(A\times B_n)$ denotes the Lebesgue measure of the set $A\times B_n$, so that ${\mathcal K}_{1,n}$ vanishes as $n$ tends to $+\infty$. As far as ${\mathcal K}_{2,n}$ is concerned, we observe that $$\begin{aligned}
{\mathcal K}_{2,n}(x,y)
=&-\frac{2in}{\ell}\int_AD_yv(x,\ell/2)\varphi(x)dx\int_{\frac{\ell}{2}-\frac{\ell}{2n}}^{\frac{\ell}{2}-\frac{\ell}{4n}}\psi'\bigg (\frac{n}{\ell}y+1-\frac{n}{2}\bigg )\sin\bigg (\frac{2k\pi}{\ell}y\bigg )dy\\
=&c\bigg [\frac{4k\pi i}{\ell}\int_{\frac{\ell}{2}-\frac{\ell}{2n}}^{\frac{\ell}{2}-\frac{\ell}{4n}}\psi\bigg (\frac{n}{\ell}y+1-\frac{n}{2}\bigg )\cos\bigg (\frac{2k\pi}{\ell}y\bigg )dy-2(-1)^k\sin\left (\frac{k\pi}{n}\right )\bigg ].\end{aligned}$$ Hence, ${\mathcal K}_{2,n}$ vanishes as $n$ tends to $+\infty$. The claim is so proved.
From and , we can now infer that $\rho^{-1}\hat v_k''=(\lambda+\rho^{-1}\lambda_k)\hat v_k-\hat v_k'-\hat f_k$. Thus, $v_k={\mathcal F}_kf$ for every $k\in{{\mathbb Z}}$ and, as a byproduct, we deduce that $v={\mathscr R}_{\lambda,\rho}f$.
\(ii) By classical results, the realization $A_{\rho}$ in $L^{\infty}({{\mathbb R}}^2;{{\mathbb C}})$ of the operator $\rho^{-1}\Delta+D_x$, with domain $D(A_{\rho})=\{u\in C^1_b({{\mathbb R}}^2;{{\mathbb C}})\cap\bigcap_{p<+\infty}W^{2,p}_{\rm loc}({{\mathbb R}}^2;{{\mathbb C}}): \Delta u\in L^{\infty}({{\mathbb R}}^2;{{\mathbb C}})\}\supset {\mathcal D}$, generates an analytic semigroup. Moreover, $\|R(\lambda,A_{\rho})\|_{L(L^{\infty}({{\mathbb R}}^2))}\le c|\lambda|^{-1}$ for every $\lambda\in\Sigma_0$ and $\|\nabla v\|_{\infty}\le c\|v\|_{\infty}^{1/2}\|A_\rho v\|_{\infty}^{1/2}$ for every $v\in D(A_{\rho})$. Note that the function $R(\lambda,A_{\rho})f^{\sharp}$ is $\ell$-periodic with respect to $y$. Indeed, $R(\lambda,A_{\rho})f^{\sharp}$ and $(R(\lambda,A_{\rho})f^{\sharp})(\cdot,\cdot+\ell)$ both solve (in $D(A_{\rho})$) the equation $\lambda u-\rho^{-1}\Delta u-u_x=f^{\sharp}$ and, by uniqueness, they coincide. Finally, since $f^{\sharp}$ is continuous in $\overline{S}$, $\Delta R(\lambda,A_{\rho})f^{\sharp}$ belongs to $C_b(\overline{S};{{\mathbb C}})$. Hence, $R(\lambda,A_{\rho})f^{\sharp}\in {\mathcal D}$ and, by $(i)$, it coincides with ${\mathscr R}_{\lambda,\rho}f^{\sharp}$. Using the above estimates, inequality follows immediately.
\(iv) Since $f\in C^{\alpha}_b(S;{{\mathbb C}})$ and $f(\cdot,-\ell/2)=f(\cdot,\ell/2)$, the function $f^{\sharp}$ belongs to $C^{\alpha}_b({{\mathbb R}}^2;{{\mathbb C}})$. Hence, $\Delta{\mathscr R}_{\lambda,\rho}f=\rho(\lambda {\mathscr R}_{\lambda,\rho}f- f^{\sharp}-D_x{\mathscr R}_{\lambda,\rho}f)$ is an element of $C^{\alpha}_b({{\mathbb R}}^2;{{\mathbb C}})$. Classical results (see e.g., [@krylov]) yield .
\[finire\] For $g\in C_b(\overline{S_0^+};{{\mathbb C}})$ and $\lambda\!\in\!{{\mathbb C}}$, such that ${\rm Re}\lambda>-{\rm Le}^{-1}({\rm Im}\lambda)^2$, the function ${\mathscr S}_{\lambda}g=
\ell^{-1}\sum_{k\in{{\mathbb Z}}}({\mathcal G}_kg)e_k$ is bounded and continuous in ${{\mathbb R}}^2$. Here, ${\mathcal G}_kg:={\mathcal F}_{k,{\rm Le}}(\overline g)$ for each $k\in{{\mathbb Z}}$, and $\overline{g}$ is the trivial extension of $g$ to ${{\mathbb R}}\times [-\ell/2,\ell/2]$. Moreover,
1. ${\mathscr S}_{\lambda}g$ belongs to $\bigcap_{p<+\infty} W^{2,p}_{\rm loc}({{\mathbb R}}^2;{{\mathbb C}})$ and $\lambda {\mathscr S}_{\lambda}g-{\rm Le}^{-1}\Delta {\mathscr S}_{\lambda}g-D_x{\mathscr S}_{\lambda}g=g$ in $S_0^+$;
2. there exists a positive constant $c_1$, independent of $\lambda$, such that $$|\lambda|\|{\mathscr S}_{\lambda}g\|_{\infty}+\sqrt{|\lambda|}\|\nabla{\mathscr S}_{\lambda}g\|_{\infty}\le c_1\|g\|_{\infty},\qquad\;\,\lambda\in\Sigma_0;
\label{braccio-3}$$
3. if further $\lim_{x\to +\infty}g(x,y)=0$ for each $y\in [-\ell/2,\ell/2]$, then $({\mathscr S}_{\lambda}g)(\cdot,y)$ vanishes as $x\to +\infty$ for each $y\in{{\mathbb R}}$;
4. if $g\in C^{\alpha}_b(S_0^+;{{\mathbb C}})$ and $g(\cdot,-\ell/2)=g(\cdot,\ell/2)$, then ${\mathscr S}_{\lambda}g$ belongs to $C^{2+\alpha}_b({{\mathbb R}}^2;{{\mathbb C}})$ for every $\lambda\in{{\mathbb C}}$ such that ${\rm Re}\lambda>-{\rm Le}^{-1}({\rm Im}\lambda)^2$. Moreover, $$\begin{aligned}
\label{cotto-11}
\|\mathscr S_{\lambda}g\|_{C^{2+\alpha}_b({{\mathbb R}}^2;{{\mathbb C}})}\leq c_{2,\lambda}\|g\|_{C^{\alpha}_b(S_0^+;{{\mathbb C}})},\end{aligned}$$ with the constant $c_{2,\lambda}$ being independent of $g$.
We split the proof into two steps: in the first one we prove properties (i), (ii) and (iii) and in the second step we prove property (iv).
[*Step 1*]{}. Let us fix $g$ and $\lambda$ as in the statement. Arguing as in the first part of the proof of Lemma \[finito\], taking the continuity of the functions ${\mathcal G}_kg$ $(k\in{{\mathbb Z}})$ into account, it can be checked that the series in the statement converges uniformly in ${{\mathbb R}}^2$, so that the function ${\mathscr S}_{\lambda}g$ is well defined and it vanishes as $x\to +\infty$ for every $y\in {{\mathbb R}}$, if $\lim_{x\to +\infty}g(x,y)=0$ for every $y\in [-\ell/2,\ell/2]$.
To check properties (i) and (ii), for each $n\in{{\mathbb N}}$ we set $g_n:=\overline{g}\star_x\psi_n$, where $\star_x$ stands for convolution with respect to the variable $x$ and $(\psi_n)$ is a standard sequence of mollifiers. Clearly, ${\mathscr S}_{\lambda}g_n={\mathscr R}_{\lambda,{\rm Le}}g_n$. The sequence $(g_n)$ converges to $\overline{g}$ pointwise in ${{\mathbb R}}^2$ as $n\to +\infty$ and $\|g_n\|_{\infty}\le\|g\|_{\infty}$ for each $n\in{{\mathbb N}}$. Thus, we can infer that ${\mathcal F}_{k,\rho}g_n$ converges to ${\mathcal G}_kg$ pointwise in ${{\mathbb R}}$ as $n$ tends to $+\infty$, for every $k\in{{\mathbb N}}$ and, by dominated convergence, ${\mathscr R}_{\lambda,{\rm Le}}g_n$ tends to ${\mathscr S}_{\lambda}g$ pointwise in ${{\mathbb R}}^2$.
Applying the classical interior $L^p$-estimates for the operator ${\rm Le}^{-1}\Delta +D_x$ and using , which allows us to write $$\begin{aligned}
& |\lambda|\|\nabla{\mathscr R}_{\lambda, {\rm Le}}g_n\|_{\infty}+
|\lambda|^{\frac{1}{2}}\|\nabla{\mathscr R}_{\lambda, {\rm Le}}g_n\|_{\infty} \leq c\|g_n\|_{\infty}\leq c\|g\|_{\infty},\qquad\;\;n\in{{\mathbb N}},
\label{nessuno}\end{aligned}$$ we can estimate $$\begin{aligned}
\|{\mathscr R}_{\lambda,{\rm Le}} g_n\|_{W^{2,p}(B(0,r);{{\mathbb C}})}\le &c_{p,r}(\|{\mathscr R}_{\lambda,{\rm Le}} g_n\|_{L^p(B(0,2r);{{\mathbb C}})}+\|{\rm Le}^{-1}\Delta{\mathscr R}_{\lambda,{\rm Le}}g_n+D_x{\mathscr R}_{\lambda,{\rm Le}}g_n\|_{L^p(B(0,2r);{{\mathbb C}})})\\
\le &c_{p,r}[(1+|\lambda|)\|{\mathscr R}_{\lambda,{\rm Le}}g_n\|_{C(\overline{B(0,2r)};{{\mathbb C}})}+\|g_n\|_{C(\overline{B(0,2r)};{{\mathbb C}})}]\\
\le &c_{p,r,\lambda}\|g_n\|_{\infty}\le c_{p,r,\lambda}\|g\|_{\infty}\end{aligned}$$ for every $p\in [1,+\infty)$ and $r>0$. Hence, by compactness, we conclude that ${\mathscr R}_{\lambda,{\rm Le}} g_n$ converges to ${\mathscr S}_{\lambda}g$ in $C^1(\overline{B(0,r)};{{\mathbb C}})$ for each $r>0$, ${\mathscr S}_{\lambda}g\in W^{2,p}_{\rm loc}({{\mathbb R}}^2;{{\mathbb C}})$ for every $p\in [1,+\infty)$ and $\lambda {\mathscr S}_{\lambda}g-{\rm Le}^{-1}\Delta {\mathscr S}_{\lambda}g-D_x{\mathscr S}_{\lambda}g=g$ in $S_0^+$. Finally, estimate follows at once from .
[*Step 2.*]{} To complete the proof, here we check property (iv), which demands some additional effort. We begin by checking that the function $\zeta={\mathscr S}_{\lambda}g(0,\cdot)$ belongs to $C^{2+\alpha}_b({{\mathbb R}};{{\mathbb C}})$. For this purpose, we set $\zeta_n=\ell^{-1}\sum_{k=-n}^n(\mathcal G_kg)(0)e_k$ for $n\in{{\mathbb N}}$. Clearly, each function $\zeta_n$ is smooth and $$\begin{aligned}
\zeta_n'=&
\frac{2\pi i}{\ell^2}\sum_{|k|<k_0}\frac{k}{Z_k}e_k\int_0^{+\infty}e^{\frac{{\rm Le}}{2}s}e^{-\frac{1}{2}Z_ks}\hat g_k(s)ds+\frac{2\pi i}{\ell^2}\sum_{k_0\le |k|\le n}\!\!\bigg (\frac{k}{Z_k}-\frac{\ell}{\pi}\bigg )e_k\int_0^{+\infty}e^{\frac{{\rm Le}}{2}s}e^{-\frac{1}{2}Z_ks}\hat g_k(s)ds\\
&+\frac{2i}{\ell}\!\sum_{|k|=k_0}^n\!\!e_k\!\int_0^{+\infty}\!e^{\frac{{\rm Le}}{2}s}\Big (e^{-\frac{1}{2}Z_ks}\!-\!e^{-\frac{\pi|k|}{2\ell }s}\Big )\hat g_k(s)ds\!
+\!\frac{2i}{\ell}\!\sum_{|k|=k_0}^n\!\!e_k\int_0^{+\infty}\!e^{\frac{\rm Le}{2}s-\frac{\pi|k|}{2\ell}s}\hat g_k(s)ds \\
=: & \mathcal I_0+\sum_{h=1}^3{\mathcal I}_{h,n},\end{aligned}$$ where $k_0\in{{\mathbb N}}$ is chosen so that $\pi k_0>\ell {\rm Le}$. As it is easily seen, $$\left |\frac{k}{Z_k}-\frac{\ell}{\pi}\right |\le \frac{c_{\lambda}}{k^2+1},\qquad\;\,k\in{{\mathbb Z}},
\label{grignani}$$ so that ${\mathcal I}_{1,n}$ converges uniformly in ${{\mathbb R}}^2$ as $n\to +\infty$. On the other hand, $$\begin{aligned}
\Big |e^{-\frac{1}{2}Z_ks}-e^{-\frac{\pi|k|}{2\ell}s}\Big |
= \bigg |\int_0^1\frac{d}{dr}e^{-\frac{1}{2}Z_k(r)s}dr\bigg |
\le s\int_0^1
\bigg |\frac{{\rm Le}^2+4\lambda{\rm Le}}{4Z_k(r)}\bigg |e^{-\frac{1}{2}{\rm Re}(Z_k(r))s}dr,\end{aligned}$$ where we have set $Z_k(r)=\left (({\rm Le}^2+4\lambda{\rm Le})r+\frac{k^2\pi^2}{\ell^2}\right )^{\frac{1}{2}}$. Note that ${\rm Re}(Z_k(r))\ge c_{\lambda}|k|$ for $|k|\ge k_0$. For such values of $k$ and for ${\rm Re}\lambda>0$ (which implies that ${\rm Re}(Z_k(r))>{\rm Le}$ for every $r\in [0,1]$ and $|k|\ge k_0$) we can estimate $$\begin{aligned}
\bigg |\int_0^{+\infty}e^{\frac{{\rm Le}}{2}s}\Big (e^{-\frac{1}{2}Z_ks}-e^{-\frac{\pi|k|}{2\ell }s}\Big )\hat g_k(s)ds\bigg |
\le &\frac{|{\rm Le}^2+4\lambda{\rm Le}|}{4|k|}\|g\|_{\infty}\int_0^1r^{-\frac{1}{2}}dr\int_0^{+\infty}se^{\frac{\rm Le}{2}s-\frac{1}{2}({\rm Re}(Z_k(r))s}ds\notag\\
\le &\frac{c_\lambda}{k^3}\|g\|_{\infty},
\label{tokyo}\end{aligned}$$ so that the sequence $({\mathcal I}_{2,n})$ converges uniformly in ${{\mathbb R}}^2$. Next, we observe that $$\begin{aligned}
{\mathcal I}_{3,n}(y)=\frac{2i}{\ell}\int_0^{+\infty}ds\int_{-\frac{\ell}{2}}^{\frac{\ell}{2}}K_n(s,\eta)g_n(s,y-\eta)d\eta,\qquad\;\,y\in{{\mathbb R}},\;\,n\in{{\mathbb N}},\end{aligned}$$ where $K_n(x,y)=H_n(x,y)+H_n(x,-y)$ and $$\begin{aligned}
H_n(x,y)=e^{\frac{\rm Le}{2}x}\frac{e^{-\frac{\pi}{2\ell}(x-4iy)(k_0-1)}-e^{-\frac{\pi}{2\ell}(x-4iy)n}}{e^{\frac{\pi}{2\ell}(x-4iy)}-1},\qquad\;\,g_n(x,y)=\ell^{-1}\sum_{k=-n}^n\hat g(x,k)e^{\frac{2k\pi i}{\ell}y}\end{aligned}$$ for $x\ge 0$, $y\in{{\mathbb R}}^2$ and $n\in{{\mathbb N}}$. We set $$\begin{aligned}
K(x,y)=e^{\left (\frac{\rm Le}{2}-\frac{\pi(k_0-1)}{2\ell}\right )x}\bigg (\frac{e^{-\frac{2\pi (k_0-1) i}{\ell}y}}{
e^{\frac{\pi}{2\ell}(x-4iy)}-1}+\frac{e^{\frac{2\pi(k_0-1) i}{\ell}y}}{
e^{\frac{\pi}{2\ell}(x+4iy)}-1}\bigg ),\qquad\;\,x\ge 0,\;\,y\in{{\mathbb R}},\end{aligned}$$ and prove that ${\mathcal I}_{3,n}$ converges pointwise in ${{\mathbb R}}$ to the function ${\mathcal I}_3$, defined by[^2] $$\begin{aligned}
{\mathcal I}_3(y)=\frac{i\ell}{\pi}\int_0^{+\infty} ds\int_{-\frac{\ell}{2}}^{\frac{\ell}{2}}K(s,\eta)g^{\sharp}(s,y-\eta)d\eta,\qquad\;\,y\in{{\mathbb R}}.\end{aligned}$$ For this purpose, we split $$\begin{aligned}
{\mathcal J}_{3,n}-{\mathcal J}_3=&\int_0^{+\infty} ds\int_{-\frac{\ell}{2}}^{\frac{\ell}{2}}(g^{\sharp}_n(s,\cdot-\eta)-g^{\sharp}(s,\cdot-\eta))K_n(s,\eta)d\eta\\
&+\int_0^{+\infty} ds\int_{-\frac{\ell}{2}}^{\frac{\ell}{2}}g^{\sharp}(s,\cdot-\eta)(K_n(s,\eta)-K(s,\eta))d\eta
=:{\mathcal A}_{1,n}+{\mathcal A}_{2,n}\end{aligned}$$ for every $n\in{{\mathbb N}}$ and observe that $$\begin{aligned}
\|{\mathcal A}_{1,n}\|_{\infty}\le \int_0^{+\infty}\|g_n(s,\cdot)-g(s,\cdot)\|_{L^2((-\ell/2,\ell/2))}\|K_n(s,\cdot)\|_{L^2((-\ell/2,\ell/2))}ds,\qquad\;\,n\in{{\mathbb N}}.\end{aligned}$$ Since $(i)$ $\|g_n(0,\cdot)-g(0,\cdot)\|_{L^2((-\ell/2,\ell/2);{{\mathbb C}})}$ vanishes as $n\to +\infty$, $(ii)~\|g_n(x,\cdot)-g(x,\cdot)\|_{L^2((-\ell/2,\ell/2);{{\mathbb C}})}\le 2\|g(x,\cdot)\|_{L^2((-\ell/2,\ell/2);{{\mathbb C}})}\le 2\ell\|g\|_{\infty}$, for $x\ge 0$ and $n\in{{\mathbb N}}$, and $(iii)~|K_n|\le 2|K|$ in ${{\mathbb R}}_+\times{{\mathbb R}}$ for every $n\in{{\mathbb N}}$, the dominated convergence theorem shows that ${\mathcal A}_{1,n}$ converges to zero pointwise in ${{\mathbb R}}$ as $n$ tends to $+\infty$. Moreover, $\|{\mathcal A}_{1,n}\|_{\infty}\le c\|g\|_{\infty}$. That theorem also shows that ${\mathcal A}_{2,n}$ converges to zero pointwise in ${{\mathbb R}}$ as $n$ tends to $+\infty$; moreover, $\|{\mathcal A}_{2,n}\|_{\infty}\le c\|g\|_{\infty}$ for each $n\in{{\mathbb N}}$. Now, writing $$\begin{aligned}
\zeta_n(y)=\zeta_n(0)+\int_0^y\bigg(\mathcal I_0(r)+\sum_{h=1}^3{\mathcal I}_{h,n}(r)\bigg)dr,\qquad\;\,y\in{{\mathbb R}},\;\,n\in{{\mathbb N}},\end{aligned}$$ and letting $n$ tend to $+\infty$, again by dominated convergence we conclude that $$\begin{aligned}
\zeta'=&\frac{2\pi i}{\ell^2}\sum_{|k|<k_0}\frac{k}{Z_k}e_k\int_0^{+\infty}e^{\frac{{\rm Le}}{2}s}e^{-\frac{1}{2}Z_ks}\hat g_k(s)ds
+\frac{2\pi i}{\ell^2}\sum_{|k|\ge k_0}\left (\frac{k}{Z_k}-\frac{\ell}{\pi}\right )e_k\int_0^{+\infty}e^{\frac{{\rm Le}}{2}s}e^{-\frac{1}{2}Z_ks}\hat g_k(s)ds\\
&+\frac{2i}{\ell}\sum_{|k|\ge k_0}e_k\int_0^{+\infty}e^{\frac{{\rm Le}}{2}s}\Big (e^{-\frac{1}{2}Z_ks}-e^{-\frac{\pi k}{2\ell }s}\Big )\hat g_k(s)ds+\frac{i\ell}{\pi}\int_0^{+\infty}ds\int_{-\frac{\ell}{2}}^{\frac{\ell}{2}}K(s,\eta)g^{\sharp}(s,\cdot-\eta)d\eta.\end{aligned}$$ Denote by $\phi_1,\ldots,\phi_4$ the four terms in the right-hand side of the previous formula. Clearly, $\phi_1$ belongs to $C^{\infty}_b({{\mathbb R}};{{\mathbb C}})$. In particular, $\|\phi_1\|_{C^{1+\alpha}_b({{\mathbb R}};{{\mathbb C}})}\le c\|g\|_{\infty}$. As far as $\phi_2$ and $\phi_3$ are concerned, using , , the same arguments here above and in the first part of the proof of Lemma \[finito\], it can be easily shown that such functions belong to $C^{1+\alpha}_b({{\mathbb R}};{{\mathbb C}})$ and $\|\phi_2\|_{C^{1+\alpha}_b({{\mathbb R}};{{\mathbb C}})}+\|\phi_3\|_{C^{1+\alpha}_b({{\mathbb R}};{{\mathbb C}})}\le c\|g\|_{\infty}$.
The function $\phi_4$ is the limit in $C_b({{\mathbb R}};{{\mathbb C}})$ of the sequence $(\phi_{4,n})$ defined by $$\begin{aligned}
\phi_{4,n}=\frac{i\ell}{\pi}\int_{\frac{1}{n}}^{+\infty}ds\int_{-\frac{\ell}{2}}^{\frac{\ell}{2}}K(s,\eta)g^{\sharp}(s,\cdot-\eta)d\eta,\qquad\;\,n\in{{\mathbb N}}.\end{aligned}$$ Clearly, each function $\phi_{4,n}$ is continuously differentiable in ${{\mathbb R}}$ and $$\begin{aligned}
\phi_{4,n}'=\frac{i\ell}{\pi}\int_{\frac{1}{n}}^{+\infty}ds\int_{-\frac{\ell}{2}}^{\frac{\ell}{2}}K_y(s,\eta)g^{\sharp}(s,\cdot-\eta)d\eta,\end{aligned}$$ where $$\begin{aligned}
K_y(x,y)=&\frac{2\pi i}{\ell}(k_0-1)e^{\left (\frac{\rm Le}{2}-\frac{\pi(k_0-1)}{2\ell}\right )x}\bigg (
\frac{e^{\frac{2\pi (k_0-1)i}{\ell}y}}{e^{\frac{\pi}{2\ell}(x+4iy)}-1}-\frac{e^{-\frac{2\pi (k_0-1)i}{\ell}y}}{e^{\frac{\pi}{2\ell}(x-4iy)}-1}\bigg )\notag\\
&+\frac{2\pi i}{\ell}e^{\left (\frac{\rm Le}{2}-\frac{\pi(k_0-1)}{2\ell}\right )x}\bigg (
\frac{e^{\frac{\pi}{2\ell}(x-4ik_0y)}}{(e^{\frac{\pi}{2\ell}(x-4iy)}-1)^2}-
\frac{e^{\frac{\pi}{2\ell}(x+4ik_0y)}}{(e^{\frac{\pi}{2\ell}(x+4iy)}-1)^2}\bigg )=:L_1(x,y)+L_2(x,y)
\label{presepe}\end{aligned}$$ for $(x,y)\in \overline{H_0^+}$. Since $K$ is $\ell$-periodic with respect to $y$, it follows that $\int_{-\ell/2}^{\ell/2}K_y(s,\eta)d\eta=0$. Hence, we can write $$\begin{aligned}
\phi_{4,n}'(y)=\frac{i\ell}{\pi}\int_{\frac{1}{n}}^{+\infty}ds\int_{-\frac{\ell}{2}}^{\frac{\ell}{2}}K_y(s,\eta)(g^{\sharp}(s,y-\eta)-g(s,y))d\eta,\qquad\;\,
y\in{{\mathbb R}},\;\,n\in{{\mathbb N}}.\end{aligned}$$ By assumptions, $g\in C^{\alpha}_b(S_0^+;{{\mathbb C}})$ and this allows us to estimate $$\begin{aligned}
|K_y(s,\eta)(g^{\sharp}(s,y-\eta)-g^{\sharp}(s,y))|\le c\min\{(s^2+\eta^2)^{\frac{\alpha}{2}-1},(e^{\frac{\pi}{2\ell}|s|}-1)^{-1}\}\|g\|_{C^{\alpha}_b([0,+\infty)\times{{\mathbb R}};{{\mathbb C}})}\end{aligned}$$ for every $(s,\eta)\in{{\mathbb R}}\times{{\mathbb R}}_+$ and $y\ge 0$. Thus, we can let $n$ tend to $+\infty$ in and conclude that $\phi_{4,n}'$ converges uniformly in ${{\mathbb R}}^2$. As a byproduct, $\phi_4$ is continuously differentiable in ${{\mathbb R}}$, $$\begin{aligned}
\phi_4'=\int_0^{+\infty}ds\int_{-\frac{\ell}{2}}^{\frac{\ell}{2}}K_y(s,\eta)(g^{\sharp}(s,\cdot-\eta)-g(s,\cdot))d\eta\end{aligned}$$ and $\|\phi_4'\|_{\infty}\le c\|g\|_{C^{\alpha}_b(S^+_0;{{\mathbb C}})}$.
To prove that $\phi_4'$ belongs to $C^{\alpha}_b({{\mathbb R}};{{\mathbb C}})$ we split $$\begin{aligned}
\phi_4'=&\int_0^{+\infty}ds\int_{-\frac{\ell}{2}}^{\frac{\ell}{2}}(L_1(s,\eta)+L_2(s,\eta)\chi_{(1,+\infty)}(s))(g^{\sharp}(s,\cdot-\eta)-g(s,\cdot))d\eta\notag\\
&+\int_0^1ds\int_{-\frac{\ell}{2}}^{\frac{\ell}{2}}L_2(s,\eta)(g^{\sharp}(s,\cdot-\eta)-g(s,\cdot))d\eta.
\label{alba}\end{aligned}$$ Since the function $L_1+L_2\chi_{(1,+\infty)}\in L^1(S^+_0;{{\mathbb C}})$, the first term in the right-hand side of belongs to $C^{\alpha}_b({{\mathbb R}};{{\mathbb C}})$ and its $C^{\alpha}_b({{\mathbb R}};{{\mathbb C}})$-norm can be estimated from above by $c\|g\|_{C^{\alpha}_b(S_0^+;{{\mathbb C}})}$. To estimate the other term, which we denote by $\Psi$, we observe that $$\begin{aligned}
\frac{e^{\frac{\pi}{2\ell}(x-4ik_0y)}}{(e^{\frac{\pi}{2\ell}(x-4iy)}-1)^2}=\frac{\ell^2}{\pi^2}\frac{4x^2+32ixy-64y^2}{(x^2+16y^2)^2}
+\psi(x,y),\qquad (x,y)\in (0,1)\times (-\ell/2,\ell/2),\end{aligned}$$ for some function $\psi\in L^1((0,1)\times (-\ell/2,\ell/2);{{\mathbb C}})$. Thus, $$\begin{aligned}
\Psi(y)=&-\frac{128i\ell^2}{\pi^2}\int_0^1ds\int_{-\frac{\ell}{2}}^{\frac{\ell}{2}}\frac{s(y-\eta)}{(s^2+16(y-\eta)^2)^2}(g^{\sharp}(s,\eta)-g(s,y))d\eta\\
&-2\int_0^1ds\int_{-\frac{\ell}{2}}^{\frac{\ell}{2}}(\psi(s,\eta)-\psi(s,-\eta))(g^{\sharp}(s,y-\eta)-g(s,y))d\eta=:\Psi_1(y)+\Psi_2(y)\end{aligned}$$ for every $y\in{{\mathbb R}}$. The function $\Psi_2$ is clearly $\alpha$-Hölder continuous in $[-\ell/2,\ell/2]$ since $\psi\in L^1((0,1)\times (-\ell/2,\ell/2);{{\mathbb C}})$. Moreover, $\|\Psi_2\|_{C^{\alpha}_b({{\mathbb R}};{{\mathbb C}})}\le c\|g\|_{C^{\alpha}_b({{\mathbb R}};{{\mathbb C}})}$. As far as the function $\Psi_1$ is concerned, we approximate it with the family of functions $\Psi_{1,h}$ defined by $$\begin{aligned}
\Psi_{1,h}(y)=-\frac{128i\ell^2}{\pi^2}\int_0^1ds\int_{-\frac{\ell}{2}}^{\frac{\ell}{2}}\frac{s(y-\eta)}{(s^2+16(y-\eta)^2+h^2)^2}(g(s,\eta)-g(s,\cdot))d\eta,\qquad\;\,h>0.\end{aligned}$$ Each of these functions is continuously differentiable in ${{\mathbb R}}$ with bounded derivative, so that, we can estimate $$\begin{aligned}
|\Psi_1(y_2)-\Psi_1(y_1)|\le &|\Psi_1(y_2)-\Psi_{1,h}(y_2)|+|\Psi_{1,h}(y_2)-\Psi_{1,h}(y_1)|+|\Psi_{1,h}(y_1)-\Psi_1(y_1)|\notag\\
\le &2\|\Psi_1-\Psi_{1,h}\|_{\infty}+\|\Psi_{1,h}'\|_{\infty}|y_2-y_1|
\label{swear}\end{aligned}$$ for $y_1,y_2\in [-\ell/2,\ell/2]$. Note that $$\begin{aligned}
\|\Psi_1-\Psi_{1,h}\|_{\infty}\le &\frac{256\ell^2}{\pi^2}\|g\|_{C^{\alpha}_b(S^+_0;{{\mathbb C}})}\int_{{{\mathbb R}}^2}\frac{|s\eta|h^2}{(s^2+16\eta^2+h^2)^2(s^2+16\eta^2)}dsd\eta\\
\le & ch^2\|g\|_{C^{\alpha}_b(S^+_0;{{\mathbb C}})}\int_0^{+\infty}\frac{\rho^{\alpha+1}}{(\rho^2+h^2)^2}d\rho= ch^{\alpha}\|g\|_{C^{\alpha}_b(S^+_0;{{\mathbb C}})}\end{aligned}$$ and $$\begin{aligned}
\|\Psi'_{1,h}\|_{\infty}\le & \frac{128\ell^2}{\pi^2}\|g\|_{C^{\alpha}_b(S_0^+;{{\mathbb C}})}\int_{{{\mathbb R}}^2}
\frac{(s^2+48\eta^2+h^2)|\eta|^{\alpha}}{(s^2+16\eta^2+h^2)^3}dsd\eta\\
\le &c\|g\|_{C^{\alpha}_b(S_0^+;{{\mathbb C}})}\int_0^{+\infty}\frac{\rho^{2+\alpha}}{(\rho^2+h^2)^2}d\rho\le c\|g\|_{C^{\alpha}_b(S_0^+;{{\mathbb C}})}h^{\alpha-1}.\end{aligned}$$ Replacing these inequalities into and taking $h=|y_2-y_1|$, we conclude that $$\begin{aligned}
|\Psi_1(y_2)-\Psi_1(y_1)|\le c\|g\|_{C^{\alpha}_b(S^+_0;{{\mathbb C}})}|y_2-y_1|^{\alpha}.\end{aligned}$$ Therefore, $\phi_4'\in C^{\alpha}_b({{\mathbb R}};{{\mathbb C}})$ and $\|\phi_4'\|_{C^{\alpha}_b({{\mathbb R}};{{\mathbb C}})}\le c\|g\|_{C^{\alpha}_b(S^+_0;{{\mathbb C}})}$. Putting everything together it follows that $\zeta\in C^{2+\alpha}_b({{\mathbb R}};{{\mathbb C}})$ and $\|\zeta\|_{C^{2+\alpha}_b({{\mathbb R}};{{\mathbb C}})}\le c\|g\|_{C^{\alpha}_b(S^+_0;{{\mathbb C}})}$.
Finally, we consider the function ${\mathscr S}_{\lambda}g-\zeta=:v\in C_b([0,+\infty)\times{{\mathbb R}};{{\mathbb C}})\cap\bigcap_{p<+\infty} W^{2,p}_{\rm loc}((0,+\infty)\times{{\mathbb R}};{{\mathbb C}})$. Since ${\rm Le}^{-1}\Delta v+v_x\in C^{\alpha}_b((0,+\infty)\times{{\mathbb R}};{{\mathbb C}})$ and by construction $v(0,\cdot)=0$, by classical results (see e.g., [@krylov]) $v\in C^{2+\alpha}_b((0,+\infty)\times{{\mathbb R}})$ and $$\begin{aligned}
\|v\|_{C^{2+\alpha}_b(H^+_0;{{\mathbb C}})}\le &c(\|v\|_{\infty}+
\|{\rm Le}^{-1}\Delta v+v_x\|_{C_b^{\alpha}(H^+_0)}+\|\zeta\|_{C^{2+\alpha}_b({{\mathbb R}};{{\mathbb C}})})\\
\le & c(\|v\|_{\infty}+\|\lambda {\mathscr S}_{\lambda}g-g\|_{C^{\alpha}_b(H^+_0;{{\mathbb C}})}
+\|\zeta\|_{C^{2+\alpha}_b({{\mathbb R}};{{\mathbb C}})})\le c\|g\|_{C^{\alpha}_b(H^+_0;{{\mathbb C}})}.\end{aligned}$$ Formula follows as once.
\[finiremo\] For each $\lambda\in {{\mathbb C}}$ such that ${\rm Re}\lambda>-({\rm Im}\lambda)^2$, $f\in C_b(\overline{S};{{\mathbb C}})$ and $g\in C_b(\overline{S_0^+};{{\mathbb C}})$, we denote by ${\mathscr T}_{\lambda}^{\pm}f:\overline{H_R^{\mp}}\to{{\mathbb C}}$ and ${\mathscr U}_{\lambda}g:\overline{H_0^+}\to{{\mathbb C}}$, respectively, the functions defined by $$\begin{aligned}
&({\mathscr T}_{\lambda}^{\pm}f)(x,y)= e^{-\frac{{\rm Le}}{2}(x-R)}\sum_{k\in{{\mathbb Z}}}e^{\pm \frac{1}{2}Y_k(x-R)}\mathcal F_{k,1}f(R)e_k(y),\qquad\;\,(x,y)\in \overline{H_R^{\mp}},\\[1mm]
&({\mathscr U}_{\lambda}g)(x,y)=e^{-\frac{\rm Le}{2}x}\sum_{k\in{{\mathbb Z}}}e^{-\frac{Y_k}{2}x}({\mathcal G}_kg)(0)e_k(y),\qquad\;\,(x,y)\in \overline{H_0^+}.\end{aligned}$$ Then, the following properties are satisfied.
1. ${\mathscr T}_{\lambda}^{\pm}f$ belongs to $C^1_b(\overline{H_R^{\mp}};{{\mathbb C}})\cap W^{2,p}_{\rm loc}(H_R^{\mp};{{\mathbb C}})$, for each $p<+\infty$, solves the equation $\lambda u-{\rm Le}^{-1}\Delta u-u_x=0$ in $H_R^{\mp}$ and, for each $M>0$, there exists a constant $c_M>0$ such that $$\begin{aligned}
\label{pasta}
|\lambda|\|{\mathscr T}_{\lambda}^{\pm}f\|_{\infty}
+\sqrt{|\lambda|}\|\nabla{\mathscr T}_{\lambda}^{\pm}f\|_{\infty}
\leq c_M\|f\|_{\infty},\qquad\;\,|\lambda|\ge M;\end{aligned}$$
2. $\displaystyle\lim_{x\to +\infty}({\mathscr T}_{\lambda}^{-}f)(x,y)=\displaystyle\lim_{x\to -\infty}({\mathscr T}_{\lambda}^+f)(x,y)=0$ for every $y\in{{\mathbb R}}$ and $f\in C_b(\overline{S};{{\mathbb C}})$;
3. if $f\in C^{\alpha}_b(S;{{\mathbb C}})$ and $f(\cdot,-\ell/2)=f(\cdot,-\ell/2)$, then the function ${\mathscr T}_{\lambda}^{\pm}f$ belongs to $C_b^{2+\alpha}(\overline{H^{\mp}_R};{{\mathbb C}})$ and $\|\mathscr T^{\pm}_{\lambda}f\|_{C_b^{2+\alpha}(H_R^{\mp})}\leq c\|f\|_{C^{\alpha}_b(S;{{\mathbb C}})}$ for each ${\rm Re}\lambda>-({\rm Im}\lambda)^2$;
4. ${\mathscr U}_{\lambda}g$ belongs to $C^1_b(\overline{H_0^+};{{\mathbb C}})\cap W^{2,p}_{\rm loc}(H_0^+;{{\mathbb C}})$, for each $p<+\infty$, solves the equation $\lambda u-{\rm Le}^{-1}\Delta u-D_xu=0$ and, for every $M>0$ there exists a positive constant $c_M'$ such that $$\begin{aligned}
|\lambda|\|{\mathscr U}_{\lambda}g\|_{\infty}
+\sqrt{|\lambda|}\|\nabla{\mathscr U}_{\lambda}g\|_{\infty}\leq c_M'\|g\|_{\infty},\qquad\;\,|\lambda|\ge M;\end{aligned}$$
5. $\displaystyle\lim_{x\to +\infty}({\mathscr U}_{\lambda}^{-}g)(x,y)=0$ for each $y\in{{\mathbb R}}$ and $g\in C_b(\overline{S_0^+};{{\mathbb C}})$;
6. if $g\in C^{\alpha}_b(S_0^+;{{\mathbb C}})$ is such that $g(\cdot,-\ell/2)=g(\cdot,-\ell/2)$, then the function ${\mathscr U}_{\lambda}g$ belongs to $C_b^{2+\alpha}(H^+_0;{{\mathbb C}})$ and $\|\mathscr U_\lambda g\|_{C_b^{2+\alpha}(H^+_0;{{\mathbb C}})}\leq c\|g\|_{C^{\alpha}_b(S_0^+;{{\mathbb C}})}$ for each $\lambda\in{{\mathbb C}}$ such that ${\rm Re}\lambda>-({\rm Im}\lambda)^2$.
\(i) The arguments as in the proof of Lemma \[finito\] show that the function ${\mathscr T}_{\lambda}^{\pm}f$ is continuous in $\overline{H_R^{\mp}}$ and smooth in its interior, where it solves the equation $\lambda u-{\rm Le}^{-1}\Delta u-u_x=0$. Further, the function $v^{\pm}={\mathscr T}_{\lambda}^{\pm}f-{\mathscr R}_{\lambda,1}f$ is bounded, vanishes on $\{R\}\times{{\mathbb R}}$ and $\lambda v^{\pm}-{\rm Le}^{-1}\Delta v^{\pm}-v^{\pm}_x=h$ in $H_R^{\mp}$, where $L^{\infty}(H_R^{\mp};{{\mathbb C}})\ni h=-{\rm Le}^{-1}f^{\sharp}+({\rm Le}^{-1}-1)(\lambda{\mathscr R}_{\lambda,1} f-D_x{\mathscr R}_{\lambda,1}f)$.
By classical results, the realization of the operator ${\rm Le}^{-1}\Delta+D_x$ in $L^{\infty}(H_R^{\mp};{{\mathbb C}})$ with homogeneous Dirichlet boundary conditions generates an analytic semigroup with domain $\{u\in C_b(\overline{H_R^{\mp}};{{\mathbb C}})\cap\bigcap_{p<+\infty}W^{2,p}_{\rm loc}(H_R^{\mp};{{\mathbb C}}): {\rm Le}^{-1}\Delta u+D_x u\in L^{\infty}(H_R^{\mp};{{\mathbb C}})\}$. In particular, for every $\lambda\in\Sigma_0$ it holds that $|\lambda|\|v^{\pm}\|_{\infty}+\sqrt{|\lambda|}\|\nabla v^{\pm}\|_{\infty}\le c\|h\|_{\infty}$. From the definition of $v^{\pm}$ and taking into account, estimate follows immediately.
\(ii) The proof of this property is immediate since the series defining ${\mathscr T}_{\lambda}^+f$ (resp. ${\mathscr T}_{\lambda}^-f$) converges uniformly in $\overline{H_R^-}$ (resp. in $\overline{H_R^+}$) and each of its terms vanishes as $x\to -\infty$ (resp. $x\to +\infty$), uniformly with respect to $y\in{{\mathbb R}}$.
\(iii) Fix $\lambda\in{{\mathbb C}}$ with ${\rm Re}\lambda>-({\rm Im}\lambda)^2$. Since $f\in C^\alpha_b(S)$ and $f(\cdot,-\ell/2)=f(\cdot,\ell/2)$, thanks to Lemma \[finito\]$(iii)$ we can infer that the function $h\in C_b^{\alpha}({{\mathbb R}}^2;{{\mathbb C}})$. Hence, ${\rm Le}^{-1}\Delta v^{\pm}+v^{\pm}_x\in C_b^\alpha(H_R^{\mp};{{\mathbb C}})$ and by classical results it follows that $v\in C_b^{2+\alpha}(H_R^{\mp};{{\mathbb C}})$ and $$\begin{aligned}
\|v^{\pm}\|_{C_b^{2+\alpha}(H_R^{\mp};{{\mathbb C}})}\leq c(\|v^{\pm}\|_{\infty}+\|{\rm Le}^{-1}\Delta v^{\pm}+v^{\pm}_x\|_{C_b^\alpha(H_R^{\mp};{{\mathbb C}})}).\end{aligned}$$ From the definition of $v^{\pm}$ and the above estimate, the assertion follows at once.
(iv)-(vi) The proof of these three properties follows applying the procedure of the first part of the proof, with ${\mathscr R}_{\lambda,1}f $ being replaced by the function ${\mathscr S}_{\lambda}g$. The details are left to the reader.
Analytic semigroups and interpolation spaces
--------------------------------------------
To state the main result of this subsection, for each $k\in{{\mathbb N}}\cup\{0\}$ we introduce the functions (the so-called dispersion relations) $$\begin{aligned}
&{\mathcal D}_k(\lambda)=({\rm Le}-Y_k)\bigg [\exp\bigg (\frac{R}{2}({\rm Le}-1-X_k-Y_k)\bigg )-1+\theta_iRX_k\bigg ],\\[1mm]
&\widetilde {\mathcal D}_k(\lambda)=({\rm Le}-Y_k)\bigg [\exp\bigg (\frac{R}{2}({\rm Le}-1-X_k-Y_k)\bigg )-\exp\bigg (\frac{R}{2}({\rm Le}-1+X_k-Y_k)\bigg )+\theta_i R X_k\bigg ],\end{aligned}$$ where $X_k=X_k(\lambda)=\sqrt{1+4\lambda+4\lambda_k}$, $Y_k=Y_k(\lambda)=\sqrt{{\rm Le}^2+4\lambda {\rm Le}+4\lambda_k}$, $\lambda_k=4\pi^2k^2\ell^{-2}$, and the sets $$\begin{aligned}
&\Omega_k=\{\lambda\in{{\mathbb C}}: {\rm Re}\lambda\ge -({\rm Im}\lambda)^2-\lambda_k\,{\rm and}\,{\mathcal D}_k(\lambda)=0\},
\label{omega-k}
\\
&\Omega_k'=\{\lambda\in{{\mathbb C}}: -{\rm Le}^{-1}(({\rm Im}\lambda)^2+\lambda_k)<{\rm Re}\lambda<-({\rm Im}\lambda)^2-\lambda_k,\,{\rm and}\,\widetilde {\mathcal D}_k(\lambda)=0\}.\notag\end{aligned}$$
\[banca\] The realization $L$ of the operator ${\mathscr L}$ in $\boldsymbol{\mathcal X}$, with domain $$\begin{aligned}
D(L)\!=\!\bigg\{&{\bf u}\in \boldsymbol{\mathcal X}: u_j(\cdot,-\ell/2)=u_j(\cdot,\ell/2), j=1,2,\, u_1^{\sharp}\in C^1_b(\overline{H_0^-};{{\mathbb C}})\cap \bigcap_{p<+\infty}W^{2,p}_{\rm loc}(H_0^{-};{{\mathbb C}})\\
&~u_1^{\sharp},u_2^{\sharp}\!\in\! C^1([0,R]\times{{\mathbb R}};{{\mathbb C}})\!\cap\! C^1_b(\overline{H_R^+};{{\mathbb C}})\!\cap\!\!\bigcap_{p<+\infty}\!\!W^{2,p}_{\rm loc}(({{\mathbb R}}_+\!\setminus\!\{R\})\!\times\!{{\mathbb R}};{{\mathbb C}}),\;{\mathscr L}{\bf u}\!\in\!\boldsymbol{\mathcal X},\,{\mathscr B}{{\bf u}}={\bf 0}\bigg\},\end{aligned}$$ $($see and $)$ generates an analytic semigroup in $\boldsymbol{\mathcal X}$. Moreover,
1. the spectrum $\sigma(L)$ of $L$ is the set $\{\lambda\in{{\mathbb C}}:{\rm Re}\lambda\le -{{\rm{{Le}}}}^{-1}({\rm Im}\lambda)^2\}\cup\bigcup_{k\in{{\mathbb N}}\cup\{0\}}(\Omega_k\cup\Omega_k')$;
2. there exist two positive constants $M$ and $c$ such that $\sqrt{|\lambda|}\|\nabla R(\lambda,L){{\bf f}}\|_{\infty}\le c\|{{\bf f}}\|_{\infty}$ for $\lambda\in{{\mathbb C}}$ with ${\rm Re}\lambda\ge M$ and ${{\bf f}}\in\boldsymbol{\mathcal X}$;
3. if ${{\bf f}}\in \boldsymbol{\mathcal X}_{\alpha}$ for some $\alpha\in (0,1)$, then for each $\lambda\in\rho(L)$ the function $R(\lambda,L){\bf f}$ belongs to $\boldsymbol{\mathcal X}_{2+\alpha}$ and $\|R(\lambda,L){\bf f}\|_{2+\alpha}\leq c_{\lambda}\|{\bf f}\|_{\alpha}$.
Since it is rather long, we split the proof into four steps. In Steps 1 and 2, we characterize $\sigma(L)$, whereas in Step 3 we prove that $L$ generates an analytic semigroup in $\boldsymbol{\mathcal X}$ as well as the estimate for the spatial gradient for the resolvent operator. Finally, in Step 4, we prove property (iii).
[*Step 1.*]{} Fix ${\bf f}\in\boldsymbol{\mathcal X}$ and $\lambda\in{{\mathbb C}}$ such that ${\rm Re}\lambda>-({\rm Im}\lambda)^2$ and $\lambda\not\in\Omega_k$ for each $k\in{{\mathbb N}}\cup\{0\}$, and assume that the equation $\lambda{{\bf u}}-{\mathscr L}{{\bf u}}={{\bf f}}$ admits a solution ${{\bf u}}=(u_1,u_2)$ in $D(L)$. The arguments in the proof of Lemma \[finito\](i) show that for every $k\in{{\mathbb Z}}$ the functions $\hat u_{1,k}$ and $\hat u_{2,k}$ (see Subsection \[sub-notation\]), solve, respectively, the differential equations $\lambda \hat u_k-\hat u_k'-\hat u''_k+\lambda_k\hat u_k=\hat f_{1,k}$ in ${{\mathbb R}}\setminus\{0,R\}$ and $\lambda \hat v_k-\hat v'_k-{\rm Le}^{-1}\hat v_k''+{\rm Le}^{-1}\lambda_k\hat v_k=\hat f_{2,k}$ in ${{\mathbb R}}_+\setminus\{R\}$. Moreover, they belong to $C^1_b({{\mathbb R}};{{\mathbb C}})$ and $C^1_b([0,+\infty);{{\mathbb C}})$, respectively. Thus, $$\begin{aligned}
\hat u_{1,k}(x)=({\mathcal F}_{k,1}f_1)(x)+c_{1,k}e^{\nu^{+}_kx}\chi_{(-\infty,0]}(x)+
(c_{2,k}e^{\nu^{-}_kx}+c_{3,k}e^{\nu^{+}_kx})\chi_{(0,R)}(x)+
c_{4,k}e^{\nu^{-}_kx}\chi_{[R,+\infty)}(x)\end{aligned}$$ for every $x\in{{\mathbb R}}$, $k\in{{\mathbb Z}}$ and $$\begin{aligned}
\hat u_{2,k}(x)=({\mathcal G}_kf_2)(x)+(d_{1,k}e^{\mu^{-}_kx}+d_{2,k}e^{\mu^{+}_kx})\chi_{[0,R)}(x)+d_{3,k}e^{\mu^{-}_kx}\chi_{[R,+\infty)}(x)
\label{u2k}\end{aligned}$$ for every $x\ge 0$ and $k\in{{\mathbb Z}}$, where $\mu^{\pm}_k=-\frac{{\rm Le}}2\pm\frac{1}{2}Y_k$, $\nu^{\pm}_k = -\frac{1}{2}\pm\frac{1}{2}X_k$ and $c_{1,k},c_{2,k},c_{3,k},c_{4,k}$, $d_{1,k},d_{2,k},d_{3,k}$ ($k\in{{\mathbb Z}}$) are complex constants determined imposing the conditions ${\mathscr B}(\hat u_{1,k},\hat u_{2,k})={\bf 0}$ that the infinitely many functions $\hat u_{1,k}$ and $\hat u_{2,k}$ have to satisfy. It turns out that the above constants are uniquely determined if and only if $\mathcal D_k(\lambda)\neq 0$, as we are assuming, and as a byproduct, $$\begin{aligned}
&u_1={\mathscr R}_{\lambda,1} f_1+\frac{1}{\ell}\sum_{k\in{{\mathbb Z}}}p_{1,k}({\mathcal F}_{k,1}f_1)(R)e_k+\frac{2}{{\rm Le}\, \ell}\sum_{k\in{{\mathbb Z}}}p_{2,k}({\mathcal G}_kf_2)(0)e_k,\qquad
{\rm in}~\overline{H_0^-},
\label{piove}\\[2mm]
&u_1={\mathscr R}_{\lambda,1} f_1+\frac{1}{\ell}\sum_{k\in{{\mathbb Z}}}p_{3,k}({\mathcal F}_{k,1}f_1)(R)e_k
+\frac{2}{{\rm Le}\,\ell}\sum_{k\in{{\mathbb Z}}}p_{4,k}({\mathcal G}_kf_2)(0)e_k,\label{piove-1} \\[2mm]
&u_2={\mathscr S}_{\lambda}f_2-\frac{{\rm Le}}{2\theta_iR\ell}{\mathscr T}_{\lambda}^{+}f_1+\frac{1}{\ell}{\mathscr U}_{\lambda}^-f_2
+\frac{{\rm Le}}{2\ell}\sum_{k\in{{\mathbb Z}}}q_{1,k}({\mathcal F}_{k,1}f_1)(R)e_k+\frac{1}{\ell}\sum_{k\in{{\mathbb Z}}}q_{2,k}({\mathcal G}_kf_2)(0)e_k,\label{vento}\end{aligned}$$ in $(0,R)\times{{\mathbb R}}$, and $$\begin{aligned}
&u_1={\mathscr R}_{\lambda,1} f_1+\frac{1}{\ell}\sum_{k\in{{\mathbb Z}}}p_{5,k}({\mathcal F}_{k,1}f_1)(R)e_k+\frac{2\theta_iR}{{\rm Le}\,\ell}\sum_{k\in{{\mathbb Z}}}p_{6,k}({\mathcal G}_kf_2)(0)e_k,\label{sole-0}\\[2mm]
&u_2={\mathscr S}_{\lambda}f_2+\frac{{\rm Le}}{2\theta_iR\ell}{\mathscr T}_{\lambda}^{-}f_1+\frac{{\rm Le}}{2\ell}\sum_{k\in{{\mathbb Z}}}q_{3,k}({\mathcal F}_{k,1}f_1)(R)e_k+\frac{1}{\ell}\sum_{k\in{{\mathbb Z}}}q_{4,k}({\mathcal G}_kf_2)(0)e_k, \label{sole}\end{aligned}$$ in $\overline{H_R^+}$, where ${\mathscr R}_{\lambda,1}f$ and ${\mathscr S}_{\lambda}g$ have been introduced in Lemmata \[finito\] and \[finire\], $$\begin{aligned}
&p_{1,k}(x)=\frac{e^{-\nu_k^{+}R}-e^{-\mu_k^{+}R}}{W_k}e^{\nu_k^+x},\qquad\;\;\;\, p_{2,k}(x)=\frac{(\theta_iRX_k-1+e^{-X_kR})Y_k}{X_kW_k(Y_k-{\rm Le})}e^{\nu_k^{+}x}\\
&p_{3,k}(x)=\frac{e^{\nu_k^{+}(x-R)}-e^{\nu_k^{-}x-\mu_k^{+}R}}{W_k},\qquad\;\,p_{4,k}(x)=\frac{Y_k[(\theta_iRX_k-1)e^{\nu^{-}_kx}+e^{-X_kR}e^{\nu^{+}_kx}]}{X_kW_k(Y_k-{\rm Le})},\\
&p_{5,k}(x)=\frac{e^{-\nu_k^{-}R}-e^{-\mu_k^{+}R}}{W_k}e^{\nu^{-}_kx},\qquad\;\;\;\, p_{6,k}(x)=\frac{Y_k}{(Y_k-{\rm Le})W_k}e^{\nu^{-}_kx},\\
&q_{1,k}(x)=\frac{e^{(\nu_k^{-}-\mu_k^+)R}-1}{\theta_iRW_k}e^{\mu_k^{+}(x-R)}-\frac{X_ke^{-\mu_k^{+}R}}{Y_kW_k}[({\rm Le}+Y_k)e^{\mu_k^{-}x}-{\rm Le}\,e^{\mu_k^{+}x}],\\
&q_{2,k}(x)=\frac{2{\rm Le}(\theta_iRX_k-1)}{W_k(Y_k-{\rm Le})}e^{\mu^{-}_kx}-\frac{e^{(\nu_k^{-}-\mu_k^+)R}}{W_k}(e^{\mu^{-}_kx}+e^{\mu^{+}_kx}),\\
&q_{3,k}(x)=\frac{1-e^{(\nu_k^{-}-\mu_k^{+})R}}{\theta_iRW_k}e^{\mu^{-}_k(x-R)}-\frac{{\rm Le}(e^{\mu^{-}_kx-\mu_k^{+}R}-e^{\mu_k^{-}(x-R)})X_k}{Y_kW_k}
-\frac{X_ke^{\mu^{-}_kx-\mu_k^{+}R}}{W_k},\\
&q_{4,k}(x)=\frac{Y_k(\theta_iRX_k-1)e^{\mu^{-}_kx}-{\rm Le}\, e^{\mu^{-}_kx+(\nu_k^{-}-\mu_k^{+})R}+(Y_k+{\rm Le})e^{\nu_k^{-}R+\mu_k^{-}(x-R)}}{W_k(Y_k-{\rm Le})}\end{aligned}$$ and $W_k=W_k(\lambda)=\theta_iRX_k-1+e^{(\nu^{-}_k-\mu_k^{+})R}$.
In view of Lemmata \[finito\] to \[finiremo\], to prove that the pair ${{\bf u}}$ defined by - belongs to $D(L)$ and $\lambda {\bf u}-\mathscr L{ \bf u}={\bf f}$ we just need to consider the series in the above formulae, which we denote, respectively by ${\mathscr P}_{\lambda,2k+j}f_j$ ($j=1,2$, $k=0,1,2$), ${\mathscr Q}_{\lambda,2h+j}f_j$ ($j=1,2$, $h=0,1$). To begin with, we observe that, by $(i)$ in the proof of Lemma \[finito\], we already know that ${\rm Re}(X_k)+{\rm Re}(Y_k)\ge c_{\lambda}|k|$ and $|X_k|+|Y_k|\le c_{\lambda}(|k|+1)$ for each $k\in{{\mathbb Z}}$. As a byproduct, taking also into account, we can infer that $|({\mathcal F}_{k,1}f_1)(R)|\le c_{\lambda}(1+k^2)^{-1}\|f_1\|_{\infty}$ and $|({\mathcal G}_kf_2)(0)|\le c_{\lambda}(1+k^2)^{-1}\|f_2\|_{\infty}$ for each $k\in{{\mathbb Z}}$. Moreover, we can also estimate $$\begin{aligned}
|W_k|&\ge \theta_iR|X_k|-1-e^{{\rm Re}(\nu_k^{-}-\mu_k^{+})R}\ge \theta_iR|X_k|-1-e^{\frac{{\rm Le}-1}{2}R}\ge c_{\lambda}|k|,\qquad\;\,k\in{{\mathbb Z}}\setminus\{0\}.
\label{aaa}\end{aligned}$$ Putting everything together, we conclude that $\|p_{1,k}\|_{C^h_b((-\infty,0];{{\mathbb C}})}\le c_1e^{-c_2k}$ for each $h\in{{\mathbb N}}$ and $$\begin{aligned}
&\|p_{2,k}\|_{C_b^j((-\infty,0];{{\mathbb C}})}+\sum_{i=1}^2[\|(p_{i+2,k},q_{i,k})\|_{C^j([0,R];{{\mathbb C}}^2)}+\|(p_{4+i,k},q_{i+2,k})\|_{C_b^j([R,+\infty);{{\mathbb C}}^2)}]
\le c_{\lambda}|k|^{j-1}\end{aligned}$$ for each $k\in{{\mathbb Z}}\setminus\{0\}$. Using these estimates, it is easy to check that ${\mathscr P}_{\lambda,1}f_1\in C^{\beta}_b(\overline{H^{-}_0};{{\mathbb C}})$ for $\beta>0$ and ${\mathscr P}_{\lambda,2}f_2\in C^{\infty}(H^{-}_0;{{\mathbb C}})\cap C^1_b(\overline{H^{-}_0};{{\mathbb C}})$. Moreover, they solve the equation $\lambda w-\Delta_xw-D_xw=0$ in $H^{-}_0$. Since the series which define ${\mathscr P}_{\lambda,1}f_1$ and ${\mathscr P}_{\lambda,2}f_2$ converge uniformly in $H^{-}_0$ and each term vanishes as $x\to -\infty$, uniformly with respect to $y\in{{\mathbb R}}$, we immediately infer that $\lim_{x\to -\infty}({\mathscr P}_{\lambda,1}f_1)(x,y)=\lim_{x\to -\infty}({\mathscr P}_{\lambda,1}f_2)(x,y)=0$ for each $y\in{{\mathbb R}}$. On the other hand, the functions ${\mathscr P}_{\lambda,3}f_1$, ${\mathscr P}_{\lambda,4}f_2$ and ${\mathscr Q}_{\lambda,1}f_1$, ${\mathscr Q}_{\lambda,2}f_2$ belong to $C^{\infty}((0,R)\times{{\mathbb R}};{{\mathbb C}})\cap C^1_b([0,R]\times{{\mathbb R}};{{\mathbb C}})$ and solve, in $(0,R)\times{{\mathbb R}}$, the equations $\lambda w_1-\Delta_xw_1-D_xw_1=0$ and $\lambda w_2-{\rm Le}^{-1}\Delta_xw_2-D_xw_2=0$, respectively. Finally, the functions ${\mathscr P}_{\lambda,5}f_1$, ${\mathscr P}_{\lambda,6}f_2$ and ${\mathscr Q}_{\lambda,3}f_1$, ${\mathscr Q}_{\lambda,4}f_2$ belong to $C^{\infty}(H^+_R;{{\mathbb C}})\cap C^1_b(\overline{H^+_R};{{\mathbb C}})$ solves, in $H^+_R$, the equations $\lambda w_1-\Delta_xw_1-D_xw_1=0$ and $\lambda w_2-{\rm Le}^{-1}\Delta_xw_2-D_xw_2=0$, respectively, and vanish as $x$ tends to $+\infty$ for each $y\in{{\mathbb R}}$. Therefore, the function ${{\bf u}}$ defined by - belongs to $\bigcap_{p<+\infty}W^{2,p}_{\rm loc}(({{\mathbb R}}\setminus\{0,R\})\times{{\mathbb R}})\times\bigcap_{p<+\infty}W^{2,p}_{\rm loc}(({{\mathbb R}}_+\setminus\{R\})\times{{\mathbb R}})$, solve the equation $\lambda{{\bf u}}-{\mathscr L}{{\bf u}}={{\bf f}}$ and $\lim_{x\to \pm\infty}u_1(x,y)=\lim_{x\to +\infty}u_2(x,y)=0$ for each $y\in{{\mathbb R}}$. Moreover, $\|\nabla{\bf u}\|_{\infty}\le c\|{{\bf f}}\|_{\infty}$. To conclude that ${{\bf u}}\in D(L)$, we have to check that ${\mathscr B}{{\bf u}}={\bf 0}$, but this is an easy task taking into account that all the series appearing in the definition of ${{\bf u}}$ may be differentiated term by term and ${\mathscr B}(\hat u_{1,k},\hat u_{2,k})={\bf 0}$ for every $k\in{{\mathbb Z}}$. We have so proved that ${{\bf u}}\in D(L)$ and that $$\begin{aligned}
\bigcup_{k\in{{\mathbb N}}\cup\{0\}}\Omega_k\subset\sigma(L)\subset \{\lambda\in{{\mathbb C}}:{\rm Re}\lambda\le -({\rm Im}\lambda)^2\}\cup\bigcup_{k\in{{\mathbb N}}\cup\{0\}}\Omega_k.\end{aligned}$$
[*Step 2.*]{} To complete the characterization of $\sigma(L)$, let us check that $\sigma(L)\supset\{\lambda\in{{\mathbb C}}:{\rm Re}\lambda\le -{\rm Le}^{-1}({\rm Im}\lambda)^2\}\cup\bigcup_{k\in{{\mathbb N}}\cup\{0\}}\Omega_k'$. Clearly, each $\lambda\in{{\mathbb C}}$ such that ${\rm Re}\lambda\le -{\rm Le}^{-1}({\rm Im}\lambda)^2$ belongs to $\sigma(L)$, since in this case $\nu_0^{\pm}$ and $\mu_0^{\pm}$ have nonpositive real parts so that the more general solution to the equation $\lambda{\bf u}-{\mathcal L}{\bf u}={\bf 0}$, which belongs to $\boldsymbol{\mathcal X}$ and is independent of $y$, is determined up to $8$ arbitrary complex constants and we have just $7$ boundary condition. Thus, the previous equation admits infinitely many solutions in $\boldsymbol{\mathcal X}$. Similarly, if $\lambda\in\Omega_k'$ for some $k\in{{\mathbb N}}\cup\{0\}$, then the pair ${{\bf u}}=(\hat u_{1,k}e_k,\hat u_{2,k}e_k)$, where $$\begin{aligned}
\hat u_{1,k}(x)=({\mathcal F}_{k,1}f_1)(x)\!+\!(c_{1,k}e^{\nu^{-}_kx}\!+\!c_{2,k}e^{\nu^{+}_kx})\chi_{(0,R)}(x)\!+\!
(c_{3,k}e^{\nu^{-}_kx}\!+\!c_{4,k}e^{\nu^{+}_kx}\chi_{[R,+\infty)}(x),\qquad\;\,x\in{{\mathbb R}},\end{aligned}$$ and $\hat u_{2,k}$ is still given by , is smooth, belongs to $\boldsymbol{\mathcal X}$ and solves the differential equation $\lambda{{\bf u}}-{\mathscr L}{{\bf u}}={\bf f}$ for every choice of $c_{i,k}$, $d_{j,k}$ ($i=1,\ldots,4$, $j=1,2,3$). Imposing the condition ${\mathscr B}{{\bf u}}={\bf 0}$, we get to a linear system of $7$ equations in $7$ unknowns whose determinant is $\widetilde {\mathcal D}_k(\lambda)$. Since $\lambda\in\Omega_k'$, the above equation admits infinitely many solutions in $D(L)$.
[*Step 3.*]{} Since the roots of the dispersion relation have bounded from above real part (see also the forthcoming computations), Step 1 shows that the resolvent set $\rho(L)$ contains a right-halfplane. Hence, to prove that $L$ generates an analytic semigroup it remains to prove that $|\lambda| \|R(\lambda,L){{\bf f}}\|_{\infty}\leq c\|{\bf f}\|_{\infty}$ for each $\lambda$ in a suitable right-halfplane. Again, in view of Lemmata \[finito\], \[finire\] and \[finiremo\], we can limit ourselves to dealing with the functions ${\mathscr P}_{\lambda,2k+j}f_j$ and ${\mathscr Q}_{\lambda,2h+j}f_j$.
For each $\lambda\in{{\mathbb C}}$ with positive real part, we can refine the estimate for ${\rm Re}(X_k)$ and ${\rm Re}(Y_k)$; it turns out that $$\begin{aligned}
&|X_k|\ge {\rm Re}(X_k)=\sqrt{\frac{|1\!+\!4\lambda\!+\!4\lambda_k|\!+\!{\rm Re}(1+4\lambda+4\lambda_k)}{2}}\ge \sqrt{2|\lambda|}\vee 1\vee 2\sqrt{\lambda_k}\ge \frac{\sqrt{3}}{3}\sqrt{|\lambda|+1+\lambda_k},\\[1mm]
&|X_k|\le 2\sqrt{1+|\lambda|+\lambda_k}\end{aligned}$$ and, similarly, $c_1\sqrt{1+|\lambda|+\lambda_k}\le {\rm Re}(Y_k)\le|Y_k|\le c_2\sqrt{1+|\lambda|+\lambda_k}$ for each $k\in{{\mathbb Z}}$ and $\lambda\in{{\mathbb C}}$ with positive real part. As a byproduct, we get $|(\mathcal F_{k,1}f_1)(R)|+|({\mathcal G}_kf_2)(0)|\leq c_R(|\lambda|+1+k^2)^{-1}\|\bf f\|_{\boldsymbol{\mathcal X}}$ for each $k$ and $\lambda$ as above. Moreover, using we can also estimate $|W_k|\ge c_R\sqrt{1+|\lambda|+\lambda_k}$ for each $k\in{{\mathbb Z}}$ and $\lambda\in\Sigma_M:=\{\lambda\in{{\mathbb C}}: {\rm Re}\lambda\ge M\}$ with $M$ large enough. Finally, $$\begin{aligned}
|Y_k-{\rm Le}|=&\bigg |\frac{4{\rm Le}\lambda+4\lambda_k}{\sqrt{{\rm Le}^2+4{\rm Le}\lambda+\lambda_k}+{\rm Le}}\bigg |
\ge \frac{{\rm Le}|\lambda|+\lambda_k}{\sqrt{{\rm Le}^2+{\rm Le}|\lambda|+\lambda_k}}
\ge \frac{{\rm Le}}{2}\sqrt{1+|\lambda|+\lambda_k}\end{aligned}$$ for each $\lambda\in\Sigma_1$ and $k\in{{\mathbb Z}}$, since ${\rm Le}\in (0,1)$. Hence, up to replacing $M$ with $M\vee 1$, if necessary, we can estimate $$\begin{aligned}
&\sum_{j=1}^2(|p_{j,k}(x)|+|q_{j,k}(x')|)+\sum_{j=3}^4(|p_{j,k}(x')|+|q_{j,k}(x'')|)+\sum_{j=5}^6|p_{j,k}(x'')|
\le c_R(1+|\lambda|+\lambda_k)^{-\frac{1}{2}}\end{aligned}$$ for each $k\in{{\mathbb Z}}$, $\lambda\in\Sigma_M$, $x<0$, $x'\in (0,R)$ and $x''>R$. We are almost done. Indeed, taking the above estimates and the fact that $$\begin{aligned}
\sum_{k\in{{\mathbb Z}}}\frac{\|{\bf f}\|_{\infty}}{(|\lambda|+1+k^2)^{3/2}}
\leq & c\|{\bf f}\|_{\infty}\int_0^{+\infty}(|\lambda|+1+r^2)^{-\frac{3}{2}}dr
\leq \frac{c}{1+|\lambda|}\|{\bf f}\|_{\infty}\end{aligned}$$ into account, we easily conclude that $$\begin{aligned}
&\sum_{k=1}^3(\|\mathscr P_{\lambda,2k-1}f_1\|_{\infty}+\|\mathscr P_{\lambda,2k}f_2\|_{\infty})
+\sum_{k=1}^2(\|\mathscr Q_{\lambda,2k-1}f_1\|_{\infty}+\|\mathscr Q_{\lambda,2k}f_2\|_{\infty})\leq c_{M}|\lambda|^{-1}\|\bf f\|_{\boldsymbol{\mathcal X}}\end{aligned}$$ for every $\lambda\in{{\mathbb C}}$ with ${\rm Re}\lambda\ge M$ and some positive constant $c_M$ independent of $\lambda$. Similarly, $$\begin{aligned}
&k^{1-j}\{|D_x^{(j)}p_{i,k}(x)|+|D_{x'}^{(j)}p_{i+2,k}(x')|+|D_{x'}^{(j)}q_{i,k}(x')|+|D_{x''}^{(j)}q_{i,k}(x'')|+|D_{x''}^{(j)}p_{i+4,k}(x'')|\}
\le c_M,\end{aligned}$$ for each $x\le 0$, $x'\in [0,R]$, $x''\ge 0$, $i=1,2$, $j=0,1$ and $$\begin{aligned}
\sum_{k\in{{\mathbb Z}}}\frac{\|{\bf f}\|_{\infty}}{|\lambda|+1+k^2}
\leq & c\|{\bf f}\|_{\infty}\int_0^{+\infty}(|\lambda|+1+r^2)^{-1}dr
\leq \frac{c}{\sqrt{1+|\lambda|}}\|{\bf f}\|_{\infty}.\end{aligned}$$ Thus, we deduce that $$\begin{aligned}
&\sum_{k=1}^3(\|\nabla\mathcal P_{\lambda,2k-1}f_1\|_{\infty}+\|\nabla\mathcal P_{\lambda,2k}f_2\|_{\infty})
+\sum_{k=1}^2(\|\nabla\mathscr Q_{\lambda,2k-1}f_1\|_{\infty}+\|\nabla\mathscr Q_{\lambda,2k}f_2\|_{\infty})\leq c_{M}|\lambda|^{-\frac{1}{2}}\|\bf f\|_{\infty}.\end{aligned}$$
[*Step 4.*]{} Finally, we show that if ${\bf f}\in \boldsymbol{\mathcal X}_{\alpha}$ then ${\bf u}\in \boldsymbol{\mathcal X}_{2+\alpha}$. Again, in view of Lemmata \[finito\]-\[finiremo\] and the estimate $\|p_{1,k}\|_{C^h_b((-\infty,0];{{\mathbb C}})}\le c_1e^{-c_2k}$ (for every $h\in{{\mathbb N}}$) in Step 3, which shows that the function ${\mathscr P}_{\lambda,1}f_1$ belongs to $C^{\beta}_b(H_0^-)$ for every $\beta>0$ and $\|{\mathscr P}_{\lambda,1}f_1\|_{C^{\beta}_b(H_0^-)}\le c_{\lambda}\|f_1\|_{\infty}$, it suffices to deal with the other functions ${\mathscr P}_{\lambda,2k+j}f_j$ and ${\mathscr Q}_{\lambda,2h+j}f_j$. We adapt the arguments in Step 2 of the proof of Lemma \[finire\]. To begin with, we consider the function ${\mathscr P}_{\lambda,2}f_2\in C^2_b(\overline{H^-_0})$ which solves the equation $\lambda {\mathscr P}_{\lambda,2}f_2-\Delta {\mathscr P}_{\lambda,2}f_2-D_x{\mathscr P}_{\lambda,2}f_2=0$ in $H^-_0$. To prove that it belongs to $C^{2+\alpha}_b(H^-_0)$, we check that $({\mathscr P}_{\lambda,2 }f_2)(0,\cdot)\in C^{2+\alpha}_b({{\mathbb R}})$. Note that $p_{2,k}(0)= \ell(\pi k)^{-1}+\widetilde p_{2,k}$ for every $k\in{{\mathbb Z}}\setminus\{0\}$, where $\widetilde p_{2,k}=O(k^{-2})$. Therefore, we can split $$\begin{aligned}
({\mathscr P}_{\lambda,2}f_2)(0,\cdot)=\frac{2}{{\rm Le}\,\pi}\sum_{k\in{{\mathbb Z}}}k^{-1}({\mathcal G}_kf_2)(0)e_k
+\frac{2}{{\rm Le}\, \ell}\sum_{k\in{{\mathbb Z}}}\widetilde p_{2,k}({\mathcal G}_kf_2)(0)e_k=\psi_1+\psi_2.\end{aligned}$$ Since $|\widetilde p_{2,k}({\mathcal G}_kf_2)(0)|\le c|k|^{-4}\|f_2\|_{\infty}$ for every $k\in{{\mathbb Z}}\setminus\{0\}$, it follows immediately that $\psi_2\in C^{2+\alpha}_b({{\mathbb R}})$ and $\|\psi_2\|_{C^{2+\alpha}_b({{\mathbb R}})}\le c\|f_2\|_{\infty}$. As far as $\psi_1$ is concerned, a straightforward computation reveals that $\psi_1'=c({\mathscr S}_{\lambda}f_2)(0,\cdot)$ so that, by Lemma \[finire\], $\psi_1'\in C^{1+\alpha}_b({{\mathbb R}})$ and $\|\psi_1'\|_{C^{1+\alpha}_b({{\mathbb R}})}\leq c\|{{\bf f}}\|_{\alpha}$. Thus, $({\mathscr P}_{\lambda,2}f_2)(0,\cdot)$ belongs to $C^{2+\alpha}_b({{\mathbb R}})$ and $\|({\mathscr P}_{\lambda,2}f_2)(0,\cdot)\|_{C^{2+\alpha}_b({{\mathbb R}})}\le c_{\lambda}\|{{\bf f}}\|_{\alpha}$. By classical results for elliptic problems (see [@krylov]), ${\mathscr P}_{\lambda,2}f_2$ belongs to $C^{2+\alpha}_b(H_0^-)$ and $\|{\mathscr P}_{\lambda,2}f_2\|_{C^{2+\alpha}_b(H_0^-)}\le c_{\lambda}\|{{\bf f}}\|_{\alpha}$.
Next, we split ${\mathscr P}_{\lambda,3}f_1$ into the sum of the functions $$\begin{aligned}
{\mathscr P}_{3,\lambda,1}f_1=\frac{1}{\ell}\sum_{k\in{{\mathbb Z}}}\frac{e^{\nu_k^+(\cdot-R)}}{W_k}({\mathcal F}_{k,1}f_1)(R)e_k,\qquad\;\,
{\mathscr P}_{3,\lambda,2}f_1=\frac{1}{\ell}\sum_{k\in{{\mathbb Z}}}\frac{e^{-\mu_k^+R}}{W_k}({\mathcal F}_{k,1}f_1)(R)e^{\nu_k^-\cdot}e_k.\end{aligned}$$ The first function belongs to $C^2_b(\overline{H^{-}_R};{{\mathbb C}})$ and $\lambda {\mathscr P}_{3,\lambda,1}f_1-\Delta {\mathscr P}_{3,\lambda,1}f_1-D_x{\mathscr P}_{3,\lambda,1}f_1=0$ in $H^-_R$. Since $$\begin{aligned}
(({\mathscr P}_{3,\lambda,1}f_1)(R,\cdot))'=c({\mathscr R}_{\lambda,1}f_1)(R,\cdot)+\frac{1}{\ell}\sum_{k\in{{\mathbb Z}}}\widetilde p_{3,k}(\mathcal F_{k,1}f_1)(R)e_k\end{aligned}$$ and $|\widetilde p_{3,k}|\le ck^{-1}$, the same arguments as above and Lemma \[finito\] allow to show first that the function $(({\mathscr P}_{3,\lambda,1}f_1)(R,\cdot))'$ belongs to $C^{1+\alpha}_b({{\mathbb R}})$ and then to conclude that ${\mathscr P}_{3,\lambda,1}f_1\in C^{2+\alpha}_b(H^-_R)$ and $\|{\mathscr P}_{3,\lambda,1}f_1\|_{C^{2+\alpha}_b(H^-_R)}\le c_{\lambda}\|{{\bf f}}\|_{\alpha}$. The smoothness of the function ${\mathscr P}_{3,\lambda,2}f_1$ is easier to prove, due to the uniform (in $[0,R]$) exponential decay to zero of the terms of the series. It turns out that $\|{\mathscr P}_{3,\lambda,2}f_1\|_{C^{2+\alpha}_b(H^-_R)}\le c_{\lambda}\|f_1\|_{\infty}$.
Let us consider the function ${\mathscr P}_{4,\lambda}f_2$, which we split it into the sum of the functions ${\mathscr P}_{4,\lambda,1}f_2$ and ${\mathscr P}_{4,\lambda,2}f_2$ defined, respectively, by $$\begin{aligned}
{\mathscr P}_{4,\lambda,j}f_2=\frac{2}{{\rm Le} \ell}\sum_{k\in{{\mathbb N}}}p_{4,j,k}(\mathcal G_kf_2)(0)e_k,\qquad\;\,j=1,2,\end{aligned}$$ where $$\begin{aligned}
p_{4,1,k}(x)= \frac{\theta_iR}{W_k}e^{\nu_k^-x}, \qquad\;\,
p_{4,2,k}(x)= \frac{\theta_iR{\rm Le}\,e^{\nu_k^-x}}{W_k(Y_k-{\rm Le})}-\frac{Y_ke^{\nu_k^-x}}{X_kW_k(Y_k-{\rm Le})}+\frac{Y_ke^{-X_kR+\nu_k^+x}}{X_kW_k(Y_k-{\rm Le})}.\end{aligned}$$ Function ${\mathscr P}_{4,\lambda,1}f_2$ belongs to $C^2_b(\overline{H_0^+})$ and $\lambda {\mathscr P}_{4,\lambda,1}f_2-\Delta {\mathscr P}_{4,\lambda,1}f_2-D_x{\mathscr P}_{4,\lambda,1}f_2=0$ in $H^+_0$. Moreover, $({\mathscr P}_{4,\lambda,1}f_2)(0,\cdot)$ is an element of $C^{2+\alpha}_b({{\mathbb R}})$ and $\|({\mathscr P}_{4,\lambda,1}f_2)(0,\cdot)\|_{C^{2+\alpha}_b({{\mathbb R}})}\le c_{\lambda}\|{{\bf f}}\|_{\alpha}$, so that ${\mathscr P}_{4,\lambda,1}f_2\in C^{2+\alpha}_b(H^+_0)$ and $\|{\mathscr P}_{4,\lambda,1}f_2\|_{C^{2+\alpha}_b(H^+_0)}\le c_{\lambda}\|{{\bf f}}\|_{\alpha}$. On the other hand, the series, which defines ${\mathscr P}_{4,\lambda,2}f_2$ is easier to analyze since it converges in $C^{2+\alpha}_b(H^-_R)$ and $\|{\mathscr P}_{4,\lambda,2}f_2\|_{C^{2+\alpha}_b(H^-_R)}\leq c_{\lambda}\|{{\bf f}}\|_{\alpha}$.
All the remaining functions ${\mathscr P}_{\lambda,2k+j}f_j$ and ${\mathscr Q}_{\lambda,2h+j}f_j$ can be analyzed in the same way. The details are left to the reader.
Now, we characterize the interpolation spaces $D_L(\alpha/2,\infty)$ and $D_L(1+\alpha/2,\infty)$. To simplify the notation, we introduce the operator ${\mathscr B}_0$, defined by $${\mathscr B}_0{\bf u}=(u_1(0^+,\cdot)-u_1(0^-,\cdot),u_1(R^+,\cdot)-u_1(R^-,\cdot)),\qquad\;\,{\bf u}\in\boldsymbol{\mathcal X},
\label{operatore-B0}$$ and the sets $\boldsymbol{\mathcal X}_{\alpha,{\mathscr B}_0}=\{{\bf u}\in \boldsymbol{\mathcal X}_{\alpha}: {\mathscr B}_0{\bf u}={\bf 0}\}$ $(\alpha\in (0,1])$ and $\boldsymbol{\mathcal X}_{2+\alpha,{\mathscr B}}=\{{\bf u}\in \boldsymbol{\mathcal X}_{2+\alpha}: {\mathscr B}{\bf u}={\bf 0}, {\mathscr B}_0L{\bf u}={\bf 0},\, \lim_{x\to \pm\infty}(L{{\bf u}})_1(x,y)=
\lim_{x\to +\infty}(L{{\bf u}})_2(x,y)=0\}$ ($\alpha\in (0,1)$), equipped with the norm of $\boldsymbol{\mathcal X}_{\alpha}$ and $\boldsymbol{\mathcal X}_{2+\alpha}$, respectively.
\[perdita\] For each $\alpha\in (0,1)$ the following characterizations hold: $$\begin{aligned}
(i)~D_L(\alpha/2,\infty)=\boldsymbol{\mathcal X}_{\alpha,{\mathscr B}_0},\qquad\;\, (ii)~D_L(1+\alpha/2,\infty)
= & \boldsymbol{\mathcal X}_{2+\alpha,\mathscr B},
\label{penna}\end{aligned}$$ with equivalence of respective norms. Moreover, $$\begin{aligned}
\label{doccia}
\boldsymbol{\mathcal X}_{1,\mathscr B_0}\hookrightarrow D_L(1/2,\infty).\end{aligned}$$
Throughout the proof, we assume that $\alpha$ is arbitrarily fixed in $(0,1)$.
[*Step 1: proof of $(i)$ and* ]{}. Given ${\bf f}=(f_1,f_2)\in\boldsymbol{\mathcal X}_{\alpha,{\mathscr B}_0}$ and $t>0$, we introduce the functions $f_{t,1}$ and $f_{t,2}$ defined by $$\begin{aligned}
&f_{t,1}(x,y)=\int_{H_0^+}f_1^{\sharp}(x+tx',y+ty')\varphi(x',y')dx'dy',\qquad\;\,(x,y)\in{{\mathbb R}}^2,\\
&f_{t,2}(x,y)=\int_{H_0^+}\widetilde{f_2}(x-tx',y+ty')\varphi(x',y')dx'dy',\qquad\;\,(x,y)\in [0,R]\times{{\mathbb R}},\\
&f_{t,2}(x,y)=\int_{H_0^+}f_2^{\sharp}(x+tx',y+ty')\varphi(x',y')dx'dy',\qquad\;\,(x,y)\in H_R^+,\end{aligned}$$ where $\varphi$ is a positive smooth function with compact support in $(0,R)\times{{\mathbb R}}$, $\|\varphi\|_{L^1({{\mathbb R}}^2)}=1$ and $\widetilde f_2:\overline{H^-_R}\to{{\mathbb C}}$ equals the function $f_2^{\sharp}\vartheta$ in $[0,R]\times{{\mathbb R}}$, whereas $\widetilde f_2(x,\cdot)=f_2^{\sharp}(0,\cdot)\vartheta(x)$ if $x<0$ and $\vartheta$ is a smooth function compactly supported in $(-1,+\infty)$ and equal to $1$ in $(-1/2,+\infty)$. Since ${\mathscr B}_0{\bf f}={\bf 0}$ and $f_j(\cdot,-\ell/2)=f_j(\cdot,\ell/2)$ for $j=1,2$, the function $f_1^{\sharp}$ belongs to $C^{\alpha}_b({{\mathbb R}}^2;{{\mathbb C}})$. On the other hand, the functions $\widetilde f_2$ and $f_2^{\sharp}$ belong to $C^\alpha(H^-_R;{{\mathbb C}})$ and to $C_b^{\alpha}(H^+_R;{{\mathbb C}})$, respectively. Moreover, $f_1^{\sharp}(\cdot,y)$ and $f_2^{\sharp}(\cdot,y)$ vanish at $\pm\infty$ and at $+\infty$, respectively, for every $y\in{{\mathbb R}}$. Hence, if we set ${\bf f}_t=(f_{t,1},f_{t,2})$, then ${{\bf f}}_t$ belongs to $\boldsymbol{\mathcal X}$, $\|{\bf f-f}_t\|_{\infty}\leq ct^\alpha\|{\bf f}\|_{\alpha}$ and $\|{\bf f}_t\|_{\infty}\le c\|{\bf f}\|_{\alpha}$ for every $t>0$. Similarly, since $\int_{H_0^+}D^{\gamma}\varphi(x',y')dx'dy'=0$ for every multi-index $\gamma$, we can write $$\begin{aligned}
D^{\gamma}f_{t,1}(x,y)=t^{-|\gamma|}\int_{H_0^+}(f_1^{\sharp}(x+tx',y+ty')-f_1^{\sharp}(x,y))D^{\gamma}\varphi(x',y')dx'dy',\qquad\;\,(x,y)\in{{\mathbb R}}^2,\end{aligned}$$ so that $|D^{\gamma}f_{t,1}(x,y)|\le ct^{\alpha-|\gamma|}\|{\bf f}\|_{\alpha}$ for every $(x,y)\in{{\mathbb R}}^2$. In the same way we can estimate the derivatives of the function $f_{t,2}$, and conclude that $t^{|\gamma|-\alpha}\|D^{\gamma}{\bf f}_t\|_{\infty}\leq c_T\|{\bf f}\|_{\alpha}$ for every $t\in (0,T]$ and $T>0$.
Now, we split $\omega^{\alpha/2}LR(\omega, L){\bf f}=\omega^{\alpha/2}LR(\omega, L)({\bf f}-{\bf f}_{\omega^{-1/2}})+\omega^{\alpha/2}LR(\omega, L){\bf f}_{\omega^{-1/2}}$ for each $\omega\in\rho(L)\cap{{\mathbb R}}$. By Theorem \[banca\], for $\omega\in{{\mathbb R}}$ sufficiently large (let us say $\omega\ge\omega_0>0$), it holds that $\|LR(\omega,L)\|_{L(\boldsymbol{\mathcal X})}\leq M$ for some positive constant $M$, independent of $\omega$. Hence, using the above estimates, we can infer that $\|\omega^{\alpha/2}LR(\omega, L)({\bf f-f}_{\omega^{-1/2}})\|_{\infty}\le c\|{{\bf f}}\|_{\alpha}$. Next, we consider the term ${\bf v}=LR(\omega, L){\bf f}_{\omega^{-1/2}}$, which belongs to $D(L)$ and solves the equation $\omega {\bf v}-L{\bf v}=L{\bf f}_{\omega^{-1/2}}$. If $\omega\ge\omega_0$, then we can estimate $\omega\|{\bf v}\|_{\infty}\leq c\|L{\bf f}_{\omega^{-1/2}}\|_{\infty}\leq c\omega^{1-\alpha/2}\|{\bf f}\|_{\alpha}$. Putting everything together, we conclude that $\omega^{\alpha/2}\|LR(\omega, L){\bf f}\|_{\infty}\leq c\|{\bf f}\|_{\alpha}$ for each $\omega\ge\omega_0$ and this shows that ${\bf f}\in D_L(\alpha/2,\infty)$ and $\|{\bf f}\|_{D_L(\alpha/2,\infty)}\leq c\|{\bf f}\|_{\alpha}$.
Formula can be proved just in the same way observing that, if ${\bf f}$ belongs to $\boldsymbol{\mathcal X}_{1,{\mathscr B}_0}$, then the functions $f_1^{\sharp}$, $\widetilde f_2$ and $f_2^{\sharp}$ are bounded and Lipschitz continuous in ${{\mathbb R}}^2$ in $[0,R]\times{{\mathbb R}}$ and in $H_R^+$, respectively.
The embedding “$\hookrightarrow$” in (i) is a straightforward consequence of two properties: $$\begin{aligned}
(a)~\|\nabla{\bf u}\|_{\infty}\leq c\|{\bf u}\|_{\infty}^{\frac{1}{2}}\|{\bf u}\|_{D(L)}^{\frac{1}{2}},\quad\;\,{\bf u}\in D(L),\qquad\;\,(b)~(\boldsymbol{\mathcal X},\boldsymbol{\mathcal X}_{1,{\mathscr B}_0})_{\alpha,\infty}\hookrightarrow \boldsymbol{\mathcal X}_{\alpha,{\mathscr B}_0}.\end{aligned}$$ Indeed, property (a) shows that $\boldsymbol{\mathcal X}_{1,{\mathscr B}_0}$ belongs to the class $J_{1/2}$ between $\boldsymbol{\mathcal X}$ and $D(L)$, so that applying the reiteration theorem, we get $D_L(\alpha/2,\infty)=(\boldsymbol{\mathcal X},D(L))_{\alpha/2,\infty}\subset (\boldsymbol{\mathcal X},\boldsymbol{\mathcal X}_{1,{\mathscr B}_0})_{\alpha,\infty}$ and conclude using (b).
[*Proof of $(a)$*]{}. It is an almost straightforward consequence of the estimate $\|\nabla R(\lambda,L){{\bf f}}\|_{\infty}\le c|\lambda|^{-1/2}\|{{\bf f}}\|_{\infty}$ for each $\lambda\in{{\mathbb C}}$ with ${\rm Re}\lambda\ge M$, contained in Theorem \[banca\]. Indeed, let $L_M=L-MI$. As it is easily seen, $\rho(L_M)\supset (0,+\infty)$ and $R(\lambda,L_M)=R(\lambda+M,L)$ for each $\lambda>0$. It thus follows that $\sqrt{\lambda}\|\nabla R(\lambda,L_M)\|_{L(\boldsymbol{\mathcal X};\boldsymbol{\mathcal X}\times{\boldsymbol{\mathcal X}})}\le c$ for each $\lambda>0$, so that we can estimate $$\begin{aligned}
\|\nabla {\bf u}\|_{\infty}=\|\nabla R(\lambda,L_M)(\lambda {\bf u}-L_M{\bf u})\|_{\infty}
\le c\lambda^{-\frac{1}{2}}\|\lambda {\bf u}-L_M{\bf u}\|_{\infty}\le c(\lambda^{\frac{1}{2}}\|{\bf u}\|_{\infty}+
\lambda^{-\frac{1}{2}}\|L_M{\bf u}\|_{\infty})\end{aligned}$$ for each ${\bf u}\in D(L)=D(L_M)$ and $\lambda>0$. Minimizing with respect to $\lambda>0$, we conclude the proof.
[*Proof of $(b)$*]{}. Let us fix a nontrivial ${\bf f}\in (\boldsymbol{\mathcal X},\boldsymbol{\mathcal X}_{1,{\mathscr B}_0})_{\alpha,\infty}$. Since $(\boldsymbol{\mathcal X},\boldsymbol{\mathcal X}_{1,{\mathscr B}_0})_{\alpha,\infty}\hookrightarrow (\boldsymbol{\mathcal X},\boldsymbol{\mathcal X}_{1,{\mathscr B}_0})_{\alpha/2,1}$ and $\boldsymbol{\mathcal X}_{1,{\mathscr B}_0}$ is dense in $(\boldsymbol{\mathcal X},\boldsymbol{\mathcal X}_{1,{\mathscr B}_0})_{\alpha/2,1}$, we immediately deduce that $f_j(\cdot,-\ell/2)=f_j(\cdot,\ell/2)$ for $j=1,2$ and ${\mathscr B}_0{\bf f}={\bf 0}$. Next, we recall that $$\begin{aligned}
\inf\left\{\|{\bf g}\|_{\infty}+t\|{\bf h}\|_{1}: {\bf f}={\bf g}+{\bf h},\,
{\bf g}\in\boldsymbol{\mathcal X}, {\bf h}\in\boldsymbol{\mathcal X}_{1,{\mathscr B}_0}\right\}
\le t^{\alpha}\|{\bf f}\|_{(\boldsymbol{\mathcal X},\boldsymbol{\mathcal X}_{1,{\mathscr B}_0})_{\alpha,\infty}},\qquad\;\,t>0.\end{aligned}$$ Fix $(x_j,y_j)\in S^-_0$, $j=1,2$, and take $t=\sqrt{|x_2-x_1|^2+|y_2-y_1|^2}$. Then, we can determine ${\bf g}\in\boldsymbol{\mathcal X}$ and ${\bf h}\in\boldsymbol{\mathcal X}_{1,{\mathscr B}_0}$ such that ${\bf f}={\bf g}+{\bf h}$ and $$\begin{aligned}
\|{\bf g}\|_{\infty}+\sqrt{|x_2-x_1|^2+|y_2-y_1|^2}\|{\bf h}\|_1\le
2\|{\bf f}\|_{(\boldsymbol{\mathcal X},\boldsymbol{\mathcal X}_{1,{\mathscr B}_0})_{\alpha,\infty}}(|x_2-x_1|^2+|y_2-y_1|^2)^{\frac{\alpha}{2}}.\end{aligned}$$ From this estimate we can infer that $$\begin{aligned}
&|f_1(x_2,y_2)-f_1(x_1,y_1)|\\
\le & |g_1(x_2,y_2)-g_1(x_1,y_1)|+|h_1(x_2,y_2)-h_1(x_1,y_1)|
\le 2\|{\bf g}\|_{\infty}+\sqrt{|x_2-x_1|^2+|y_2-y_1|^2}\|{\bf h}\|_1\\
\le & 4\|{\bf f}\|_{(\boldsymbol{\mathcal X},\boldsymbol{\mathcal X}_{1,{\mathscr B}_0})_{\alpha,\infty}}(|x_2-x_1|^2+|y_2-y_1|^2)^{\frac{\alpha}{2}}.\end{aligned}$$ Hence, $f_1\in C_b^{\alpha}(S^-_0)$ and $\|f_1\|_{C_b^{\alpha}(S^-_0)}\le 5\|{\bf f}\|_{(\boldsymbol{\mathcal X},\boldsymbol{\mathcal X}_{1,{\mathscr B}_0})_{\alpha,\infty}}$.
The same argument can be used to prove that $f_1, f_2\in C^{\alpha}((0,R)\times (-\ell/2,\ell/2);{{\mathbb C}})\cap C^{\alpha}_b(S_R^+;{{\mathbb C}})$ and $$\begin{aligned}
&\|f_1\|_{C^{\alpha}_b(S_R^+;{{\mathbb C}})}+\sum_{j=1}^2\|f_j\|_{C^{\alpha}((0,R)\times (-\ell/2,\ell/2);{{\mathbb C}})}+\|f_2\|_{C^{\alpha}_b(S_R^+;{{\mathbb C}})}\le c
\|{\bf f}\|_{(\boldsymbol{\mathcal X},\boldsymbol{\mathcal X}_{1,{\mathscr B}_0})_{\alpha,\infty}}.\end{aligned}$$
The proof of (b) is now complete.
[*Step 2: proof of $(ii)$.*]{} The embedding “$\hookleftarrow$” easily follows from the first part of the proof. The other embedding follows from Theorem \[banca\]. Indeed, fix ${\bf u}\in D_L(1+\alpha/2,\infty)$ and $\lambda\in\rho(L)$. Then, the function ${\bf f}:=\lambda {\bf u}-L{\bf u}$ belongs to $D_L(\alpha/2,\infty)=\boldsymbol{\mathcal X}_{\alpha,{\mathscr B}_0}$ and $\|{\bf f}\|_{\alpha}\le c\|{\bf u}\|_{D_L(1+\alpha/2,\infty)}$ for some positive constant $c$, independent of ${\bf u}$. Since ${\bf u}=R(\lambda,L){\bf f}$, Theorem \[banca\] implies that ${\bf u}\in\boldsymbol{\mathcal X}_{2+\alpha}$ and $\|{\bf u}\|_{2+\alpha}\le c\|{\bf f}\|_{\alpha}\le c\|{\bf u}\|_{D_L(1+\alpha/2,\infty)}$. Clearly, ${\mathscr B}{\bf u}={\bf 0}$, ${\mathscr B}_0L{\bf u}={\bf 0}$ and $\lim_{x\to\pm\infty}(L{{\bf u}})_1(x,y)=\lim_{x\to +\infty}(L{{\bf u}})_2(x,y)=0$, and this completes the proof.
\[rmk-5.6\] [From the classical theory of analytic semigroups (see e.g., [@lunardi]) and Proposition \[perdita\] it follows that the part $L_{\alpha}$ of $L$ in ${\boldsymbol {\mathcal X}}_{\alpha,{\mathscr B}_0}$, i.e., the restriction of $L$ to ${\boldsymbol {\mathcal X}}_{2+\alpha,{\mathscr B}_0}$, generates an analytic semigroup for each $\alpha\in (0,1)$.]{}
The lifting operators {#subsection-lifting}
---------------------
In this subsection we introduce some lifting operators which are used in the proof of the Main Theorem and Theorem \[soliera\].
To begin with, we consider the operator $\mathscr M$ defined by $\mathscr M\boldsymbol\psi=(0,M\psi_1+\frac{1}{2}(M\psi_2)(\cdot-R,\cdot))$ on functions $\boldsymbol\psi\in C([-\ell/2,\ell/2];{{\mathbb R}}^2)$ such that $\boldsymbol{\psi}(-\ell/2)=\boldsymbol{\psi}(\ell/2)$, where $$\begin{aligned}
(M\psi_1)(x,y) & :=|x|\eta (x)\int_{{{\mathbb R}}}\varphi(\xi)\psi_1^{\sharp}(y+\xi x)d\xi,\qquad\;\,(x,y)\in{{\mathbb R}}^2.\end{aligned}$$ Here, $\eta$ and $\varphi$ are smooth functions such that $\chi_{(-R/4,R/4)}\le\eta\le\chi_{(-R/2,R/2)}$, $\varphi$ is an even nonnegative function compactly supported in $(-1,1)$ with $\|\varphi\|_{L^1({{\mathbb R}})}=1$. As it is easily seen, ${\mathscr M}\boldsymbol\psi\in\boldsymbol{\mathcal X}_{2+\alpha}$, $\mathscr B\mathscr M\boldsymbol\psi=(0,0,0,\psi_1,0,0,\psi_2)$ for each $\boldsymbol\psi$ as above.
Next, we introduce the operator ${\mathscr N}$ defined by $$\begin{aligned}
(\mathscr N\boldsymbol h)_1=&N_1h_1+(N_1h_2)\circ\tau_R+\frac{N_2h_3}{2{\rm Le}}+\frac{(N_2h_5)\circ\tau_R}{2\theta_i R}+N_3\bigg (h_8-\frac1{{\rm Le}}h_3-h_1''\bigg )\\
&+\bigg [N_3\bigg (h_9-\frac{1}{\theta_i R}h_5-h_2''\bigg )\bigg ]\circ\tau_R,\\
({\mathscr N}\boldsymbol h)_2=& \bigg [N_1\bigg (h_6-\frac{\rm Le}{\theta_i R}h_5\bigg )\bigg ]\circ\tau_R+N_2h_4+\frac{1}{2}\bigg [N_3\bigg (h_7-{\rm Le} h_6+\frac{{\rm Le}^2}{\theta_i R}h_5\bigg )\bigg ]\circ\tau_R\end{aligned}$$ on smooth enough functions $\boldsymbol h:[-\ell/2,\ell/2]\to{{\mathbb R}}^9$, where $\tau_R(x,y)=(x-R,y)$, $$\begin{aligned}
(N_1\zeta)(x,y)=&\frac{1}{2}\eta(x)[\chi_{[0,+\infty)}(x)-\chi_{(-\infty,0]}(x)]\int_{{{\mathbb R}}}\varphi(\sigma)\zeta^\sharp(y+\sigma x^3)d\sigma,\\[1mm]
(N_2\zeta)(x,y)=&|x|\eta (x)\int_{{{\mathbb R}}}\varphi(\sigma)\zeta^{\sharp}(y+\sigma |x|)d\sigma,
\qquad (N_3\zeta)(x,y)=\frac{x}{4}(N_2\zeta)(x,y),\end{aligned}$$ for each $(x,y)\in{{\mathbb R}}^2$. Moreover, we set ${\mathscr B}_*{\bf v}=(\widetilde{\mathscr B}{\bf v},{\mathscr B}_0{\mathscr L}{\bf v})$ for each ${\bf v}\in\boldsymbol{\mathcal X}_{2+\alpha}$, where $\widetilde{\mathscr B}$ is the operator in Remark \[rmk-opB\] and the operator ${\mathscr B}_0$ is defined in .
In the next lemma, we deal with real valued spaces. In particular, by $\widetilde D(L_{\alpha})$ we denote the subset of $D(L_{\alpha})$ of real valued functions.
\[lemma-5.7\] The following properties are satisfied.
1. The operator ${\mathscr N}$ is bounded from the set $(\boldsymbol{\mathcal X}_{2+\alpha})^2\times (\boldsymbol{\mathcal X}_{1+\alpha})^5\times (\boldsymbol{\mathcal X}_{\alpha})^2$ into $\boldsymbol{\mathcal X}_{2+\alpha}$. Moreover, the operator $P=I-{\mathscr N}{\mathscr B}_*:\boldsymbol{\mathcal X}_{2+\alpha}\to \boldsymbol{\mathcal X}_{2+\alpha}$ is a projection onto the kernel of $B_*$ which coincides with $\widetilde D(L_\alpha)$[^3].
2. Let $\boldsymbol{\mathcal I}$ denote the set of all functions ${{\bf u}}\in {\boldsymbol {\mathcal X}}_{2+\alpha}$ such that ${\mathscr B}{\bf u}=\mathscr H({\bf u})$, $\mathscr B_0(\mathcal L{\bf u}+\mathscr F({\bf u}))={\bf 0}$. Then, there exist $r_0, r_1>0$ such that $\boldsymbol{\mathcal I}\cap B(0,r_0)$ is the graph of a smooth function $\Upsilon:B(0,r_1)\subset\widetilde D(L_{\alpha})\to (I-P)(\boldsymbol{\mathcal X}_{2+\alpha})$ such that $\Upsilon({\bf 0})={\bf 0}$.
\(i) Showing that ${\mathscr N}$ is a bounded operator is an easy task. Some long but straightforward computations reveal that ${\mathscr B}_*{\mathscr N}=I$ on $(\boldsymbol{\mathcal X}_{2+\alpha})^2\times (\boldsymbol{\mathcal X}_{1+\alpha})^5\times (\boldsymbol{\mathcal X}_{\alpha})^2$ and allow to prove that $P$ is a projection onto of ${\rm Ker}(B_*)=\widetilde D(L_{\alpha})$. In particular, we can split $\boldsymbol{\mathcal X}_{2+\alpha}=\widetilde D(L_{\alpha})\oplus (I-P)(\boldsymbol{\mathcal X}_{2+\alpha})$. The details are left to the reader.
\(ii) Let ${\mathscr K}:B(0,r_0)\subset\widetilde D(L_\alpha)\oplus (I-P)(\boldsymbol{\mathcal X}_{2+\alpha})\to (\boldsymbol{\mathcal X}_{2+\alpha})^2\times (\boldsymbol{\mathcal X}_{1+\alpha})^5\times (\boldsymbol{\mathcal X}_{\alpha})^2$ be the operator defined by ${\mathscr K}({\bf u},{\bf v})=(\mathscr B({\bf u}+{\bf v})-\mathscr H({\bf u}+{\bf v}), \mathscr B_0(\mathcal L{\bf u}+\mathcal L{\bf v}+\mathscr F({\bf u}+{\bf v})))$ for each $({{\bf u}},{\bf v})\in B(0,r_0)$, with $r_0$ small enough to guarantee that ${\mathscr K}$ is well defined. Since the functions $\mathscr F$ and $\mathscr H$ are quadratic at ${\bf 0}$, it follows that ${\mathscr K}({\bf 0},{\bf 0})={\bf 0}$ and ${\mathscr K}$ is Fréchet differentiable at $({\bf 0},{\bf 0})$, with ${\mathscr K}_{\bf u}({\bf 0},{\bf 0})={\mathscr B}_*$. In view of (i), ${\mathscr B}_*$ is an isomorphism from $(I-P)(\boldsymbol{\mathcal X}_{2+\alpha})$ to $(\boldsymbol{\mathcal X}_{2+\alpha})^2\times (\boldsymbol{\mathcal X}_{1+\alpha})^5\times (\boldsymbol{\mathcal X}_{\alpha})^2$. Thus, we can invoke the implicit function theorem to complete the proof.
Solving the nonlinear problem {#sect-5}
==============================
Now we are able to solve the nonlinear Cauchy problem $$\begin{aligned}
\left\{
\begin{array}{lll}
D_t{\bf u}={\mathscr L}{\bf u}+\mathscr F({\bf u}),\\[1mm]
\mathscr B{\bf u}=\mathscr H({\bf u}), \\[1mm]
D^{\gamma_1}_xD^{\gamma_2}_y{\bf u}(\cdot,\cdot,-\ell/2)=D^{\gamma_1}_xD^{\gamma_2}_y{\bf u}(\cdot,\cdot,\ell/2), &&\gamma_1+\gamma_2\le 2,
\end{array}
\right.
\label{asta1}\end{aligned}$$ for the unknown ${\bf u}=(u,w)$. Also in this section we assume that the function spaces that we deal with are real valued ones.
\[soliera\] Fix $\alpha\in(0,1)$ and $T>0$. Then, there exists $r_0=r_0(T)>0$ such that, for each ${\bf u}_0\in B(0,r_0)\subset \boldsymbol{\mathcal X}_{2+\alpha}$ satisfying the compatibility conditions $\mathscr B{\bf u}_0=\mathscr H({\bf u}_0)$, $\mathscr B_0({\mathscr L}{\bf u}_0+\mathscr F({\bf u_0}))={\bf 0}$ and $D^{\gamma}{\bf u}_0(\cdot,-\ell/2)=D^{\gamma}{\bf u_0}(\cdot,\ell/2)$ for each multi-index $\gamma$ with length at most two, Problem admits a unique solution ${\bf u}\in\boldsymbol{\mathcal Y}_{2+\alpha}$ with ${\bf u}(0,\cdot)={\bf u}_0$. Moreover, $\|{\bf u}\|_{\boldsymbol{\mathcal Y}_{2+\alpha}}\leq c\|{\bf u}_0\|_{2+\alpha}$.
The proof can be obtained arguing as in the proof of [@lorenzi-1 Theorem 4.1]. For this purpose, we just sketch the main points.
We first need to prove optimal regularity results for the linear version of Problem , i.e., with the problem $$\label{wallace}
\left\{
\begin{array}{lll}
D_t{\bf u}(t,\cdot,\cdot)={\mathscr L}{\bf u}(t,\cdot,\cdot)+{\bf f}(t,\cdot,\cdot), &t\in [0,T],\\[1mm]
\mathscr B({\bf u}(t,\cdot,\cdot))={\bf h}(t,\cdot), &t\in [0,T],\\[1mm]
D^{\gamma_1}_xD^{\gamma_2}_y{\bf u}(t,\cdot,-\ell/2)=D^{\gamma_1}_xD^{\gamma_2}_y{\bf u}(t,\cdot,\ell/2), &t\in [0,T], &\gamma_1+\gamma_2\le 2,\\[1mm]
{\bf u}(0,\cdot)={\bf u_0},
\end{array}
\right.$$ when ${\bf f}\in \boldsymbol{\mathcal Y}_{\alpha}$, ${\bf u_0}\in\boldsymbol{\mathcal X}_{2+\alpha}$, when $h_j\equiv 0$ if $j\neq 4,7$, $h_4=\psi_1$, $h_7=\psi_2$, $\boldsymbol{\psi}=(\psi_1,\psi_2)\in C^{(1+\alpha)/2,1+\alpha}((0,T)\times (-\ell/2,\ell/2);{{\mathbb R}}^2)$ satisfy the compatibility conditions
- $\mathscr B{\bf u_0}={\bf h}(0,\cdot)$, $\mathscr B_0({\mathscr L}{\bf u_0}(0,\cdot)+{\bf f}(0,\cdot))={\bf 0}$;
- ${\bf f}(0,\cdot,-\ell/2)={\bf f}(0,\cdot,\ell/2)$, $D^{\gamma}{\bf u_0}(\cdot,-\ell/2)=D^{\gamma}{\bf u_0}(\cdot,\ell/2)$ and $D_y^{(j)}\boldsymbol\psi(\cdot,-\ell/2)=D_y^{(j)}\boldsymbol\psi(\cdot,\ell/2)$ for every multi-index $\gamma$ with length at most two and $j=0,1$.
We also need to show the estimate $$\begin{aligned}
\|{\bf u}\|_{\boldsymbol{\mathcal Y}_{2+\alpha}}\leq c_0(\|{\bf f}\|_{\boldsymbol{\mathcal Y}_{\alpha}}
+\|{\bf u_0}\|_{2+\alpha}+\|\boldsymbol{\psi}\|_{C^{(1+\alpha)/2,1+\alpha}((0,T)\times (-\ell/2,\ell/2);{{\mathbb R}}^2)}).
\label{argine}\end{aligned}$$ for its unique solution ${\bf u}\in\boldsymbol{\mathcal Y}_{2+\alpha}$. This is the content of Steps 1 to 3.
[*Step 1.*]{} To begin with, we note that $Mh\in C^{(1+\alpha)/2,2+\alpha}_b((0,T)\times H_0^-)\cap C^{(1+\alpha)/2,2+\alpha}_b((0,T)\times H_0^+)$ for all $h\in C^{(1+\alpha)/2,1+\alpha}((0,T)\times (-\ell/2,\ell/2))$ such that $D_y^{(j)}h(\cdot,-\ell/2)=D_y^{(j)}h(\cdot,\ell/2)$ ($j=0,1$), and $$\begin{aligned}
\|Mh\|_{C^{(1+\alpha)/2,2+\alpha}_b((0,T)\times H_0^-)}
+\|Mh\|_{C^{(1+\alpha)/2,2+\alpha}_b((0,T)\times H_0^+)}\le c\|h\|_{C^{(1+\alpha)/2,1+\alpha}((0,T)\times (-\ell/2,\ell/2))}.\end{aligned}$$ Thus, the function ${\bf f}+{\mathscr L}{\mathscr M}\boldsymbol\psi$ belongs to $C^{\alpha/2}([0,T];\boldsymbol{\mathcal X})$. Moreover, by Proposition \[perdita\] and the compatibility conditions on ${\bf u}_0$ it follows that ${\bf u}_0-{\mathscr M}(\boldsymbol{\psi}(0,\cdot))\in D(L)$, ${\mathscr L}{\bf u}_0+{\bf f}(0,\cdot)\in D_L(\alpha/2,\infty)$. The theory of analytic semigroups (see e.g., [@lunardi Chapter 4]), Theorem \[banca\] and Proposition \[perdita\] show that there exists a unique function ${\bf v}\in C^{1+\alpha/2}([0,T];\boldsymbol{\mathcal X})\cap C([0,T];D(L))$ which solves the equation $D_t{\bf v}={\mathscr L}{\bf v}+{\bf f}+{\mathscr L}\mathscr M\boldsymbol{\psi}$ and satisfies the condition ${\bf v}(0,\cdot)={\bf u}_0$. In addition $D_t{\bf v}$ is bounded with values in $\boldsymbol{\mathcal X}_{\alpha,{\mathscr B}_0}$ (which implies that $D_t{\bf v}\in \boldsymbol{\mathcal Y}_{\alpha}$) and ${\mathscr L}{\bf v}\in C^{\alpha}([0,T];\boldsymbol{\mathcal X})$. By difference, ${\mathscr L}{\bf v}=L{\bf v}$ is bounded in $[0,T]$ with values in $\boldsymbol{\mathcal X}_{\alpha}$ and, in view of Theorem \[banca\], ${\bf v}$ is bounded in $[0,T]$ with values in $\boldsymbol{\mathcal X}_{2+\alpha}$. In particular, ${\mathscr B}{\bf v}={\bf 0}$ and $D^{\gamma}{\bf v}(\cdot,\cdot,-\ell/2)=D^{\gamma}{\bf v}(\cdot,\cdot,\ell/2)$ for every $|\gamma|\le 2$. Further, $\|{\bf v}(t,\cdot)\|_{2+\alpha}+\|D_t{\bf v}\|_{\boldsymbol{\mathcal Y}_{\alpha}}\le
c(\|{\bf f}\|_{\boldsymbol{\mathcal Y}_{\alpha}}
+\|{\bf u_0}\|_{2+\alpha}+\|\boldsymbol{\psi}\|_{C^{(1+\alpha)/2,1+\alpha}((0,T)\times (-\ell/2,\ell/2);{{\mathbb R}}^2)})$ for every $t\in [0,T]$.
[*Step 2.*]{} Let $\widetilde{\bf w}$ be the function defined by $\widetilde{\bf w}(t,\cdot,\cdot)=\int_0^te^{(t-s)L}({\mathscr M}\boldsymbol\psi(s,\cdot)-{\mathscr M}\boldsymbol\psi(0,\cdot))ds$ for every $t\in [0,T]$, where $\{e^{tL}\}$ is the analytic semigroup generated by $L$ in $\boldsymbol{\mathcal X}$. Taking into account, it follows that $\|\mathscr M\boldsymbol{\psi}\|_{C^{(1+\alpha)/2}([0,T], D_L(1/2,\infty))}
\leq c\|\boldsymbol\psi\|_{C^{(1+\alpha)/2,1+\alpha}((0,T)\times (-\ell/2,\ell/2);{{\mathbb R}}^2)}$. Again by the theory of analytic semigroups we infer that $\widetilde{\bf w}\in C([0,T];D(L))$, $D_t\widetilde{\bf w}$ is bounded in $[0,T]$ with values in $\boldsymbol{\mathcal X}_{2+\alpha,{\mathscr B}}$, $L\widetilde{\bf w}\in C^{1+\alpha/2}([0,T];\boldsymbol{\mathcal X})$, $D_t\widetilde{\bf w}=L\widetilde{\bf w}+{\mathscr M}\boldsymbol\psi-{\mathscr M}\boldsymbol\psi(0,\cdot)$ and $\|D_t\widetilde{\bf w}\|_{C^{1+\alpha/2}([0,T];D(L))}+\sup_{t\in [0,T]}\|LD_t\widetilde{\bf w}(t,\cdot)\|_{\boldsymbol {\mathcal X}_{\alpha,{\mathscr B}_0}}\le c\|\boldsymbol{\psi}\|_{C^{(1+\alpha)/2,1+\alpha}((0,T)\times (-\ell/2,\ell/2);{{\mathbb R}}^2)}$. From these properties and using the same arguments as above, it can be easily checked that the function ${\bf w}=-L\widetilde{\bf w}+{\mathscr M}(\boldsymbol{\psi}(0,\cdot))$ is as smooth as ${\bf v}$ is. Moreover, $D_t{\bf w}={\mathscr L}{\bf w}-{\mathscr L}{\mathscr M}\boldsymbol\psi$, ${\mathscr B}{\bf w}=(0,0,0,\psi_1,0,0,\psi_2)$, ${\bf w}(0,\cdot,\cdot)=\mathscr M(\boldsymbol{\psi}(0,\cdot))$ and $D^{\gamma}{\bf w}(\cdot,-\ell/2)=D^{\gamma}{\bf w}(\cdot,\ell/2)$ for every $|\gamma|\le 2$.
[*Step 3.*]{} Clearly, the function ${\bf u}={\bf v}+{\bf w}\in \boldsymbol{\mathcal Y}_{2+\alpha}$ solves the Cauchy problem , satisfies and it is the unique solution to the above problem in $C^1([0,T];\boldsymbol{\mathcal X})\cap C([0,T];D(L))$. Moreover, $$\sup_{t\in [0,T]}\|{\bf u}(t,\cdot)\|_{2+\alpha}+\|D_t{\bf u}\|_{\boldsymbol{\mathcal Y}_{\alpha}}\le
c(\|{\bf f}\|_{\boldsymbol{\mathcal Y}_{\alpha}}
+\|{\bf u_0}\|_{2+\alpha}+\|\boldsymbol{\psi}\|_{C^{(1+\alpha)/2,1+\alpha}((0,T)\times (-\ell/2,\ell/2);{{\mathbb R}}^2)}).
\label{negramaro}$$
To conclude that ${\bf u}\in \boldsymbol{\mathcal Y}_{2+\alpha}$ and it satisfies estimate , we use an interpolation argument. It is well known that $\|\cdot\|_{C^2_b(H^+_R;{{\mathbb R}}^2)}\le c\|\cdot\|_{C^{\alpha}_b(H_R^+;{{\mathbb R}}^2)}^{\alpha/2}\|\cdot\|_{C^{2+\alpha}_b(H^+_R;{{\mathbb R}}^2)}^{1-\alpha/2}$. Using this estimate and the formula $$\begin{aligned}
{{\bf u}}(t,x,y)-{{\bf u}}(s,x,y)=\int_s^tD_t{{\bf u}}(r,x,y)dr,\qquad\;\,s,t\in [0,T],\;\,(x,y)\in H_R^+,\end{aligned}$$ to show that $\|{{\bf u}}(t,\cdot)-{{\bf u}}(s,\cdot)\|_{C^{\alpha}_b(H_R^+;{{\mathbb R}}^2)}\le\|D_t{{\bf u}}\|_{\boldsymbol{\mathcal Y}_{\alpha}}|t-s|$, we get $\|{{\bf u}}(t,\cdot)-{{\bf u}}(s,\cdot)\|_{C^2_b(H^+_R;{{\mathbb R}}^2)}\le c\|D_t{{\bf u}}\|_{\boldsymbol{\mathcal Y}_{\alpha}}^{\alpha/2}\sup_{r\in [0,T]}\|{{\bf u}}(r,\cdot)\|_{2+\alpha}^{1-\frac{\alpha}{2}}|t-s|^{\alpha/2}$.
In the same way, we can show that $u_1\in C^{\alpha/2}([0,T];C^2_b(\overline{H_R^-};{{\mathbb R}}^2))$, ${{\bf u}}\in C^{\alpha/2}([0,T];C^2_b([0,R]\times [-\ell/2,\ell/2];{{\mathbb R}}^2))$ and $$\begin{aligned}
\|{{\bf u}}\|_{C^{\alpha/2}([0,T];C^2_b([0,R]\times [-\ell/2,\ell/2];{{\mathbb R}}^2))}+\|u_1\|_{C^{\alpha/2}([0,T];C^2_b(\overline{H_R^-};{{\mathbb R}}^2))}
\le c\bigg (\|D_t{{\bf u}}\|_{\boldsymbol{\mathcal Y}_{\alpha}}+\sup_{t\in [0,T]}\|{{\bf u}}(t,\cdot)\|_{2+\alpha}\bigg ).\end{aligned}$$
Taking into account we complete this step of the proof. In particular, from all the above results it follows that $${{\bf u}}(t,\cdot,\cdot)=e^{tL}{{\bf u}}_0+\int_0^te^{(t-s)L}[{\bf f}(s,\cdot,\cdot)+{\mathscr L}{\mathscr M}(\boldsymbol\psi(s,\cdot))]ds
-L\int_0^te^{(t-s)L}{\mathscr M}(\boldsymbol\psi(s,\cdot))ds,\qquad\;\,t\in [0,T].
\label{repr-formula}$$
[*Step 4.*]{} Let $r>0$ and $\boldsymbol{\mathcal C}_r$ be the space of all ${\bf u}\in\boldsymbol{\mathcal Y}_{2+\alpha}$ such that $D^{\gamma_1}_xD^{\gamma_2}_y{\bf u}(\cdot,\cdot,-\ell/2)=D^{\gamma_1}_xD^{\gamma_2}_y{\bf u}(\cdot,\cdot,\ell/2)$, for every $0\le\gamma_1, \gamma_2$ such that $\gamma_1+\gamma_2\le 2$, $\|{\bf u}\|_{\boldsymbol{\mathcal Y}_{2+\alpha}}\le r$ and ${\bf u}(0,\cdot,\cdot)={\bf u}_0$.
In view of Steps 1-3, for every ${\bf u}_0\in B(0,r_0)\subset \boldsymbol{\mathcal Y}_{2+\alpha}$ satisfying the compatibility conditions in the statement of the theorem, we can define the operator $\Gamma$, which to every ${\bf u}\in \boldsymbol{\mathcal C}_r$ (with $r$ sufficiently small norm to guarantee that the nonlinear terms ${\mathscr F}({\bf u}(t,\cdot,\cdot))$ and ${\mathscr H}({\bf u}(t,\cdot,\cdot))$ are well defined for every $t\in [0,T]$) associates the unique solution ${\bf v}$ of the Cauchy problem with ${\bf f}={\mathscr F}({\bf u})$ and $\boldsymbol{\psi}={\mathscr H}({{\bf u}})$. Since the maps ${{\bf u}}\mapsto {\mathscr F}({{\bf u}})$, ${{\bf u}}\mapsto {\mathscr H}({{\bf u}})$ are smooth in $\boldsymbol{\mathcal C}_r$ and quadratic at ${{\bf u}}={\bf 0}$, we can estimate $$\begin{aligned}
\begin{array}{l}
\|{\mathscr F}({{\bf u}})\|_{\boldsymbol{\mathcal Y}_{\alpha}}\le c\|{{\bf u}}\|_{\boldsymbol{\mathcal Y}_{2+\alpha}}^2,\qquad\;\, \|{\mathscr F}({{\bf u}})-{\mathscr F}({\bf v})\|_{\boldsymbol{\mathcal Y}_{\alpha}}\le cr\|{{\bf u}}-{\bf v}\|_{\boldsymbol{\mathcal Y}_{\alpha}},\\[1mm]
\|{\mathscr H}({{\bf u}})\|_{C^{(1+\alpha)/2,1+\alpha}((0,T)\times (-\ell/2,\ell/2);{{\mathbb R}}^2)}\le c\|{{\bf u}}\|_{\boldsymbol{\mathcal Y}_{2+\alpha}}^2,\\[1mm]
\|{\mathscr H}({{\bf u}})\!-\!{\mathscr H}({\bf v})\|_{C^{(1+\alpha)/2,1+\alpha}((0,T)\times (-\ell/2,\ell/2);{{\mathbb R}}^2)}\le cr\|{{\bf u}}-{\bf v}\|_{\boldsymbol{\mathcal Y}_{\alpha}}.
\end{array}
\label{vita}\end{aligned}$$ These estimates combined with show that $r$ and $r_0$ can be determined small enough such that $\Gamma$ is a contraction in $\boldsymbol{\mathcal C}_{\rho}$.
Uniqueness of the solution ${\bf u}$ to follows from standard arguments, which we briefly sketch here. At first, for every $t_0\in[0,T]$, $R,\delta>0$ and ${\bf u}_1\in\boldsymbol {\mathcal X}_{2+\alpha}$, which satisfies the compatibility conditions ${\mathscr B}({\bf u}_1)={\mathscr H}({\bf u}_1)$, ${\mathscr B}_0({\mathscr L}{\bf u}_1+{\mathscr F}({\bf u}_1))=0$ and $D^{\gamma}{\bf u}_1(\cdot,-\ell/2)=D^{\gamma}{\bf u}_1(\cdot,\ell/2)$ for each multi-index $\gamma$ with length at most two, we set $$\begin{aligned}
\boldsymbol{\mathcal Z}_{\delta,R}^{t_0}({\bf u}_1):=\{{\bf u}\in \boldsymbol{\mathcal Y}_{2+\alpha}(t_0,t_0+\delta):{\bf u}(t_0,\cdot)={\bf u}_1, \ \|{\bf u}-{\bf u}_1\|_{\boldsymbol{\mathcal Y}_{2+\alpha}(t_0,t_0+\delta)}\leq R\}.\end{aligned}$$ Given $R>0$ we can determine $r_1>0$ and $\delta>0$ (independent of $t_0$) with $\delta^{\alpha/2}R$ sufficiently small such that, if ${\bf u}_1$ belongs to $B(0,r_1)\subset\boldsymbol{\mathcal X}_{2+\alpha}$, then the Cauchy problem $$\begin{aligned}
\left\{
\begin{array}{lll}
D_t{\bf w}={\mathscr L}{\bf w}+\mathscr F({\bf w}),\\[1mm]
\mathscr B{\bf w}=\mathscr H({\bf w}), \\[1mm]
D^{\gamma_1}_xD^{\gamma_2}_y{\bf w}(\cdot,\cdot,-\ell/2)=D^{\gamma_1}_xD^{\gamma_2}_y{\bf w}(\cdot,\cdot,\ell/2), &&\gamma_1+\gamma_2\le 2 \\ [1mm]
{\bf w}(t_0,\cdot)={\bf u}_1,
\end{array}
\right.
\label{pitenji}\end{aligned}$$ admits a unique solution ${\bf w}\in \mathscr Z_{\delta,R}^{t_0}({\bf u}_1)$. We are almost done. Indeed, let ${\bf u}\in\boldsymbol{\mathcal C}_r$ be the unique fixed point of $\Gamma$, and take $r_0$ small enough such that ${\bf u}\in B({\boldsymbol 0},\rho_1)\subset \boldsymbol{\mathcal Y}_{2+\alpha}$. Assume that ${\bf v}\in\boldsymbol{\mathcal Y}_{2+\alpha}$ is another solution to , and let $t_0>0$ denote the supremum of the set $\{\tau\in[0,T]:{\bf u}(t,\cdot)={\bf v}(t,\cdot), t\in[0,\tau]\}$. Suppose by contradiction that $t_0<T$. Then, both ${\bf u}$ and ${\bf v}$ are solutions in $\boldsymbol{\mathcal Y}_{2+\alpha}(t_0,t_0+\delta)$ to the Cauchy problem , with ${\bf u_1}:={\bf u}(t_0,\cdot)={\bf v}(t_0,\cdot)$. Taking $R\geq 2\max\{\|{\bf u}\|_{\boldsymbol{\mathcal Y}_{2+\alpha}}, \|{\bf v}\|_{\boldsymbol{\mathcal Y}_{2+\alpha}}\}$ large enough and $\delta>0$ small enough, it follows that ${{\bf u}}$ and ${\bf v}$ both belong to $\boldsymbol{\mathcal Z}_{\delta,R}^{t_0}({\bf u}_1)$, so that they do coincide, leading us to a contradiction.
\[remark-nonlinear\] [Since Problem is autonomous, under the same assumptions as in Theorem \[soliera\], for each $a>0$ and $T>0$ there exists a unique solution ${{\bf u}}\in\boldsymbol{\mathcal Y}_{2+\alpha}(a,a+T)$ such that ${{\bf u}}(a,\cdot)={{\bf u}}_0$.]{}
Proof of the main result {#sect-6}
========================
Study of the dispersion relation and the point spectrum
-------------------------------------------------------
Since, we are interested in the instability of the travelling wave solution $(\Theta^{(0)},\Phi^{(0)})$ to Problem -, we need to determine a range of Lewis numbers ${{\rm{{Le}}}}$ which lead to eigenvalues of $L_{\alpha}$ (see Remark \[rmk-5.6\]) with positive real part. In view of Theorem \[banca\], such eigenvalues will lie in $\Omega_k$ (see ). For simplicity, we will look for positive real elements of $\Omega_k$. Note that ${{\rm{{Le}}}}-Y_k$ vanishes for no $\lambda$’s with positive real part. To determine elements of $\sigma(L_{\alpha})$ with positive real part we need to analyze the reduced dispersion relation $$\begin{aligned}
{\mathcal D}_{0,k}(\lambda, {{\rm{{Le}}}})=\exp\bigg (\frac{R}{2}({\rm Le}-1-X_k(\lambda)-Y_k(\lambda,{{\rm{{Le}}}}))\bigg )-1+\theta_iRX_k(\lambda).\end{aligned}$$ We recall that $X_k(\lambda)=\sqrt{1+4\lambda+4\lambda_k}$, $Y_k(\lambda,{{\rm{{Le}}}})=Y_k(\lambda)=\sqrt{{\rm Le}^2+4\lambda {\rm Le}+4\lambda_k}$ and $\lambda_k=4\pi^2k^2\ell^{-2}$ for each $k\in{{\mathbb N}}\cup\{0\}$. Throughout this subsection we assume $\theta_i$ is fixed in $(0,1)$; so is $R>0$ via .
\[leading\_mode\] There exists ${\ell}_0(\theta_i)$ such that, for all $\ell >{\ell}_0(\theta_i)$, ${\mathcal D}_{0,1}(0, {{\rm{{Le}}}})=0$ has a unique root ${{\rm{{Le}}}}_c={{\rm{{Le}}}}_c(1) \in (0,1)$. Moreover, for each $\ell$ fixed as above, there exists a maximal integer $K \geq 1$ such that, for $k\in\{1,\ldots,K\}$, ${\mathcal D}_{0,k}(0, {{\rm{{Le}}}})=0$ has a unique root ${{\rm{{Le}}}}_c(k) \in(0,1)$. Finally, it holds: $$\begin{aligned}
\label{Le_decreasing}
0< {{\rm{{Le}}}}_c(K) \leq \ldots \leq {{\rm{{Le}}}}_c(2) \leq {{\rm{{Le}}}}_c(1).\end{aligned}$$
An easy but formal computation shows that, if ${{\rm{{Le}}}}_c(k)$ is a root of ${\mathcal D}_{0,k}(0,\cdot)$, then $$\begin{aligned}
{{\rm{{Le}}}}_c(k) = 1 + X_k(0) +Y_k(0, {{\rm{{Le}}}}_c(k)) + 2R^{-1}\ln(1-\theta_i R X_k(0)),\end{aligned}$$ or equivalently: $$\label{critic1}
{{\rm{{Le}}}}_c(k) = \displaystyle\frac{(1+X_k(0))[R^2 +2R \log(1-\theta_i R X_k(0))] +2|\log(1-\theta_i R X_k(0))|^2}{R^2(1+ X_k(0)) + 2R \log(1-\theta_i R X_k(0))}.$$ However, Formula makes sense only if $1-\theta_i R X_k(0) >0$. Hence, for each fixed $\ell>0$ there exists $K\in{{\mathbb N}}$ such that $1-\theta_i R X_k(0) >0$ if and only if $k\le K$.
Further, ${{\rm{{Le}}}}_c(k)$ is required to meet the physical requirement that $0<{{\rm{{Le}}}}_c(k)<1$. In this respect, $\ell$ should be large enough: $$\begin{aligned}
{{\rm{{Le}}}}_c(1) = {{\rm{{Le}}}}_0 + \frac{16 \pi^2 \theta_i e^R (1 - \theta_i e^R)}{(2 \theta_i e^R -1) \ell^2} + o(\ell^{-2})\end{aligned}$$ as $\ell\to +\infty$, where ${{\rm{{Le}}}}_0 = R(2e^R-R-2)^{-1}$ belongs to $(0,1)$ see [@BGKS15 Formula (43), p. 2083]. Thus, there exists $\ell_0(\theta_i)>0$ such that, if $\ell>\ell_0(\theta_i)$, then ${{\rm{{Le}}}}_c(1)\in (0,1)$.
Finally, it remains to prove property , for a fixed $\ell>\ell_0(\theta_i)$ which in turn defines the integer $K \geq 1$. The latter property follows from the following estimate (see [@BGZ Proposition 3.1]): $$\begin{aligned}
\left (\sqrt{({{\rm{{Le}}}}_c (k))^2+4\lambda_k}-{{\rm{{Le}}}}_c (k)\right )\frac{d{\hspace{1pt}}{{\rm{{Le}}}}_c(k)}{d\lambda_k} < 4\bigg(1-\frac{\theta_i}{1-\theta_i R}\bigg ).\end{aligned}$$ Obviously, $ \displaystyle \frac{\theta_i}{1-\theta_i R}= \frac{1-e^{-R}}{Re^{-R}}=\frac{e^{R}-1}{R} >1$, which implies that $\displaystyle\frac{d{\hspace{1pt}}{{\rm{{Le}}}}_c(k)}{d\lambda_k}<0$.
In view of Lemma \[leading\_mode\] our focus will be on the case when ${{\rm{{Le}}}}\in (0,{{\rm{{Le}}}}_c(1))$. Hereafter we will simply denote the critical value ${{\rm{{Le}}}}_c(1)$ by ${{\rm{{Le}}}}_c$, keeping in mind that ${{\rm{{Le}}}}_c$ at fixed $0<\theta_i<1$ depends on $\ell > {\ell}_0(\theta_i)$.
\[lem-pasqua\] The function ${\mathcal D}_{0,1}$ is smooth in $[0,\sqrt{\lambda_1}]\times [0,{{\rm{{Le}}}}_c]$. Moreover, $\displaystyle\frac{\partial {\mathcal D}_{0,1}}{\partial \lambda}$ is positive in $[0,\sqrt{\lambda_1}]\times [0,{{\rm{{Le}}}}_c]$, $\displaystyle\frac{\partial {\mathcal D}_{0,1}}{\partial {{\rm{{Le}}}}}$ is positive in $[0,\sqrt{\lambda_1})\times [0,{{\rm{{Le}}}}_c]$ and vanishes on $\sqrt{\lambda_1}\times [0,{{\rm{{Le}}}}_c]$.
The proof of the positivity of $\displaystyle\frac{\partial {\mathcal D}_{0,1}}{\partial {{\rm{{Le}}}}}$ is straightforward and based on the observation that $Y_1(\lambda, {{\rm{{Le}}}}) - \rm Le - 2 \lambda >0$ for $\lambda> \sqrt{\lambda_1}$ and $Y_1(\sqrt{\lambda_1}), {{\rm{{Le}}}})={{\rm{{Le}}}}+2\sqrt{\lambda_1}$. On the other hand, we observe that $$\begin{aligned}
\frac{\partial {\mathcal D}_{0,1}}{\partial \lambda}(\lambda, {{\rm{{Le}}}}) &= -R\exp\bigg (\frac{R}{2}({\rm Le}-1-X_1(\lambda)-Y_1(\lambda, {{\rm{{Le}}}})) \bigg)(X_1^{-1}
+{{\rm{{Le}}}}\, Y_1^{-1})+2(1-e^{-R})X_1^{-1}.\end{aligned}$$ Since ${{\rm{{Le}}}}\, Y_1^{-1}<(X_1)^{-1}$ and ${\rm Le}-1-X_1(\lambda)-Y_1(\lambda, {{\rm{{Le}}}})<-2$, we can estimate $$\begin{aligned}
\frac{\partial {\mathcal D}_{0,1}}{\partial \lambda}(\lambda, {{\rm{{Le}}}}) > 2(1-(R+1)e^{-R})X_1^{-1},\qquad\;\,(\lambda,{{\rm{{Le}}}}_c)\in [0,\sqrt{\lambda_1}]\times [0,{{\rm{{Le}}}}_c],\end{aligned}$$ and the positivity of $\displaystyle\frac{\partial {\mathcal D}_{0,1}}{\partial \lambda}$ in $[0,\sqrt{\lambda_1}]\times [0,{{\rm{{Le}}}}_c]$ follows immediately.
We can now prove the following result.
\[thm-spectrum\] Under the assumptions of Lemma $\ref{leading_mode}$, there exist $\lambda_*\in (0,\sqrt{\lambda_1})$ and a decreasing, continuously differentiable function $\widetilde{\varphi}:(0,{{\rm{{Le}}}}_c) \to (0,\lambda_*)$ such that ${\mathcal D}_{0,1}(\widetilde{\varphi}({{\rm{{Le}}}}), {{\rm{{Le}}}})=0$ for all ${{\rm{{Le}}}}\in (0,{{\rm{{Le}}}}_c)$.
From Lemma \[lem-pasqua\] it follows that $\frac{\partial {\mathcal D}_{0,1}}{\partial {{\rm{{Le}}}}}(0, {{\rm{{Le}}}}_c)>0$. We can thus apply the implicit function theorem, which shows that there exist $\delta_1, \delta_2>0$ and a unique function $\varphi\in C^1([{{\rm{{Le}}}}_c-\delta_1,{{\rm{{Le}}}}_c+\delta_1])$ such that, if $(\lambda,{{\rm{{Le}}}})\in [{{\rm{{Le}}}}_c-\delta_1, {{\rm{{Le}}}}_c+\delta_1]\times [-\delta_2,\delta_2]$ is a root of ${\mathcal D}_{0,1}$, then $\lambda=\varphi({{\rm{{Le}}}})$. In view of the previous lemma, $\varphi$ is a decreasing function. As a byproduct, taking the restriction of $\varphi$ to $[{{\rm{{Le}}}}_c-\delta_1, {{\rm{{Le}}}}_c]$, we have constructed a (small) branch $\lambda=\varphi ({{\rm{{Le}}}})$ of positive roots of ${\mathcal D}_{0,1}(\lambda, {{\rm{{Le}}}})=0$. We may reiterate the implicit function theorem and continue this branch up to a left endpoint $(\lambda_*,{{\rm{{Le}}}}_*) \in [0,\sqrt{\lambda_1}]\times [0,{\rm Le}_c]$. By continuity, ${\mathcal D}_{0,1}(\lambda_*,{{\rm{{Le}}}}_*)=0$. This maximal extension of $\varphi$ is a non-increasing, $C^1$-function $\widetilde{\varphi}:({{\rm{{Le}}}}_*,{{\rm{{Le}}}}_c]\to [0,\lambda_*)$.
To complete the proof, we need to show that ${{\rm{{Le}}}}_*=0$. If ${{\rm{{Le}}}}_*>0$ then $\lambda_*=\sqrt{\lambda_1}$ otherwise, applying the implicit function theorem again, we could extend $\widetilde\varphi$ in a left-neighborhood of ${{\rm{{Le}}}}_*$, contradicting the maximality of $\widetilde\varphi$. On the other hand, it is not difficult to check that ${\mathcal D}_{0,1}(\sqrt{\lambda_1},{{\rm{{Le}}}}_*)\neq 0$ whenever $R>0$. Indeed, using condition we can easily show that $$\begin{aligned}
{\mathcal D}_{0,1}(\sqrt{\lambda_1},{{\rm{{Le}}}}_*)=e^{-R(1+2\sqrt{\lambda_1})}-1+\theta_iR(1+2\sqrt{\lambda_1})=
e^{-R(1+2\sqrt{\lambda_1})}+2\sqrt{\lambda_1}-e^{-R}(1+2\sqrt{\lambda_1})\end{aligned}$$ and the function $x\mapsto f(x)=e^{-x(1+2\sqrt{\lambda_1})}+2\sqrt{\lambda_1}-e^{-x}(1+2\sqrt{\lambda_1})$ vanishes at $x=0$ and its derivative is positive in $(0,+\infty)$.
![Numerical computation of the implicit curve $\lambda = \widetilde{\varphi}({{\rm{{Le}}}})$ for ${{\rm{{Le}}}}\in (0,{{\rm{{Le}}}}_c)$, extended beyond ${{\rm{{Le}}}}_c$. Here $\theta_i=0.75$, $\ell=100$, ${{\rm{{Le}}}}_c\simeq 0.5641$, $\lambda_* \simeq 0.0315$. Note that $\sqrt{\lambda_1}=\pi/50 \simeq 0.0628$.[]{data-label="Implicit_curve"}](implicit.pdf){height="5.5cm" width="8cm"}
\[coro-spettro\] The spectrum of the operator $L$ contains elements with positive real parts. Moreover, the part of $\sigma(L)$ in the right halfplane $\{\lambda\in{{\mathbb C}}: {\rm Re}\lambda\ge 0\}$ consist of $0$ and a finite number of eigenvalues.
By Theorem \[banca\] we know that if $\lambda\neq 0$ is an element in the spectrum of $L$ with nonnegative real part, then it is an eigenvalue and it belongs to $\Omega_k$ for some $k\in{{\mathbb N}}\cup\{0\}$. Hence, $D_k(\lambda)=0$ or, equivalently, ${\mathcal D}_{0,k}(\lambda)=0$ for some $k\in{{\mathbb N}}\cup\{0\}$. As it is immediately seen, each function $\lambda\mapsto {\mathcal D}_{0,k}(\lambda)$ is holomorphic in the halfplane $\Pi=\{\lambda\in{{\mathbb C}}:{\rm Re}\lambda\ge 0\}$ and it does not identically vanish in it. Therefore, its zeroes in $\Pi$ are at most finitely many. Moreover, for each $\lambda\in\Pi$ and $k\in{{\mathbb N}}\cup\{0\}$ we can estimate $$\begin{aligned}
{\rm Re}X_k(\lambda)\ge \sqrt{\frac{1}{2}+2\lambda_k},\qquad\;\,{\rm Re}Y_k(\lambda,{\rm Le})\ge \sqrt{\frac{Le^2}{2}+2\lambda_k},\end{aligned}$$ so that the real part of ${\mathcal D}_{0,k}(\cdot,{\rm Le})$ diverges to $+\infty$, as $k\to +\infty$, uniformly with respect to $\lambda\in\Pi$. As a byproduct, we deduce that there exists $k_0\in{{\mathbb N}}$ such that the nontrivial eigenvalues $\lambda\in\Pi$ lie in $\bigcup_{k=0}^{k_0}\Omega_k$ and this completes the proof.
To prove the main result of this section, we also need the following result which is a variant of [@H80 Theorem 5.1.5] and [@BL00 Theorem 4.3].
Let $X$ be a complex Banach space, $r>0$ and $T_n:B(0,r)\subset X\to X$ $(n\in{{\mathbb N}})$ be a bounded operator such that $T_n(x)=Mx+O(\|x\|^p)$ as $\|x\|\to 0$, for some $p>1$ and some bounded linear operator $M$ on $X$ with spectral radius $\rho>1$. Further, assume that there exists an eigenvector $u$ of $M$ with eigenvalue $\lambda\in{{\mathbb C}}$ such that $|\lambda|^p>\rho$ and that there exists $x'\in X'$ such that $x'(u)\neq 0$. Then, there exist $c>0$ and, for any $\delta>0$, $x_0\in B(0,\delta)$ and $n_0\in{{\mathbb N}}$ $($depending on $\delta)$ such that the sequence $x_0,\ldots,x_{n_0}$, where $x_n=T_n(x_{n-1})$ for any $n=1,\ldots,n_0$, is well defined and $|x'(x_{n_0})|\ge c|x'(u)|$. \[pisapia\]
Without loss of generality, we assume that $\|u\|= 1$ and $\|x'\|\leq 1$. Moreover, we choose $a,b>0$ such that $\|T_n(x)-Mx\|\leq b\|x\|^p$ for each $x\in B(0,a)\subset X$ and $n\in{{\mathbb N}}$. Since $|\lambda|^p>\rho$, we can fix $\eta>0$ such that $|\lambda|^p>\rho+\eta$ and, from the definition of the spectral radius of a bounded operator, we can also determine a positive constant $K$ such that $\|M^n\|_{L(X)}\leq K(\rho+\eta)^n$ for any $n\in{{\mathbb N}}$. Finally, we fix $\delta>0$, choose $n_0\in{{\mathbb N}}$ be such that $|\lambda|^{-n_0}<\delta$, and set $x_0:=\sigma u |\lambda|^{-n_0}$, where $\sigma\in (0,1)$ is chosen so as to satisfy the conditions $$\begin{aligned}
\sigma\leq \frac{a}{2}, \qquad\;\, \frac{2^p b K}{|\lambda|^p-\rho-\eta}\sigma^{p-1}\leq \frac{1}{2}|x'(u)|.
\label{brasile-mex}\end{aligned}$$
To begin with, we prove that the sequence $x_0,\ldots,x_{n_0}$ is well defined. For this purpose, in view of the condition in it suffices to check that, if $x_k$ is well defined, then $\|x_k\|\le 2 \sigma |\lambda|^{k-n_0}$. We prove by recurrence. Clearly, $x_0$ satisfies this property. Suppose that the claim is true for $k=0,\ldots,n-1$. Then, $x_n$ is well defined and it easy to check that $$x_n=M^nx_0+\sum_{k=0}^{n-1}M^{n-1-k}(x_{k+1}-Mx_k)=M^nx_0+\sum_{k=0}^{n-1}M^{n-1-k}(T_{k+1}(x_k)-Mx_k).
\label{form-1}$$ Thus, we can estimate $$\begin{aligned}
\label{riverdale}
\|x_n\|\le |\lambda|^n\|x_0\|+Kb\sum_{k=0}^{n-1}(\rho+\eta)^{n-1-k}\|x_k\|^p.\end{aligned}$$ Let us consider the second term in the right-hand side of , which we denote by $S_n$. Since, we are assuming that $\|x_k\|\le 2 \sigma |\lambda|^{k-n_0}$ for each $k=0,\ldots,n-1$, we can write $$\begin{aligned}
S_n\le & 2^pKb\sigma^p |\lambda|^{p(n-n_0-1)}\sum_{k=0}^{n-1}\bigg (\frac{\rho+\eta}{|\lambda|^p}\bigg )^{n-1-k}\le \sigma |\lambda|^{n-n_0}\frac{2^pKb}{|\lambda|^p-\rho-\eta}\sigma^{p-1}\end{aligned}$$ and, using the second condition in and the fact that $x'$ has norm which does not exceed $1$, we conclude that $$\begin{aligned}
\label{suits}
S_n \le \frac{1}{2}\sigma|\lambda|^{n-n_0}|x'(u)|\leq \frac{1}{2}\sigma |\lambda|^{n-n_0}.\end{aligned}$$ Since $|\lambda|^n\|x_0\|\leq \sigma|\lambda|^{n-n_0}$, from and the claim follows at once.
To conclude the proof, it suffices to use with $n=n_0$, as well as and again, to estimate $$\begin{aligned}
|x'(x_{n_0})|
\geq & |x'(M^{n_0}x_0)|-|x'(S_{n_0})|\geq \sigma |x'(u)|-\frac{1}{2}\sigma|x'(u)|=\frac{1}{2}\sigma|x'(u)|.\end{aligned}$$ The assertion follows with $c=\sigma/2$.
Now, we can state and prove the following theorem.
\[thm-main-1\] Let $0<\theta_i<1$ be fixed, $\ell>\ell_0(\theta_i)$ as in Lemma $\ref{leading_mode}$, ${{\rm{{Le}}}}_c= {{\rm{{Le}}}}_c(1)$ defined by . Then, for each ${{\rm{{Le}}}}\in (0,{{\rm{{Le}}}}_c)$, the null solution ${{\bf u}}$ of Problem is poinwise unstable with respect to small perturbations in $\boldsymbol{\mathcal X}_{2+\alpha}$. More precisely, there exists a positive constant $C$ such that for each $y_0\in{{\mathbb R}}$ and $\delta>0$ there exist $ \widetilde {\bf u}_0, {\bf u}_0^*\in B(0,\delta)\subset\boldsymbol{\mathcal X}_{2+\alpha}$ and $\widetilde n, n_*\in{{\mathbb N}}$ depending on $\delta$ such that $\min\{|\widetilde u_2(\widetilde n,0,y_0)|, |u_1^*(n_*,R,y_0)|\}\ge C$, where $\widetilde{\bf u}=(\widetilde u_1,\widetilde u_2)$ and ${\bf u}_*=(u_1^*,u_2^*)$ denote the solution to the Cauchy problem with initial datum $\widetilde {\bf u}_0$ and ${\bf u}_0^*$, respectively.
We split the proof into two steps. The first one is devoted to prove an estimate which will allow us to apply Lemma \[pisapia\]. Then, in Step 2, we prove the pointwise instability.
. The smoothness of $\Upsilon$ implies that there exists $c>0$ such that $\|\Upsilon({\bf v}_0)\|_{2+\alpha}\leq c\|{\bf v}_0\|_{2+\alpha}$, for each ${\bf v}_0\in D(L_{\alpha})$ with sufficiently small norm. Here, $\Upsilon$ is defined in Lemma \[lemma-5.7\]$(ii)$. Fix $r$ so small such that $\|{\bf v}_0+\Upsilon({\bf v}_0)\|_{2+\alpha}\le r_0$ for each ${\bf v}_0\in \overline{B(0,r)}\subset D(L_\alpha)$, where $r_0=r_0(1)$ is defined in the statement of Theorem \[soliera\].
For $n\in{{\mathbb N}}$, let $R_n:B(0,\rho)\subset \widetilde D(L_\alpha)\to \widetilde D(L_\alpha)$ be the map defined by $R_n({\bf v}_0)=P({\bf u}_n(n,\cdot,\cdot))$ for each ${\bf v_0}\in B(0,\rho)$ (see Lemma \[lemma-5.7\]), where ${\bf u}_n$ is the solution to problem with initial condition ${\bf u}_0={\bf v}_0+\Upsilon({\bf v}_0)$ at $t=n-1$. (Note that, by Lemma \[lemma-5.7\]$(ii)$, ${\bf u}_0$ satisfies the compatibility conditions in Theorem \[soliera\]. Further, by Remark \[remark-nonlinear\], ${\bf u}_n$ is well defined in the time domain $[n-1,n]$.) We claim that $$\begin{aligned}
\label{treno}
\|R_n({\bf v_0})-e^{L_\alpha}{\bf v}_0\|_{2+\alpha}\leq c\|{\bf v}_0\|^2_{2+\alpha},\qquad\;\,{\bf v}_0\in B(0,r).\end{aligned}$$ Estimate follows from the integral representation of the solution of Problem and estimates . Indeed, again by Remark \[remark-nonlinear\], ${\bf u}_n(n,\cdot,\cdot)$ is the value at $t=1$ of the solution ${{\bf u}}$ to Problem , with ${{\bf u}}(0,\cdot)={{\bf u}}_0$ and, by the proof of Theorem \[soliera\] (see, in particular, formula ), $$\begin{aligned}
{{\bf u}}(1,\cdot,\cdot)-e^L{{\bf u}}_0=\int_0^1e^{(1-s)L}[{\mathscr F}({{\bf u}}(s,\cdot,\cdot))\!+\!{\mathscr L}{\mathscr M}({\mathscr H}({{\bf u}}(s,\cdot,\cdot))]ds\!-\!
L\int_0^1e^{(1-s)L}{\mathscr M}({\mathscr H}({{\bf u}}(s,\cdot,\cdot))ds.\end{aligned}$$ Since ${\mathscr F}$ and ${\mathscr H}$ are quadratic at $0$, it follows immediately that $\|{\bf u}(1,\cdot,\cdot)-e^L{\bf u}_0\|_{2+\alpha}\leq c\|{\bf v}_0\|^2 _{2+\alpha}$. Noting that $P({\bf u}(1,\cdot,\cdot)-e^{L_{\alpha}}{\bf u}_0)=
R_n({\bf v}_0)-e^{L_{\alpha}}{\bf v}_0$, formula follows at once.
[*Step 2.*]{} Let us begin by proving that there exists $C>0$ such that for any $y_0\in{{\mathbb R}}$ and $\delta>0$ there exists ${\bf u}_0\in B(0,\delta)\subset \boldsymbol{\mathcal X}_{2+\alpha}$ and $n_0\in{{\mathbb N}}$ depending on $\delta$ such that $|u_2(n_0,0^+,y_0)|\geq C$, where ${\bf u}=(u_1,u_2)$ is the solution to with initial datum ${\bf u}_0$ at time $t=0$. For this purpose, we want to apply Lemma \[pisapia\] with $X=D(L_{\alpha})$ endowed with the norm of $\boldsymbol{\mathcal X}_{2+\alpha}$. To begin with, we observe that, by Corollary \[coro-spettro\], there exists only a finite number of eigenvalues of $L$ (and hence of $L_{\alpha}$) with positive real part. From the spectral mapping theorem for analytic semigroups it thus follows that the spectral radius $\rho$ of the operator $M=e^{L_{\alpha}}$ is larger than one and there exists an eigenvalue $\lambda$ such that $|\lambda|=\rho$. Let us fix $y_0\in{{\mathbb R}}$ and $\delta>0$. It is not difficult to show that a corresponding eigenfunction is the function ${\bf w}=(w_1 e_1(\cdot - 2\pi\ell^{-1}y_0)), w_2 e_1(\cdot - 2\pi\ell^{-1}y_0)))$, where $$\begin{aligned}
&w_1(x)=e^{\nu_1^+x}\chi_{(-\infty,0]}(x)+(c_1e^{\nu_1^- x}+c_2e^{\nu_1^+x})\chi_{(0,R)}(x)+c_3e^{\nu_1^-x}\chi_{[R,+\infty)}(x),\\
&w_2(x)=(d_1e^{\mu_1^- x}+d_2e^{\mu_1^+ x})\chi_{[0,R)}(x)+d_3 e^{\mu_1^-x}\chi_{[R,+\infty)}(x),\end{aligned}$$ for every $x\in{{\mathbb R}}$ and $$\begin{aligned}
\begin{array}{llll}
&\displaystyle c_1=\frac{e^{(X_1+\mu_1^+)R}(\theta_iRX_1-1)}{e^{\mu_1^+R}-e^{\nu_1^+R}},\qquad\;\,
&\displaystyle c_2=\frac{e^{\mu_1^+R}}{e^{\mu_1^+R}-e^{\nu_1^+R}},\qquad\;\, &\displaystyle c_3 = \frac{\theta_iRe^{(X_1+\mu_1^+)R}X_1}{e^{\mu_1^+R}-e^{\nu_1^+R}}\vspace{2mm} \\
&\displaystyle d_1=-\frac{{\rm Le}({\rm Le}+\mu_1^+)e^{\nu_1^+R}X_1}{(e^{\mu_1^+R}-e^{\nu_1^+R})Y_1},\qquad\;\, &\displaystyle d_2 = -({{\rm{{Le}}}}+\mu_1^-)d_1,\qquad\;\,
&\displaystyle d_3 = (1-e^{Y_1R})d_1, \vspace{2mm} \\
& \displaystyle \nu_1^{\pm}=-\frac12\pm\sqrt{1+4\tilde\lambda+4\lambda_1},
& \displaystyle \mu_1^{\pm}=-\frac{\rm le}2\pm\sqrt{{\rm Le}^2+4{\rm Le}\tilde\lambda+4\lambda_1}.
\end{array}\end{aligned}$$ Note that $$\begin{aligned}
w_2(0^+,y_0)=(d_1+d_2)=\frac{{\rm Le}X_1e^{\nu_1^+R}X_1}{(e^{\mu_1^+R}-e^{\nu_1^+R})Y_1}\neq 0.\end{aligned}$$ Hence, if we set $x'({\bf f})=f_2(0^+,y_0)$ for any ${\bf f}\in D(L_\alpha)$, then $|x'({\bf w})|\neq 0$. As in Step 1, we fix $r>0$ such that $\|{\bf v}_{0,j}+\Upsilon({\bf v}_{0,j})\|_{{2+\alpha}}\le r_0$ for $j=1,2$ for each ${\bf v}_0={\bf v}_{0,1}+i{\bf v}_{0,2}\in B(0,r)\subset D(L_{\alpha})$. By Theorem \[soliera\] both ${\bf u}(n,{\bf v}_{0,1}+\Upsilon({\bf v}_{0,1}),n-1)$ and ${\bf u}(n,{\bf v}_{0,2}+\Upsilon({\bf v}_{0,2}),n-1)$ are well defined for any $n\in{{\mathbb N}}$. We can thus introduce the operator $T_n:B(0,r)\subset D(L_{\alpha})\to D(L_{\alpha})$ ($n\in{{\mathbb N}}$) by setting $T_n({\bf v}_0)=P{\bf u}(n,{\bf v}_{0,1}+\Upsilon({\bf v}_{0,1}),n-1)+iP{\bf u}(n,{\bf v}_{0,2}+\Upsilon({\bf v}_{0,2}),n-1)$ for each ${\bf v}_0\in B(0,r)$, where $P$ is the projection in Lemma \[lemma-5.7\](i). By the arguments in Step 1 we deduce that $\|T_n({\bf v}_0)-e^{L_\alpha}{\bf v}_0\|_X\leq C\|{\bf v}_0\|_X^2$ for some positive constant $C$ and each ${\bf v}_0\in B(0, r)$. We can thus apply Lemma \[pisapia\] with $M=e^{L_\alpha}$, $p=2$ and conclude that there exist $c>0$ and, for each $\delta>0$, a function ${\bf v}_0={\bf v}_{0,1}+i{\bf v}_{0,2}\in B(0,\delta)\subset D(L_{\alpha})$ and $n_0\in{{\mathbb N}}$ such that ${\bf v}_n=T_n({\bf v}_{n-1})$ is well defined for each $n\in\{1,\ldots,n_0\}$ and $|x'({\bf v}_{n_0})|\ge c$. Since ${\bf v}_{n_0}=P{\bf u}(n_0,{\bf v}_{0,1}+\Upsilon({\bf v}_{0,1}),0)+iP{\bf u}(n_0,{\bf v}_{0,2}+\Upsilon({\bf v}_{0,2}),0)$, where ${\bf u}(n_0,{\bf u}_0,0)$ denotes the value at $n_0$ of the unique solution to problem with initial datum ${\bf u}_0$ at time $t=0$, we have so proved that $$\begin{aligned}
|(P{\bf u}(n_0,{\bf v}_{0,1}+\Upsilon({\bf v}_{0,1}),0))_2(0^+,y_0)|^2+|(P{\bf u}(n_0,{\bf v}_{0,2}+\Upsilon({\bf v}_{0,2}),0))_2(0^+,y_0)|^2\ge c^2.
\label{russia}\end{aligned}$$ By definition, $P=I-{\mathscr N}{\mathcal B}_*$ (see Lemma \[lemma-5.7\](i)) and $({\mathscr N}{\bf u})_2(0^+,\cdot)=0$ for any function ${\bf u}$. Hence, $(P{\bf u}(n_0,{\bf v}_{0,j}+\Upsilon({\bf v}_{0,j}),0))_2(0^+,y_0)=({\bf u}(n_0,{\bf v}_{0,j}+\Upsilon({\bf v}_{0,j}),0))_2(0^+,y_0)$ for $j=1,2$. From it thus follows that there exists $\bar j$ such that $|({\bf u}(n_0,{\bf v}_{0,\bar j}+\Upsilon({\bf v}_{0,\bar j}),0))_2(0^+,y_0)|\ge c/2$ and the thesis follows with $C=c/2$, $\widetilde n=n_0$ and $\widetilde {\bf u}={\bf u}(\widetilde n,{\bf v}_{0,\bar j}+\Upsilon({\bf v}_{0,\bar j}),0)$.
To prove the existence of ${\bf u}_*$ as in the statement of the theorem, it suffices to take as $x'$ the functional defined by $x'({\bf f})=f_1(R,y_0)$ for each ${\bf f}\in D(L_\alpha)$. The missing easy details are left to the reader.
[Clearly, Theorem \[thm-main-1\] implies the $C^{2+\alpha}$-instability of the null solution ${{\bf u}}$ to .]{}
[The $C^{2+\alpha}$-instability of the null solution ${{\bf u}}$ to can be directly obtained by applying [@H80 Theorem 5.1.5], taking advantage of Step $1$ of Theorem \[thm-main-1\] and arguing as in [@BL00 Corollary 4.5]. Finally, it can also be proved in a slightly different way adapting the arguments in [@lorenzi-3 Theorem 3.4].]{}
From Theorem \[thm-main-1\] we can now easily derive the proof of the main result of this paper.
Taking the changes of variables and unknown in Subsections \[fix-domain\] and \[sect-3.2\] into account, the result in Theorem \[thm-main-1\] allows us to conclude easily that the normalized temperature $\Theta$ and the normalized concentration of deficient reactant in problem - are unstable with respect to two dimensional $C^{2+\alpha}$ perturbations. Similarly, using formulae and and again Theorem \[thm-main-1\], we can infer that there exist initial data $(\widetilde \Theta,\widetilde \Phi)$ and $(\Theta_*,\Phi_*)$ with $C^{2+\alpha}$-norm, arbitrarily close to the travelling wave solution such that the trailing front $G$ (resp. the ignition front $F$) to problem - with initial datum $(\Theta(0,\cdot),\Phi(0,\cdot))=(\widetilde \Theta,\widetilde \Phi)$ (resp. $(\Theta(0,\cdot),\Phi(0,\cdot))=(\Theta_*,\Phi_*)$) is not arbitrarily close to $0$ (resp. $R$).
Numerical simulation {#sect-7}
====================
In this section, we are going to use some high resolution numerical methods, including Chebyshev collocation and Fourier spectral method (see, e.g., [@STW; @WWQ; @WZW]). We consider the problem in the finite domain $\Omega = \Omega_- \cup {\hspace{1pt}}\Omega_0 \cup {\hspace{1pt}}\Omega_+ = ([-A,0] \cup [0,R] \cup [R, B]) \times [-\ell/2,\ell/2]$, where $A>0$ and $B>0$ are large enough, see Figure \[domain\]. The independent variables are $-A\le\xi\le B$, $-\ell/2 < \eta < \ell/2$.
![Computational domain.[]{data-label="domain"}](domain.pdf){height="3cm" width="7cm"}
The linear system
-----------------
The linearized system around the null solution of System reads: $$\begin{aligned}
\left\{
\begin{array}{ll}
u_\tau = u_\xi + u_{\xi\xi} + u_{\eta\eta}, & \mbox{in} \quad \Omega,\\[1mm]
w=0, & \mbox{in} \quad \Omega_-, \\[1mm]
w_\tau = w_\xi + {{\rm{{Le}}}}^{-1}(w_{\xi\xi} + w_{\eta\eta}), &\mbox{in} \quad \Omega_0 \cup \Omega_+,\\[1mm]
\end{array}
\right.
\label{linear}\end{aligned}$$ with $$\begin{aligned}
{\mathscr B}(u,w)=0.
\label{IC-0R}\end{aligned}$$
We map $\Omega_-, \Omega_0$ and $\Omega_+$ to $\mathbb{D} = [-1,1] \times [0, 2 \pi]$. Then, we consider in $\mathbb{D}$ the system for the three pairs of unknowns $(u_1, w_1)$, $(u_2, w_2)$ and $(u_3, w_3)$, corresponding respectively to $(u,w)$ in $\Omega_-, \Omega_0$ and $\Omega_+$. The new independent variables are denoted by $x\in [-1,1]$ and $y \in [0, 2 \pi]$.
Therefore, System - is equivalent to: $$\begin{aligned}
\left\{
\begin{array}{ll}
D_\tau u_{1} = \frac{2}{A} D_x u_{1} + \frac{4}{A^2} D_{xx} u_{1} + \frac{4\pi^2}{\ell^2} D_{yy} u_{1}, \; w_1 \equiv 0, & \\[1mm]
D_\tau u_{2} = \frac{2}{R} D_x u_{2} + \frac{4}{R^2} D_{xx} u_{2} + 4\frac{\pi^2}{\ell^2} D_{yy} u_{2}, & \\[1mm]
D_\tau w_{2} = \frac{2}{R} D_x w_{2} + \frac{4}{{{\rm{{Le}}}}R^2} D_{xx} w_{2}+ \frac{4 \pi^2}{{{\rm{{Le}}}}{\hspace{1pt}}\ell^2} D_{yy} w_{2}, & \\[1mm]
D_\tau u_{3} = \frac{2}{B-R} D_x u_{3} + \frac{4}{(B-R)^2} D_{xx} u_{3} + \frac{4\pi^2}{\ell^2} D_{yy} u_{3}, & \\[1mm]
D_\tau w_{3} = \frac{2}{B-R} D_x w_{3} + \frac{4}{{{\rm{{Le}}}}\,(B-R)^2} D_{xx} w_{3}+ \frac{4 \pi^2}{{{\rm{{Le}}}}\,\ell^2} D_{yy} w_{3} ,&
\end{array}
\right. \label{systemin3domain}\end{aligned}$$ together with the boundary conditions: $$\begin{aligned}
\left\{
\begin{array}{ll}
u_1(-1) = u_3(1) = w_3(1) = 0, \quad u_1(1) = u_2(-1), \\[1mm]
w_2(-1) = \frac{2{{\rm{{Le}}}}}{A} D_x u_{1}(1) -\frac{2{{\rm{{Le}}}}}{R} D_x u_{2}(-1), \quad D_x w_{2}(-1)= -\frac{{{\rm{{Le}}}}R}{2} w_2(-1),\\[1mm]
D_x w_{2}(1) = \frac{R}{B-R} D_x w_{3}(-1)+\frac{{{\rm{{Le}}}}R}{2} ( w_3(-1)- w_2(1) ), \\[1mm]
D_x u_{3}(-1) = \frac{B-R}{R} D_x u_{2}(1)-\frac{B-R}{2{{\rm{{Le}}}}} ( w_3(-1)-w_2(1)), \\[1mm]
u_2(1) = -\frac{2\theta_i R}{B-R} D_x u_{3}(-1) + 2 \theta_i D_x u_{2}(1) , \quad u_3(-1) = u_2(1).
\end{array}
\right. \label{systemin3domain-IC}\end{aligned}$$
Let us give a brief overview of the numerical method. Hereafter, we denote by $(u,w)$ any pair of unknowns $(u_i, w_i)$, $1\leq i \leq 3$. We discretize System - using a forward-Euler explicit scheme in time. Then, we use a discrete Fourier transform in the direction $y \in (0, 2 \pi)$, namely: $$\begin{aligned}
u(x,y) = \sum_{k=-N_y/2}^{N_y/2} \hat{u}_k(x) e^{ik y}, \qquad\;\, w(x,y) = \sum_{k=-N_y/2}^{N_y/2} \hat{w}_k(x) e^{ik y},\end{aligned}$$ and $$\begin{aligned}
D_{yy}u(x,y) = -\sum_{k=-N_y/2}^{N_y/2} k^2 \hat{u}_k(x) e^{ik y}, \qquad\;\, D_{yy}w(x,y) = - \sum_{k=-N_y/2}^{N_y/2} k^2 \hat{w}_k(x) e^{ik y}.\end{aligned}$$ Finally, we use a Chebyshev collocation method in $x \in (-1, 1)$. Let $\{ l_j(x)\}_{j=0}^{N_x}$ be the Lagrange polynomials based on the Chebyshev-Gauss-Lobatto points $\{ x_j\}_{j=0}^{N_x} = \{ \cos(j\pi/N_x)\}_{j=0}^{N_x}$. We set: $$\begin{aligned}
\hat{u}_k(x) = \sum_{j=0}^{N_x}\hat{u}_{kj} l_j(x), \qquad\;\, \hat{w}_k(x) = \sum_{j=0}^{N_x}\hat{w}_{kj} l_j(x).\end{aligned}$$ Denoting the differential matrix of order $m$ associated to $\{ x_j\}_{j=0}^{N_x} $ by $D^m = (d_{ij}^{(m)})_{i,j = 0,\cdots,N_x}$, where $d_{ij}^{(m)} = l_j^{(m)}(x_i)$ and $l_j(x_i) = \delta_{ij}$, we eventually obtain $$\begin{aligned}
& \hat{u}_k(x_i) = \sum_{j=0}^{N_x}\hat{u}_{kj} \delta_{ij}, \qquad\;\, D_x \hat{u}_k(x_i) = \sum_{j=0}^{N_x}\hat{u}_{kj} d_{ij}^{(1)}, \qquad\;\, D_{xx} \hat{u}_k(x_i) = \sum_{j=0}^{N_x}\hat{u}_{kj} d_{ij}^{(2)},\\
& \hat{w}_k(x_i) = \sum_{j=0}^{N_x}\hat{w}_{kj} \delta_{ij}, \qquad\;\, D_x\hat{w}_k(x_i) = \sum_{j=0}^{N_x}\hat{w}_{kj} d_{ij}^{(1)}, \qquad\;\, D_{xx}\hat{w}_k(x_i) = \sum_{j=0}^{N_x}\hat{w}_{kj} d_{ij}^{(2)}.\end{aligned}$$
As initial data, we take $w_2(0,\cdot)= \varepsilon \big(1+\sin^2(y)\big)$, $y \in [0, 2 \pi] $, which corresponds to $\xi = {R}/{2}$; the other unknowns are taken as $0$. The following pictures are for $\varepsilon = 10^{-2}, A=B=10, \ell =100, \Delta t = 10^{-3}$. As expected, the two profiles blow up for Lewis number below critical.
The fully nonlinear system
--------------------------
By treating the nonlinearities explicitly, we can use the same algorithm as in the linear case. In the coordinates $(\xi,\eta)$, we approximate the mollifier $\beta(\xi)$ by the following trapezoid, see Figure \[fig-mollifiers\]: $$\begin{aligned}
\beta(\xi) = \left\{
\begin{array}{ll}
2 + {\xi / \delta}, \quad & -2 \delta <\xi< -\delta, \\[1mm]
1, \quad & -\delta \le \xi \le \delta, \\[1mm]
2 - {\xi / \delta}, \quad &\delta <\xi< 2 \delta, \\[1mm]
0, \quad &\text{elsewhere,}
\end{array}\right.\end{aligned}$$
![Approximation of the mollifier $\beta(\xi)$.[]{data-label="fig-mollifiers"}](mollifiers.pdf){height="3cm" width="10cm"}
Then, the fully nonlinear terms in System , namely ${\mathscr F}_1$, ${\mathscr F}_2$, ${\mathscr G}_1$, ${\mathscr G}_2$, ${\mathscr G}_3$, as well as ${\varrho}_\tau, \ {\varrho}_{\tau\xi}$, have to be computed separately in eight intervals, as they are zero elsewhere: $[-2 \delta, - \delta] \cup \cdots \cup [R+\delta, R+2 \delta]$, see Figure \[fig-mollifiers\]. We refer to the Appendix for the formulas.
Hereafter, we present some typical numerical results for the fully nonlinear problem. Simulations were performed using a standard pseudo-spectral method with small time step $\Delta t = 10^{-5}$ and small amplitude of initial perturbations (of order $10^{-4}$ to $10^{-3}$), to ensure sufficient accuracy.
We consider the situation when ignition temperature is fixed at $\theta_i=0.75$ and $\ell=100$, in such a case ${{{\rm{{Le}}}}}_c \simeq 0.5641 $. Three significant values of the Lewis number have been chosen in the interval $(0,{{\rm{{Le}}}}_c)$, namely ${{\rm{{Le}}}}=0.10$, ${{\rm{{Le}}}}=0.20$ and ${{\rm{{Le}}}}=0.50$. Figures \[ignition-trailing-fronts-1050\] and \[ignition-trailing-temperature-1050\] represent the interface patterns and temperature levels. Numerically, we observe that, after a rapid transition period, a steady configuration consisting of “two-cell” patterns for the ignition and trailing interfaces is established. These simulations confirm the theoretical analysis, that is instability of the planar fronts for ${{\rm{{Le}}}}\in (0,{{\rm{{Le}}}}_c)$.
\
\
\
Acknowlegments {#acknowlegments .unnumbered}
==============
We are grateful to Grisha Sivashinsky for bringing reference [@BGKS15] to our attention. We also thank Peter Gordon for fruitful discussions and advices. The work of Z.W. was supported by National Natural Science Foundation of China (Grants No. 11761007, 11661004), Research Foundation of Education Bureau of Jiangxi Province (Grant No. GJJ160564) and Doctoral Scientific Research Foundation (Grant No. DHBK2017148).
[11]{}
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Appendix
========
[^1]: $\dag$ Corresponding author
[^2]: ${\mathcal I}_3$ belongs to $C_b({{\mathbb R}};{{\mathbb C}})$. Indeed, $H\in L^1(S^{+}_0;{{\mathbb C}})$ as it follows observing that $|e^{\frac{\pi}{2\ell}(x\pm 4iy)}-1|\ge e^{\frac{\pi}{2\ell}x}-1$, which implies the inequality $|K(x,y)|\le e^{\left (\frac{\rm Le}{2}-\frac{\pi(k_0-1)}{2\ell}\right )x}\min \{2(e^{\frac{\pi}{2\ell}x}-1)^{-1},c(x^2+y^2)^{-1/2}\}$ for every $(x,y)\in S_0^{+}$. This shows that ${\mathcal I}_3$ is bounded in ${{\mathbb R}}^2$. Moreover, ${\mathcal I}_3$ is the uniform limit as $\varepsilon\to 0^+$ of the function $\int_{\varepsilon}^{+\infty}ds\int_{-\ell/2}^{\ell/2}K(s,\cdot-\eta)g(s,\eta)d\eta$, which is clearly continuous in ${{\mathbb R}}$ thanks to the above estimate for $K$. Hence, ${\mathcal I}_3$ is itself continuous in ${{\mathbb R}}$.
[^3]: see Remark \[rmk-5.6\]
| null | minipile | NaturalLanguage | mit | null |
You've heard people say that film is a collaborative art. Boy, is it. And we want these new ideas. You'll have a visual effects producer for a visual effects company go, 'Hey, you know, you asked for this but there's an idea to maybe tweak it a little bit.' And then you get something that is a thousand times better. And it's great and goes in the movie. I love the notion that everybody playing in this Marvel Studios sandbox feels an ownership and feels excited to be a part of it - none more so than myself. That's another reason I love the tags. It forces people to actually look and acknowledge the thousands of people responsible for the experience you just had.
Of course it's: Do people like it? Do people have an experience that they are still talking about a day later? A week later? A month later? And secretly, I've always hoped for years later. I've always thought you can't really tell an impact of a film until years have gone by. Now, years have gone by for us. Ten years since Iron Man. And the fact is that all of these characters are still...well, are more popular than they've ever been. And people want to see them again and see how they've changed, like Thor did in Ragnarok, like Iron Man has over the movies, like Captain America has...that's exciting for us.
To me, that is a true testament. I think of the movies that I loved as a kid - like Back to the Future, like the Star Wars movies - that still feel as relevant today as they've ever been. That's the true test. It's been 30-plus, 40 years. We've got another 30 years to figure out what our impact has been. But 10 years on, it feels pretty good.
As one of the biggest, most successful and most beloved film franchises in history, the MCU has a profound impact on Hollywood. So much so that many people know to stay all the way till the very end of the credits to not miss any scenes that tease what is to come in a future film.Marvel Studios Chief Kevin Feige spoke about why he enjoys having post credits sequences at the end of each of his films. Ultimately, the scenes allow Marvel to spotlight creativity and highlight all of the talented people that helped create the film.Feige also spoke about how he judges whether a Marvel film has been successful. Yes, the movies are box office juggernauts but its more than just money for Feige as he values the impacts that the films have on audiences. If people are still talking long term about the films after their release, then they have achieved true success.As Feige notes, the characters in the MCU such as the core Avengers have only become more popular as time has gone by since the start of the franchise. He hopes the MCU will have the same longevity and cultural relevance as some of his favorite films growing up, noting that so far the MCU is at a great place after a decade.Do you think the MCU will have the same success 30 years from now? Be sure to share your thoughts below about what Feige said. | null | minipile | NaturalLanguage | mit | null |
Yu and his associates, from a diverse range of businesses, have finally given up on attempts to buy Birmingham after six months of negotiations with Carson Yeung.
Yu, who was once Yeung’s main football advisor at St Andrew’s, said: ‘The consortium and myself maybe will now consider a takeover of another English club instead.’
And that looks likely to be Wednesday for around £15m.
Mandaric, who bought Wednesday for £1 in 2010 having spent more than £7m clearing debts, reports the club are losing £5m a year.
Former England coach Duncan Fletcher discarded world-class bowler Graeme Swann early in his career because he thought off-spinners were finished unless they could deliver a doosra — the delivery that spins the other way.
That is according to another ex-England boss, Mickey Stewart, who says in an interview with The Cricket Paper on Wednesday: ‘I have a lot of respect for Fletch but I think the only way you bowl a doosra is to throw it.’
Ruthless? Duncan Fletcher discarded Graeme Swann
Sky cricket pundit Sir Ian Botham has an important family date before the third Ashes Test.
His son Liam, a former county cricketer for Hampshire as well as a professional in both rugby codes, is getting married for the second time in Mustique to Lisa Harrison, a Belfast magazine publisher.
Liam’s best man is business tycoon David Ross, an independent director of the British Olympic Association.
Cause to celebrate: Sir Ian Botham's son is getting married
Geoff Boycott, who was paid £180,000 in 1987 — the biggest advance at the time for a sports book — by publishers Macmillan for his autobiography is to write another life story. Simon & Schuster have agreed a contract with Boycott, although not on comparative terms, for his second memoirs that will be published in October 2014, which is 50 years on from his England Test debut against Australia.
The first book was written before Boycott became cricket’s best-known pundit.
Beeb stay in running
BBC Sport have shown faith in drugs-tainted track and field by agreeing a six-year renewal of their TV contract with UK Athletics.
This will see the Beeb have rights to all UKA events, including two Diamond League meetings a season, until 2020.
The Olympic Stadium will host one of those showcase events every year, making the estimated £25m deal that much more attractive.
Controversial: But the doping scandal surrounding Tyson Gay and Asafa Powell has not deterred the BBC
And the Beeb — who have become increasingly selective in their bidding strategy — staying with athletics after the doping scandal involving big-name sprinters Tyson Gay and Asafa Powell, is a major boost for the sport.
The current BBC contract, which is also a six-year agreement, expires after next year’s Commonwealth Games in Glasgow.
BT Sport, who have already signed up Robin van Persie, Gareth Bale, Rio Ferdinand, Alex Oxlade-Chamberlain, Daniel Sturridge and Marouane Fellaini as ambassadors, are talking to Chelsea’s Frank Lampard about joining their team. However, all of BT’s ambassador payments won’t add up to the £20m Sky are paying David Beckham.
Major attraction: Sky spent big money to recruit David Beckham
Liverpool’s controversial former communications director Jen Chang has made a success of his new media role at New York Cosmos, with the management there praising him for his social media expertise.
Chang left Anfield after the club were forced to apologise for threats he made in a restaurant to supporter Sean Cummins over his blog and Liverpool insider Twitter postings. | null | minipile | NaturalLanguage | mit | null |
Liberal goo-goos and “good citizens” of all stripes are fond of saying that “We must continue to obey the law while we work to change it.” Every day I become more convinced that this approach gets things precisely backwards. Each day’s news demonstrates the futility of attempts at legislative reform, compared to direct action to make the laws unenforceable.
The principle was stated most effectively by Charles Johnson, one of the more prominent writers on the libertarian Left (“Counter-economic Optimism,” Rad Geek People’s Daily, Feb. 7, 2009):
“If you put all your hope for social change in legal reform … then … you will find yourself outmaneuvered at every turn by those who have the deepest pockets and the best media access and the tightest connections. There is no hope for turning this system against them; because, after all, the system was made for them and the system was made by them. Reformist political campaigns inevitably turn out to suck a lot of time and money into the politics—with just about none of the reform coming out on the other end.”
Far greater success can be achieved, at a tiny fraction of the cost, by “bypassing those laws and making them irrelevant to your life.”
Johnson wrote in the immediate context of copyright law. In response to an anti-copyright blogger who closed up shop in despair over the increasingly draconian nature of copyright law, he pointed to the state’s imploding ability to enforce such laws. The DRM of popular music and movie content is typically cracked within hours of its release, and it becomes freely available for torrent download. Ever harsher surveillance by ISPs in collusion with content “owners” is countered by the use of anonymizers and proxies. And the all-pervasive “anti-songlifting” curriculum in the publik skools, in today’s youth culture, is met with the same incredulous hilarity as a showing of “Reefer Madness” to a bunch of potheads.
The weakest link in any legal regime, no matter how repressive on paper, is its enforcement.
I saw a couple of heartening news items this past week that illustrate the same principle. First, a judge in Missoula County Montana complained that it would soon likely become almost impossible to enforce anti-marijuana laws because of the increasing difficulty of seating juries. In a recent drug case, so many potential jurors in the voir dire process declared their unwillingness to enforce the pot laws that the prosecution chose to work out a plea deal instead. The defendant’s attorney stated that public opinion “is not supportive of the state’s marijuana law and appeared to prevent any conviction from being obtained simply because an unbiased jury did not appear available under any circumstances …” The same thing happened in about sixty percent of alcohol cases under Prohibition.
Public agitation against a law may be very fruitful indeed — but not so much by creating pressure to change the law as by creating a climate of public opinion such that it becomes a dead letter.
Another morale booster is the rapidly improving technology for recording cops, which Radley Balko (a journalist whose chief bailiwick is police misbehavior) describes in the January issue of Reason Magazine (“How to Record the Cops“). Miniaturized, unobtrusive video cameras with upload capability can instantly transmit images for storage offsite or stream content directly to the Internet — which means that the all-too-frequent tendency of thuggish cops to seize or destroy cameras will result only in video of the very act of seizure or destruction itself being widely distributed on the Internet. “Smile, Officer Friendly — you’re on Candid Camera!”
The practical implication, according to Balko, is this:
“Prior to this technology, prosecutors and the courts nearly always deferred to the police narrative. Now that narrative has to be consistent with independently recorded evidence. And as examples of police reports contradicted by video become increasingly common, a couple of things are likely to happen: Prosecutors and courts will be less inclined to uncritically accept police testimony, even in cases where there is no video, and bad cops will be deterred by the knowledge that their misconduct is apt to be recorded.”
As such technology becomes cheap and ubiquitous, police will increasingly operate in an atmosphere where such monitoring is expected — and feared — as a routine part of their job. Even the most stupid and brutal of cops will always carry, in the backs of their minds, the significant possibility that this might be one of the times they’ve got an audience.
New technology, empowering the individual, will soon deter cops in a way that decades of civilian review boards and police commissions failed to achieve.
So the goo-goos have it backwards. Don’t waste time trying to change the law. Just disobey it.
23 comments
Interesting take on politics in general and political theory in particular. The classic liberal vs the traditional conservative. Read it, leave a comment, follow, tell a friend to do the same. Become part of the solution by joining the revolution http://confederateunderground.blogspot.com/2010/1…
Good article which makes several valid points. However, many people have difficulty breaking the law because most of us were conditioned from birth to obey, and because they overestimate the risk of being caught. But in fact, laws are nothing more than the whims of a handful of politicians, and most laws can be easily, routinely, and safely broken with no more effort than that required to coordinate one's wardrobe or operate an automobile safely. And, like anything, the more you do it, the easier it gets.
"I saw a couple of heartening news items this past week that illustrate the same principle. First, a judge in Missoula County Montana complained that it would soon likely become almost impossible to enforce anti-marijuana laws because of the increasing difficulty of seating juries. In a recent drug case, so many potential jurors in the voir dire process declared their unwillingness to enforce the pot laws that the prosecution chose to work out a plea deal instead. The defendant’s attorney stated that public opinion “is not supportive of the state’s marijuana law and appeared to prevent any conviction from being obtained simply because an unbiased jury did not appear available under any circumstances…” The same thing happened in about sixty percent of alcohol cases under Prohibition."
Have you a link to this? I'd like to forward it to New Zealand's marijuana activism lists.
I personally don't think the division between working with and outside the system is necessarily so stark. You are almost undoubtably right in an American context, where civil rights exist on paper only and the system is hopeless. But in New Zealand MPs aren't at an out-of-touch exalted social level, radicals can get into Parliament, and minor parties are hopeless. There's simply a great deal more functioning democracy, and this is understood to extend to political civil disobedience. I've participated in several public smoke-ins on Parliament's steps, and the same people organising the march also try to influence the government through the Green Party and the Law Commission- and they're quite optimistic about success. It's just a different socio-political atmosphere.
I agree that working within the system pretty much guarantees ceding the playing field to your opponent, who after all is also the referee, ruling body and rulemaking body. Politics isn't a sanctioned athletic competition. Neither is business. I see a similar weakness in the goo-goo's economic analog, the 'social entrepreneurship' fad. You can't tame the business world into something humane by running a business. Inevitably your principals (i.e. financiers) will claim veto power over your principles.
In addition to the parliamentary system, which is more democratic as you say, you've also got the advantage of a smaller sized polity.
In the case you mention it sounds like they're optimistic about success precisely because there is a wing of the movement that is willing to engage in direct action. The pro-smoke movement is essentially good cop/bad cop-ing the government. Actionistas flouting the law on one hand and causing a legitimacy crisis and then offering legislators a way to save face by "compromising" with the political wing on the other. That's great movement strategy. (although a bit different than what Kevin is outlining above.)
But in order to do something like that you need a movement that is very clear about whose team they're on and what the teams even are for that matter. In the US the goo-goo's don't even know what they're fighting for, much less who the enemy is, and at the end of the day perpetuation of state power is alway more important to them than any particular policy change. They're company men to the bitter end.
Don't forget the example of homeschooling. It is not legal in all 50 states because a legislature controlled by the teacher's union wanted to help kids, but because parents decided to homeschool, despite laws previously making it illegal. Eventually the law caught up with reality (although the attempt to co-opt homeschooling was also there – but a healthy population of parents not even compliant with the modest regulation of state approved homeschooling also dooms that strategy to failure).
Laws are made by thugs, to turn people into sheep. They should be treated with the contempt they deserve.
Aster: I'm glad Josh supplied the link, because I deleted it from my email and emptied the trash after I got through writing this.
You're probably right that the state is less virulent. Although coercion is the defining feature of the state qua state, the state is made up of human beings, some of them doing pretty much what they'd be doing if things were organized voluntarily. The state is really just a bunch of human beings doing stuff, who can do it in a more or less "statelike" manner.
Lori: That would be true if dependence on outside finance were a constant. But one of the most revolutionary changes in the world today is the imploding capital outlays required to undertake production. If it's possible to run a business on a hundredth the capital it took to perform the same function twenty years ago, and laws for suppressing competition from such businesses are becoming unenforceable, then it should have a serious taming effect on the business world.
My recent post Homebrew Industrial Revolution in Kindle Format
I disagree with the idea that most people are inherently obedient; they will break ANY law that they feel is 'pointless'.
We all know 'law and order' types (like my own dear mum) who
* exceed the 'speed limit'; or
* overtake on double-whites; or
* do a U-turn where it's prohibited, or
* run a yellow-turning-red light.
Likewise, almost everybody I know will admit to having driven while "probably over the limit" at some point in their lives.
So the 'germ' is there; all we have to do is convince people to widen their perceived sert of laws that are 'ridiculous' (my mother's defence when I pointed out she was overtaking on double-white lines – and exceeding the speed limit while doing so – when she picked me up from the rail station on Dec 23rd!)
At the end of the day, people will not let 'bad law' interfere with the expression of their preferences: hence folks who want drugs, get them (albeit at prices that embody a huge risk premium to compensate suppliers); just as folks who wanted a drink during Prohibition, got one.
Radley Balko's point about recording the cops is something that we of the sousveillance movement have written about since micro-cams became cost-effective (e.g., the MD80); uploadability makes it so much better (which is why my home surveillance system streams onto non-detelable web storage).
On reading the headline, here was I thinking that C4SS was about to embrace Jim Bell's "undermine the operational effectiveness of state drones by running Death pools on bad cops" mechanism… perhaps y'all think it's still a little early to be there yet.
I certainly don't – which is why I think everybody ought to read http://jya.com/ap.htm to get a handle on the most efficient (in terms of 'bang for the bang') mechanism for helping undermine the ability of state goons to tyrannise us Mundanes.
GT, one problem – both ethical and tactical – I have with direct action against public servants is that, at least in Australia, they comprise a curious blend of good (ethical, competent) and bad (unscrupulous, sometimes incompetent) people. The lower levels have more of the good ones, who often act to make a bad system treat people right, finding loopholes etc. in a botched and far too rapidly changing framework of laws and regulations – they are anti-Kevin Rudds, since it is the Kevin Rudds who cause all that and unknowingly rely on the others to bail them out. In my experience, nearly all branches of the public service mostly have these good people at the interface with the public, with the principal exceptions being the police (who are about half good and half bad) and social services (who until recently were a dumping ground for the other branches, but have been improving). This means not only that targeting low level public servants – whether with prejudice or with lesser forms of harassment, e.g. boycotts and/or poll taxes when and where “join and sabotage” could take over local governments – could remove potential allies but also that it could alienate and polarise them against the public they currently identify with.
As well as the earlier Irish “join and sabotage” methods (that used Westminster), one parallel we could learn from is what the Irish did from 1916 on: they targeted low and middle level public servants who couldn’t be protected cost-effectively in sufficient numbers (violently, as suited their particular circumstances), but selectively, using inside information they gained from those public servants who were of their persuasion – even getting information from at least one middle level police officer who could get at the files within Dublin Castle itself.
I wholeheartedly agree with you except for one little quibble:which is why my home surveillance system streams onto non-detelable web storage
Unless you own the servers and they are in the possession of someone you trust, don't bet on the information remaining yours for all time. The Wikileaks/Amazon fiasco clearly highlighted the dangers of "the cloud" with regard to information security.
That's a very good point, Todd S.; were it possible to uplod in real time to freenet or some other such distributed mechanism, I would do that instead. Like quite a few people, I have cancelled my use of AMZN's AWS/S3 services for static site components: they are free to lick the whip, but I am free to refuse to deal with them when they make it clear that they will roll over the moment uncle Stupid tells them to… while providing an implausible rationale – in the best traditions of US-CEO bullshit.
Given that fully a half-dozen of my close family are in the 'Defence' establishment (4 at EL1 or higher, 2 in the SES), I hope that they are part of a 'join and sabotage' operation… but my hopes aren't high. The idea of getting information on PS types is a good one, too – that is used all the time by the intelligence community, and forms a good part of the basis for Wikileaks' 'insurance file'.
The problem with the 'good guys at the bottom' idea is that it's only ever been half-true; one might be the beneficiary from time to time of inexplicable leniency from those of the Fat Blue Whine, but those who join it have a very similar psychotype to those who join the military (ignore the officer caste, whose motivations are different).
The history of police corruption inquiries is also replete with the 'primrose way' – whereby apparently-upright police are suborned to perjury on behalf of a corupt colleague, or 'forced' to participate in corruption as some form of 'loyalty test'.. where the loyalty being tested is loyalty to a corrupt conspiracy, not to the 'ideals' they are supposed to represent. ('Ideals' being PR for 'enforcement of whatever set of dicates spew out of the collective deliberations of the tax-parasites of the political class').
Also, consider the mindset of a man who is prepared to wield deadly force against one of his own class, at the whim of the political class (who would be SHATTERED if their daughters married constables – although the constable would look forward to rapid rise through the ranks).
As Thoreau famously said in 'Civil Disobedience': "The mass of men serve the state thus, not as men mainly, but as machines, with their bodies. They are the standing army, and the militia, jailers, constables, posse comitatus, etc. In most cases there is no free exercise whatever of the judgment or of the moral sense; but they put themselves on a level with wood and earth and stones; and wooden men can perhaps be manufactured that will serve the purpose as well. Such command no more respect than men of straw or a lump of dirt."
In any case, the true 'OrgA' version of the Liberty Pool concentrates solely on those members of the state's droog squads who have done wrong to somebody and have gone unpunished by the State's review processes (there are hundreds of examples in the US, every year; the example of Jean Charles de Manezes is a good one for the UK). The core aim of the limited violence – and subsequent information campaign to the colleagues of the target – is a 'chilling effect' whereby
(1) the price of being a brutal jerk are raised such that the marginal would-be brutal jerk decides it is in his best interest to refrain from brutality; and
(2) the perceived 'blind side' career risk to being a cop per se is raised, thereby forcing a tilt in the cost-benefit nalysis undertaken by would-be recruits (although many of them are stupid enough to be ATTRACTED by the risk – there being, as Adam Smith said, "scarce a man alive who, when in tolerable health and spirits, does not over-estimate his chances of sucess").
The darker, 'OrgB' pools simply respond to prices; it doesn't matter if the target has done nothing at all, it's a simple matter that somehow their name ends up "in the hat" with sufficient money behind it for the market to do its work (currently that's about $500 for a brutal beating, and $2500 for a dirt-nap; psychotics abound and can be found in vengeance portals all over the world).
There are even folks who hold the 'Little Eichmann' view of anybody who so much as lifts a pen on behalf of the machinery of tyranny; thus clerks, cleaning staff – in fact the whole government workforce – become 'valid' objects (and since they are 'soft targets' they are easier, too). I don't hold to that view, but only because I think that concentrating on the brutalisers will suffice (government finances being in the shape they are) and the brutalisers are far more vulnerable than they think (e.g., when they're not wearing their Kevlar storm-trooper dressups; the adress of many of them can be had at low cost).
As to whether little Daniel Miralles (who should not be allowed anywhere near children: anyone who sends their kids to LILA should be warned) is a concern: if he was genuinely doing me harm, the 'OrgB' discussion above applies. I have no qualms about massively asymmetric retaliatory violence – my own military days sociopathised me nicely, and I know precisely where to procure OrgB services should it cross my mind to do so.
That said, I hold the view that intellectual property is fiction, and that therefore my reputation is not my property – so there is nothing of mine to which I can claim violence is being done. Principles are often annoying; in this specific instance the urge to abandon them surfaces sporadically.
Further to the "reputation not being one's own" idea, when thinking about conducting a campaign of that nature, you have to consider the likelihood that your adversary will participate, then escalate.
For example, let's assume that Miralles can convince someone whose opinion I care about that everything he claims is true; the odds of any serious consequences are minimal. The interests of 'non-aligned' individuals will not lead them to do anything except perhaps write something on the internet. In the scheme of things, that's not going to change the taste of my toast.
Now, let's assume that someone became convinced that Daniel Miralles was, say, a pedophile… those close to him might (and should) dismiss it out of hand, but they're not the target audience. The number of pedophiles being beaten to twitching messes as a result of OrgB pools is growing faster than Wikileaks document repositories. All it requires, is a plausible narrative in the right receptacle.
As I've said to Daniel before, it's silly to expect a 5'5" man to understand the 'long' game when he can't see far unless standing on a chair. This is precisely the sort of thing that happens when the French take on the Saxons; the Gallic 'sneak attack' feels good, but eventually it's always humiliation for poor Froggo.
hahaha.
A reasonably-spec'd low-wattage server machine at a reasonable price. Never gonna run much on it, obv… but good enough that TOR recommends it for those wanting to set up a node. (It's about the spec of a spare machine I have running my freenet node on Ubuntu).
Here is the email we received from Geoffrey Transom's girl friend parents after they received a pack of evidences from us of what they did in France. Fact.
DM
Patrick Varney
to me
show details Mar 19
Dear Daniel and Jennifer,
No, sad to say, after her initial emotional response, Sarah has not been able to raise the subject and we are reluctant to see her so upset again. She didn't speak to us for some weeks and is still very guarded with us.
I believe that she will have to face up to what she has been involved iin before she can get on with her life. But that would mean she sees Geoff for what he is, and I can see that this is too much to ask for the moment.
It is heartening to see some actual discussion of the "enforcement" aspect of laws. Unfortunately it has been my observation that the majority of online discussion is aimed at the origination and/or continuation of legislation (and likely TV and print media also, when the "problem" of some law is discussed at all) .
For more than 5 years online I have written – in my own articles and comments online (as well as verbally in person) – that the enforcers are the key element in maintaining/growing the State, and therefore in bringing about its withering away.
I strongly urge that everyone please keep the following in mind always. The politicians and bureaucrats – rulers – do *not* get out into the field and enforce their own legislation/decrees/mandates/etc or even their own opinions. Instead they – including Obama together with his chief underlings – depend on the enforcers to do the dirty work, both domestically via the enormous numbers of agencies federal, state and local and also the branches of the military. Therefore the enforcers are the key!! Politicians and bureaucrats simply talk and write, even when it is to give orders – whether to enforcers or directly to ordinary citizens (via Internet blogs/vids, phone, snail and email pronouncements for the latter).
Strong negative Social Preferencing – withdrawal or refusal of voluntary association with the reasons made public – towards government enforcers who continue in that role after attempts to logically persuade them of their errors (offers of assistance in obtaining new productive jobs is a good idea for those known personally), is the needed step. Public identification (photos, names and location) of these continuing government enforcers will enable others to also Socially Preference against them, thereby increasing the social pressure to change their employment, their major interaction behavior – use cellphone cameras and the Internet to the fullest. This selective (discriminating) association to exclude those who cause harm is a potentially *very* powerful method of non-violent action, referred to as ostracism by many down through the ages. It is included in Gene Sharp's 2nd volume (of 3), "The Politics of Nonviolent Action", Chapter 4, "The Methods of Social Noncooperation".
Even in the current very unfree societies (of which the US is a major one), negative Social Preferencing can be effectively used to influence individual social behavior and the actions of the State. I wrote about this practice in general most recently (April 2009) in "Tax/Regulation Protests are Not Enough: Relationship of Self-Responsibility and Social Order" – http://selfsip.org/focus/protestsnotenough.html
So I hope in the very near future to see more articles and comments from others making use of the understanding that government enforcers are *the* key to turn – pun intended – for the withering away of the State.
Hey Kevin, absolutely correct and many are practicing the various free market approaches such as boycotts, civil disobedience etc. to correcting free market problems of government interventions. I want everyone however to look at a proposal that I have been working on. A little deception is in place however, as I'm am trying to appeal to everyone, not just anarchists and mini-archists. My goal is to create a replacement judicial system that is superior to our current injustice system. Shouldn't be very difficult from a systems approach. Once created, the system would be tested using real life Supreme Court cases. We will publish our opinion in sequence with theirs. That's when the fund should start. Getting Public appeal to enbrace the our opinions over the crappy Supreme Court.
My recent post Cost to Produce Web Based Judicial System | null | minipile | NaturalLanguage | mit | null |
Q:
Does calling the constructor directly outside the class implicitly create an object?
class A
{
public:
A() { cout << "constructor" << endl; };
A(const A &) { cout << "copy constructor" << endl; };
void operator()(int a, int b) { cout << "a+b" << endl; }
};
void set(A visit) {
visit(1, 2);
}
int main()
{
set(A());
return 0;
}
The A() constructor directly creates an access to an object that is passed to the set parameter by an implicit object. The constructor also has the ability to create objects?
A:
Yes. The manner in which you are constructing the object in the line:
set(A());
is called functional notation type conversion. This is specified in the section on constructors in the CPP standard draft.
A constructor is used to initialize objects of its class type. Because constructors do not have names, they are never found during name lookup; however an explicit type conversion using the functional notation will cause a constructor to be called to initialize an object. [ Note: The syntax looks like an explicit call of the constructor. — end note ]
complex zz = complex(1,2.3);
cprint( complex(7.8,1.2) );
An object created in this way is unnamed.
| null | minipile | NaturalLanguage | mit | null |
Albert Gnaegi Center for Health Care Ethics
The Albert Gnaegi Center for Health Care Ethics is an independent health sciences academic unit of Saint Louis University. The center has a high academic output and offers Doctorate of Philosophy programmes in Health Care Ethics and clinical bioethics. The current director, Jeffrey Bishop, joined the Center in July 2010 from Vanderbilt University and was previously at the Peninsula College of Medicine and Dentistry in the United Kingdom and the University of Texas. He is the author of The Anticipitory Corpse: Medicine, Power, and the Care of the Dying and sits on the editorial board of The Journal of Medicine and Philosophy and The Journal of Christian Bioethics, both Oxford Journals. Other notable staff include Griffen Trotter, M.D., PhD, and Tobias Winright, PhD, who holds the Hubert Maeder Endowed Chair in Health Care Ethics and is Associate Professor of Theological Ethics in the Department of Theological Studies at Saint Louis University.
References
External links
Category:Bioethics
Category:Bioethics research organizations
Category:Medical ethics
Category:Saint Louis University | null | minipile | NaturalLanguage | mit | null |
Definitive childlessness in women with multiple sclerosis: a multicenter study.
The frequency of definitive childlessness in women with multiple sclerosis (MS) may be higher than in the general population. MS may also affect decisions on the delivery procedure and on breast-feeding issues. Aim of the study was to assess the frequency of childlessness and its possible causes, the proportion of cesarean deliveries (CD), and the frequency of breast-feeding in patients and controls who have reached the end of their reproductive period. Female MS patients (>43 years) and controls (>45 years) filled out a questionnaire. We enrolled 303 patients and 500 controls. MS was associated with a higher frequency of childlessness (22 vs 13%) and less patients were in a stable relationship (83 vs 89%). There was no difference in the reported rates of infertility and miscarriages, while elective abortions were more frequent in patients (20 vs 12%). MS did not significantly affect the frequency of CD or of breast-feeding. MS-related reasons for childlessness, reported by 16% of childless patients, included disability/fear of future disability, fear of genetically transmitting MS, fear of not starting/discontinuing treatments, and discouragement by physician. Definitive childlessness is more frequent in women with MS compared to controls. A portion of voluntary childlessness may be avoided through correct/tailored information to patients. | null | minipile | NaturalLanguage | mit | null |
Q:
Trouble with AsyncTask: where have I gone wrong?
I'm trying to use AsyncTask to connect to a web service. When I launch the app it closes unexpectedly. What am I doing wrong? Here's my code:
public class Main extends Activity {
private static final String CONSUMER_KEY = "dsfjfsdksfdjl322342";
private static final String CONSUMER_SECRET = "fwefer234242424";
private TextView mText;
ProgressDialog progress = null;
TextView mTextView = null;
/** Called when the activity is first created. */
@Override
public void onCreate(Bundle savedInstanceState) {
super.onCreate(savedInstanceState);
setContentView(R.layout.main);
mTextView = (TextView)findViewById(R.id.textView);
//new ConnectThinkNearTask().execute();
new ConnectToThinkSomeAPITask().execute("http://api-someapi.api.com/1.0/offers/40.7144,-74.0060.json");
}
private class ConnectToSomeAPITask extends AsyncTask<String, Void, String> {
@Override
protected String doInBackground(String...urls) {
String response = "";
for(String url: urls) {
OAuthConsumer consumer = new DefaultOAuthConsumer(CONSUMER_KEY,
CONSUMER_SECRET);
consumer.setTokenWithSecret("", "");
try {
URL url1 = new URL(url);
HttpURLConnection request = (HttpURLConnection) url1.openConnection();
// sign the request
consumer.sign(request);
// send the request
request.connect();
if(request.getResponseCode()==200) {
response = "Sorry, failed to connect to x";
// mText.setText("Sorry, failed to connect to x");
// } else if(request.getResponseCode()==401) {
response = "Congrats, you're connected to x!";
//mText.setText("Congrats, you're connected to x!");
} else
response = "Whatever you're asking for, it ain't a valid HTTP request...";
// mText.setText("Whatever you're asking for, it ain't a valid HTTP request...");
} catch (Exception e) {
e.printStackTrace();
}
} return response;
}
@Override
protected void onPostExecute(String result) {
mTextView.setText(result);
}
}
}
And here's the log:
05-09 21:34:00.219: ERROR/AndroidRuntime(29878): FATAL EXCEPTION: AsyncTask #1
05-09 21:34:00.219: ERROR/AndroidRuntime(29878): java.lang.RuntimeException: An error occured while executing doInBackground()
05-09 21:34:00.219: ERROR/AndroidRuntime(29878): at android.os.AsyncTask$3.done(AsyncTask.java:200)
05-09 21:34:00.219: ERROR/AndroidRuntime(29878): at java.util.concurrent.FutureTask$Sync.innerSetException(FutureTask.java:273)
05-09 21:34:00.219: ERROR/AndroidRuntime(29878): at java.util.concurrent.FutureTask.setException(FutureTask.java:124)
05-09 21:34:00.219: ERROR/AndroidRuntime(29878): at java.util.concurrent.FutureTask$Sync.innerRun(FutureTask.java:307)
05-09 21:34:00.219: ERROR/AndroidRuntime(29878): at java.util.concurrent.FutureTask.run(FutureTask.java:137)
05-09 21:34:00.219: ERROR/AndroidRuntime(29878): at java.util.concurrent.ThreadPoolExecutor.runWorker(ThreadPoolExecutor.java:1068)
05-09 21:34:00.219: ERROR/AndroidRuntime(29878): at java.util.concurrent.ThreadPoolExecutor$Worker.run(ThreadPoolExecutor.java:561)
05-09 21:34:00.219: ERROR/AndroidRuntime(29878): at java.lang.Thread.run(Thread.java:1096)
05-09 21:34:00.219: ERROR/AndroidRuntime(29878): Caused by: java.lang.NoClassDefFoundError: oauth.signpost.basic.DefaultOAuthConsumer
05-09 21:34:00.219: ERROR/AndroidRuntime(29878): at org.androidstack.thinknear.Main$ConnectToThinkNearTask.doInBackground(Main.java:45)
05-09 21:34:00.219: ERROR/AndroidRuntime(29878): at org.androidstack.thinknear.Main$ConnectToThinkNearTask.doInBackground(Main.java:1)
05-09 21:34:00.219: ERROR/AndroidRuntime(29878): at android.os.AsyncTask$2.call(AsyncTask.java:185)
05-09 21:34:00.219: ERROR/AndroidRuntime(29878): at java.util.concurrent.FutureTask$Sync.innerRun(FutureTask.java:305)
05-09 21:34:00.219: ERROR/AndroidRuntime(29878): ... 4 more
A:
Ok, well the logcat was helpful. This happens to me a lot, when I just add libraries to the build path by configuring the build path by selecting user libraries. For some reason, I need to always created a folder called libs in my project, then add the jars and click to add to build path. It's working now!
| null | minipile | NaturalLanguage | mit | null |
Containers for light-sensitive strip or sheet materials, especially those for photographic films and papers are known. By way of example, such containers are illustrated and descried in U.S. Pat. No. 5,860,613 to Holland the teachings of which are incorporated by reference as if fully set forth herein. Such containers commonly comprise an elongated opening or slit through which the strip or sheet material of film is withdrawn from the container. To prevent visible light from entering the container and prematurely exposing the light-sensitive material, light-locking or light-shielding members long have been provided on either side of the opening, between which the material is drawn. | null | minipile | NaturalLanguage | mit | null |
Wednesday, February 23, 2011
ICESPIKE™ Product Review
As a winter runner, I am faced with many challenges when I step out the door. Weather conditions can be dressed for, reflective gear can be worn for safety, but a slip and fall is by far the biggest worry. Some shoes are made with winter running in mind, using differing densities of rubber on the sole to create lugged soles that give you traction in most conditions. Most conditions, but not all. Thus, winter runners add traction devices for an added layer of security. These devices range from strap on spikes to wire coils to sheet metal screws. All can be effective, yet all have their drawbacks.ICESPIKE™ is a spin on the traditional screw shoe.
“ICESPIKE™ is for runners who want to keep their winter training on track and outdoors. With ICESPIKE™ you have non-slip grip shoes without the additional pressure on the top of the foot or the extra bulk underneath the shoe associated with traditional strap-on traction devices. ICESPIKE™ creates non-slip running shoes with unparalleled traction on ice and snow. When your winter running shoes are outfitted with ICESPIKE™, you retain your natural feeling of freedom and range of movement! The ICESPIKE™ system is so lightweight and effective you don’t even know you’re wearing them! ICESPIKE™ is the only traction system light enough to put in 20+ mile long runs without causing injury, muscle fatigue or alterations in running gait.
Using "screw shoes"? ICESPIKE™ makes these far superior upgrades: ICESPIKE™ ice spikes are 10 times more durable, have researched layout pattern for proper gait and engineered design for extreme traction. Simply installing ICESPIKE™ ice spikes on the sole of any shoe creates an ice running shoe, ice walking shoe, or ice hiking boot that is unsurpassed in safety and durability by any other product. The long-lasting hardness and integrity of the grip of ICESPIKE™ ice spikes has no equal. Our cold-rolled, tool quality steel of the ICESPIKE™ shoe system ice spike will outlast your shoes, yet still can be removed at any time using the ICESPIKE™ installation tool. ICESPIKE™ will not damage the shoe sole.
ICESPIKE™ is for walkers, hikers, and trail runners who want to enjoy the outdoor beauty of winter. Any footwear can be converted into non-slip walking shoes, non-slip hiking boots, or winter trail running shoes with ICESPIKE™. Having ICESPIKE™ on your winter walking shoes gives you peace of mind on the most treacherous terrain.
ICESPIKE™ gives you the freedom and security to participate in any winter outdoor activity. ICESPIKE™ is a premium product made and packaged in the USA.”
As someone who loves to run on trails, this is an absolute necessity for winter running. I was interested first and foremost how simple this product was to use, then how well it worked, and finally how it felt on pavement.
Installation is quite simple. Since the shoes I chose had a dark sole, I looked around the house for a white paint marker or correction fluid, but to no avail. I found some lime green stickers (the “it” color this season) and used those instead.
Apart from some initial muscle necessary to get the screws started, getting the screws in is not difficult. I left the stickers in place in order to have the ICESPIKEs™ show up more clearly on the dark sole.
To make this a realistic test, I installed the ICESPIKEs™ on one shoe only and headed out. First part of the run was on cement and pavement, as I ran to a forest preserve near home, beautiful place to run.
There was a marked difference between the two shoes. First, it felt strange on the cement to have the additional screws underfoot, and it took a while to get used to the sound. I purposefully ran across ice patches and some of the trails were nothing but ice, giving a good test of the efficacy of this product. The foot with the spikes did not slip a single time, while the other did. Simply put, this product works.
The ICESPIKEs™ give three benefits: safety, options, and confidence. After running on a variety of surfaces, I felt safer planting the shoe with spikes, especially on ice, though the shoe felt slightly more grippy on asphalt and cement as well. Options comes from the fact that, with this product installed, a winter run can be undertaken in even the most difficult conditions, so one does not have to opt for the treadmill unless so desired. Confidence came from knowing that planting the spikes on the iciest uphill or downhill would result in traction.
Now I wouldn't recommend anyone else attach the spikes to one shoe only - the height difference made it somewhat awkward to run. Also, I've done the hard work for you - the spikes made an apparent difference.
If you are a winter runner and want safety, options, and confidence, this is a product that truly delivers.
Disclaimer: This product was sent to me for review purposes, courtesy of ICESPIKES via Mesh Marketing. I was not compensated in any other way for the review, was not obligated to give it a positive review, and all opinions are my own. Some information in this review was taken from the company website.
I'd suggest drilling a shallow couple mm starter hole during installation of the spikes. I'd also modify it with a small washer to absolutely make sure that the spike was not able to force itself all the way through the sole of the shoe.
Haven't pulled them out, but from what I've read the soles somewhat heel themselves. Adding a washer might make it more likely for the spikes to pull out. Inserting them in a lug makes it seemingly unlikely that they would penetrate the sole.
I like this idea ... plus if you carry the screwdriver with you ... you can jab it through the head of the snow-plow driver deciding not to finish plowing the road you are running on and instead turning off the road to buy himself a coffee.
Kovas, thanks SO much for your wonderful & honest review of our product!
FYI, for you warm weather dwellers, Icespikes are handy on tracks & dirt trails as well, as illustrated by Dani Ashford, who just became the fastest woman to run a marathon on an indoor track (using Icespikes) (http://danerunsalot.blogspot.com/2011/02/sandbox-indoor-trail-marathon-recap.html)
@Patrick (regarding adding a washer) Icespike was created with a reverse, sharp, serrated washer under the spike to grip against the rubber soles, thus not allowing the spike to loosen and be lost. If you add a washer you are taking away an important part of the system. You are also, by adding the washer, shortening the distance that the threads penetrate, thus taking much of the holding power of the spike. The design of the spike length will not penetrate to the foot if directions on the package are followed; this has been tested in 100 of shoes and if installed correctly won’t damage the shoes or the wearer.
Running, Skiing, and Endurance Sports - Patagonia.com
REI: Gear for the Great Outdoors
UnderArmour - I WILL
Outdoor DIVAS - Adventure Gear for Active Women
Rock 'n' Roll Race Series Discount!
Many Ways to Connect!
Kovas Palubinskas
I’m a transplanted Southern Californian living in the Midwest. It’s taken some time, and plenty of searching, to appreciate the nature we have here. In the Midwest we don’t have the jagged peaks or deep canyons others do, but we do have trails to run and hike, rivers to canoe and kayak, and forests to explore. In winter we make do with ski hills, ride fat tire bikes, and snowshoe to revisit favorite spots.
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If you would like a product reviewed or become a blog sponsor, shoot me an email. Multisport, outdoor, winter sports, nutrition, food, book and kid-friendly products are my mainstays, but all pitches are welcome.
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Midwest Multisport Life Blog Disclosure Policy
This blog is a personal blog written and edited by me. For questions about this blog, please contact Kovas Palubinskas at [email protected]. This blog accepts forms of cash advertising, sponsorship, paid insertions or other forms of compensation. This blog abides by word of mouth marketing standards. We believe in honesty of relationship, opinion and identity. The compensation received may influence the advertising content, topics or posts made in this blog. That content, advertising space or post will be clearly identified as paid or sponsored content. The owner(s) of this blog is compensated to provide opinion on products, services, websites and various other topics. Even though the owner(s) of this blog receives compensation for our posts or advertisements, we always give our honest opinions, findings, beliefs, or experiences on those topics or products. The views and opinions expressed on this blog are purely the bloggers' own. Any product claim, statistic, quote or other representation about a product or service should be verified with the manufacturer, provider or party in question. This blog does not contain any content which might present a conflict of interest. This policy is valid from 20 April 2010. To get your own policy, go to http://www.disclosurepolicy.org/ | null | minipile | NaturalLanguage | mit | null |
This invention relates to a nickel-base superalloy article having a protective layer containing aluminum and a reactive element deposited on its surface, with the carbon content reduced by decarburizing.
In an aircraft gas turbine (jet) engine, air is drawn into the front of the engine, compressed by a shaft-mounted compressor, and mixed with fuel. The mixture is burned, and the hot exhaust gases are passed through a turbine mounted on the same shaft. The flow of combustion gas turns the turbine by impingement against an airfoil section of the turbine blades and vanes, which turns the shaft and provides power to the compressor and fan blades. The hot exhaust gases flow from the back of the engine, driving it and the aircraft forwardly.
The hotter the combustion and exhaust gases, the more efficient is the operation of the jet engine. There is thus an incentive to raise the combustion and exhaust gas temperatures. The maximum temperature of the combustion gases is normally limited by the materials used to fabricate the turbine vanes and turbine blades of the turbine, upon which the hot combustion gases impinge. In current engines, the turbine vanes and blades are made of nickel-based superalloys, and can operate at temperatures of up to about 1800-2100xc2x0 F.
Many approaches have been used to protect the turbine blades and vanes against the highly aggressive combustion-gas environment and to increase the operating temperature limit of the turbine blades and vanes. For example, the composition and processing of the base materials themselves have been improved. Physical cooling techniques may also be used.
In another approach, the surfaces of the turbine blades and vanes are coated with aluminum-containing protective coatings that protect the articles against the combustion gas, and in some cases insulate the articles from the temperature of the combustion gas. The articles are thereby able to run cooler and be more resistant to environmental attack.
The addition of selected elements to the protective coatings may improve the mechanical and environmental properties of the coatings. However, those results have not always been consistent, and there is a large scatter in the data. Even though there has been an indication of improved performance as a result of the presence of such elements, those improvements cannot be relied upon in all cases.
There is a need for an approach to improving the properties obtained in nickel-base superalloys having a protective coating. The present invention fulfills this need, and further provides related advantages.
The present invention provides a procedure that improves the performance of a nickel-base superalloy having a protective coating applied to its surface, and an article having this improved performance. The protective coating contains aluminum and a reactive element such as hafnium, zirconium, yttrium, lanthanum, and/or cerium. The procedure is readily performed with available apparatus, and may be integrated into the coating process. The coating protects the surface of the article against environmental damage, as in the case of conventional protective coatings.
A method for preparing a surface-protected article includes providing an article substrate having a surface and having a nominal bulk composition comprising a nickel-base superalloy. The nickel-base superalloy has more nickel than any other element, and a nominal bulk composition of carbon. The method further includes depositing a protective layer overlying the surface of the article substrate, wherein the protective layer comprises aluminum and a reactive element selected from the group consisting of hafnium, zirconium, yttrium, lanthanum, and cerium, and combinations thereof. The step of depositing a protective layer includes the steps of decarburizing locations where the carbon may serve as a barrier to the mobility of the reactive elements within the protective layer by scavenging the reactive elements and preventing their diffusion in the protective layer, and depositing the protective layer overlying the substrate. The protective layer may be an overlay coating or a diffusion coating. A ceramic layer may be deposited over the protective layer.
The reactive elements (hafnium, zirconium, yttrium, lanthanum, and cerium, and combinations thereof) present in the protective layer yield their greatest benefits when they are in solid solution and free to diffuse within the coating. The impurity element carbon chemically combines with the reactive elements to form compounds that remove the reactive elements from solid solution and thence prevent them from moving within the protective layer. The result is that their advantageous effects are reduced or lost. In the present approach, the carbon which may chemically combine with the reactive elements of the protective layer is partially removed so as to lessen its concentration. The carbon is preferably removed not only from the protective layer itself, but also from the surface region of the substrate, because it may diffuse from the substrate into the protective layer during service.
In practicing the method, the reducing of the carbon content is preferably accomplished by contacting a decarburizing agent to the surface of the substrate to decarburize to a depth of from about 5 to about 100 micrometers, decarburizing a platinum-containing layer after deposition (where the protective layer is a platinum aluminide), depositing the aluminum-containing layer from an atmosphere containing a reducing agent, and/or decarburizing the substrate and protective layer after it is deposited. The decarburizing agent is preferably a reducing gas such as hydrogen or carbon dioxide. Particularly in the case of the overlay protective layer, the starting materials of the protective layer may be decarburized prior to deposition.
The present approach provides a low-carbon region in the protective layer and in the substrate adjacent to the surface where the protective layer is deposited. The low carbon content of the protective layer results in the reactive elements not chemically combining with carbon to produce carbides of the reactive elements, so that the reactive elements remain free to diffuse throughout the protective layer. Such carbides reduce the level of the solute reactive element that is available to strengthen and improve the environmental properties of the coating. However, it is desirable to remove carbon from the surface region of the substrate as well, so that this surface region cannot serve as a diffusion source of carbon into the protective layer during service. The result is improved performance of the coating during service.
Other features and advantages of the present invention will be apparent from the following more detailed description of the preferred embodiment, taken in conjunction with the accompanying drawings, which illustrate, by way of example, the principles of the invention. The scope of the invention is not, however, limited to this preferred embodiment. | null | minipile | NaturalLanguage | mit | null |
.. _clfbase:
The Random Forests Classifier in MRIQC
^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
MRIQC is shipped with a random-forests classifier, using the combination of the
`ABIDE <http://fcon_1000.projects.nitrc.org/indi/abide/>`_ and
`DS030 <https://openfmri.org/dataset/ds000030/>`_ datasets as training sample.
To predict the quality labels (0="accept", 1="reject") on a features table
computed by ``mriqc`` with the default classifier, the command line
is as follows:
::
mriqc_clf --load-classifier -X aMRIQC.csv
where ``aMRIQC.csv`` is the file ``T1w.csv`` generated by the ``group`` level run of
``mriqc``.
.. _clfcustom:
Building your custom classifier
^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
Custom classifiers can be fitted using the same ``mriqc_clf`` tool in fitting
mode:
::
mriqc_clf --train aMRIQC_train.csv labels.csv --log-file
where ``aMRIQC_train.csv`` contains the IQMs calculated by ``mriqc`` and ``labels.csv`` contains
the matching ratings assigned by an expert.
The labels must be numerical (``-1``= exclude, ``0``= doubtful, ``1`` = accept).
With the flat ``--multiclass`` the flags are not binarized.
Otherwise ``0`` and ``1`` will be mapped to ``0`` (accept) and ``-1`` will be mapped
to ``1`` (reject).
Removing all arguments of the ``--train`` flag we instruct ``mriqc_clf`` to run cross-validation
for model selection and train the winner model on the ABIDE dataset:
::
mriqc_clf --train --log-file
Model selection can be followed by testing on a left out dataset using the flag ``--test``.
If test is provided empty (without paths to samples and labels), then the default
features and labels for ds030 are used:
::
mriqc_clf --train --test --log-file
The trained classifier can be then used for prediction on unseen data with
the command at the top, indicating now which classifier should be used:
::
mriqc_clf --load-classifier myclassifier.pklz -X aMRIQC.csv
Predictions are stored as a CSV file, containing the BIDS identifiers as
indexing columns and the predicted quality label under the ``prediction`` column.
Usage of ``mriqc_clf``
^^^^^^^^^^^^^^^^^^^^^^
.. argparse::
:ref: mriqc.bin.mriqc_clf.get_parser
:prog: mriqc
:nodefault:
:nodefaultconst:
| null | minipile | NaturalLanguage | mit | null |
1. Field
The following description relates to a simulator for CPR and defibrillator training, and more particularly, to a simulator including a human body model having various types of actuators and sensors, a simulator control device and a monitoring device having a built-in training procedure program so that a general person, an unskilled person, and emergency medical technicians such as a doctor, a nurse and an emergency medical technician can perform CPR and defibrillator training in the same manner in which CPR and a defibrillator are used on an actual human body.
2. Description of the Related Art
Cardiopulmonary resuscitation (CPR) is an emergency treatment of assisting a person suffering from a heart attack with blood circulation and breathing. A defibrillator is a medical device that generates and delivers an electric shock to revive a person's heart, and may be classified as a manual defibrillator used by emergency medical technicians or an automated external defibrillator (AED) used by general people. CPR and the use of the defibrillator allow a person's blood to be circulated during a heart attack, so that it is possible to delay brain damage and to resuscitate the person from the heart attack. Statistics show that if an acute heart attack patient receives CPR within four minutes, the resuscitation rate can be increased to 50% or higher. As such, it is important to use CPR and the defibrillator as soon as a heart attack patient is discovered, and this necessity has been increasingly recognized.
CPR is performed through basic steps including a consciousness identification step of identifying a patient's state, a help and report step of immediately asking neighbors for help in a loud voice and dialing an emergency number when it is identified that the patient is unconscious, an airway security step of pulling back the patient's head and lifting the patient's chin so as to prevent the patient's airway from being blocked due to tongue muscle relaxation, a pulse identification step of identifying the pulse of the carotid artery, a breast pressure step of applying pressure to the heart of the patient whose heart has stopped, a respiration identification step of observing whether or not the patient's breast goes up and down while identifying the patient's respiration so as to start artificial respiration, an artificial respiration step of inspiring air into the patient's lungs in which the respiration has stopped, and a repetition step of repeatedly providing the breast pressure and artificial respiration to the patient until emergency medical technicians arrive on the spot.
In addition to CPR, the use of the defibrillator, which restores an irregular electrocardiogram signal of a patient's heart to a normal electrocardiogram signal through electric shock treatment, has recently increased. The defibrillator is a medical device that generates and delivers an electric shock to revive a person's heart. The defibrillator determines whether the waveform state of the patient's heart is normal or abnormal and forcibly applies an electric shock to the patient's heart based on the determined result, thereby restoring the abnormal waveform of the patient's heart to the normal waveform. Because a bill for installing the defibrillator in many facilities, etc. has recently been approved, the installation of AEDs easily available for general people is tending to increase, and the AED instructs people on its usage in voice so that anyone can easily use the AED.
According to the related art, there are many products in which CPR training and evaluation are possible but the training and evaluation using the defibrillator are impossible. The products have configurations that do not satisfy main functions of CPR.
In a related art simulator, the function of a pupil reaction test does not exist, or a pupil is manually replaced for each case. The function of pulse generation does not exist either, or a manual pulse generation method of pumping with a hand is used.
Since the related art simulator is configured in such a manner that if the breast of the related art simulator is pressed, one compression spring moves up and down, the related simulator is different from an actual human body. The breast pressure is possible only when pressure is applied vertically to the breast of the related art simulator. However, the breast pressure is possible when pressure is applied vertically or diagonally to the breast of an actual human body. The related art simulator is manufactured in such a manner that the position of the breast pressure is identified with the naked eye or a button switch. However, if the position of the breast pressure is identified with the naked eye, a trainer has difficulty in deciding the exact position of the breast pressure, and may subjectively decide the position of the breast pressure. If the position of the breast pressure is identified with the button switch, the measurement of the breast pressure is possible only at the position of the button switch, and therefore, the decision of accuracy may be inaccurate. The depth of the breast pressure is identified using a photointerrupter method or using a method in which the breast pressure is mechanically stuck when the breast pressure reaches a certain depth. However, when identifying an appropriate depth of the breast pressure recommended in the guidelines of the International CPR Institute or Korean Association of CPR, the mechanical method can measure only the certain depth of the breast pressure. Therefore, when the breast pressure is weak or strong, the decision of the depth of the breast pressure is impossible. The photointerrupter method is a method of detecting a position by passing or blocking light transmitted through holes formed at a certain interval in a plate. In the photointerrupter method, the unit of height becomes an interval between the holes. However, since it is difficult to implement a hole interval of a few millimeters (mm), the measurement accuracy is low, and there is a limitation in improving the measurement accuracy.
In the related art simulator, the detection of a flow rate of air in training for artificial respiration is performed by identifying, with the naked eye, that if air is injected into the related art simulator, a lung-shaped bag expands and the height of the lung-shaped bag increases or by measuring a change in height using a photointerrupter in the same manner as the breast pressure. However, as described above, a trainer has difficulty in detecting an exact flow rate of air, and may subjectively decide the flow rate of air. Therefore, it is difficult to perform training for the flow rate of air in the artificial respiration recommended in the guidelines of the International CPR Institute or Korean Association of CPR.
In the related art simulator, an input/output display device displays a training course in such a manner that LEDs are simply turned on/off. Since there exists no scenario program for training or the reaction of the related art simulator is manually reproduced when the scenario program is executed, the reality of the related art simulator is deficient. | null | minipile | NaturalLanguage | mit | null |
Characterization of apolipoproteins from chicken plasma.
Although functionally similar, the lipoprotein systems of birds and mammals differ in composition. The major apolipoproteins, apo A-I and apo B, are common to all vertebrates; however apo A-II and apo E, functionally important components of mammalian lipoproteins, are absent from chicken plasma. Chicken apo A-I and apo B have been characterized, and several minor apolipoprotein components have been observed in electrophoretic patterns of chicken lipoproteins. In this study a single density gradient ultracentrifugation was used to isolate and subfractionate chicken lipoproteins into density classes. Isolated lipoproteins were delipidated with hexane-isopropanol (3:2). Apolipoproteins were then solubilized at pH 8.5 in 3 M guanidine hydrochloride and chromatographed on a 25 X 0.4 cm C4 reversed-phase column using 0.1% trifluoroacetic acid in a gradient of acetonitrile in water. Molecular weights estimated by sodium dodecyl sulfate-polyacrylamide gel electrophoresis and amino acid compositions were compared with those of apolipoproteins from other species in a search for functional similarities. Similarities in composition between the major chicken apolipoprotein and several human apolipoproteins were observed. | null | minipile | NaturalLanguage | mit | null |
How is Blockchain in Asia? BlockShow Meetup Conclusions
After the tremendous success of BlockShow Europe in April, we were all set and excited for our next adventure – BlockShow Asia 2017!
As the continents changed, so did the cities, people and cultures – but one thing stayed the same – the BlockShow vibe. This time BlockShow comes with a new motto: “Window to the Global Blockchain World.” This means that in a few short weeks we are going to build a bridge between the Eastern and the Western parts of the Global Blockchain ecosystem.
That’s why we traveled across five cities in five countries to meet the actual Blockchain community out there and also bring together the most promising and innovative Blockchain startups. Wherever we went, we were welcomed by the local Blockchain enthusiasts and experts with open arms and valuable knowledge.
In all the countries we were joined by the movers and shakers of the industry who have been involved with Blockchain for many years. For instance, Nicolas Cary, co-founder of Blockchain.info, accompanied us at our New Delhi meetup and shared his valuable insights on the Blockchain industry.
Meetups across the continent
Malaysia
Our journey began from the capital city of Malaysia, Kuala Lumpur. Followed by Shanghai, Singapore, Hong Kong and finally the most colorful one, New Delhi.
Currently in Malaysia, Bitcoin is not considered as a currency. Players in Malaysia think that regulation is to blame for the slow adoption of Blockchain in Malaysia. Gary Goh of Metlife says:
“Anything that touches customer information or privacy would stoke lots of discussions, especially when the technology stems from Bitcoin, which is not largely accepted in Malaysia.”
MiGHT, the Malaysian Industry-Government Group for High Technology is planting roots in Malaysia and providing a beautiful future for Blockchain in the country. The MiGHT Blockchain Programme director Mastura Ishak commented:
“We are seeing an encouraging response from foreign players who are interested in setting up centers of excellence for Blockchain here to educate the younger generation.”
During our visit to Malaysia, we were lucky to speak with Mastura Ishak from the Malaysian government, who gave us insight on how Malaysia is planning to go about ICO and cryptocurrency regulations in the country.
When asked what people think of ICO’s at BlockShow Meetup in Kuala Lumpur Mastura Ishak commented: “It feels very good, it’s very refreshing, and there’s a lot of potentials that you guys (BlockShow) are showing here.”
Mastura Ishak hold lots of promise in Malaysia in educating and allowing Blockchain to thrive in the future. We are grateful that we can provide some new refreshing perspectives to her as well to breathe more life into the Blockchain industry in Malaysia.
Singapore
In Singapore, we got a chance to catch up with Anson Zeall, Chairman of ACCESS – Singapore’s Cryptocurrency and Blockchain Industry Association. He provided valuable insight on how ready Singapore is to play a major role in the future of the Blockchain industry. Zeall believes Blockchain should be seen as a useful tool to aid current business processes instead of replacing them.
Hong Kong
Hong Kong was filled with people from the Blockchain industry. Simon Dixon, CEO and Founder of BnkToTheFuture.com, was also present and told us how Japan and Korean markets are benefiting because to Chinese traders moving out of the country.
According to Simon, Bitcoin, Blockchain and crypto will have the biggest impact in Asia.
“I think it’s gonna be the first country or region, we are here in Hong Kong, and its going across Singapore and everywhere else in Asia. I believe Bitcoin and Blockchain have the most to offer and will be adopted first in Asia.”
If he saw Satoshi Nakamoto, he told us he would “thank him for remaining anonymous and thank him for actually creating the technology that I believe is the most significant invention since the Internet. And the application of the Internet that changes everything across Wall Street, across banking, across the financial sector.”
We agree with Dixon in his view, as without this technology none of the remarkable ideas that have come to life as of recently would be possible. Satoshi Nakamoto has given a great gift to change the world in how we know in ease of use and changing many sectors outside of the financial sector in ways we never imagined before. The team is excited to host BlockShow Asia in Singapore as it holds a significant strategic point in terms of Blockchain development in Asia.
India
Nicolas Cary, co-founder at Blockchain Labs, was super stoked to come to our New Delhi meetup. He shared the stage with Addy Creeze, CEO of BlockShow, to discuss how Blockchain technology will evolve in the coming years.
When he was asked whether or not regulation is good for the industry, Nic said:
“To me, regulation is all about protecting consumers, and in some situations it definitely makes sense. Governments have an important responsibility to create an open space for people to be able to experiment and test in a safe way. So we welcome thought for regulation to drive innovation and we think this will happen in India. We’ve seen it happen in various countries already.”
There was a very diverse crowd in Delhi, from investors to developers to college students. It was a one of a kind experience to see them all interact one place. We were fortunate enough to reach so many individuals and are hopeful that Nic’s thoughts will bring revolutionary thinking and a change in regulation along with further development in the region.
The takeaway
On our Meetup journey, we were joined by more than 1,000 guests from around the world in five different venues. Looking back, some special moments will stay with us forever.
For instance, when Philip McMaster, one of the well-known personalities of the Republic of Conscience, came to the event with his famous Panda. It gave the event a special vibe, which is hard to describe in words.
The Q&A session between the startups and the audience at the end of every pitch was something beyond our expectations. It was stellar that the audience was well aware of how Blockchain products work and provided us with relevant issues and questions about the Blockchain.
All the prominent speakers have led us to the conclusion that the Blockchain will thrive and have a huge positive impact on Asia. Anson Zeall has high hopes for the Blockchain in Singapore and its implementation in business. Simon Dixon and Nicolas Cary both believe regulation and adoption of cryptocurrencies will help the Blockchain thrive throughout Asia and India. We even have high hopes that China will also hop on board to embrace the Blockchain in the future.
Dixon hit the nail on the head with his interpretation of our slogan of BlockShow: the window of the global Blockchain world as Blockchain being the technology that allows people to gain access the world without borders. Without Satoshi’s revolutionary idea this would not be remotely possible and we are grateful for that.
We have learned from all our speakers that Blockchain, crypto will have the biggest impact and influence in Asia. Even the infamous “Chinese ICO ban” won’t make any difference – Asia and China, in particular, will stay one of the biggest hubs of the Blockchain disruption for quite a while. That’s why we are excited to see what the future hold, we know it will be bright for the Blockchain!
The winners
From every city, we chose one startup that stood out from the rest. The audience chose the winning startup through a straightforward voting process. The winners were:
These startups will join us at the BlockShow Singapore conference to compete for the final winning prize of $20,000. In addition to that, the winning startup will get support from our sponsor Waves to run an ICO.
What’s next?
BlockShow Singapore!
We are bracing ourselves for the main conference in Singapore on Nov. 29-30 where we are expecting 1,000+ attendees from all over the world, 50+ high-class speakers and 40+ exhibitors.
There will also be some popular Blockchain-focused YouTubers and Bloggers such as Ameer Rosic and Ivan on Tech. You will have an opportunity to meet them in person at our special Media corner, and perhaps get a chance to take that selfie! | null | minipile | NaturalLanguage | mit | null |
Given who had delivered the killing blow, however, all she could really do was muster a little smile.
For reasons unknown even to her, Annette had left the TV on for hours, so as to watch the going-ons in the mansion. If nothing else, it intrigued her.
She was also amused at just how many times Marigold almost came across the Omnipotent interloper in her home.
The Builder turned away from the current camera, kneeling forward on to the floor. Annette tried to see what they could possibly be doing.
On screen, The Builder took hold of the now unseen head with both hands. About the same moment she sat onto the bed, their head snapped forward, then jerked back.
Annette looked on as they spat a chunk of flesh and bone out onto the floor. Then they looked back and began to messily devour what she presumed to be Max’s brain matter.
Another camera screen displayed Marigold on her way to the same room. The Builder then snapped their head in the direction of the hallways.
Even without sound, Annette could’ve sworn she heard them cursing as they ran into the bathroom.
…
Standing at the entrance to the hospital now, Shark and Sinbad waited in silence. Neither of them knew what they would say when he inevitably confronted them with questions.
“He’s taking a long time. Is he making sure he and Uncle Dudley really were blood-related to that baby?”
Sinbad put his hands into his pockets and shrugged. “DNA don’t lie, man. Personally, I’m surprised you ain’t freaked out more by knowing you’d have the same half-sibling as them.”
“I know I should be, but…” Shark huddled closer to Sinbad, so as to try and warm up. “I guess I just feel bad for that child. Imagine if it were born in prison. That’s no place for a baby.”
“Yeah, I highly doubt your uncle would have left that kid in there. Maybe he’d have put it in foster care or something.” He wrapped an arm around his fiance. “Guess it doesn’t matter now, though. That kid was born dead, apparently.”
They they said nothing. In an effort to lighten the situation, Shark tried changing the subject.
“Did you leave the TV on for Sagebear when we left? I don’t like having to leave her alone, and we don’t have a working radio, so…”
Knowing what the dog was watching when they left the house got Sinbad to let out a snort. “She’ll be fine. As long as she doesn’t try selling crystal meth with a guy named Jesse after today.”
…
The Builder slid down into the bathtub, attempting to remain undetected in case Marigold wandered into the bathroom. Trying to maneuver around, they stuck their hand back into the head, and pulled out a handful of grey matter.
“Max! What were you doing on the floor? Wake up! Now don’t you pretend to be asleep! I know all your tricks!” Her ranting and raving made The Builder wish they still had eyes to roll.
“Addled old bat,” they muttered to themself before shoving Max’s brains into their mouth, and swallowing. Marigold continued going on, seemingly oblivious to the fact her husband no longer had a head.
“Come on now! Put on your fancy clothes! We’re going to dinner! That reminds me, I should wash up lest The Terminator can detect me through scent!”
The bathroom door opened, causing The Builder to drop Max’s head onto themself. The remainder of his head innards now splashed onto their clothing.
“Ah, shit! That ain’t gonna come out, is it?” Grabbing his now empty head, they jumped out of the tub and flung it across the room. Their actions were most definitely noticed.
…
Marigold fell to the ground, and pointed at her uninvited guest while shrieking like a demented banshee. They tried playing it cool, it looked to her.
“Uh…hey,” they said over her screaming. “Nice house you got here. Um…You’re probably curious as to why I just threw your husband’s head at the wall.”
Her response was as disjointed as could be expected. “Are you the Angel of Death?!” This accusation made them tilt their head, as if they were trying to find the right countering words.
“No,” they said while drawing the word out almost sarcastically, “I’m…Charles Lee Ray. Yeah, that’s me. Just don’t ask why I’m not in that creepy-ass doll.”
Their face lit up. “Hey! You know what could be fun?” They went off to think again. “You got access to a shotgun somewhere in this place, right?”
…
When Shark and Sinbad came home, Sagebear jumped up on them in a most jubilant welcome back.
“Hi, puppy! Did you enjoy your show while we were gone?” Shark made sure that the itch behind her ears was properly scratched.
“Gotta go out? Gotta go to the bathroom?” Sagebear barked to answer him, and began doing what he assumed was a potty dance.
Sinbad grabbed her leash for Shark, and handed it to him. He then checked the couch to make sure she didn’t have an accident.
Keeping Sagebear happy proved to be good for keeping the two men from worrying. Taking his seat on the couch, Sinbad watched the rest of the current episode. Outside, Shark could be heard laughing and calling Sagebear’s name.
By the time he managed to be properly engrossed in the show, Sinbad felt himself tackled onto his back. He then felt Sagebear snuffling and licking his face excitedly.
“Turns out she didn’t have to go to the bathroom. She just wanted to play.” Seeing that she was already covered with dirt, Sinbad believed him.
“I can see that,” he managed to sputter out in between dog kisses.
…
Horse-Face located the hidden passageway in the back of the cellar. He hoped his companion wouldn’t somehow notice that he’d been crying over him a lot as of recently.
It hadn’t taken long to find him, sitting against a wall. The young man appeared to be spacing out, quietly singing to himself.
“Hey,” Horse-Face said to get his attention. This resulted in the singing abruptly being ended. Apparently, his sobs were evident when he said the word.
“Are you thinking about what happened?”
Horse-Face shrugged. “In a way. I don’t blame her for wanting to shank me.” He sat down next to him, pulling his knees into his chest. “Maybe it would have been better if I did get my head blown off that day…”
The young man rested his head onto Horse-Face’s arm. “If you did, there might have been someone else more willing to have done it.” He sighed. “And they might not have tried to help me.”
“Yeah.” Without really thinking, he took hold of his hand, and weaved their fingers together.
“And I don’t hate you for following orders. I hate the person who gave them to you.”
A floor or two above them, Horse-Face then heard Moony indistinctly shouting at Ox-Head. Then followed Ox-Head’s hysterical jabbering.
“…You want to spend some time out of the passageways? The Builder isn’t here, and it sounds like shit’s going down, so…”
…
Annette was startled by the sound of the bedroom being thrown open.
“Is it that bad to talk about the time I exposed myself in a Denny’s?!” Ox-Head dove under the bed she was sitting on. Annette couldn’t help but notice he was holding a severed arm.
Mere seconds later, she could hear Moony shouting and spewing insults. His accent seemed to be coming out as he did so. | null | minipile | NaturalLanguage | mit | null |
Restaurants
BatterFish
When Guy tasted the authentic fish and chips that Dublin-native Chef Jason Killalee was dishing out he said, “If you can a make a living off a fish and chips truck in a town like Los Angeles, you must be doing it right.” | null | minipile | NaturalLanguage | mit | null |
When you need pest-control services at your home or place of business, you want to forget about the creepy-crawly invasion as quickly as possible. You want those pests – no matter their species – out completely and for good! You don’t want a single critter escaping to restore its numbers and plague your space again.
To ensure that no pests escape the net, call A & A Pest Control now! No matter the species, whether it has four legs or six, whether it crawls or flies, A & A Pest Control can stop the invasion in its tracks, sending pests of all kinds packing. We’ve got the experience and skill to handle pest control in homes, commercial locations and industrial facilities alike. | null | minipile | NaturalLanguage | mit | null |
Explore this issue
Also by this Author
If you love lakes, gophers, and exemplary food safety infrastructure and leadership, Minnesota is the place for you. Welcome to the incomparable Land of 10,000 Lakes, the indomitable Gopher State, a state that showcases strong and enviable food safety priorities and relationships among regulatory agencies, academia, agriculture, industry, and consumers that are arguably second to none.
How did this seemingly idyllic situation come to be?
Simply stated, Minnesota citizens have a historical and inspiring commitment to public health, according to William Hueston, DVM, MS, PhD, ACVPM, a professor of veterinary medicine, public health, and public affairs who directs global leadership initiatives for the Center for Animal Health and Food Safety (CAHFS) at the University of Minnesota (UMN) College of Veterinary Medicine.
For starters, there has long been close collaboration between the Minnesota Department of Health’s (MDH) foodborne disease epidemiologists and the MDH laboratory that handles samples from sick people, Dr. Hueston says. “The MDH Infectious Disease section focuses on aggressive scientific investigation of disease outbreaks and documentation of findings in refereed scientific journals,” he relates. “Not only do they rapidly respond to foodborne disease, they also continue to raise the bar on the methods used to investigate and respond effectively to outbreaks.”
Another food safety plus in Minnesota is the collocation of the MDH, the Minnesota Department of Agriculture (MDA), and the Board of Animal Health in the Orville Freeman Building in downtown St. Paul, the state capital. “The MDH lab for human samples and the MDA lab for food samples are side by side,” Dr. Hueston says, “and there is direct connection between the laboratory building and the epidemiologists. Working side by side builds collaboration and teamwork for more effective investigations and response.”
Dr. Hueston is also quick to extol the food safety benefits of what he calls the unique partnership between Minnesota government agencies and UMN.
“There is a very strong epidemiology training program at the UMN School of Public Health (SPH) so that many, if not most, of the MDH epidemiologists have training and graduate degrees from UMN,” he begins. “And there is active recruitment of MDH and MDA employees as adjunct faculty, so that UMN students get ‘real’ insights into the way government works and how to handle current foodborne illness challenges.”
Minnesota boasts one of the most effective foodborne disease surveillance systems in the U.S., Dr. Hueston points out, thanks to an infrastructure whereby the investigation of the food consumption history of ill people begins immediately, as samples are received at the lab from physicians.
To that end, UMN public health students (who are available to work nights and weekends outside the normal work window of state government employees), Golden Gophers affectionately known as “team diarrhea,” conduct phone interviews with consumers stricken with foodborne illness in a timely fashion.
“The MDH was early to adopt molecular tools for ‘fingerprinting’ the bacteria recovered from diarrhea, allowing identification of the connections between patients and between sick people and contaminated food,” Dr. Hueston adds. “And MDH was an early contributor to PulseNet, the CDC clearinghouse for pulse field electrophoresis ‘fingerprints’ across the nation. The PulseNet partnership has enabled Minnesota to crack several high visibility national foodborne illness outbreaks where other states and federal government agencies were stumped.”
Center of Excellence
Acknowledging the excellence and leadership exuded by the MDH and UMN SPH, on Aug. 31, 2012 the CDC designated Minnesota as an Integrated Food Safety Center of Excellence (CoE) to help fulfill its role in the Food Safety Modernization Act (FSMA).
About Linda L. Leake, MS
Linda L. Leake, doing business as Food Safety Ink, is a food safety consultant, auditor, and award-winning freelance journalist based in Wilmington, N.C. Specializing in agriculture, food, food safety, and travel, her articles have appeared in some 89 print and online publications. Along with garnering awards for her articles and photographs, she holds the prestigious Master Writer status with American Agricultural Editors’ Association. Majoring in Dairy Science, she completed a BS in Agriculture at University of Wisconsin and an MS in Food Safety at Michigan State University. She’s an active member of IAFP, Toxicologists Without Borders, Inc., and the National Dairy Shrine. She’s currently enrolled in the International Development Doctoral Program at University of Southern Mississippi Gulf Coast. Reach her at [email protected].
Food Quality & Safety (formerly Food Quality) is the established authority in delivering strategic and tactical approaches necessary for quality assurance, safety, and security in the food and beverage industry. | null | minipile | NaturalLanguage | mit | null |
About:
HumanixR is a decentralized application (DApp) built on Ethereum Blockchain. HumanixR is designed to disrupt and replace the traditional methods of social human interaction over current famous social media networks
HumanixR is a completely transparent platform where each user can help any other user and get rewarded for the help they provide.
HumanixR is the next stage in the evolution of human interaction over the internet. With the advent of Blockchain technology combined with the trust of smart contracts we are trying to give back what every human deserves -control and rewards for their effort.
We are at a time in the history of humankind when we have the opportunity to use technology for the greater good of the community. For too long the data pertaining to people's interaction over the internet has been monetized and used by corporations while the real content creators or users get nothing but only a platform to connect with each other.
HumanixR intends to disrupt this traditional model and give back to the users or the community through our unique revolutionary DApp.
At HumanixR we are trying to leverage the power of people and the information each individual possesses. We at HumanixR will give people a means to create dynamic content in real time based on the needs of their peers while getting rewarded for creating information within the network.
To demonstrate the platform and be profitable from the word “Go”, we will build a DApp over the human network, “HumanixR”. HumanixR is a mobile application, which facilitates human interaction incentivizes it.
HumanixR is a self-sustaining dynamic content ecosystem where the whole community is in charge and gets paid for the information they create.
The basic economic motive of HumanixR is to reward all the users for the valuable content they create. The revenue model revolves around Targeted and unobtrusive Advertisements. | null | minipile | NaturalLanguage | mit | null |
BIG 'THALA 56' SURPRISE ON AJITH'S BIRTHDAY
Today, May 1 is Thala Ajith's 44th birthday as we all know and his fans have been busy organizing welfare activities, blood donation camps and the likes. They have also been mighty active in the social media with countless tweets, messages, tags and display pictures. Celebrities have also been pouring in with their wishes for Ajith.
Anirudh, the composer for Ajith's ongoing 56th film with director Siva, wished the star and also updated about his composing status for the film.
#HappyBirthdayThalaAjith .. #Thala56 update - Theme Music and Introduction song is ready
We had already updated the same a few weeks back after an exclusive interview with Anirudh, but we now have it straight from the horse's mouth in a public platform. The excitement is building up for sure. | null | minipile | NaturalLanguage | mit | null |
Q:
Disable console mess in CMUSphinx4
I'm trying CMUSphinx but I'm having a hard time. I have included core and data jars to the project in Eclipse which I have downloaded from here
I tried to do Live speech recognition as mentioned in the tutorial but when I try to run it, the console is filled with some kind of errors like The dictionary is missing a phonetic transcription for the word '....' and many.
Code:
import edu.cmu.sphinx.api.Configuration;
import edu.cmu.sphinx.api.LiveSpeechRecognizer;
import edu.cmu.sphinx.api.SpeechResult;
public class Hello {
public static void main(String[] args) throws Exception {
Configuration configuration = new Configuration();
configuration.setAcousticModelPath("resource:/edu/cmu/sphinx/models/en-us/en-us");
configuration.setDictionaryPath("resource:/edu/cmu/sphinx/models/en-us/cmudict-en-us.dict");
configuration.setLanguageModelPath("resource:/edu/cmu/sphinx/models/en-us/en-us.lm.bin");
LiveSpeechRecognizer recognizer = new LiveSpeechRecognizer(configuration);
while(true)
{
System.out.println("Start speaking...");
recognizer.startRecognition(true);
SpeechResult result = recognizer.getResult();
System.out.println("result: "+result.getHypothesis());
recognizer.stopRecognition();
}
}
}
Console:
23:15:07.552 INFO unitManager CI Unit: *+NSN+
23:15:07.556 INFO unitManager CI Unit: *+SPN+
23:15:07.556 INFO unitManager CI Unit: AA
23:15:07.556 INFO unitManager CI Unit: AE
23:15:07.556 INFO unitManager CI Unit: AH
23:15:07.556 INFO unitManager CI Unit: AO
...
... /* Similar to above messages */
...
...
23:15:08.447 INFO autoCepstrum Cepstrum component auto-configured as follows: autoCepstrum {MelFrequencyFilterBank, Denoise, DiscreteCosineTransform2, Lifter}
2016-02-22 23:15:08.649 java[68475:2340128] 23:15:08.649 WARNING: 140: This application, or a library it uses, is using the deprecated Carbon Component Manager for hosting Audio Units. Support for this will be removed in a future release. Also, this makes the host incompatible with version 3 audio units. Please transition to the API's in AudioComponent.h.
Start speaking... /* My Output */
23:15:08.653 INFO dictionary Loading dictionary from: jar:file:/Users/sriharish/Java/sphinx4-data-5prealpha-20151218.160017-5.jar!/edu/cmu/sphinx/models/en-us/cmudict-en-us.dict
23:15:08.786 INFO dictionary Loading filler dictionary from: jar:file:/Users/sriharish/Java/sphinx4-data-5prealpha-20151218.160017-5.jar!/edu/cmu/sphinx/models/en-us/en-us/noisedict
23:15:08.786 INFO acousticModelLoader Loading tied-state acoustic model from: jar:file:/Users/sriharish/Java/sphinx4-data-5prealpha-20151218.160017-5.jar!/edu/cmu/sphinx/models/en-us/en-us
23:15:08.787 INFO acousticModelLoader Pool means Entries: 16128
23:15:08.787 INFO acousticModelLoader Pool variances Entries: 16128
23:15:08.787 INFO acousticModelLoader Pool transition_matrices Entries: 42
23:15:08.787 INFO acousticModelLoader Pool senones Entries: 5126
23:15:08.787 INFO acousticModelLoader Gaussian weights: mixture_weights. Entries: 15378
23:15:08.787 INFO acousticModelLoader Pool senones Entries: 5126
23:15:08.787 INFO acousticModelLoader Context Independent Unit Entries: 42
23:15:08.788 INFO acousticModelLoader HMM Manager: 137095 hmms
23:15:08.788 INFO acousticModel CompositeSenoneSequences: 0
23:15:08.789 INFO trieNgramModel Loading n-gram language model from: jar:file:/Users/sriharish/Java/sphinx4-data-5prealpha-20151218.160017-5.jar!/edu/cmu/sphinx/models/en-us/en-us.lm.bin
23:15:09.821 INFO dictionary The dictionary is missing a phonetic transcription for the word '3-d'
23:15:09.821 WARNING trieNgramModel The dictionary is missing a phonetic transcription for the word '3-d'
23:15:09.830 INFO dictionary The dictionary is missing a phonetic transcription for the word 'adjustors'
23:15:09.830 WARNING trieNgramModel The dictionary is missing a phonetic transcription for the word 'adjustors'
...
... /* Similar to above messages */
...
23:15:11.231 INFO lexTreeLinguist Max CI Units 43
23:15:11.232 INFO lexTreeLinguist Unit table size 79507
23:15:11.234 INFO speedTracker # ----------------------------- Timers----------------------------------------
23:15:11.235 INFO speedTracker # Name Count CurTime MinTime MaxTime AvgTime TotTime
23:15:11.237 INFO speedTracker Compile 1 1.1360s 1.1360s 1.1360s 1.1360s 1.1360s
23:15:11.238 INFO speedTracker Load Dictionary 1 0.1330s 0.1330s 0.1330s 0.1330s 0.1330s
23:15:11.238 INFO speedTracker Load AM 1 2.0880s 2.0880s 2.0880s 2.0880s 2.0880s
23:15:11.238 INFO speedTracker Load LM 1 1.3080s 1.3080s 1.3080s 1.3080s 1.3080s
23:15:16.674 INFO speedTracker This Time Audio: 0.37s Proc: 4.12s Speed: 11.14 X real time
23:15:16.676 INFO speedTracker Total Time Audio: 0.37s Proc: 4.12s 11.14 X real time
23:15:16.676 INFO memoryTracker Mem Total: 738.00 Mb Free: 378.28 Mb
23:15:16.676 INFO memoryTracker Used: This: 359.72 Mb Avg: 359.72 Mb Max: 359.72 Mb
23:15:16.676 INFO trieNgramModel LM Cache Size: 870 Hits: 153862 Misses: 870
result: none /* My Output */
I have downloaded recommended jars and included it in the project.So what is wrong with it? Did they failed to provide a dictionary? Can I add the missing phonetic transcription? If yes, provide a link please. And finally how can I disable all the console warnings etc and only have output that I provide.
A:
I have the same problem where the only classes I'm using are StreamSpeechRecognizer and Configuration. After looking through the sphinx source for ConfigurationManagerUtils I've managed to get the follow code in my own classes to disable all console logging from edu.cmu.sphinx packages. Place this code before instantiating the recognizer.
Logger cmRootLogger = Logger.getLogger("default.config");
cmRootLogger.setLevel(java.util.logging.Level.OFF);
String conFile = System.getProperty("java.util.logging.config.file");
if (conFile == null) {
System.setProperty("java.util.logging.config.file", "ignoreAllSphinx4LoggingOutput");
}
Hardly elegant but it works for me, unless someone has a cleaner approach for disabling logging at runtime through the sphinx configuration/context classes.
As with any third party packages I need to consume in my own project, I have a valid reason for doing this directly in code because 1/ I don't care to expose any third party logging and 2/ I don't wish to ship customised logging configuration files for third party code.
| null | minipile | NaturalLanguage | mit | null |
UAE to host global knowledge economy summit
Dubai, November 20, 2012
The United Arab Emirates will host for the first time a major international conference on the future of global knowledge economies, part of the British Council’s prestigious annual Going Global series.
The world’s leaders of education and academics will gather in Dubai with government ministers and business leaders from more than seventy countries to discuss the role of higher education and skills in developing global knowledge economies.
The conference takes place from the March 4 to 6, 2013 at Dubai World Trade Centre.
This year’s theme is Global education: knowledge-based economies for 21st century nations. The programme is built around three strands:
• Research and innovation - the role of international collaboration;
• Developing skilled knowledge workers: the role of international collaboration; and
• Internationalising tertiary education structures and systems.
Highlights of the Going Global 2013 programme include the world’s experts in their field participating in sessions on ‘Soft Power’: higher education and cultural diplomacy; creating globally skilled young entrepreneurs and conditions for jobs and growth in a global economy, among others.
In addition, Going Global will launch five new pieces of major research:
• An exploration across nine countries of the cultural skills global employers require, from British Council, Booz Allen Hamilton and IPSOS
• A report on the social and economic impact of transnational education on host countries and index of future transnational opportunities by the British Council
• A report on how the vision for higher education in the UK will affect future graduate’s employability, by the UK’s University Alliance;
• A study of student’s perceptions of barriers to their outward mobility in the UK, Australia and USA by the British Council
• The official launch of the Times Higher Education’s 2013 global university reputation rankings
Martin Davidson CMG, the British Council’s chief executive, said: “We believe international collaboration is vital for the UK’s growth and prosperity and Going Global provides the perfect forum for debate, discussion and networking.
“We have created a diverse programme to respond to some of the current key challenges affecting higher education and skills around the world, and I look forward to the stimulating discourse.”
Richard Cotton, the British Council’s director in UAE said: “We are delighted to be bringing Going Global to Dubai for the first time, and I believe that the rapidly transforming middle east, north African and south Asian regions Dubai is an excellent location for this meeting of global education leaders.” – TradeArabia News Service | null | minipile | NaturalLanguage | mit | null |
The Araca Group presents the return of AracaWorks, a week-long series of play readings celebrating new work for the theatre, Dec. 5-9 at Ars Nova.
John Doyle
Photo by Joseph Marzullo/WENN
The free readings, scheduled to begin at 7:30 PM, are new plays at an early stage of development. AracaWorks pairs each playwright with a director and a group of New York actors. In addition, the series presents a play by the winner of the AracaWorks graduate playwriting award.
Kicking off the series Dec. 5 is Edgar and Annabel, written by Sam Holcroft (While You Lie at the Traverse, Edinburgh; Pink at the Tricycle) and directed by Raz Shaw. Here's how the play is described: "A young married couple prepares dinner in a smartly furnished kitchen. Annabel is composed, intelligent, in love. Edgar is professional, successful, assured. She’s chopping vegetables, he’s brought the wine. But something isn’t right. In a city not so different from our own capital, a group of freedom fighters attempt to stand up to an Orwellian establishment in increasingly perilous circumstances. The story that unfolds brings into question relationships, identities and the very nature of reality itself."
Following Edgar and Annabel will be Acapulco, held Dec. 6. Written by Jacquelyn Reingold (String Fever at Ensemble Studio Theatre, Girl Gone at MCC), the play is directed by Tony winner John Rando (Urinetown). Acapulco, described as "a swinging 60s kind of comedy with a 2012 point of view," centers around "almost divorced Doris, her lying husband, her scheming laundress mother-in-law, her chest-hair-dying boyfriend, a mysterious woman who's really a man, and the Whore of Brooklyn."
Kenneth Lin's (Fallow, Intelligence-Slave, Life on Paper) Warrior Class will be held Dec. 7 and is directed by Tony winner John Doyle (Sweeney Todd). Here's how Warrior Class is billed: "Julius Lee is a fast-rising Asian politician in the New York state assembly. But will an incident from his past be his undoing as he seeks the US Congress?"
Recall, written by Eli Clark (Edgewise, "The Killing") and directed by Adrienne Campbell-Holt, will be held Dec. 8. In Recall, "Lucy makes people uncomfortable," according to press notes. "There’s something about her eyes. There’s something about the way her mother’s boyfriends keep disappearing. And there’s something about the government agents on her trail. Boundaries of love and trust are pushed in this fierce and funny play." Concluding the series will be Caroline McGraw's (winner of the AracaWorks Graduate Playwriting Award) The Vaults, directed by Sarah Benson. Here's how the play is described: "In the twisting tunnels below Edinburgh, four American friends take a tour of the city's famous vaults. When one of them vanishes without a trace, those left behind must unravel his disappearance, grappling with unfriendly ghosts, suspicious police, fringe festival actors—and their own dark natures."
AracaWorks premiered in December 2010, showcasing both established playwrights and emerging voices. Past slates have featured the New York premieres of Suicide, Incorporated by Andrew Hinderaker (Roundabout Underground); How the World Began by Catherine Trieschmann (South Coast Repertory); and Disgraced by Ayad Akhtar (American Theater Company).
The Ars Nova building is located at 511 West 54th Street. For more information and reservations, visit www.aracaworks.com. | null | minipile | NaturalLanguage | mit | null |
Audiovisual integration in low vision individuals.
Behavioral and neurophysiological studies have shown an enhancement of visual perception in crossmodal audiovisual stimulation conditions, both for sensitivity and reaction times, when the stimulation in the two sensory modalities occurs in condition of space and time congruency. The purpose of the present work is to verify whether congruent visual and acoustic stimulations can improve the detection of visual stimuli in people affected by low vision. Participants were asked to detect the presence of a visual stimulus (yes/no task) either presented in isolation (i.e., unimodal visual stimulation) or simultaneously with auditory stimuli, which could be placed in the same spatial position (i.e., crossmodal congruent conditions) or in different spatial positions (i.e., crossmodal incongruent conditions). The results show for the first time audiovisual integration effects in low vision individuals. In particular, it has been observed a significant visual detection benefit in the crossmodal congruent as compared to the unimodal visual condition. This effect is selective for visual stimulation that occurs in the portion of visual field that is impaired, and disappears in the region of space in which vision is spared. Surprisingly, there is a marginal crossmodal benefit when the sound is presented at 16 degrees far from the visual stimulus. The observed crossmodal effect seems to be determined by the contribution of both senses to a model of optimal combination, in which the most reliable provides the highest contribution. These results, indicating a significant beneficial effect of synchronous and spatially congruent sounds in a visual detection task, seem very promising for the development of a rehabilitation approach of low vision diseases based on the principles of multisensory integration. | null | minipile | NaturalLanguage | mit | null |
There is no doubt that wheel-chairs have given mobility to the handicapped and to invalids. Nevertheless, such wheel-chairs suffer from various drawbacks due to the fact that their users can occupy a sitting position only, which position is also generally maintained for relatively long periods of time.
Such a position is unsuitable for providing readaptation to ordinary life and it does not facilitate social contacts. In addition, when a sitting position is maintained for relatively long periods of time, it causes physical deterioration, such as the loss of angular amplitude in the lower limbs, defective blood circulation, slowing down of the digestive and intestinal functions, bone fragility, etc.
To remedy the above drawbacks, proposals have been made for chairs each having a chassis that supports a hinged structure comprising a seat back, a seat, and a footrest. Such a structure is mounted in hinged relationship to the seat on a front horizontal axis, extending perpendicularly to the vertical plane of symmetry of the chassis. The hinged structure can be controlled with full motorization or with motorization for power assistance to cause the seat to pass from a low position to a high position, and vice versa. Such chairs are often referred to as "verticalizing chairs".
Regardless of whether the source of power that controls raising and lowering of the hinged structure or that enables it to be controlled is based on electricity, or on elastic actuators, in particular gas actuators, or is purely manual, chairs of the above type have certainly made it possible to a large extent to solve the drawbacks that stem from using a conventional chair.
That is doubtless why such chairs have been such a success over several years. By way of reference, mention can be made of French patent FR 2 529 456 which specifically relates to a design for such a verticalizing chair.
Although they give satisfaction, it appears that such chairs give rise to objections concerning comfort relating in particular to the nature of the hinged structure for raising and lowering the invalid or handicapped person in a position of maximum safety.
Account needs to be taken of the hinged nature of the structure which is capable of passing from a traditional seated position to an elevated or verticalizing position in which the various segments making up the structure are substantially in alignment one after another, in a pseudo-vertical direction.
To satisfy anatomical requirements in the various positions it can occupy, the hinged structure is made up of a seat element, a back element, and a footrest element which must therefore be capable of occupying a relative position that is generally of the seat type, and also of being placed in line with one another in the verticalization position.
Substantially parallel relative hinge planes are therefore necessarily established which are situated between the seat and the footrest, and between the seat and the back.
Although these various hinged segments do not give rise to major problems of comfort in supporting and holding the body of a handicapped subject or an invalid when they are in the traditional seated position, on passing into the verticalized position these various segments are generally subjected to displacement in which they slide relative to the body of the subject. This relative displacement is not good for maintaining maximum comfort, and in particular it requires appropriate settling back into the chair when in the seated position, so that the subject is again bearing comfortably against the back, against the seat, and against the footrest.
On reflection, it appears that this problem which, even if it is secondary, needs nevertheless to be addressed, stems from the fact that the hinged structures fitted to chairs for verticalizing purposes provide no scope for adjusting the depth of the seat as a function of the morphology of the subject.
Thus, apart from the ideal case where the depth of the seat is right, it can be considered that as a general rule this depth is either too deep or not deep enough to provide the subject with maximum comfort, whether in the sitting position or in the verticalization position. | null | minipile | NaturalLanguage | mit | null |
Sites in Reuse in Idaho
Click on the image above
to see a
larger map with site locations.
Bunker Hill Mining & Metallurgical Complex
The Bunker Hill Mining & Metallurgical Complex Superfund site is located in Idaho’s Silver Valley, one of the largest historical mining districts in the world. Mining operations began in the area in 1883 and continue to this day. When the Bunker Hill lead smelter and several of the associated mines closed in the 1980s, the economy of the surrounding area nearly collapsed. Thousands were jobless and heavy metals had contaminated the countryside. Local tests found high blood lead levels in area children. In response, EPA added the 25-square-mile area around the old smelter to the National Priorities List (NPL) in 1983. Cleanup and ecological restoration around the lead smelter have included the removal of lead-contaminated soil from lawns and parks, the containment of tons of mine tailings and the planting of thousands of trees. Lead levels in children have fallen dramatically to levels equivalent to national averages. The Panhandle Health District, the State of Idaho and EPA continue to educate Silver Valley children to avoid lead-contaminated areas and accidental lead ingestion. Panhandle Health District, the state and EPA also developed a comprehensive Institutional Controls Program for the site, which provides safe and clear procedures for developing property in the Silver Valley. EPA is cleaning up the site in three main areas, or operable units. In 2009, the site received $16.8 million in American Reinvestment and Recovery Act funding to expedite the Coeur d'Alene Basin residential cleanup program. This work is a top priority for the site and key to protecting public health. In August 2012 issued an amended cleanup decision for the Upper Basin. This decision document calls for $635 million in additional cleanup actions in this area of the site over the next 30 years. Starting in 1987, the City of Kellogg began to pursue redevelopment opportunities at cleaned up portions of the site. The site is now home to the Silver Mountain Resort, which includes a mixed residential neighborhood, commercial development and 18-hole golf course; the Silver Valley Business Center, which supports industrial and commercial development; and light manufacturing, outdoor recreation, telecommunications, workforce training, and environmental remediation businesses. As of 2012, workers have cleaned up over 6,300 residential and commercial properties. EPA has finished converting nearly 400 acres of agricultural property near Medimont to healthy wetland habitat. The area is now a clean feeding habitat for swans, ducks and other wetland birds.
Updated 12/2012
Idaho National Engineering Laboratory (USDOE)
The 890-square-mile Idaho National Engineering Laboratory (USDOE) Superfund site is located in a remote and lightly populated area of southeast Idaho. Established in 1949, the U.S. Department of Energy (DOE)-managed reservation has been devoted to energy research and related work. The laboratory currently supports DOE’s missions in nuclear and energy research, science and national defense. In 1986, investigators detected contaminants in ground water. In response, DOE identified hazardous waste disposal areas at the site that could pose unacceptable risks to health, safety or the environment. In 1989, EPA listed the site on the National Priorities List (NPL). In 1992, DOE signed a Federal Facilities Agreement with EPA and the state to address site contamination. DOE has since undertaken a number of cleanup actions. Additional cleanup actions and ground water monitoring continue. In 2003, DOE defined two business units, one for laboratory research and development missions (Idaho National Laboratory (INL)) and one for remediation (Idaho Cleanup Project). DOE renamed the 890-square mile facility the INL site. The site currently supports facility and program operations. DOE has reserved parts of the central area for the Idaho Cleanup Project (ICP) and INL operations. DOE conducts environmental research as well as ecological and sociocultural preservation on the remaining land within the site’s core. This area is largely undeveloped. Public highways and the Experimental Breeder Reactor I (EBR-I) National Historic Landmark are the only parts of the INL site with unrestricted access. The federal Bureau of Land Management manages livestock grazing leases within undeveloped portions of the site perimeter. DOE also collaborates with the Idaho Department of Fish and Game to permit controlled hunting within half a mile of the boundary. Though uncertain, future land use most likely will remain essentially unchanged, with research facilities within site boundaries and agricultural and open land surrounding the site. DOE expects to retain ownership and control of the site until at least 2095, and will continue to manage portions that cannot be released for unrestricted land use beyond 2095.
Updated 12/2012
Monsanto Chemical Co. (Soda Springs Plant)
The Monsanto Chemical Co. (Soda Springs Plant) Superfund site is located outside the city limits of Soda Springs, Idaho. The 800-acre site includes the 540-acre Monsanto plant operating area as well as 260 acres of buffer area owned in part by Monsanto and in part by various farmers. About 400 employees and 200 contract employees work at the facility. In 1984, Monsanto started evaluating ground water impacts from past and current operations. In 1990, EPA listed the site on the National Priorities List (NPL) after the identification of contamination in ground water and soils at the site. EPA required that Monsanto place restrictions on its property surrounding the plant. Other property owners with soil contamination could either have their soil cleaned up or have land use restrictions placed on their property. Affected property owners chose to sell their rights, allowing Monsanto to place land use restrictions on their property. Monsanto began carrying out these requirements in 1998. EPA did not require Monsanto to take actions to address contaminated source piles and materials. This was because Monsanto's past cleanup actions and ongoing efforts have reduced potential sources of worker exposure. Contaminant migration to surrounding soils appeared to be at acceptable levels under current land use. The soil sampling completed for the third Five-Year Review during summer 2012 indicates that one parcel without land use restrictions may exceed the cleanup level. Depending upon the results from recent follow-up sampling, EPA may require land use restrictions for this property. EPA selected monitored natural attenuation to address ground water contamination. Follow-up ground water monitoring indicated that natural attenuation may not be occurring for some of the contaminants as predicted and that the area of ground water contamination is larger than originally defined. Ground water monitoring will continue and EPA is in the process of determining the need for a new ground cleanup approach for the site.
Updated 12/2012
Mountain Home Air Force Base
Established in 1943, Mountain Home Air Force Base (AFB) is located on nine square miles of land on a plateau southwest of Mountain Home, Idaho. The base has been under the control of the Tactical Air Command since 1965. The U.S. Department of Defense established the base in 1943 as a training base for several bombardment groups during World War II. In addition to supporting military operations, current land use within the base includes a residential area with approximately 7,500service men and women and their dependents. Past practices resulted in contaminated soil and ground water. In 1990, EPA placed the site on the National Priorities List (NPL). The U.S. Air Force signed a Federal Facilities Agreement with EPA and the state in 1992 to address site contamination. The U.S. Air Force has since undertaken a number of cleanup actions on the base. Additional cleanup actions as well as long-term monitoring of ground water continue.
Updated 12/2012
Pacific Hide & Fur Recycling Co.
The 17-acre Pacific Hide & Fur Recycling Co. Superfund site is located in Pocatello, Idaho. From 1950 to 1983, the McCarty family owned and operated gravel mining and metal salvaging businesses at the site. Metals from site activities seeped into the soil, and in 1983 the EPA found soils on site and in the surrounding area with high lead concentrations. The EPA removed highly contaminated soils and added the site to the National Priorities (NPL) in 1984. Working with the owners, the EPA led removal and treatment of soils contaminated with lead and polychlorinated biphenyls (PCBs) throughout the site. The EPA deleted the site from the NPL in 1999. Operations at the site continue under Pacific Steel and Recycling, Inc.
Updated 10/2013
Union Pacific Railroad Co.
The Union Pacific Railroad Co. (UPRR) Superfund site is located in Pocatello, Idaho. The 1-acre site is also known as the UPRR Sludge Pit site. From 1961 until 1983, UPRR dumped approximately 2,500 cubic yards of sludge from its wastewater treatment plant into a 1-acre unlined sludge pit. In 1983, EPA found that seepage from the UPRR’s sludge pit and from a nearby railroad tie treating facility contributed to the contamination of the underlying aquifers. EPA added the site to the National Priorities List (NPL) in 1984. UPRR performed cleanup activities under a legal agreement with EPA. UPRR completed the cleanup actions in 1994. Cleanup activities included of the excavation and off-site disposal of 13,821 tons of sludge and soil. In addition, UPRR pre-treated of over 62 million gallons of ground water before discharging the water to the city’s water treatment plant. After confirming the success of the cleanup, EPA deleted the site from the NPL in 1997. Amtrak currently uses the site as a train station.
Updated 12/2012 | null | minipile | NaturalLanguage | mit | null |
M. Night Shyamalan Will Work with Bruce Willis Again
If the year were 1999, there might be a reason to be excited for an M. Night Shyamalan and Bruce Willis collaboration. Alas, the year is 2014 so news that they are getting back together is met with a few more laughs and a hearty dose of skepticism.
Deadline's Mike Fleming Jr. reported that the two are in talks to make Labor of Love, a film about a book store owner who, after losing his wife, decides to walk across the country to prove how much he loved her. So Willis, we don't think, will be actually dead in this one. (Yes, SPOILER, but are we referring to The Sixth Sense or Unbreakable?) If anything, it should maybe be encouraging that Shyamalan sold the script way back in 1993, before his string of failures made him out to be something of a hack.
This could also be just not so good.
This article is from the archive of our partner The Wire.
We want to hear what you think about this article. Submit a letter to the editor or write to [email protected]. | null | minipile | NaturalLanguage | mit | null |
Recognition and separation of single particles with size variation by statistical analysis of their images.
Macromolecules may occupy conformations with structural differences that cannot be resolved biochemically. The separation of mixed molecular populations is a pressing problem in single-particle analysis. Until recently, the task of distinguishing small structural variations was intractable, but developments in cryo-electron microscopy hardware and software now make it possible to address this problem. We have developed a general strategy for recognizing and separating structures of variable size from cryo-electron micrographs of single particles. The method uses a combination of statistical analysis and projection matching to multiple models. Identification of size variations by multivariate statistical analysis was used to do an initial separation of the data and generate starting models by angular reconstitution. Refinement was performed using alternate projection matching to models and angular reconstitution of the separated subsets. The approach has been successful at intermediate resolution, taking it within range of resolving secondary structure elements of proteins. Analysis of simulated and real data sets is used to illustrate the problems encountered and possible solutions. The strategy developed was used to resolve the structures of two forms of a small heat shock protein (Hsp26) that vary slightly in diameter and subunit packing. | null | minipile | NaturalLanguage | mit | null |
Losing face: impaired discrimination of featural and configural information in the mouth region of an inverted face.
Given that all faces share the same set of features-two eyes, a nose, and a mouth-that are arranged in similar configuration, recognition of a specific face must depend on our ability to discern subtle differences in its featural and configural properties. An enduring question in the face-processing literature is whether featural or configural information plays a larger role in the recognition process. To address this question, the face dimensions task was designed, in which the featural and configural properties in the upper (eye) and lower (mouth) regions of a face were parametrically and independently manipulated. In a same-different task, two faces were sequentially presented and tested in their upright or in their inverted orientation. Inversion disrupted the perception of featural size (Exp. 1), featural shape (Exp. 2), and configural changes in the mouth region, but it had relatively little effect on the discrimination of featural size and shape and configural differences in the eye region. Inversion had little effect on the perception of information in the top and bottom halves of houses (Exp. 3), suggesting that the lower-half impairment was specific to faces. Spatial cueing to the mouth region eliminated the inversion effect (Exp. 4), suggesting that participants have a bias to attend to the eye region of an inverted face. The collective findings from these experiments suggest that inversion does not differentially impair featural or configural face perceptions, but rather impairs the perception of information in the mouth region of the face. | null | minipile | NaturalLanguage | mit | null |
628 P.2d 646 (1981)
Rad Lee PAYNE, Plaintiff and Appellant,
v.
Billie BUECHLER, Defendant and Respondent.
No. 80-227.
Supreme Court of Montana.
Submitted on Briefs December 17, 1980.
Decided May 26, 1981.
Dissenting Opinion May 27, 1981.
*647 Gerald J. Neely, Billings, for plaintiff and appellant.
Berger, Anderson, Sinclair & Murphy, Billings, for defendant and respondent.
HASWELL, Chief Justice.
This is an action by a real estate broker to collect a commission under a written contract granting him the exclusive right to sell the property. During the term of the listing, the property owner canceled the listing and sold the property herself. From a judgment of the District Court of Yellowstone County denying recovery of the commission, the broker appeals.
Plaintiff and appellant is Rad Lee Payne, a licensed real estate broker in Billings, Montana. Defendant and respondent is Billie Buechler, the owner of the Red Rooster Bar in Shepherd, Montana. On July 5, 1977 the owner and broker entered into a written agreement whereby the broker was employed to sell the owner's bar, liquor license, furniture and fixtures, a three-bedroom residence and four lots for $139,000 on a 10-year installment basis at 8 1/2% interest. The written agreement provided, among other things:
"THIS LISTING IS AN EXCLUSIVE LISTING and you hereby are granted the absolute, sole and exclusive right to sell or exchange the said described property. In the event of any sale by me or any other person, or of exchange or transfer of said business, personal property, lease(s), if any, or any part thereof, during the term of your exclusive employment, or in case I withdraw the authority hereby given prior to said expiration date, I agree to pay you the said commission just the same as if a sale had actually been consummated by you."
The agreement provided that the commission was 10% of the selling price. The expiration date of the agreement and listing was January 1, 1978. The agreement provided for reasonable attorney fees in case of suit on the contract.
The broker proceeded to advertise the property in the Billings Gazette, prepared and mailed brochures including the property which went to approximately 3,000 potential out-of-state buyers, and showed the property to a number of people. He sent further information to those making inquiries and responded to telephone inquiries. He expended $1,120 in attempting to sell the property.
Thereafter on September 15, 1977, the owner sent a letter to the broker as follows:
*648 "Dear Rad:
"Having decided to keep the bar, I wish to take it off the market and cancel my listing.
"If in the future I want to list it I will give you first chance.
"Yours truly, "Billie Buechler "Red Rooster Bar "Shepherd, Mont. 59079"
Twelve days later on September 27, the owner entered into an agreement to sell the bar to a third party for $120,000.
The broker filed his complaint to collect his 10% commission, interest from the date of sale, attorney fees and costs. The owner answered denying the contract was exclusive and alleging that the broker's authority was terminated prior to sale and that the purchaser was not procured through any efforts of the broker.
Pretrial discovery consisted of interrogatories and answers of the broker and owner, depositions of the broker, his father who was associated in business with him, and the owner.
The case came on for trial on March 26, 1980 before the District Court sitting without a jury. Admitted in evidence without objection were the deposition and exhibits thereto of the broker; the deposition of his father; an exhibit concerning the broker's attorney fees; the letter from the owner to the broker canceling the listing; an exhibit relating to the broker's costs and expenses; the interrogatories and answers of the broker and the owner; and a real estate listing agreement on the bar between the owner and another broker. The broker also moved for admission in evidence of the deposition of the owner excepting therefrom certain parts which the broker contended were parol evidence and inadmissible; the owner sought admission in evidence of her entire deposition; and the court reserved a ruling on the admissibility of those parts objected to and admitted the rest.
At the trial the broker, the owner and a Mr. Van Lueschene testified in person, albeit briefly.
The District Court entered findings of fact, conclusions of law and judgment in favor of the owner. The substance of the court's findings was that the written agreement granting the broker the exclusive right to sell the bar was entered into by the broker and owner on July 5, 1977; that the owner did not intend to grant the broker the exclusive right to sell the bar as she had at least two other listings with other real estate agencies on the same property in effect on July 5, 1977; that no consideration flowed from the broker to the owner other than their mutual contemplation that the broker would attempt to attract prospective purchasers for his own benefit; that the owner advised the broker on September 15, 1977, that the agreement was canceled; and that the broker had nothing to do with attracting the subsequent purchasers to contract the owner or to buy the property.
From these findings the court concluded that the written contract of July 5 lacked consideration and mutuality and the owner had the right to revoke it at any time; that the agreement was not an exclusive agreement to sell the owner's property; and that the owner acted in good faith in terminating the written contract and did not perpetrate a fraud on the broker. Judgment for the owner was entered accordingly.
We frame the issues on appeal in this manner:
(1) Did the written contract between the broker and owner lack consideration and mutuality?
(2) Was there sufficient evidence to support the finding that the written contract was not intended to and did not give the broker the exclusive right to sell the property?
(3) Did the owner have the right to cancel the written contract during its term without liability for the commission?
Lack of consideration was not raised as a defense to the written contract by the owner but becomes an issue on appeal by reason of the District Court's findings and conclusions. It has been regularly *649 held that a broker's expenditure of time and money to find a purchaser is sufficient consideration for the promise to pay a commission and upon such expenditure of time and money, the agreement becomes bilateral and binding upon the owner. Kimmel v. Skelly (1900), 130 Cal. 555, 62 P. 1067; Garrett v. Richardson (1962), 149 Colo. 449, 369 P.2d 566. Here the owner employed the broker on a commission basis and the broker's expenditure of his time and $1,120 of his money to attract a purchaser constituted consideration for the owner's agreement to pay a commission.
The District Court further found that the agreement lacked mutuality. Mutuality of obligation was created by the efforts of the broker to find a purchaser for the property on the owner's terms and the broker's expenditure of time and money in this effort. Harris v. McPherson (1922), 97 Conn. 164, 115 A. 723, 24 A.L.R. 1530.
We are next faced with the issue of whether the evidence is sufficient to support the court's finding that the owner did not intend to give the broker the exclusive right to sell the property and the written contract was not an exclusive agreement to sell. The written contract plainly states on its face that the broker is granted the exclusive right to sell the property to the exclusion of the owner or any other person. The District Court's findings and conclusions to the contrary are clearly based on the testimony of the owner that she did not intend to give the broker an exclusive listing; that she had previously given listings to other agencies which were still in effect, one of which was produced and admitted in evidence; and that the broker had written "nonexclusive" on her copy of the contract. She also called a Mr. Van Lueschene who testified that "nonexclusive" was written on her copy of the agreement. Her copy of the written agreement was never produced; she testified that she had destroyed it after her home had been vandalized and molasses and ketchup had been poured on it. With the exception of the copy of a prior and existing listing of the property with another real estate agency, all this evidence was objected to under the parol evidence rule and the objection was taken under advisement by the court.
The parol evidence rule generally provides that the terms of a written agreement cannot be altered or contradicted by oral testimony subject to certain well-recognized exceptions. See section 28-2-905, MCA. The written contract supersedes all oral negotiations or stipulations which preceded or accompanied its execution. Section 28-2-904, MCA. In accord, Danielson & Ward v. Danielson & Neu (1977), 172 Mont. 55, 560 P.2d 893; Batey Land & Livestock Co. v. Nixon (1977), 172 Mont. 99, 560 P.2d 1334; Larson v. Burnett (1972), 158 Mont. 421, 492 P.2d 921.
The owner contends that the foregoing parol evidence is admissible pursuant to section 26-1-103, MCA, which provides:
"Where the declaration, act, or omission forms part of a transaction which is itself the fact in dispute or evidence of that fact, such declaration, act, or omission is evidence as part of the transaction."
Not so. This statute is simply an exception to the hearsay rule, Callahan v. C B & Q Ry. Co. (1913), 47 Mont. 401, 133 P. 687. It has nothing to do with the parol evidence rule which is a rule of substantive law.
The owner also asserts that Rule 106 of the Montana Rules of Evidence and Rule 32(a)(4), M.R.Civ.P., render this parol evidence admissible. These are rules on admissibility of evidence. Parol evidence cannot be introduced because as a matter of substantive law the written agreement constitutes the entire transaction between the parties.
The owner also argues that the evidence is admissible under an exception to the parol evidence rule set forth in section 1-4-102, MCA:
"For the proper construction of an instrument, the circumstances under which it was made, including the situation of the subject of the instrument and of the parties to it, may also be shown so that the judge be placed in the position of those whose language he is to interpret."
*650 This statute relates to construction and interpretation of written instruments but is irrelevant here. The language of the contract is plain and unambiguous. Under such circumstances, the language alone controls and there is nothing for the Court to interpret or construe. Section 28-3-401, MCA and section 28-3-303, MCA. The quoted statute only applies where an ambiguity exists in the language of the contract.
We have examined the cases cited by the owner which she claims support the admissibility of parol evidence that she did not intend to give the broker an exclusive listing and did not give him such a listing. None support the admissibility of such parol evidence in this case. Brown v. Homestake Exploration Co. (1934), 98 Mont. 305, 39 P.2d 168, involved a lengthy written contract ambiguous on its face and parol evidence was admitted as an aid to interpretations, a clear exception to the parol evidence rule. In Platt v. Clark (1963), 141 Mont. 376, 378 P.2d 235, parol evidence was admitted, not to vary or alter the terms of a written contract, but to show that a condition precedent to an otherwise valid and binding lease had not occurred and therefore the written lease never became effective. See generally Anno.: Applicability of Parol Evidence Rule to Written Listing Agreement of Real Estate Broker, 38 A.L.R.2d 542.
Here the parol evidence directly contradicts the plain and unambiguous language of the written instrument; it does not fall within any recognized exception permitting its admission in evidence, and is clearly inadmissible. Although the District Court did not rule on the broker's objection to its admission in evidence, its findings and conclusions clearly reflect that the court based them on this inadmissible evidence. This was error.
The last issue concerns whether the owner had the right to cancel the written listing agreement during its term and thereby deny the broker a commission. The written agreement plainly gave the broker an exclusive right to sell the property during the term of the agreement; provided that if the owner or any other person sold or transferred the property, the owner would pay the broker the commission; and finally provided that if the owner withdrew the broker's exclusive authority to sell the property, the owner would pay the broker the commission. The District Court held that the owner had the right to revoke the agreement at any time and denied the broker any commission.
As we have previously indicated, once the broker began performance under the written agreement by expenditure of his time, efforts and money to attract a purchaser on the owner's terms, the written agreement became bilateral and binding on both parties. It could not be unilaterally terminated by the owner without payment of the broker's commission. Piatt & Heath Co. v. Wilmer (1930), 87 Mont. 382, 288 P. 1021; McDonald & Co. v. Fishtail Creek Ranch (1977), 175 Mont. 53, 572 P.2d 195; Anno.: 88 A.L.R.2d 938, 966.
Flinders v. Hunter (1922), 60 Utah 314, 208 P. 526 is cited by the owner for the proposition that an agency relationship is revocable by the owner unless the broker has an interest in the property. The broker's contention in this case is not that the owner lacks the right to terminate the broker's authority. The broker's contention is that if the owner does revoke, he is nonetheless liable for the broker's commission by the clear language of the written agreement. Flinders does not support the owner's contention that she is not liable for the commission.
In summary, the District Court's findings and conclusions that the written agreement lacked consideration and mutuality and that the agreement was not an exclusive agreement to sell the property were error as a matter of law.
The judgment of the District Court denying the broker his commission is reversed. The cause is remanded to the District Court for entry of judgment for plaintiff broker in the amount of 10% of the price at which the property was sold by defendant owner to Eugene F. Schaul and Karen M. Schaul, his wife, under the agreement dated September *651 27, 1977, plus reasonable attorney fees and costs.
DALY, HARRISON and SHEEHY, JJ., concur.
SHEA, J., dissents.
SHEA, Justice, dissenting:
If compelled to make a choice, I would affirm the judgment. The majority is clearly in error by invoking the parol evidence rule to prevent the defendant from proving that she had an agreement with the plaintiff that the real estate listing contract was nonexclusive. But even though I believe the majority is in error, this case was so poorly tried, and the findings and conclusions are so inadequate that I cannot in good conscience vote to affirm the judgment. It would be an injustice to do so. Justice requires that the judgment be vacated, and that the case be tried again.
REASONS WHY THE JUDGMENT MUST BE VACATED AND THE CASE TRIED AGAIN
This case took no more than an hour to try. Attorneys for both sides stipulated that three depositions (that of the plaintiff, Rad Lee Payne, that of his father, Carl Payne, and that of the defendant, Billie Buechler) be admitted in evidence. The only exception was that plaintiff's attorney reserved an objection to a part of the defendant's deposition testimony on her claim that she had a listing agreement with plaintiff with the word "nonexclusive" written on it in the defendant's handwriting. Plaintiff's attorney claimed that this testimony was barred by the parol evidence rule.
In addition to the depositions, the parties agreed that all pretrial interrogatories and their answers be admitted in evidence.
The record does not disclose the answer, but I surmise that the parties were pressured into speeding up the trial but it was hurried up so much that it was hardly a trial at all.
The trial transcript of testimony covers a total of sixteen pages. The plaintiff did not testify to the circumstances surrounding the execution of the listing agreement. He only testified to the amount of work he put in trying to sell the defendant's property after he had obtained the listing. Nothing in the record discloses why he did not testify on the circumstances surrounding the execution of the listing agreement on July 5, 1977. In effect, plaintiff's case-in-chief on the main legal question was based entirely on depositions.
Defendant's case was not much longer. Defendant was called to the witness stand and her counsel asked her if she understood that "your deposition has been introduced into evidence in lieu of a great deal of your testimony ..." and she replied "yes." She then identified another listing agreement that was in existence and still in effect when she signed the listing agreement with plaintiff on July 5, and this listing agreement was introduced into evidence with no objection from the plaintiff. She then testified that she refused to sign an exclusive listing agreement with the plaintiff.
She testified that Mr. Van Lueschene was present when she signed the listing agreement with plaintiff. Next, she testified that she sold the property on her own and that plaintiff did nothing to find the buyer. Her testimony on direct examination covers four pages.
Plaintiff's attorney then cross-examined her by asking her if she had found her copy of the listing agreement that she had with the plaintiff, and she again explained how her copy was destroyed as a result of vandalism. This cross-examination covers one-and-a-half pages of trial transcript.
Next, defendant called Mr. Van Lueschene as a witness, and he testified that he was present when plaintiff and defendant signed the listing agreement, that he had seen the plaintiff filling out the listing agreement and that he had seen the word "nonexclusive" written on the defendant's copy of the listing agreement. Before Van Lueschene testified to the circumstances surrounding the execution of the listing agreement, plaintiff's attorney asked for a continuing objection on the ground that this *652 testimony should be barred by the parol evidence rule. The entire testimony (direct and cross) covers five pages of the trial transcript.
In rebuttal, plaintiff called the defendant as an adverse witness. She was asked if she agreed with Van Lueschene's testimony to the effect that the word "nonexclusive" was written on the front of her copy of the listing agreement, and she replied yes. He then attempted to impeach her by revealing that at her deposition, she had testified that the word "nonexclusive" was handwritten on the back side of her copy of the listing agreement. Both sides then rested.
Plaintiff then submitted proposed findings and conclusions to the court and the defendant, who did not have any prepared, was given time to get them submitted. Later, the court adopted verbatim the defendant's proposed findings and conclusions, and entered judgment for the defendant. These findings and conclusions are absolutely inadequate, and I could not put the stamp of approval on a judgment based on them.
The findings and conclusions contain no reference to the claimed "nonexclusive" listing agreement. The findings are absolutely silent as to whether plaintiff knew that defendant had one or more listing agreements in effect when the listing agreement was signed on July 5, 1977. In fact, there is no finding which covers the main issue of this appeal whether parol evidence of the claimed "nonexclusive" listing agreement should be admitted.
The only finding which remotely bears on this issue, finding no. 3 states:
"That the contract was not intended by the defendant to be an exclusive right of the plaintiff to sell her property; that she had at least two other listing agreements with other real estate agencies in effect on July 5, 1977, whereby they also had the right to sell the property."
This finding is in essence, the entire basis for the trial court's decision, for the question of consideration and the question of the letter of termination are not dispositive of this case.
The only way that the judgment should be affirmed on the basis of the findings and conclusions is to invoke the doctrine of implied findings, and I am not about to do so. Too often it is used to get trial courts off the hook who have simply not done their job. If the case was well-tried and there was a good evidentiary record in existence, I would remand only for further findings. But here, because there was hardly a trial at all, I would order a new trial.
I add that neither the parties nor the trial court have pointed out a glaring discrepancy as to who was present on July 5, 1977, when the listing agreement was signed. In their depositions, the plaintiff, Rad Lee Payne, and his father, Carl Payne, testified that both of them were with the defendant when they discussed the listing agreement and when it was signed. Both of them also testified that Mr. Van Lueschene was not present on July 5. On the other hand, the defendant and Van Lueschene testified that only the defendant, the plaintiff, and Van Lueschene were present on July 5 when the listing agreement was signed. Both of them unequivocally testified that Carl Payne was not there. Someone is not telling the truth, and who was present is vital to the question of whether plaintiff wrote "nonexclusive" on defendant's copy of the listing agreement.
For these reasons, I would vacate the judgment and remand for a full trial before a different judge. I turn now to a discussion of why the majority is wrong in reversing the case and ordering judgment to be entered for the plaintiff.
Before proceeding to an analysis of the applicable rules, I think it necessary to expand on the facts stated in the majority opinion, and also to put this case in a procedural perspective as to how it was tried. The procedural question will be considered first.
As previously indicated, the depositions of plaintiff, of defendant, and of Carl Payne, were admitted in evidence. Plaintiff did reserve an objection, however, to defendant's claim that she had a "nonexclusive" *653 listing the objection was apparently aimed at keeping out of evidence her discussions with the plaintiff where she claims to have insisted on a "nonexclusive" listing.
The fact is, however, that the depositions of plaintiff and of Carl Payne, cover their version of discussions they had with defendant as to whether the listing was to be exclusive or nonexclusive, and as to whether plaintiff or anyone else had written "nonexclusive" on defendant's copy of the listing agreement. Their deposition testimony also covers their claim that they believed that defendant did not have the property listed with anyone else at the time she signed their listing agreement, even though they knew she had previously had listing agreements in effect with other real estate agencies.
For reasons of fairness alone, the trial court should also have been allowed to consider the testimony of the plaintiff and of Van Lueschene on their version of the signing of the listing agreement. See Rule 32(a)(4), M.R.Civ.P. But aside from this, the majority has missed the point in applying the parol evidence rule to bar the evidence. The evidence clearly falls within the exceptions to the parol evidence rule (set out in the statute itself, section 28-2-905, MCA), but unfortunately, the majority opinion does not discuss the exceptions other than to state that the evidence "does not fall within any recognized exception permitting its admission in evidence ..." The exceptions, however, clearly apply to this case.
ADDITIONAL FACTS NOT MENTIONED IN THE MAJORITY OPINION
I proceed next to an expansion of the facts beyond those stated in the majority opinion, because it must be done in order to reach the legal issues not covered in the majority opinion.
In her deposition, the defendant explained what had happened to her copy of the listing agreement:
"And on my exhibit, or on my contract which was destroyed when my house was broken into, I lost all my bank statements, lost all my records from my bar, I lost all my bank statements from last year, everything that I had, because when they broke into my bar or into my house, they poured ketchup, mustard, molasses, anything they could find in my house, all over everything, I mean there was no way that I could salvage anything ..."
At trial, plaintiff's attorney did not object to this testimony introduced through defendant's deposition.
He did object, however, to the remainder of the defendant's answer in the deposition, which states:
"I told him right then, I said I would not list this exclusively; he said: `Well, we really don't do this this way, but' he said, `we will make an exception.' So on the top of my listing he wrote `non-exclusive.' Well, I have nothing to prove that he wrote this, except that people that were sitting there listening to me say this."
The basis of the plaintiff's objection was that the parol evidence rule barred admission of this testimony.
Further, at trial, plaintiff's counsel, in his first question to the defendant on cross-examination, opened up the question of the defendant's missing copy of the listing agreement. He asked her if she had ever found her copy, and she again repeated what she said in her deposition. She said that vandals had broken into her house and, among other things, had poured ketchup, mustard and molasses over many of her papers, including the listing agreement, and that the papers had been ruined. In explaining how she threw out many of the items she said:
"I didn't even realize what I was throwing out, because there was so much stuff there that I couldn't salvage. They had poured molasses and ketchup and everything out of my cupboards all over my papers."
By her deposition testimony, and by her trial testimony, admitted without objection, defendant laid a foundation under Rule 1004(1), M.R.Evid., for proof of the contents of a lost or destroyed document here *654 the listing agreement. The next question, then, is whether defendant and Van Lueschene were entitled to testify that plaintiff had written "nonexclusive" on her copy of the listing agreement. I have no doubt that such testimony is admissible.
Contrary to the implications of the majority opinion, we are not dealing with only the plaintiff's copy of the listing agreement. Rather, we are dealing with the plaintiff's copy and defendant's copy, and the defendant claims that her copy contradicts the contents of the plaintiff's copy. Although the effect of defendant's offered evidence is clearly to alter the terms of the listing agreement copy held by the plaintiff, the evidence was offered to show that another copy of the same listing agreement existed, and that it contradicted the plaintiff's copy. Proof of the contents of a lost or destroyed document is permitted under Rule 1004(1), M.R.Evid. The parol evidence rule certainly would not bar evidence that defendant's copy contained the word "nonexclusive" in the plaintiff's handwriting. Once this evidence was admitted, the parol evidence rule would not bar the evidence needed to explain the patent contradiction in the listing agreements. Both versions cannot be right.
The logic of the majority opinion means that the outcome would always have to be controlled by the copy of the contract held by the broker. If a broker actually wrote different terms on the seller's copy of the listing agreement, he would never have to worry about being bound by what he had written on the seller's copy.
If this case were sufficiently tried to bring out the facts and if sufficient findings had been entered, I would affirm the judgment for two reasons. First, the two copies of the listing agreement (one of which the contents had been proved by first laying a foundation of a lost or destroyed document), not only created an ambiguity in need of explanation, they created a patent contradiction for which the explanation was vital. Because the actual copy of defendant's listing agreement was not produced, the question becomes one of credibility: Either the trial court believes plaintiff and his father that neither of them wrote "nonexclusive" on defendant's copy of the listing agreement, or the trial court believes the defendant and Van Lueschene, that plaintiff did write "nonexclusive" on defendant's copy of the listing agreement.
Second, assuming that "nonexclusive" was never written on defendant's copy of the listing agreement, defendant still could claim on the basis of illegality, invalidity, or fraud (section 28-2-905, MCA) that the listing was nonexclusive. If the defendant told plaintiff that she did not want an exclusive listing agreement, and if she told plaintiff that she already had a listing agreement in effect, what right would plaintiff have to claim that he had an exclusive listing agreement and be entitled to a real estate commission under the facts here? Assuming these facts, plaintiff would have known he had no right to receive an exclusive listing because defendant had no right to give him one. In such event, the exclusive listing would be invalid (an exception to the parol evidence rule listed in section 28-2-905(1)(b), MCA), and any attempt by the plaintiff to recover on this basis of an exclusive listing could well be given the name of fraud (another exception to the parol evidence rule listed in section 28-2-905(2), MCA). My explanation follows.
THE MAJORITY RATIONALE FOR EXCLUSION OF THE EVIDENCE
Without any analysis, the majority collects the following evidence under one umbrella and implicitly declares it inadmissible by application of the parol evidence rule:
"... The District Court's findings and conclusions are clearly based on the testimony of the owner that she did not intend to give the broker an exclusive listing; that she had previously given listings to other agencies which were still in effect, one of which was produced and admitted in evidence; and that the broker had written `nonexclusive' on her copy of the contract. She also called Mr. Van Lueschene who testified that `nonexclusive' was written on her copy of the agreement. Her copy of the written *655 agreement was never produced; she testified that she had destroyed it after her home had been vandalized and molasses and ketchup had been poured on it. With the exception of a prior and existing listing of the property with another real estate agency, all this evidence was objected to under the parol evidence rule and the objection was taken under advisement by the court." (Emphasis added.)
Only once does the majority refer to the parol evidence rule (section 28-2-905, MCA), and never to the exceptions contained in the same statute. They state: "The parol evidence rule generally provides that the terms of a written agreement cannot be altered or contradicted by oral testimony subject to certain well-recognized exceptions. See section 28-2-905, MCA." (Emphasis added.) After summarizing the evidence that apparently violates the parol evidence rule (previously discussed), and never setting out the exceptions to the parol evidence rule, the opinion states: "Here the parol evidence directly contradicts the plain and unambiguous language of the written instrument; it does not fall within any recognized exception permitting its admission in evidence, and is clearly inadmissible ..." (Emphasis added.)
I find such analysis and rationale to be seriously defective.
THE PAROL EVIDENCE RULE AND ITS APPLICATION HERE
All of the evidence (excluded by the majority opinion) is plainly admissible by the very terms of the entire parol evidence statute, section 28-2-905. Section 28-2-905 (cited but neither quoted nor applied by the majority) states:
"(1) Whenever the terms of an agreement have been reduced to writing by the parties, it is to be considered as containing all those terms. Therefore, there can be between the parties and their representatives or successors in interest no evidence of the terms of the agreement other than the contents of the writing except in the following cases:
"(a) when a mistake or imperfection of the writing is put in issue by the pleadings;
"(b) when the validity of the agreement is the fact in dispute.
"(2) This section does not exclude other evidence of the circumstances under which the agreement is made or to which it relates, as defined in 1-4-102, or other evidence to explain an extrinsic ambiguity or to establish illegality or fraud.
"(3) The term `agreement' ... includes deeds ... as well as contracts between parties." (Emphasis added.)
The validity of the agreement is "the fact in dispute" and therefore falls within the exception listed in section 28-2-905(1)(b), MCA. That is, the plaintiff broker contends he had an exclusive listing and was therefore entitled to a commission. But the defendant owner contends the broker had only a "nonexclusive" listing, and that he wrote "nonexclusive" on her copy of the listing agreement. Assuming the defendant to be correct, the listing agreement being held by the plaintiff broker would be invalid as an "exclusive" listing. Applied here, it would mean that under the facts here, the plaintiff could not collect a commission.
Further, although the pleadings were not so framed, the clear intent of the defendant was to show that the plaintiff broker was holding an illegal exclusive listing agreement, that in fact, when all the facts are considered, including her lost or destroyed copy of the listing agreement, the broker was holding a "nonexclusive" listing agreement. This being so, the evidence would be admissible under section 28-2-905(2). If defendant prevailed (that is, if the trial court believed plaintiff and the witness Van Lueschene), the plaintiff broker would be holding an illegal exclusive listing agreement, and he could not recover a commission based on the facts of this case.
The essence of defendant's case is that she tried to show that the exclusive listing held by the broker did not constitute the entire transaction that her copy of the listing agreement stating "nonexclusive", flatly contradicted plaintiff's copy of the *656 listing agreement. If plaintiff had actually written "nonexclusive" on the listing agreement, or otherwise agreed to a nonexclusive listing, I have no doubt he would be guilty of fraud against the defendant by then trying to collect a commission based only on his copy of the listing agreement showing that he had an "exclusive" listing. For this reason, all of the testimony would be admissible under the fraud exception set out in section 28-2-905(2).
Under all of the circumstances here, admitting the evidence would mean that the trial court was faced with a question of fact as to who to believe. If he believed plaintiff and his father, he could still order that the commission be paid. But if he believed the defendant and the witness Van Lueschene, he could rule (as he did here) that the plaintiff held only a "nonexclusive" listing and was therefore not entitled to collect a commission under the facts of the case.
I have set forth the reasons why I cannot affirm the District Court judgment, and I have also set forth the reasons why I cannot abide by the majority opinion. One more factor, however, must be addressed and that is the evidentiary question posed by the need to introduce evidence of the contents of the destroyed listing agreement.
PROOF OF CONTENTS OF THE DESTROYED COPY OF THE DEFENDANT'S LISTING AGREEMENT
An implied assumption of the majority decision is that even if the defendant produced a copy of her listing agreement with "nonexclusive" written in the plaintiff's handwriting, that defendant would still be bound exclusively by the plaintiff's copy of the listing agreement. Such decision has behind it neither logic nor justice.
The first question on defendant's listing agreement is: could the listing agreement be introduced in evidence if it still existed. I have no doubt that it could, for her copy was every bit as much an original as was the plaintiff's copy. Further, if her copy contained the word "nonexclusive" written on it, in plaintiff's handwriting, it would defeat plaintiff's right to recover the real estate commission. Simple fairness requires that defendant be permitted to introduce her copy of the agreement, just as simple fairness requires that plaintiff be permitted to introduce his copy of the listing agreement. The question then becomes one of whether plaintiff, who claims to have thrown away her copy of the listing agreement because it was virtually destroyed by vandals, should be permitted to establish the contents of this document anyway. The law permits her to do so.
The applicable rule of evidence to prove the contents of an original or copy where it has been lost or destroyed, Rule 1004(1), M.R.Evid., provides:
"The original is not required, and other evidence of the contents of a writing, recording, or photograph is admissible if: ... (1) All originals are lost or have been destroyed, unless the proponent lost or destroyed them in bad faith; ..." (Emphasis added.)
The defendant's copy of the listing agreement was an original within the meaning of the rules of evidence. Rule 1001(3), M.R. Evid., defines original as follows:
"(3) An original of a writing or a recording is the writing or recording itself, of any counterpart intended to have the same effect by a person executing or issuing it ..."
The evidence establishes that the listing agreements were form listings, with a carbon in between the top and the second copy. With the exception of the word "nonexclusive" being written on defendant's copy by the plaintiff (according to the plaintiff's testimony) after she received her copy, the listing agreement was filled out all in one motion. The defendant's copy was also an original.
Under Rule 1004(1), supra, the defendant could prove the contents of her copy of the listing agreement by her testimony and by Van Lueschene, who testified that he saw plaintiff write "nonexclusive" on the defendant's copy. The only question for the trial court to determine under this rule was whether defendant threw away her copy in bad faith after she claimed it was made *657 worthless by the vandals. If the court ruled she did it in bad faith, it would rule that she could not introduce through her testimony and that of Van Lueschene, evidence of the contents of this listing agreement. But if it ruled that she was in good faith when she threw away the listing agreement, evidence of the contents of her listing would be permitted. In any event, this would be a question of fact for the trial court to first determine.
If the trial court ruled defendant was not in bad faith in throwing away her copy of the listing agreement, she could testify, and so could witness Van Lueschene, to its contents that plaintiff wrote the word "nonexclusive" on her copy. The trial court was not, of course, required to believe either the defendant or Van Lueschene. But if he did, it would be a devastating admission against plaintiff's interest, for it would flatly contradict his claim that he had an "exclusive" listing. Nonetheless, once evidence of the contents of both these writings were in evidence, a flat contradiction arose. In this situation, the trial court undoubtedly would have the right to hear evidence from both sides as to the circumstances surrounding the execution of the listing agreements. The trial court would well decide either way depending on whose version of the facts it believed. In any event, the parol evidence rule would not prevent introduction of evidence to explain the contradiction existing between the two listing agreements. Both of them could not be right.
For the reasons stated, I would vacate the judgment and remand for a new trial.
| null | minipile | NaturalLanguage | mit | null |
A Middlebury blog
Sarah Barnhart
Dog-Driven Therapy: A glimpse into the life of Lissie Heminway
Eleven sled dogs started howling in an off-key chorus as I approached. Their claws scrabbled at the chain link fence as they stared at me, their mouths stretched into grins. One dog with a black-and-white face snarled at her colleagues, instigating small skirmishes. Later I learned that the little spitfire, Ziggy, has an attitude problem. “Don’t worry,” Lissie assured us, “their bite is worse than their bark.” I stared at her with concern until she laughed and corrected her mistake. As I stood outside the door that held back the eager dogs, Lissie instructed me on how to best greet them, but in reality she had only one piece of advice: keep walking.
Having grown up around animals I know how to keep calm; animals can smell nerves or fear on a person and will take advantage of the situation. Keeping this in mind, while eyeing one husky as tall as my waist, I took a deep breath and pushed open the gate. The dogs immediately pounced, sniffing every orifice of my body. The dogs even jumped onto the roofs of their shelters in order to have easier access to my ears and face. I couldn’t help but smile as they slammed their bodies into mine, leaping onto me from all sides, desperate for some attention. Lissie deftly moved through the throng of dogs, until we reached the field. She pointed to each dog individually, listing off their names. Although I had met Lissie only ten minutes before, already she started to teach me. I listened as she began with the basics in order to clear up my confusion about the interchangeability of the terms dog sledding and mushing.
Mushing refers to any form of transport powered by dogs, including dog sledding, pulking, carting, skijoring, freighting and weight pulling. The term mushing is believed to derive from the French word marche, which means to run. The practice of using sled dogs as transport began with the Inuit and Eskimo populations in the Arctic region. The European colonists that arrived in the region adopted the practice and in the 1920s gold miners from Alaska brought sled dogs to New England where the art of mushing gained popularity as a sport, rather than as a practical means of transportation. Famous long-distance dog sledding races, such as the Iditarod in Alaska, entered the common vernacular, even becoming the focus for multiple Disney movies. Even now, when I mention to my peers that I am learning how to dog sled, they cheer, “BALTO!” The animators of the children’s movie would be thrilled to know that in my generation their work remains our primary association with dog sledding. With the growing popularity of the sport, finding dog sledding opportunities in Vermont proved to be easier than expected. Luckily, I called the number of Lissie Heminway.
The family friend could never have known that by handing six-year old Lissie a book, he changed the course of her life. The book on working dogs had one page on competitive dog team racing. She obsessed over the page, and several years later started to train her family’s dogs — golden retriever mutts. She harnessed up the retrievers and strapped into her cross-country skis. Unfortunately, determination does not always guarantee success. An active imagination, however, allowed Lissie to pretend her highly uncoordinated golden retrievers were a pack of athletic huskies. Later, at the age of sixteen, she boarded a plane bound for Alaska to participate in a National Outdoor Leadership School program (NOLS). Although the course centered on hiking, during one excursion she passed a dilapidated sled on the side of the road. Most people walked by the sled without a second thought. That ancient sled, however, struck Lissie. When describing the moment to me she explained, “I realized that dog sledding was never going to go away.” And it didn’t.
Sometimes the fates seem to be speaking. They drop hint after hint, crying for someone to notice the obvious signs that provide answers to life’s important decisions. Lissie heeded the fates after finding a husky patiently waiting in her locked car after a hike in Olympia, Washington. No, magic did not happen. Only a desperate dog; so desperate that it had climbed through a partially open car window. A freshman at Evergreen State College, at first Lissie did not want to own a dog. New to the rigors of college academics and still navigating the waters of the social scene, she assumed that a dog would negatively transform her lifestyle at school. The search ensued. With no tags and only a trailing length of a broken chain, which suggested that the husky lived outside, the owners proved untraceable. The dog was Lissie’s, whether she wanted him or not.
Although Olympia, Washington is notorious for rain not snow, the idea of owning a dog team had remained in the back of her mind. The husky in her car pushed the idea to the front. In a manner of months, Lissie went from not wanting a dog to buying another husky. With a team of two she began her own unique overland training method. Dog biking. Every day the dogs would pull Lissie to the main campus on her bicycle. Only when she graduated and moved to Vermont to work with Ed Blechner, a dog musher in the region, did her true exposure to the sport begin. Initially she planned to mush for only a season in order to fulfill her childhood dream and continue with her life. A mushing detox, if you will. Eighteen years later, she owns a full dog team and is committed to keeping dog sledding an integral part of her life.
“Come in, come in!” Lissie Hemingway called when I arrived at her house for the first time. When organizing the details of our initial meeting over the phone, her warm voice made me picture a kind, down-to-earth, calm woman. She retained this calm even after I had to call her twice for directions to her house. People may exalt Google Maps but Google’s directions for Vermont back-country roads leave much to be desired. Despite my frustration at being hopelessly lost between the towns of Shoreham and Whiting, on the drive up to her house I turned off the music and fell silent.
Trees lined the dirt road, their branches forming a natural roof. In the summer, this drive would be divine. In the winter, in the eerie gray that comes before the first real snowstorm, it seemed like the beginning of a horror movie. Despite this creepy entrance we stumbled upon a picturesque wooden house, with smoke (literally) curling out of the chimney. I learned later that Lisse and her husband built this house with their own hands about ten years ago. The house remains off the power grid and all the utilities and machines run off batteries and solar energy.
We let ourselves in and found her busy at the stove, pulling a tray out of the oven. The smell of chocolate chip cookies filled the wood kitchen. A half-made piñata sat in the corner of the room in preparation for her oldest daughter’s birthday party. The wood stove crackled in the corner and a large window framed the length of the kitchen. Through the glass, I could see the various pens and paddocks encircling her animals. Even inside, Lissie could keep an eye on her small farm. Never before had my expectations of a person matched with reality so perfectly. Her voice, which had sounded kind and warm, was exactly the woman I met. Dressed in worn green carhartts and a red wool sweater, she offered us appropriate jackets for the rain and mud and, after perfunctory introductions, led us out the door and into her backyard.
Lissie’s passion for her dogs and other animals is illuminated immediately upon stepping onto her land; she exudes the zeal required to tend to four horses, six cows, seven cats, eleven sled dogs, one house dog and an uncertain number of chickens. Imagine how much time it would take to complete the simple chore of daily meals. I quickly discovered, however, that Lissie is a person of many passions. When I broached the topic of education on the tour of her property, her enthusiasm peaked. Seeing her soft blue-grey eyes light up and hearing the ardor in her voice as she described her work with the Vermont school system and youth camps revealed how much the students, not just the dogs, are a part of her identity. It took time, however, for me to learn how she blends these two aspects of her identities into one.
Gilly, Willow, Mochi, Uma, Little Bear, Illick, Petra, Ziggy, Glassy and Tinder. On my fourth visit I could identify all the sled dogs, slip on their harnesses and correctly position them on the line. Line placement requires a certain amount of artistic license. The pairings form naturally. Like any human, the dogs want to run with their best friends on the team. On the line, the placement of the pairs depends upon a combination of brawn and brains. The biggest and strongest dogs stay close to the sled (or in this case, the ATV), serving two purposes: in the back they can pull the most weight and be easily controlled by the musher. In the front are the lead dogs, the most intelligent and enthusiastic members of the team. “Your team will only be as good as the leads,” Lissie explains. “They set the pace and direction. I can’t give you a rubric of qualities you are looking for in a lead dog, but I follow my instincts.” The rest of the dogs on the line are a part of the team. Of course, there is the odd-one-out. Wiley, the house dog. A lithe border collie, he runs alongside the team, distracting the dogs whenever he bounds into the woods to chase an unsuspecting woodland creature.
By this time, the dogs recognized my scent. Although they still barked when I approached to prepare them for a run, they weren’t quite as raucous and calmed with a touch. I quickly grew to love the dogs and would have been happy to spend hours in the pen giving each one special attention and learning each dogs’ personality. By watching the dogs run, I knew the slackers and the hard-workers. By witnessing their interactions in the pen, I knew the alphas and omegas of the pack. Slowly, I began to clue into the dynamics of the team and understand more about the dogs and the sport.
Lissie owns three different breeds of huskies: Alaskan, Yukon and Inuit. Of the ten sled dogs, only the two Inuit huskies, Ziggy and Petra, were born in their native environment: the north. The roots of the Inuit, or Qimmiq, breed have been traced back 4,000 years to present-day Mongolia. When humans migrated across the Bering Strait between 900-1100 AD, the dogs traveled with their human counterparts as their transport. The populations dispersed throughout Canada and Greenland, and today Inuit dogs can still be found in those regions. Inuit huskies face a tough life in the Arctic Circle. Bred as working dogs, the Inuit people use them as transport during the winter, but as the snow and ice begins to thaw, they leave the huskies to fend for themselves on islands during the summer months. Whichever dogs survive the summer rejoin the team the next winter. Therefore, Inuit breeds are known for their hearty, aggressive behavior.
Although they are both Inuit dogs, Ziggy originated from Baffin Island and Petra from Greenland. The owners of Ziggy’s mother intended to kill her while Ziggy gestated in the womb in order to prevent the birth of an entire litter. The owners could not afford to feed their family, let alone a whole litter of new pups. The obvious solution: destroy the source of the problem. When biologists working in the area heard of this plan they offered to take the mother off their hands. When this idea did not face any resistance, the scientists flew the mother and the pups to the United States. The scientists knew Lissie and her growing dog sledding team. They offered one of the pups to Lissie as a gift — Ziggy. When I first met Ziggy, I assumed she had an attitude problem. Now, I realize that her aggression is the result of the hard-wiring of the instinctual survival of the fittest.
I also began to understand why people become addicted to the sport. Although the snow-less winter kept us on the ATV, it still felt like flying. With three people on a four-hundred-pound ATV, the dogs pulled us at approximately 8-18mph, flying around trees and over plywood bridges with such speeds that sometimes made me glad I sat in a sturdy four wheeler and not a light, maneuverable sled. I felt reassured, however, when Lissie informed me that she would never run ten dogs with a sled. “There would be too much power! Even on the ATV I use the brake continuously. Sled dogs can pull about three times their weight,” Lissie crowed, “in comparison, horses can only pull their own weight!” This fun fact, however, requires a disclaimer. Lissie admits she has no idea where this statistic originated, but it sounds impressive so she spreads this information onto her students and customers. Although the fact may have no scientific basis, I believe her. On one run, I jumped onto the moving ATV. Although the dogs started pulling the ATV with two passengers on an upward slope, I still had to run a fast clip in order to catch the accelerating vehicle. Even with that weight, the dogs could run faster than me, even though at that moment I fancied myself a female version of Jason Bourne.
Before I saw the dogs in action, I jokingly agreed with my father when he suggested I do an exposé on dog sledding. The number of times the musher needed to whip the dogs would decrease “the humanity quotient.” Although I never believed that the dogs were literally whipped into shape, I equally did not consider how much they love and yearn to pull. The dogs exude a frenzied energy before the run, jumping all over one another and snapping at their partners. Unlike horses, dogs run with the most energy at the outset and start to lose energy as they near home. The four-mile training trail snakes through the land surrounding the house, cutting through woods, fields and even Shacksboro Rd. While approaching the road I looked both ways to check for incoming traffic and I thought I noticed the dogs whipping their heads from side to side. I strongly believe that dogs are intelligent, sentient beings. Believe it or not, but I assumed that they, like me, were checking for cars. When the coast was clear the team barreled down an expansive, brown field. The lack of snow doesn’t stop these dogs, even though Lissie says that they seem a little confused. “This is supposed to be their season. I imagine they must be wondering if we are still in the fall or they accidentally slept through the winter.”
One day I experienced the full power of one of the strongest dogs on the team, Uma. The chicken appeared out of nowhere. The dogs headed straight up to their pen after a run and I jumped off the ATV to take off their harnesses in order to release them into the pen. I started leading Uma over to the gate to take off her harness when she lunged after the squawking feather-ball, determined to sink her teeth into the stupid creature. I held onto Uma, skidding across the icy mud. I yelled a sharp command as the chicken took off in an explosion of feathers. Once the chicken found a safe perch on a branch, Uma allowed me to lead her back to the pen. I started laughing as Uma looked at me with a mischievous glint in her eye. That experience taught me quickly that though these dogs are adorable, lovable and friendly, they are also powerful and, if you are not careful, can pull your arm out of your socket.
Occasionally, my friend Heather would accompany me to Lissie’s house. We would ride side-saddle on the ATV, gripping each other around corners. Although the lack of snow hindered the true experience of dog sledding, Heather and I found a way around that small dilemma. One day after returning on the ATV, unsnapping the dogs and releasing them into the pen Lissie helped us pull out two sleds. One for Heather and me and the other for her youngest son, Elliot. Although the two inches of snow would be too shallow for the snow hook, required for steering while actually dog sledding, there was plenty to muck around with in the field behind the house. We took turns playing the role of sled dog. “Hike! Hike!” Heather yelled at me, and off I went running, pulling her in the rutted, ice-encrusted snow.
After taking turns we took out Willow and Gilly, the old women of the team, and hooked them up to the two sleds. With much coaxing and cajoling they walked in a semi circle while we helped push the sled. If we jumped onto the runners at the back they stopped in their tracks and turned to glare at us. Only the six year old Elliot appeared graceful on his sled, unperturbed by the slow moving dogs, while the two college students whooped, hollered and jerked around on the back of the sled. Lissie only laughed at our attempts and tried to start teaching us the physics of dog sledding. Her answers revealed her training as a teacher, but in the end she admitted, “Once you are on the sled, it will all become clear. Whenever I take a customer or friend out for a run, inevitably they will crash. Luckily, after just a short amount of time they learn which way to lean to correctly maneuver the trail.”
About fifteen years ago Lissie began a dog sledding business, running trips in the mountains for schools, colleges, families, or any other customer. It seemed like the perfect set up — being paid to do what she loved. However, the job lacked satisfaction. People wished to do dog sledding as an alternative to skiing. They arrived, experienced the adventure and left never to return. While running the business, Lissie continued her twelve-year stint working as a teacher in the special needs program and stumbled upon the idea to combine her two passions. Now two different programs bring a group of students every week to work with Lissie and her animals. Although Lissie volunteers her time and dogs to interact with the students, she hopes to transform the volunteer program. The dream is to run a fully fledged dog sledding program targeted at special needs and at-risk students to provide them with a source of therapy. “Being fulfilled is what makes a successful business. It’s not about the money,” she explains, sitting on the couch, her framed by her dirty blonde hair pulled back in a messy ponytail.
Currently, students enrolled in a program called Diversified Occupations (DO) at the Middlebury Union High School arrive at Lissie’s home every Friday where they learn basic skills required for managing a small farm, interact with the dogs and learn about mushing. The high school program focuses on teaching non-academic skills, similar to vocational training. In the past, the program only taught students how to clean dishes, funneling them into Middlebury College’s kitchens. Now, the revamped program works with many local volunteers to teach the students skills that will help them find employment outside of the kitchen post graduation. These include building a boat for the Marine Museum, banding birds with a local biologist, working at the local daycare and traveling to D.C. to learn about the American government. Although Lissie briefly mentions the benefits of learning how to work on a small hobby farm, she concentrates primarily upon the therapeutic benefits of working with animals.
Lissie informed me that she also works with students in an Eckerd Youth Alternative program in Vermont called Camp Ewanaki. Her last batch of students graduated from the program one week before my arrival. Established in 1968 in Brooksville, FL, Eckerd Youth Academies take troubled teenagers into either residential or short-term outdoor camps in order to provide the troubled teens with an outlet that allows them to redirect their behavior. Before the 1960s, youth suffering from behavioral issues or troubled homes were hospitalized. Thankfully, centers and homes run by various organizations, including the Eckerd Youth program, provide an alternative option. One of twelve sites in the United States, Camp Ewanaki focuses upon the therapeutic benefits of spending time in the outdoors and working with animals.
For centuries, the fact that animals can improve a person’s well being has been a piece of common knowledge. Finally scientific evidence can back up this universal truth. Studies indicate that pet owners enjoy the benefits of lower blood pressure and a stronger immune system in addition to the emotional benefits that derive from the bond between pet and owner. The medical field has begun to incorporate this knowledge into their rehabilitation programs, developing animal-assisted therapies (AAT) in order to alleviate mental and physical maladies. Although caring for animals may aid in physical rehabilitation by exercising particular muscles, the real advantage of AATs are the mental benefits. Emotional bonds can form between animals and humans, forming an unconditional friendship and trust that sometimes cannot be found in human-human relationships. Additionally, caring for the animals requires maturity and responsibility. Patients suffering from learning disabilities, developmental disorders, emotional trauma or delinquency benefit from AAT programs. Lissie works with these types of students weekly, styling her volunteer program after AAT programs.
After two weeks of waiting, the opportunity to meet Greg, Christa, Patrick and Monty, students in the Diversified Occupations (DO) program from Middlebury Union high school finally arrived. The week before I arrived at the house early Friday morning, but unfortunately the students could not attend due to poor weather conditions. The rain/snow mixture rendered the roads treacherous. One week later under blue skies the driveway proved to be equally dangerous, but this time the students braved the elements. Two days before, rain from warmer temperatures froze when the temperature dropped to negative numbers. The winding dirt road that leads to Lissie’s house turned into a skating rink. The bus could not make it past the 180 degree turn in the driveway, so the students disembarked and trekked up to the house on foot.
When the group of four students and their teacher walked up the ice shrouded driveway I didn’t know what to expect. Although I have worked with children with severe developmental disorders in Jordan and have a cousin with severe autism, dealing with students with learning disabilities barely younger than myself poses a whole new dilemma and requires a different set of skills. I shouldn’t have worried. If Lissie hadn’t informed me that the group of students attended the DO program, I might not have known that any four of them suffered from learning disabilities. Only after some one-on-one conversations with the students did characteristics associated with OCD and other disorders manifest themselves.
After introducing me, Lissie informed the group that they would watch and learn how to harness the dogs. While we ran the dogs on the five-mile trail in the land surrounding her house, the students would move wood from the chopping block to the back of the house. Upon our return, we would play with the dogs and perhaps pull out some sleds. One by one, Lissie and I took out the dogs and hooked them onto the line. I learned that this was the first time the students had seen the dogs outside of the pen and their enthusiasm was palpable. Although they interacted with the dogs in the enclosure and knew them by name, they had never seen the excitement and crazy energy that they project prior to setting out for a run. With so many people around, the dogs could barely control themselves. The students scratched their ears while we clipped each dog to the line, forming a crushing mass of dogs and humans. I could barely lift each the legs of each dog to slip on the harnesses. Finally, we were ready to go. The dogs reared against their halters, attempting to pull the parked ATV. Lissie and I climbed on-board and took off, leaving our audience of students behind.
The trail, similar to the driveway, turned into one large sheet of ice overnight. “Ahhhh! Hold on! Or maybe you should jump,” Lissie yelled as we skidded around corners. I gripped the iron bars on the back of the ATV as we swung dangerously close to the trees lining the dirt path. The tires couldn’t grip the slick surface, leaving the ATV at the mercy of momentum. The dogs’ legs slipped out from under their bodies as they struggled to keep moving forward. I held my breath. Lissie never sounded worried before. “We made it through the worst bit!” she cheered and I started to breathe again. Lissie and I made it back in one piece, and the students stood waiting for us.
Now, it was time for dog sledding 101. The first lesson, however, did not require any dogs. The students raced the sled down a miniature hill, learning how to turn and playing with the brake. We cheered as they sped down, although occasionally I had to sidestep a sled headed in my direction. After a lunch break, we all headed back out to do some chores around the farm. Many of the students work on dairy farms, and while I mucked out horse stalls one of the students, Patrick, joined to help me out. “I’d rather clean out the cow paddock,” he grumbled. Never before had I ever heard anyone say that. Compared to the stalls, the cow shelter looked like a mess. I quickly learned that these students do not need to learn skills required for running a farm. Working on a farm is part of their life. I wondered why the students came to Lissie, other than to expose them to the art of mushing. When she handed out the various tasks, I understood.
Lissie directed the students to the barn and asked them to create a system for storing the strings from the bales of hay. She left them with a hammer and a box of nails, believing that they could successfully complete the job. By allowing the students to work independently on a project, not just a simple chore, she showed them that she trusted their abilities, placing responsibility on their shoulders and boosting their self-confidence. Interacting with the dogs provided an outlet for these students in a society where teenagers with disabilities may be shunned. However, Lissie herself also provided a form of therapy.
Warming up after a run, Lissie, Heather and I gripped our tea mugs full of delicious black tea, mixed with milk and honey. For seventeen years, Lissie worked for Kirk Webster, a premier beekeeper in Middlebury, VT whose work gained international recognition after he bred a species of honeybee resistant to mites. For all those years, Lissie has been blessed with a constant supply of fresh honey. Moving on from the glories of unpasteurized, fresh honey we fell into easy conversation while flipping through Lissie’s old photos.
“Did you know I am allergic to dogs, cats and horses?” Lissie casually threw into the conversation. Heather and I stopped. We stared at her for a moment. Took a deep breath, and then stared for a little longer.
“Wait. What?”
“I kept going to my allergist with lists of what could possibly be the culprit for my congestion. Coffee, wheat, pollen, etc. He decided to run a few tests to get to the bottom of the issue immediately. When he told me that I was allergic to dogs and cats all I could do was laugh and almost walk out of the room.” We started barraging her with questions:
“So, the baseline for your health is a constant allergic reaction?”
“I mean, it’s not a severe reaction. Just irritation.”
“Do you feel different when you leave for holiday?”
“I guess a little better. Most of the time we travel to places where there are animals.” She said this so nonchalantly. Lissie’s passion, in fact her entire life, revolves around animals that make her physically ill. That is true dedication.
This dedication extends beyond Lissie and into her family. Owning a dog team and all of the other farm animals requires a specific lifestyle, a lifestyle that Lissie’s whole family embraces. I finally met the whole Heminway clan when, for the first time, I visited the house on a weekend. Heather and I had baked fresh pumpkin bread in order to thank Lissie for her willingness to let us work with her. Most of the family was outside as we drove up. We watched as Lissie pitched a ball to Elliot, the youngest son, who swung his plastic bat and missed spectacularly. Owen, the middle son, crouched over tree stumps while his father, Bill, sawed the wood into logs for the fire. As we opened the car door, Lissie called us over and Wiley hurtled himself onto us. He proceeded to sprint behind the house in order to alert the pack of huskies that we had arrived. The whole family headed into the house where Polly, the eldest daughter, sat at the table reading a book. We settled onto the benches around the kitchen table and dug into the bread. The two boys could only sit still for twenty minutes, before they pulled out their brass instruments and serenaded us.
Earlier, Lissie mentioned that managing her dogs and farm animals had gotten harder as the kids grew older. This countered my assumption. I thought that older and more mature children would have greater independence and therefore would be able to help with the daily chores associated with keeping animals. Although each child does their share to keep the small farm running, Lissie concedes that when they were younger she could keep them on her schedule. Now they have other commitments and activities, ranging from basketball practice to music lessons. In all of my interactions with Lissie, Bill and her children, however, I got the impression that the dogs and other animals are embedded into family life and are here to stay, no matter the challenges.
Lissie and Bill dream about transforming the land into a functional farm and fully fledged school. “We want to leave our children with something valuable. Either they could sell the farm or come back and run it themselves,” Lissie explains. Both Lissie and her husband are educators and they would both teach at the school, which would combine academics with the therapeutic benefits garnered from working with animals and the land. This would not be the first time Lissie and Bill worked together. In the past, Bill acted as the financial manager of the dog sledding business. Although he had no independent investment in the sport, he adopted it as his own. Based upon their track record, building a home and managing a business together, I believe that one day this dream will be a reality. The idea seemed even more grounded in reality when we spoke about the possibility of hiring a summer intern from Middlebury College to start the first part of the project — the working farm.
An hour and half a loaf of bread later, Heather and I slid into our seats to drive back to the Middlebury campus feeling rejuvenated. On a college campus, we interact primarily with people our own age. Trapped within a bubble of our peers, we rarely interact with adults, children or animals. The Heminway household provided all three. As we neared the paved road, Heather happened to glance in the rear-view window. A small, black-and-white figure tore down the road at a full sprint. Wiley had chased us from the house all the way down the dirt path. We stopped the car and loaded him into the backseat. He promptly decided that he wanted to ride shot-gun so I, ever obliging, clambered into the back. I scratched behind his ears, definitely rewarding him for his bad behavior. We led him back into the house and gave the boys strict instructions not to open the door for at least ten minutes.
When Wiley chased us down, I had mixed feelings. Fear about what would have happened if we hadn’t looked in the mirror and annoyance about the inconvenience of driving back to the house. Mostly, however, I felt loved. A dog wouldn’t chase a stranger down the road, I hoped. After only three weeks, I felt embraced by Lissie and her entire family, which includes the dogs and other animals. I may not have been a student from one of the many programs Lissie works with, but working and learning from her provided a form of therapy unavailable on the college campus, reminding me that life exists beyond the rigors of academics. Insha’allah (god willing) my relationship with Lissie and her dogs will extend beyond my time at Middlebury College and someday I will return, when snow covers the ground, to dog sled past a bustling farm and renowned school. | null | minipile | NaturalLanguage | mit | null |
A comparison of the approaches to assess the abdominal aortic stiffness using M-mode ultrasonography, tissue tracking and strain rate imaging.
Which kind of ultrasound imaging technique is suitable for the assessment of the abdominal aortic stiffness are seldom reported. The purposes of this study were to explore a reliable method to evaluate the abdominal aortic stiffness in patients with hypertension among the following ultrasound imaging techniques: M-mode ultrasonography (M-mode), tissue tracking and strain rate imaging. Fifty patients with hypertension and fifty age and sex-matched healthy volunteers were involved in this study. The displacement (d), the peak strain (ε) and the peak strain rate (s) were obtained from the long-axis images of the abdominal aorta using tissue tracking and strain rate imaging, respectively. The pressure strain elastic modulus (Ep), β stiffness index and distensibility were calculated according to the conventional formulas using M-mode combined with the blood pressure. Compared to the normal subjects, the difference between systolic diameters and diastolic diameters (∆diameter), the displacement of posterior wall (d-posterior), the difference of the displacement between anterior and posterior wall (∆ displacement), and the distensibility decreased and the Ep and β stiffness index increased in the hypertension patients There were no significant differences between the patients with hypertension and the normal subjects according to the ε, s. Among ∆diameters, d-posterior, ∆displacement, the ε and s, only ∆diameters significantly correlated with the Ep, β stiffness index and the distensibility in hypertension patients. Strain rate imaging cannot sensitively discriminate the difference of the abdominal aortic stiffness between patients with hypertension and the normal subjects. M-mode ultrasonography is still a classical method for accessing the aortic elasticity. | null | minipile | NaturalLanguage | mit | null |
[Neuroleptics and Cognition]
OBJECTIVE: The aim of this study was to evaluate the effects of atypical antipsychotics on cognitive function in schizophrenic patients under clinical routine conditions. METHOD: Schizophrenic patients (n = 78) were evaluated on neuropsychological tests of attention, short-term- and working memory, learning, long-term memory (retention) and executive function. Data were analyzed according to medication, severity of illness and age. RESULTS: We observed that treatment with atypical antipsychotics compared to conventional neuroleptics was significantly associated with a more favorable effect on cognitive function. Especially in short-term memory and retention a clear advantage of atypical antipsychotics could be seen. CONCLUSION: Results from this study suggest that even under clinical routine conditions atypical antipsychotics have an advantage on cognitive function when compared with conventional neuroleptics. | null | minipile | NaturalLanguage | mit | null |
Screenshot : All images Capcom
Kotaku East East is your slice of Asian internet culture, bringing you the latest talking points from Japan, Korea, China and beyond. Tune in every morning from 4am to 8am. Prev Next View All
I’ll be honest. When Project Resistance was announced, I was skeptical. A multiplayer Resident Evil? Surely, that cannot work, I thought. At the Tokyo Game Show, I got hands-on with Project Resistance, and from what I played, I realized that, yes, yes it can.
Project Resistance is a four-on-one asymmetrical co-op multiplayer game. Four players team up in hopes of solving puzzles and escaping from locked rooms. The other player does everything possible to make sure that doesn’t happen.
While Project Resistance is a temporary title, the game itself felt fully fleshed out. The four different Survivors each have a special ability. The character January Van Sant, for instance, can disable the cameras that exist throughout the rooms and hallways in Project Resistance. The character Tyrone Henry specializes in defensive moves; he’s also able to easily kick down doors and rally the others. Valerie Harmon can heal the injured, and Samuel Jordan is good on the attack. These are the Survivors. Each of these specialized skills means that players must work together to solve puzzles and escape from the rooms.
What makes Project Resistance interesting is that there is a fifth player who takes the role of the Mastermind, who tries to prevent the Survivors from escaping. The Mastermind can access the CCTV cameras to monitor the rooms and hallways. When a camera is selected, the Mastermind can then spawn zombies and creatures in that area as well as do other things to make life difficult for the Survivors such as lock doors, turn out the lights and set traps. Since the character January can hack the cameras, that means the Mastermind must toggle from camera to camera to prevent—or slow down—the Survivors.
The Mastermind can also take control of a zombie that’s been spawned, which keeps things interesting. When Mr. X is spawned for a limited time things get really interesting because he packs so much brute strength.
I found that playing as the Survivors was a solid Resident Evil experience, but with the added element of everyone working together, trying to evade the zombies and escape. Because the maps are cramped, however, sometimes it seemed like all the Survivors could get clumped up together in confined areas. The co-op experience is good, and there is the same sense of dread in traditional Resident Evil games. The way that is recreated in a multiplayer co-op was impressive.
What I really liked was how difficult the Mastermind experience was. Shooting a zombie in the face is satisfying in Resident Evil, but when you are the Mastermind, doing something as simple as turning out the lights was also equally satisfying—and for the players, unnerving and frightening. The Project Resistance demo understood what makes horror so effective for those who are being pursued and for those doing the persuing. What could be a deeper understanding of what Resident Evil is than that? | null | minipile | NaturalLanguage | mit | null |
Holocaust Studies and Materials
Zagłada Żydów. Studia i Materiały () is a Polish academic yearbook published by a group of historians and researchers from the Polish Center for Holocaust Research created in 2003 in Warsaw. It is an annual devoted to the topics connected with the broadly understood Holocaust research. The target audience could include academics dealing with the Holocaust, but also college and university students, as well as broader public interested in this topic. Each volume forms individual and self-contained monograph. Authors of the articles represent various generations and scholarly approaches. The common characteristic is their frequent reevaluation of primary and secondary sources as well as the popular perception of truth. Important part of the journal consists of book reviews.
Since 2008 the biannual or triennial English language edition titled Holocaust Studies and Materials is also being published. Editorial board includes Dariusz Libionka (editor-in-chief), Barbara Engelking, Jacek Leociak, Jan Grabowski, and Agnieszka Haska. The managing editor is Jakub Petelewicz.
The work is divided into nine sections, as follows: Studies, Profiles, Materials, From the research workshops, Points of view, Book Reviews, Events, Curiosa, and Letters.
References
External links
Journal homepage.
Articles available online.
Polish Center for Holocaust Research homepage.
Category:2003 establishments in Poland
Category:English-language magazines
Category:History magazines
Category:Holocaust studies
Category:Magazines established in 2003
Category:Media in Warsaw
Category:Polish magazines
Category:Polish-language magazines
Category:History books about the Holocaust
Category:Polish Academy of Sciences academic journals | null | minipile | NaturalLanguage | mit | null |
Q:
CHDK: Timelapse/intervalometer with gradually adjusting exposure or overexposure detection
I'm trying to tweak a CHDK script for my Canon PowerShot G12 to take pictures at a regular interval. The goal is to make smooth time lapse videos.
I've previously burned the sensor of a G11 because of an unexpected lighting change, and would like to prevent this from happening again. So now I'm looking for a way to either:
Adjust the exposure slowly over the span of several pictures, or
Detect a certain level of overexposure and abort the script
Here is what I have so far:
@title Timelapse photo fixed
@param m Interval (min)
@default m 0
@param s Interval (sec)
@default s 5
@param h Number of takes (x100)
@default h 10
@param x Initial delay (sec)
@default x 10
set_raw 0
t=100*h
if t<100 then t=100
i=60000*m+1000*s
if i<100 then i=100
if x<5 then x=5
sleep 1000*x-2000
print "Pressing shutter halfway..."
press "shoot_half"
sleep 2000
for j=1 to t
cls
print "Taking photo",j,"of",t
click "shoot_full"
press "shoot_half"
print " Waiting",m;":";s,"..."
print " "
sleep i
next j
release "shoot_half"
end
(The shoot_half bits are there to prevent the exposure/focus from changing during a shoot; if there is a better way to do that I'd also love to about it.)
Where should I go from here?
A:
These problems were encountered when trying to compensate from the bright sunlight to the dark of night for sunset and sunrise events. There've been a couple of "sunset" scripts created long ago to circumvent these problems, as well as trying to adjust for exposures when the light levels are far too low and the camera's own exposure meter can no longer function. Done by polling the data directly from the RAW sensor data when needed.
Look into these scripts on these two links:
http://chdk.setepontos.com/index.php?topic=2156.0
http://chdk.setepontos.com/index.php?topic=3079.0
| null | minipile | NaturalLanguage | mit | null |
Shou Leather Bracelet
Shou Leather Bracelet
Fashioned from soft calf leather, this masculine leather bracelet has a contrast colour on the reverse side, and is embellished with a chic geometric interpretation of the Chinese shou motif, which is palladium alloy plated. The bracelet is made in Italy.
Colors: Black, Navy, Brown, Dark Brown, Blue
Made in Italy | null | minipile | NaturalLanguage | mit | null |
West Footscray railway station
West Footscray railway station is located on the Sunbury line in Victoria, Australia. It serves the western Melbourne suburb of Footscray. Two dual gauge tracks run north of the station forming the South Kensington – West Footscray freight line, linking the Port of Melbourne and other freight terminals to the rest of the state. The tracks also form part of the Melbourne – Sydney standard gauge line. The Regional Rail Link lines run to the south of the station.
The station opened on the 1 October 1888 as Footscray West being renamed West Footscray on 1 September 1912. The original stations buildings were reconstructed in 1976. The station has since been rebuilt 200 metres to the west of its original location.
A third platform is to be built to the north of the existing platform to allow terminating services to operate. Work was due to commence in late 2018.
2012–13 Regional Rail Link Rebuild
In 2013 the station was rebuilt 200 metres further west to accommodate two lines to the south as part of the Regional Rail Link project. The new station opened on 14 October 2013 and the old station was demolished.
Facilities, platforms & services
West Footscray has one island platform with two sides. Access to the platforms is provided by stairs, lifts and ramps from an overhead footbridge which also contains a cycling path.
It is serviced by Metro Trains' Sunbury line services.
Platform 1:
Sunbury line: all stations services to Flinders Street (Pakenham & Cranbourne after 2025)
Platform 2:
Sunbury line: all stations services to Sunbury
'''Turnback platform:
Terminating services from Sunbury, Pakenham and Cranbourne after 2025
Transport links
CDC Melbourne operate three routes via West Footscray station:
411: Footscray station – Laverton station
412: Footscray station – Laverton station
414: Footscray station – Laverton station
Sita Buslines operates one route via West Footscray station:
472: Williamstown – Moonee Ponds Junction
References
External links
Rail Geelong gallery
Category:Former rail freight terminals in Victoria (Australia)
Category:Railway stations in Melbourne
Category:Railway stations opened in 1888 | null | minipile | NaturalLanguage | mit | null |
Factors associated with mortality in older people.
A random sample of older people from Edinburgh (215 men and 272 women aged 62--90 years) was examined clinically and by a questionnaire. Various measurements were made. Five years later, mean values of measurements were compared in those who had died and in survivors. Where significant differences occurred, regression techniques were used to separate age and mortality effects. Variables in which death was the predominant independent variable in the regressions were body weight, bi-iliac diameter, FEV1.0, transverse chest diameter, index of kyphosis, leucocyte ascorbic acid and some nutrient intakes in men plus transverse cardiac diameter and leucocyte ascorbic acid in women. Apart from index of kyphosis in men and cardiac diameter in women, mean values were significantly larger in survivors. Dichotomous variables from questionnaire and examination significantly related to mortality were 'possible' ischaemic heart disease in women, diastolic hypertension in men, persistent cough in men and dyspnoea worse than grade 2 in men and women. Cigarette smoking had no mortality effects in this study. | null | minipile | NaturalLanguage | mit | null |
Menu
Safe Staffing
Appropriate staffing is essential to the delivery of safe and effective patient care, and it helps to ensure efficient throughput processes in the emergency department. Evidence supports that appropriate staffing leads to better patient outcomes.
Efforts are increasing across the country to enact legislation mandating specific nurse-to-patient ratios for all units within hospitals, including emergency departments. However, mandating one-size-fits-all nurse-to-patient ratios is limited in scope and does not consider the variables that affect the consumption of nursing resources. They don’t take into account differing factors at hospitals across a state, such as typical patient acuity and the level of training and experience of the registered nurses at each hospital.
ENA believes every health care facility should work with its nursing staff to implement a system that ensures appropriate nurse staffing levels. That is why we support legislation requiring all hospitals to form staffing committees comprised of 55% direct care nurses. The committees would be tasked with developing unit-specific nurse staffing plans to ensure high-quality care focused on the medical needs of the patient. Factors the committee would consider include input from direct care RNs, the number of patients a hospital has, and the level of education, training, and experience of the registered nurses providing the care.
This type of mandatory committee staffing plan has been shown to improve patient safety and nurse satisfaction levels, while reducing costs, saving lives, and reducing the incidence of medication errors, dangerous infections, and patient falls within hospitals nationwide. Emergency nurses must stand up and support these plans, while opposing efforts to mandate one-size-fits-all nurse-to-patient ratios.
Schools also share school psychologists: seven each in Pender County Schools and Brunswick County Schools. School psychologists often provide more intensive intervention services than counselors, including referrals to outside health care. At the elementary and middle school level, 13 additional clinical therapists provided by the New Hanover County Health Department split their time between 19 schools.
In 1989, it had 134 fighter squadron across active duty, guard and reserves. The base also houses the 192nd Fighter Wing, a group of experienced Virginia Air National Guard maintainers and pilots who also fly the stealth jet. Col. The Air Force typically wants 65 percent of eligible pilots to accept retention bonuses. | null | minipile | NaturalLanguage | mit | null |
Clinical guidelines: breast cancer.
Case histories are based on actual medical negligence claims, however, certain facts have been omitted or changed by the author to ensure the anonymity of the parties involved. Failure to diagnose breast cancer is a relatively common cause of complaints and claims involving general practitioners. This article examines the use of evidence based clinical guidelines and outlines some risk management strategies for GPs to minimise the possibility of a complaint or claim arising from an allegation of failure to diagnose breast cancer. | null | minipile | NaturalLanguage | mit | null |
TORONTO -- Fresh from a weekend off and sporting player reinforcements, struggling Toronto FC returns to action Saturday against the visiting Columbus Crew.
In recent days, Toronto has added Scottish defender Steven Caldwell, American midfielder Bobby Convey and New Zealand forward Jeremy Brockie. They join 21-year-old Argentine midfielder Matias Laba, who already has two league games under his belt.
Caldwell, formerly of Birmingham City, is expected to start in central defence, possibly alongside Doneil Henry with captain Darren O'Dea shifted to left back and Ryan Richter at right back.
Brockie and Convey could see action from the bench.
Both teams need a change of fortune.
Columbus (3-4-3) has lost its last two league matches and has just one win in its last six starts.
Toronto (1-5-4) has lost four straight in all competitions during which it has been outscored 11-2. And it is suffering through an eight-game winless streak in MLS play.
The struggling club has just two wins in 12 outings in all competitions this season and is 1-15-8 in its last 24 MLS matches dating back to July 28, 2012.
"The next four games are massive now for us and they're all against teams around us," O'Dea said of upcoming games against Columbus, New England, Philadelphia and D.C. United. "It starts with (Saturday). It's early in the season but it looks like a must-win game for us."
"This team is going forward," said Welsh striker Robert Earnshaw. "We're not thinking about going sideways and thinking about what's happened in the last few weeks. It's about getting better all the time."
Toronto has dropped nine points in six matches due to goals conceded after the 80th minute. In three instances, the goals came in stoppage time.
Despite Toronto's recent rocky road, manager Ryan Nelsen says the size of the challenge of remaking the franchise has not been as big as he expected.
The coach says his club has been in almost every single game, save the season opener in Vancouver.
"You could probably say we competed and were in the game or we could have won or drawn. I didn't think I would be able to say that after 10 games," said the former New Zealand international.
"When I first came in, I knew it was a huge job. But (look at) the improvements that these guys have made over a couple of months," he added. "And this takes time. When you've had seven years of mediocrity, it's very hard to just flip it like a light switch."
But asked if Toronto is a better team than the record suggests, O'Dea said no.
"The table doesn't lie. You are where you are and that's it," said the Irish international. "But the confidence and self-belief in the team has to remain ... We just need to cut out stupid errors and we'll be higher up in the table.
Both teams come into the game with injury problems.
The Crew are missing stud defender Chad Marshall while midfielder/defender Danny O'Rourke and midfielder Agustin Viana are questionable.
Columbus will also be without forward Jairo Arrieta, suspended for two games and fined an undisclosed amount by the league for violent conduct against Colorado Rapids defender Drew Moor in a May 11 game. Arrieta got a yellow card for a swinging arm that connected with Moor's head.
Toronto, meanwhile, is missing defenders Richard Eckersley, Logan Emory, Gale Agbossoumonde and Darel Russell, and midfielders Hogan Ephraim and Terry Dunfield.
The good news is Dutch striker Danny Koevermans, a long-term injury absence after knee surgery, is due to take apart in a reserve match Saturday.
Toronto will have excellent intel on the visitors given that former Crew assistant Duncan Oughton is now on Nelsen's coaching staff.
Nelsen says the revolving door on personnel moves will keep turning if the club can improve its roster. There has been talk of the signing of a designated player in midfield in the summer.
"My job is to make this team better," the coach said.
NOTES -- Columbus has never lost at BMO Field, posting a record of 1-0-5 there. Toronto has beaten the Crew just once in 16 career meetings, a 4-2 decision in September 2011 at Columbus Crew Stadium. TFC's all-time record against the Crew is 1-8-7 ... Saturday's game marks Italian Heritage Night ... Toronto FC has Brazilian centre back Gleason Pinto dos Santos on trial. Known as Santos, he has played in Brazil, Greece and Italy ... Ephraim has confirmed he will be returning to Queens Park Rangers. | null | minipile | NaturalLanguage | mit | null |
64 F.Supp. 162 (1945)
ERIE R. CO. et al.
v.
UNITED STATES.
Civil Action No. 768.
District Court, S. D. Ohio, E. D.
December 8, 1945.
J. P. Canney, of Cleveland, Ohio, Samuel P. Delisi, of Pittsburgh, Pa., and Leo P. Day, of Chicago, Ill., for complainants.
Nelson Thomas, Atty. I. C. C., and E. M. Reidy, Asst. Chief Counsel I. C. C., both of Washington, D. C., and Wendell Berge, Asst. Atty. Gen., for defendant.
Before ALLEN, Circuit Judge, and UNDERWOOD and DRUFFEL, District Judges.
PER CURIAM.
This suit was originally brought to set aside an order of the Interstate Commerce Commission which required the plaintiffs to reduce their rates for the transportation of scrap iron in carload lots from South Bend, Indiana, to Mansfield, Ohio; from Hastings, Michigan, to Massillon, Ohio, and from Lansing, Michigan, to Sandusky, Ohio, to "70% of the basic scale of rates on iron and steel articles." In oral argument counsel for the Commission conceded that no evidence was introduced which tended to establish the unreasonableness of the plaintiffs' rates, and the sole basis of *163 reduction in the rates was the fact that in other litigated cases involving other parties the Commission had established the 70% basis as a reasonable rate. This court enjoined enforcement of the order upon the ground that it was arbitrary and sustained by no evidence whatever. 59 F.Supp. 748. Thereafter the Commission, on its own motion, reopened the case for rehearing and took further evidence. The plaintiffs appeared, specially objecting to the jurisdiction of the Commission, and introduced no evidence. The majority of the Commission in its report and order on rehearing reinstated the requirements of its previous order that the plaintiffs establish the 70% basis on scrap iron between the points involved on or before July 9, 1945. The report was concurred in by six members of the Commission, and two other members concurred as to rates for the future. Two members of the Commission disagreed with the finding of the majority that the assailed rates were or will be unreasonable in the future, and one commissioner dissented. The plaintiffs, by leave of court, filed a supplemental bill of complaint for the purpose of setting aside the Commission's order upon rehearing, contending that the issues on rehearing are res judicata; that no evidence supports the order on rehearing, and that the order is arbitrary and capricious because it is not sustained by the record.
As to the point that the controversy is res judicata, the plaintiffs urge that the Commission has no further jurisdiction in this case since the court has reviewed and set aside and enjoined the enforcement of the Commission's original order. It cites in support of this contention Title 49 U.S. C. § 15(2), 49 U.S.C.A. § 15(2), which declares that orders of the Commission shall continue in force until its further order "unless the same shall be suspended or modified or set aside by the commission, or be suspended or set aside by a court of competent jurisdiction." Since this court set aside the Commission's order the plaintiffs contend that the Commission had no authority to proceed in the case.
We think this contention has no merit. The statute specifically provides that the Commission is authorized to "suspend or modify its orders upon such notice and in such manner as it shall deem proper." Section 16(6). This provision invests the Commission with a continuing jurisdiction, and the provision in § 15(2) above quoted does not create nor contemplate any limitation upon the continuing jurisdiction. It provides that after an order of the Commission is suspended or set aside by a court, it is no longer effective. But neither expressly nor by necessary implication does it provide that after the court has set aside one of its orders the Commission can take no further action with reference to the subject matter of the order. In this case the construction contended for would result in the absurd conclusion that when a court has determined that the Commission erred in issuing an order not based on evidence, the Commission is not empowered to acquiesce in the court's ruling and to reopen the case for the taking of evidence. Such a result is neither required nor authorized by the statute.
We are strengthened in this conclusion by the fact that the Supreme Court has sustained the action of the Commission in instances where an order has been set aside because of inadequate findings and thereafter the Commission has heard additional evidence and made additional findings. Cf. United States v. Capital Transit Co., 65 S.Ct. 278; United States v. Griffin, Receiver, 303 U.S. 226, 58 S.Ct. 601, 82 L.Ed. 764; Baltimore & Ohio R. Co. v. United States, D.C. 15 F.Supp. 674; Thompson v. United States, D.C., 20 F.Supp. 827.
We conclude that the Commission was clearly authorized to proceed with the rehearing in this case.
The order issued by the Commission awarded reparation to the claimant in the sum of $525.48 with interest, and also prescribed a lower basis of rates for the future. The complainant before the Commission has acquiesced in our ruling reversing the Commission's original order, and does not appear here, the Commission having reopened the case of its own motion. This fact is urged by the plaintiffs as requiring vacation of the present order of the Commission. But the three-judge court has no jurisdiction in the case of reparation orders. Brady v. Interstate Commerce Commission, D.C., 43 F.2d 847. Under the Interstate Commerce Act, Title 49 U.S.C. § 16(2), 49 U.S.C.A. § 16(2), after the Commission has made a reparation order the complainant is authorized to file a petition in the District Court, which proceeds as other civil suits for damages, with exception of the fact that the findings and orders of the Commission are prima *164 facie evidence of the facts therein stated. The statute makes a clear distinction between reparation orders and orders made by the Commission in the exercise of its jurisdiction to regulate interstate commerce. This court has no authority to proceed with reference to reparation matters. Its jurisdiction extends only to adjudicating cases brought for the benefit of the public. Baltimore & Ohio R. Co. v. United States, supra.
It follows that the absence of the complainant from these proceedings has no bearing on the propriety of the order.
A more serious question is presented as to whether facts have been adduced which support the order.
While the Commission, in its report, details the additional evidence showing the relation of the rates on scrap iron and steel to points in Ohio and the iron and steel scale, it made the following statement:
"Neither at the original hearing nor at the rehearing did the complainant herein undertake to present evidence relating to traffic and transportation conditions within all of official territory or within that territory as compared with other territories. In the circumstances here disclosed, we do not believe such evidence is indispensable or that its absence precludes a finding that the assailed rates are unreasonable. Transportation and traffic conditions throughout official territory have been the subject of exhaustive inquiry by us in numerous proceedings of broad scope, such as Iron and Steel Articles, supra; Eastern Livestock Cases of 1926, 144 I.C.C. 731; Eastern Class Rate Investigation, 164 I. C.C. 314; Furniture, 177 I.C.C. 5; Newsprint Paper Investigation, 197 I.C.C. 738; Eastern Fertilizer Cases, 198 I.C.C. 483; Consolidated Stone Cases, 200 I.C.C. 65; Eastern Brick Rates, 218 I.C.C. 59; No. 28190, New Automobiles in Interstate Commerce, ___ I.C.C. ___. In the cases cited we have drawn the general conclusion that official territory is a homogenous rate region in which the conditions of transportation and traffic require, or at least justify, uniform levels of rates, with certain exceptions and qualifications which are explained in the reports but which are not here pertinent."
It added:
"Evidence to establish in a satisfactory manner the nature of the transportation and traffic conditions throughout an extensive territory is necessarily of an elaborate character and requires the services of numerous transportation experts; i. e., those experienced and skilled in the fields of traffic, operation, engineering, cost finding and statistics. To assemble and produce such evidence is expensive and time-consuming. Having examined such evidence carefully and announced our conclusions thereon, we believe that we are entitled to take notice of those conclusions in cases arising thereafter, and until, upon further broad investigation, such conclusions are shown to be wrong."
With reference to these views, so emphatically reiterated by the Commission, we adhere to our position expressed on the original review of this case. We think it is not the law that carriers are compelled to adjust their rates to the 70% basis in official territory because in other cases involving other rates the Commission has found rates in excess of 70% unreasonable.[1]
The Commission states in effect that within official territory, which, roughly speaking, is the area in the United States east of the Mississippi River and north of the Ohio and Potomac Rivers, also embracing all of West Virginia and all except the extreme southern part of Virginia, it intends to apply the conclusion reached in one group of litigated cases to other cases in which different issues and different litigants are involved. It thus reiterates the holding which we reversed in the original case, that proof need not be offered of relevant material facts because the Commission can assume the existence of these facts from other litigated cases.
This reasoning, as formerly held, is entirely erroneous. "Facts conceivably known to the Commission, but not put in evidence, will not support an order." The Chicago Junction Case, 264 U.S. 258, 263, 44 S.Ct. 317, 319, 68 L.Ed. 667; United States v. Abilene & Southern R. Co., 265 U.S. 274, 44 S.Ct. 565, 68 L.Ed. 1016. There *165 has been no relaxation of this rule except in cases where the evidence was used against litigants who had been parties to the proceedings in which the evidence was originally presented. Crichton v. United States, 323 U.S. 684, 65 S.Ct. 559; Pittsburgh Plate Glass Co. v. National Labor Relations Board, 313 U.S. 146, 157, 158, 61 S.Ct. 908, 85 L.Ed. 1251. However, the order of the Commission, though endeavored to be sustained by erroneous legal reasoning, is supported by the evidence. Testimony was given with reference to various rates in central territory, which is one of the sub-territories of official territory, raising a presumption that conditions of traffic and transportation are the same. These rates supported the conclusion that the present carriers' rates are unreasonable. Since this testimony was not rebutted by the carriers, its authenticity stands as admitted.
The order is affirmed.
NOTES
[1] In its opinion on the original hearing the Commission stated: "When we have so clearly indicated that in our opinion rates in excess of the seventy percent basis in official territory are unreasonable, carriers should not be permitted to refuse to adjust their rates to that basis until they are ordered so to do."
| null | minipile | NaturalLanguage | mit | null |
Establishment of a heteroplasmic mouse strain with interspecific mitochondrial DNA haplotypes and improvement of a PCR-RFLP-based measurement system for estimation of mitochondrial DNA heteroplasmy.
Mitochondrial DNA segregation is one of the characteristic modes of mitochondrial inheritance in which the heteroplasmic state of mitochondrial DNA is transmitted to the next generation in variable proportions. To analyze mitochondrial DNA segregation, we produced a heteroplasmic mouse strain with interspecific mitochondrial DNA haplotypes, which contains two types of mitochondrial DNA derived from C57BL/6J and Mus spretus strains. The strain was produced on a C57BL/6J nuclear genomic background by microinjection of donor cytoplasm into fertilized eggs. The PCR-RFLP semi-quantitative analysis method, which was improved to reduce the effect of heteroduplex formation, was used to measure the proportion of heteroplasmic mitochondrial DNA in tissues. Founder mice contained up to approximately 14% of exogenous Mus spretus mitochondrial DNA molecules in their tails, and the detected proportions differed among tissues of the same individual. Heteroplasmic mitochondrial DNA is transmitted to the next generation in varying proportions under the maternal inheritance mode. This mitochondrial heteroplasmic mouse strain and the improved PCR-RFLP measurement system enable analysis of the transmission of heteroplasmic mitochondrial DNA variants between tissues and generations. | null | minipile | NaturalLanguage | mit | null |
Q:
Creating Custom Elements in HTML
I am trying to create my own HTML element. Now, before you mark this as a duplicate and downvote it, let me tell you that I am not doing it for styling. I have created my own styling tags, and even answered a stack overflow question about that.
I HAVE seen sites like:
Is there a way to create your own html tag in HTML5?
and
html5 - how can i create my own HTML tag?
but they do not answer my question.
I am attempting to make a "widget" tag. E.G:
<submit value='Submit form'></submit>
Instead of
<input type='submit' value='Submit form'></input>
I deal with forms a lot and it is very unorganized and tedious to use the input and type.
Is there any way I could do this?
I would prefer not to use a third-party script like jquery but I can use it if needed.
A:
You can use <button>Submit form</button> instead. That uses even fewer characters than your widget.
<button> submits the containing form by default. Reading the MDN documentation: "The button submits the form data to the server. This is the default if the [type] attribute is not specified, or if the attribute is dynamically changed to an empty or invalid value." – Blazemonger 45 mins ago
| null | minipile | NaturalLanguage | mit | null |
30-Yr Mortgage Rate Edges Higher To 4.33% – Freddie Mac
By Michael Aneiro
Mortgage rates rose a bit in tandem with Treasury yields in the latest week, with the average 30-year fixed-rate mortgage rate climbing to 4.33% in the week ended today from a six-week low 4.27% a week ago, according to Freddie Mac’s (FMCC) latest weekly Primary Mortgage Market Survey. A year ago that rate averaged 3.40%. The average 15-year fixed-rate mortgage rate rose to 3.39% from 3.33% a week earlier and 2.61% a year ago.
A similar 30-year mortgage rate measured by the Mortgage Bankers Association’s latest weekly survey rose to 4.49% in the latest week from 4.47% a week earlier. The average rate for jumbo loans of more than $417,000 remains below the rate for smaller loans, rising to 4.41% from 4.39% a week ago. That survey also showed mortgage applications fell by 3.3% in the latest week.
Amey Stone is Barron’s Income Investing blogger and Current Yield columnist. She was formerly a managing editor at CBS MoneyWatch, MSN Money and AOL DailyFinance. Her responsibilities included overseeing market coverage and personal finance topics. Prior to those roles, she was a senior writer at BusinessWeek where she authored the Street Wise column online and contributed to the magazine’s Inside Wall Street column. Topics covered included economics, corporate finance, Fed policy, municipal bonds, mutual funds and dividend investing. She co-authored King of Capital, a biography of Citigroup Chairman Sandy Weill. She is a graduate of Yale University and Columbia University’s Graduate School of Journalism. | null | minipile | NaturalLanguage | mit | null |
Love West Virginia?
This Small Town Is Like A Little Piece Of India Right Here In West Virginia
Most of West Virginia’s cities have some foreign influences, as many American cities do, but there is one place that is so unlike West Virginia, you may think you have stumbled into another country entirely.
New Vrindaban was founded by His Divine Grace A.C. Bhaktivedanta Swami Prabhupada, or Srila Prabhupada, as he is known to his followers.
His ultimate vision of seven temples on seven hills, just like in Vrindavan, India, is still under way, but Prabhupada passed away before he can see this vision fulfilled. The second of the seven temples is currently under construction.
In 1973, Prabhupada's followers decided to build a home for their worshipped leader.
What was originally going to be a fairly modest home turned into what is known today as the Palace of Gold. Apparently, no blueprints were drawn up and none of the devotees who built the palace were experienced carpenters. Yet, somehow, through much trial and error, the palace was completed. Followers of Prabhupada and his teachings believe that the miraculous structure was divinely guided by Krishna, the Indian name for God.
Today, New Vrindaban is a spiritual community through which the teaching of both Krishna and Prabhupada are strong.
People who wish to visit New Vrindaban can stay at the Palace Lodge, the temple, or the cabins around the lake. Combined, the three locations have 62 rooms where guests may stay, from single accommodations to full suites.
As you tour the beautiful town will come across stained glass, murals, temples and statues. | null | minipile | NaturalLanguage | mit | null |
Officials of the Pakistan Customs claimed on Monday to have foiled smuggling of five kilogrammes (kg) of gold to Karachi.The action was taken by the preventive staff of the Pakistan Customs posted at Jinnah International Airport, Karachi. A Customs spokesperson said that the action was taken by officials after Customs Deputy Collector Yousuf Ali Khan provided information regarding possible smuggling of gold. "We had a tip-off about the arrival of three passengers with smuggled gold," said the spokesperson. "The passengers were arriving from Dubai on Emirates airline flight EK-602. We intercepted the passengers and foiled the smuggling bid."The passengers, who were intercepted and later arrested, were identified as Muhammad Imran, Anila Haroon and Maheen Patel. The Customs officials also claimed to have recovered a huge cache of memory cards from their possession. According to the spokesperson, 5kg of gold jewellery worth nearly Rs20 million and 24,000 memory cards worth Rs2.5 million were recovered from the suspects during physical checking. "The suspects had hidden jewellery and memory cards in the clothes they were wearing," revealed the spokesperson. A case was registered against them and further investigations are under way. | null | minipile | NaturalLanguage | mit | null |
Sega, the No. 3 Player in Games,
Hunts for Chip-Making Partners
TOKYO -- The ailing underdog of the video-game business is plotting a comeback, with help from some big friends.
Sega Enterprises Ltd. is in talks to license its Dreamcast video-game technology to other companies, in a bid to keep up in its bruising race with industry leader Sony Corp. Sega also is negotiating with U.S., Japanese and European semiconductor companies about creating a joint venture that would build an advanced chip to power Dreamcast, Sega's Internet-capable game machine, and other Internet devices, people... | null | minipile | NaturalLanguage | mit | null |
Lot 43: Russian Deck Watch
$275.00
Moscow First Watch Company, steel case, 22-jewels, lever watch, a sweep second Naval deck watch used on board ship for navigation and general purpose timekeeping on the quarterdeck, winds, sets and ticks, 62mm case.
Estimate: $200-300
All property is sold “AS IS”. We have exercised reasonable care to catalogue and describe correctly the property to be sold. We do not warrant authenticity of authorship, signatures, historical relevance, correctness of description, genuineness, provenance, age or condition of said property. It is the bidders responsibility to inspect the property to determine its condition and authenticity. No statement made at the Sale, or by any other method, shall be deemed an assumption of liability or warranty with respect to the above. We and the Consignors are not responsible for errors and omissions in the catalog. | null | minipile | NaturalLanguage | mit | null |
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