text:
Car Buying Platform Carwow Secures £4.6M Series A To Step On The Gas
Steve O'Hear	@sohear / 7 years
Just ten months after disclosing a £1.3 million seed round, UK car buying platform Carwow has made another fundraising pitstop. The London-based company has raised a £4.6 million Series A round, led by Balderton Capital, with participation from Episode 1 Ventures, and Samos Investments.
All three VCs are previous backers of Carwow, so we can flag this up as a follow-on round for the young startup, which now claims to have moved into pole-position as the leading site for buying a new car in the UK after launching in May last year.
Aiming to reduce the friction involved in buying a new car, Carwow's website allows consumers to compare offers online and buy directly from "trusted" dealers that are registered with the platform, specifically avoiding the arduous but otherwise necessary requirement to haggle over price, and in a way that potentially introduces a lot more transparency.
It does this via what is essentially a reverse marketplace: users "build" the car they would like to buy — specifying make and model/features etc. — and then receive offers direct from dealers. They can then quickly compare offers by price, location of the dealer, reviews of the dealer and what's actually in stock. It's then up to the buyer to decide whether to contact a dealer based on their offer, either by anonymous messaging through the Carwow platform or by giving the dealer a call.
To back up its claim to be the number one new car buying site in the UK, Carwow says it has driven "more than" £150 million in car sales for dealerships (saving users a claimed combined total of £15 million). That's up from £30 million in sales, as reported in February.
Meanwhile, today's Series A round is all about stepping on the gas. The injection of fuel will be used to "expand the team further, roll out above-the-line advertising campaigns and start exploring new markets," says Carwow's 27-year old CEO James Hind in a statement.
That will likely see Carwow's brand attempt to reach additional mainstream audiences via traditional newspaper and television advertising as it attempts to race further in front of local competitors.


text:
vhansen@nesl.edu
Lawyering Skills
Professional Responsibility (Lawyers)
National Security Law
Military Tribunals
Equal Protection
Criminal Procedure I
Criminal Procedure II
Victor M. Hansen
Professor of Law and Director of Criminal Practice and Procedure Certificate Program
LLM, The Judge Advocate General's School
JD, Lewis and Clark Law School
BA, Brigham Young University
Before joining New England Law | Boston faculty, Professor Hansen served a 20-year career in the US Army, as a Judge Advocate General's Corps (JAG) officer.
As a JAG officer, he served as a regional defense counsel for the US Army Trial Defense Service, a military prosecutor, and supervising prosecutor. He has been involved in military capital litigation as a prosecutor and as a defense attorney.
Professor Hansen also served as an associate professor of law at The Judge Advocate General's Legal Center and School in Charlottesville, Virginia. He is the author of several articles and books on criminal and military law, evidence, and national security issues.
Congress Gets to Check the President – That's What the Framers Had in Mind , New England Law Review Forum (2021)
U.S. Military Renews Effort To Root Out Extremists Following Deadly Capitol Riot , The Wilmington Journal (2021)
We Need a 1/6 Commission , Jurist (2021)
Key allies watching closely to see how Canada handles sexual misconduct claims against Vance, McDonald , CBC (2021)
Do Military Courts Play a Role in the Wake of January 6? , PBS (2021)
25 Investigates: The Murder of Amanda Dabrowski , (2020)
Deploying Federal Officers to Local Protests: Can the President Send Agents to New England? , (2020)
'We got nothing from the Army': Grieving family of Brockton soldier demands answers , Boston Globe (2020)
Justices to Hear First #MeToo-Era Assault Case from Military , Bloomberg Law (2020)
In the Matter of a Grand Jury Investigation, 485 Mass. 641 (2020) , Boston Bar Journal (2020)
The House as Prosecutor: Speaker Pelosi on Impeachment After the Mueller Report , The National Law Journal (2019) (with Lawrence Friedman)
9/11 families still waiting on justice for Khalid Sheikh Mohammed , New York Post (2019)
Trump allies charge Lt. Col. Vindman is disrespecting chain of command with testimony , Washington Examiner (2019)
US Marine Officials Identify Marine Allegedly Involved In Emerson Student Death , WGBH (2019)
Congress needs to take a look at constitutional use of military force , The Hill (2018) (with Lawrence Friedman)
Aftermath of 9/11 attacks shows American justice system at work , The Hill (2018) (with Lawrence Friedman)
Enlarge player → Chelsea Manning subject to prosecution for Senate run? , (2018)
25 Investigates: Local challenge to FBI use of child porn site could have national impact , Boston 25 News (2018)
Even If It Ain't Broke, Why Not Fix It? Three Proposed Improvements to the Uniform Code of Military Justice , 6 Journal of National Security Law & Policy 447 (2013) (with Geoff Corn)
The Law of Armed Conflict: An Operational Approach , (2013) (with Geoffrey S. Corn, M. Christopher Jenks, Richard Jackson, Eric Talbot Jensen, James A. Schoettler, Jr.)
The Impact of Military Justice Reforms on the Law of Armed Conflict: How to Avoid Unintended Consequences , Michigan State International Law Review (2013)
The Value of the Military Commissions Act as Non-Judicial Precedent in the Context of Litigation Over National Security Policymaking , 53 South Texas Law Review 1 (2012)
Secrecy, Transparency, and National Security , 38 William Mitchell Law Review (2012)
Avoiding the Extremes: A Proposal for Modifying Court Member Selection in the Military , 44 Creighton Law Review 911 (2011)
Use and Misuse of Evidence Obtained During Extraordinary Renditions: How to Avoid Diluting Fundamental Protections , 35 Nova Law Review 281 (2011)
Book Review: Death by Moderation: The U.S. Military's Quest for Useable Weapons by David A. Koplow , American Journal of Legal History (2011)
The Ghailani Terrorism Case: Military Lawyers in Federal Court? , Jurist Forum (2009)
The Case for Congress: Separation of Powers and the War on Terror , (2009) (with Professor Lawrence Friedman)
Can We be Less Secure and Less Free? Answers to Ten Questions in National Security, Journal on National Security , 35 William Mitchell Law Review 5041 (2009)
Understanding the Role of Military Lawyers in the War on Terror: A Response to the Perceived Crisis in Civil-Military Relations , 50 South Texas Law Review 617 (2009)
The Usefulness of a Negative Example: What We Can Learn About Evidence Rules from the Government's Most Recent Efforts to Construct a Military Commissions Process , 35 William Mitchell Law Review 1480 (2009)
The War on Terror and the Laws of War: A Military Perspective , (2009) (with Geoffrey Corn, Eric Jensen, Richard Jackson, Michael Lewis, and James Schoettler).
Changes in Modern Military Codes and the Role of the Military Commander: What Should the United States Learn from this Revolution? , 16 Tulane Journal of International and Comparative Law 419 (2008)
Extraordinary Renditions and the State Secrets Privilege: Keeping Focused on the Task at Hand , 33 North Carolina Journal of International Law and Commercial Regulation 629 (2008)
The End of History: The Vice President and the Presidential Records Act , Jurist Forum (2008) (with Lawrence Friedman)
Pulling Rank: Reinforcing an Independent JAG Corps , Jurist Forum (2008)
Developing Empirical Methodologies to Study Law of War Violations , 16 Willamette Journal of International Law and Dispute Resolution 342 (2008)
What's Good for the Goose is Good for the Gander – Lessons from Abu Ghraib: Time for the United States to Adopt a Standard of Command Responsibility Towards its Own , 42 Gonzaga Law Review 335 (2007)
Conservative Bastion or Progressive Problem Solver: The Evolving Face of Military Jurisprudence and International Law , 13 ILSA Journal of International and Comparative Law 329 (2007)
Commonwealth v. Dwyer and the New Protocol Governing Access to Privileged Information , 51:3 Boston Bar Journal 13 (2007)
Military Crimes and Defenses , (2007) (with David A. Schlueter, Charles H. Rose, III, and Christopher Behan)
The Case Against Secret Evidence , 12 Roger Willliams University Law Review 772 (2007) (with Lawrence Friedman)
The Jordan Abu Ghraib Verdict: Command Responsibility in the UCMJ , Jurist Forum (2007)
Going Native at Guantanamo: When Government Lawyers Lose Their Way , Jurist Forum (2006)
The Army and the Constitution: Time for Congress to Step In , Jurist Forum (2006) (with Lawrence Friedman)
Congress Should Champion the Advice of Military Lawyers , Jurist Forum (2006) (with Lawrence Friedman)
The Haditha Double Standard , Jurist Forum (2006)
No Secret Rules on Torture , The Washington Post (2005)
Walking on Unfamiliar Ground: A Primer for Defense: Counsel Representing Clients in an Inspector General Investigation , Army Law (2005)
The Art of Trial Advocacy-Effective Motions Practice , (2001)
New Developments in Evidence 2000 , Army Law (2001)
The Supreme Court Provides a Framework for Reliability Determinations; Rule of Evidence 702 , 162 Military Law Review 1 (1999)
Scholarly Work
SSRN Profile: http://ssrn.com/author=607476

redpajama_set_name: RedPajamaCommonCrawl


SANTA CLARA | MUMBAI, September 27, 2017: Tata Consultancy Services (BSE: 532540, NSE: TCS), a leading global IT services, consulting and business solutions organization, today announced that it will set up an artificial intelligence (AI) Center of Excellence (CoE) with Intel to accelerate adoption of AI solutions.
The new AI CoE will be equipped with the latest technologies and experts to assist Intel® Nervana™ AI Academy members, including developers, data scientists, students, professors, and startups, in developing AI solutions. In addition, the CoE will collaborate with academic communities to develop new algorithms for AI, support AI research projects and enhance teaching content delivered in the classroom. The CoE will have a strategic physical and online presence across Americas, Asia Pacific, PRC, and Europe, to seamlessly collaborate with global customers to support and propel their initiatives from conceptualization to implementation. These solutions will utilize machine Learning and deep learning libraries optimized for the Intel® portfolio of technologies, including Intel® Xeon® Scalable processors, Intel® Xeon Phi™ processors, and Intel® Nervana™ technologies.
"AI is already beginning to play an important role in helping enterprises re-imagine their products and services, drive revenue, realize business efficiencies, and enrich the customer experience. Harnessing the power of AI as part of a digital transformation strategy will be key in creating the intelligent enterprise of tomorrow," said V Rajanna, Vice President and Global Head, Technology Business Unit, TCS.
There has been an explosion of data as organizations have started harnessing machine data from Internet of Things (IoT) devices in addition to traditional data sources, and this has laid the foundation for big data platforms. Organizations are focusing on building intelligent systems that can learn from data and conventional insights and make automated, smarter, real-time, human-like decisions to solve large-scale problems across industries and to address societal needs. TCS' contextual knowledge of various industries and deep AI expertise, coupled with Intel's state-of-the-art AI product portfolio, offers the necessary ecosystem for helping enterprises rapidly adopt AI technologies to differentiate themselves and achieve a strategic advantage in the marketplace.
"Intel is dedicated to bringing the latest AI technology and education to developers and universities around the world. Collaborating with leaders in the industry allows us to further support the expanding global community of developers, students, professors, and startups who are working on groundbreaking research, new algorithms, and disruptive products and solutions. TCS is an excellent company to join forces with as a pioneer in innovation, with its deep industry knowledge, experience, and technical expertise," said Morris Beton, Vice President of Software and Services Group and General Manager of Developer Relations Division at Intel Corporation.


text:
When I wrote my first message for the Transformation Project webpage three years ago, it felt as if I was writing about a vision for a distant future. But the realisation of the dream is swiftly drawing near. I even dare to speak about approaching the final stage, now we have raised approximately two thirds of the money needed for completion of the whole project.
The vision behind the project is still the same. We had experienced how the lack of facilities such as a dedicated meeting space and kitchen, and also the inflexible church layout, were a hindrance in our aspirations to grow. So we started by installing energy-efficient heating, a state of the art sound and audio-visual system, and new energy-friendly lighting. Within months we saw growth, a greater interest in the life of the Parish church, and a response from the community that illustrated the potential of the site as a place for meaningful and fine worship, for education, culture, music and drama.
Despite these positive signs after the first improvements, we felt that we could do better as a church at the heart of the community. There is a great need for catering for residents, and in particular the vulnerable, isolated, disadvantaged, elderly, bereaved and young. But not only small gatherings, but also running big events of a liturgical, educational, cultural, social and entertaining nature will be made much easier with the new facilities. As an example, when we will have completed the project, a large organisation such as the Diocese will never again have to decline our offer to run a training day at St Mary's due to the restrictions of our pews or outdated toilets which are inaccessible for wheelchair users. We have missed out in many other ways on great opportunities to reach out to all groups in the community.
Bringing the project to completion by raising the remaining funds will make us more sustainable, efficient and environmentally friendly. It will be a new start not only for church people but for the entire community of all backgrounds. This is no longer just a dream, it is the reality of taking up our responsibility as a church to serve the society, care for the weak, protect and endorse our heritage, strengthen unity and promote tolerance.
We are very grateful to all those who have given so generously since the launch of our appeal. If you haven't had the opportunity to donate and would like to find out how to contribute to this wonderful cause click on the link below.
Rev Erwin Lammens
Rector, The Parish of St Mary's, Wivenhoe


Kyocera's horizontal Y peeler features an ultra-sharp advanced ceramic blade that won't rust, pit or corrode. The dual eye corners can remove imperfections from any fruit and vegetable and the comfortable, ergonomic handle makes the task of repetitive peeling a breeze. The ceramic blade is made from Kyocera proprietary zirconia material produced in Japan and stays razor sharp at least 10 times longer than traditional peelers making them ideal for everyday use. Kyocera advanced ceramic cutlery and kitchen tools are fully warranted to be free of defects in material and craftsmanship.
The KitchenAid Classic Euro Peeler is a sleek and stylish design that makes peeling fruits and vegetables simple. The Euro Peeler features a swivel head blade constructed of tough, rust-resistant steel. The peeler's ergonomic handle provides a sure, comfortable grip. This peeler's compact size makes it easy to store in the kitchen. The peeler measures 7-1/2 inches in length. The Euro Peeler is completely dishwasher safe for easy cleanup.One year hassle-free replacement and lifetime limited warranty. This fine product is warranted to be free from defects in material and workmanship. For one year from date of purchase, under normal use and care, KitchenAid will replace these products free of charge, if they are found to be defective in material or workmanship. In addition, from year two through the life of the products, any piece found to be defective under conditions of normal use and care will be repaired at no charge or replaced with the same item or an item of equal or better value. For more information call the KitchenAid Customer Satisfaction Center toll-free at 1-800-541-6390 or visit our web site at www.KitchenAid.com.
Now in available four vibrant colors, our Peeler will make peeling any fruit or vegetable an easy job and reduce prep time! Great for tomatoes and eggplants. The micro-serrated edge enables effortless peeling. Ergonomically designed handle and lightweight, this sharp blade glides through even the toughest fruit and vegetable skins with ease. This swivel peeler has a pivoting blade allowing it to self-adjust based on the pressure applied and without digging into the fragile produce. The micro-serrated blade edge on the peeler helps remove skins off of waxy and slippery fruits and vegetables.
This KitchenAid swivel peeler is the perfect tool for peeling all kinds of fruits and vegetables. The cutlery grade stainless steel swivel blade provides better maneuvering. This handy kitchen tool peels fruits and vegetables and forms cheese curls with precision and speed. Dishwasher safe. Swivel peeler measures 8.5 inches long.


Jerry,You healed me dunrig the winter of 2006 and I a still alive and healthy thanks to you. Although physically all right, I have had many bad things happen in my life since then including the loss of my wife in 2007. I relocated to the New Orleans area shortly after that to be a volunteer and help people still affected by the Katrina disaster. I try so very hard to think positive but with all that keeps happening it's tough to do. Thanks for my life. Can you help me with the quality of it now? Also, I have a very dear friend that I love very much who is very sick. She is 50 years old, recently had open heart surgery, and now may require a liver transplant. Although very faithful she is to the point where she has about given up hope. I told her about you and your healing powers and she believes you can help her. Please contact me and let me know if you can help. Although I am living on a fixed income and still disabled, I will do all that I can to be of assistance.Thank you Jerry.
List and still be happy after reading this.


This Sub- Component will provide laser guided land leveling equipment and associated deep ripping equipment, and will facilitate training in the use of deep ripping equipment. These will help farmers save irrigation water, curtail irrigation time, improve efficiency of agriculture Inputs such as fertilizers, and achieve uniform seed germination, resulting in increased crop yields.
This Sub- Component will facilitate training to farmers to improve crop production and irrigation agronomy practices.
This Sub-component will provide Technical Assistance and Training (TAT) Consultants to provide training to field team, public sector staff and farmers in HEIS installation, operation and repair, crop diversification, crop production under HEIS, soil measurement and fertilizer application, and to support piloting of activities under Sub- component B2 and training of trainers for operators in laser land leveling and deep ripping under Sub- Components C1.
Prevention of water wastage during crop production process is obligatory for appropriate production and maximum utilization of available resources. This will not only enhance production but also saved water will be used for harvesting more soil on national level.
Pakistan's agrarian system is one of the oldest, broadest and unique systems of the world though. Yet it very complex in terms of its compatibility. Only 40 million, out of 142 million-acre feet derived from our rivers, is available for crop production. Whereas throughout its flow, by one way or another, it comes across such circumstances that it is wasted, causes harm and also rendering our soil water logging & salinity.
Unleveled soil is one of those reasons which cause wastage of water & less agriculture production.
More water is required for upper level of soil.
In the wake of limited resources of water, irrigation is insufficient on upper level of soil. On the contrary, soil at lower level gets more than required water. So upper part of soil becomes saline due to salty substances and lower part becomes under water logging due to unwanted still water.
Because of unleveled soil, both planting and growth of seed are affected, thus leaving a negative effect on production.
Many parts of unleveled soil remain deprived of proper effect of fertilizer due to its unevenness which consequently wastes farmer's capital as well.
Sindh Irrigated Agriculture Productivity Enhancement Project (SIAPEP) has been initiated by Government of Sindh in Collaboration with World Bank. Under this scheme, 1073 Land Levelers have been distributed on a subsidy rate in various district of Sindh so far.
In this connection, farmers and landowners of Sindh are advised to get benefitted on a great extent by this program and get Land Leveler on a considerably low rate. In this scheme, farmer will bear only 55% of the cost and the rest 45% will be borne by the government.
If soil is leveled at standard level, it will not only save water but also will change characteristics of soil.
Even supply of water in whole piece of soil and increase the potentially of irrigation.
Conservation of water by 25-30%, result in more production.
زمين ۾ مٿانهن حصن تائين پاڻي پهچائڻ لاءِ زمين کي گهڻي مقدار ۾ اضافي پاڻي فراهم ڪرڻو پوندو آهي.
پوري زمين ۾ هڪ جههڙو پاڻي بيهڻ سان زمين ۾ موجود ڳرڻ جوڳا نقصان ڪار لوڻياٺ پاڻي ۾ حل ٿي ڪري ٻوٽن جي پاڙن کان هيٺ هليا ويندا آهن ، جنهن ڪري فصل جي پئداوار ۾ ڪافي واڌارو ٿيندو آهي.


text:
require 'spec_helper'

describe "rearview/dashboards/show" do
  let(:dashboard) {
    parent = create(:dashboard)
    child = FactoryGirl.create(:dashboard,parent: parent)
    parent
  }
  let(:dashboard_keys) {
    ["id",
     "userId",
     "name",
     "createdAt",
     "modifiedAt",
     "description",
     "children"]
  }
  it "renders dashboard json" do
    assign(:dashboard,dashboard)
    render :template => "rearview/dashboards/show", :formats => :json, :handler => :jbuilder
    json = JSON.parse(rendered)
    expect(json).to be_a_kind_of(Hash)
    dashboard_keys.each { |k| expect(json).to include(k) }
    expect(json.keys.size).to eq(dashboard_keys.size)
  end
end

redpajama_set_name: RedPajamaGithub


CONTENTS CONCRETE REPAIR CORROSION CONTROL GROUTS & ANCHORS PROTECTIVE COATINGS INDUSTRIAL FLOORING SURFACE TREATMENTS JOINT SEALANTS CONCRETE JOINTING JOINT FILLERS WATERPROOFING WATERSTOPS ROOFING TIMBER FLOORING ALPHABETICAL INDEX Fosroc® Construction Grout Important notice A Safety Data Sheet (SDS) and Technical Data Sheet (TDS) are available from the Parchem website or upon request from the nearest Parchem sales office. Read the SDS and TDS carefully prior to use as application or performance data may change from time to time. In emergency, contact any Poisons Information Centre (phone 13 11 26 within Australia) or a doctor for advice. Product disclaimer This Technical Data Sheet (TDS) summarises our best knowledge of the product, including how to use and apply the product based on the information available at the time. You should read this TDS carefully and consider the information in the context of how the product will be used, including in conjunction with any other product and the type of surfaces to, and the manner in which, the product will be applied. Our responsibility for products sold is subject to our standard terms and conditions of sale. Parchem does not accept any liability either directly or indirectly for any losses suffered in connection with the use or application of the product whether or not in accordance with any advice, specification, recommendation or information given by it. Page 4 103 *Manufactured and sold under license from Fosroc International Limited. Fosroc and the Fosroc logo are trade marks of Fosroc International Limited, used under license. *Denotes a trade mark of Fosroc International Limited. Parchem Construction Supplies Pty Ltd 7 Lucca Road, Wyong NSW 2259 Phone: 1300 737 787 www.parchem.com.au ABN 80 069 961 968 Distributed in New Zealand by: Concrete Plus Ltd 23 Watts Rd, Sockburn 8042 Ph: 03 343 0090 Estimating Supply Construction Grout is supplied in a 20 kg moisture resistant bags. Yield Allowance should be made for wastage when estimating quantities required. The approximate yields for different consistencies are: Consistency (AS 1478.2-2005 Table 4.1.2.2) Yield (Litres of mixed material) Stiff 10.4 Plastic 10.7 Flowable 10.8 Fluid 10.9 Storage Construction Grout has a shelf life of 12 months if kept in a dry store in sealed bags. If stored in high temperature and high humidity locations, the shelf life may be reduced.


We invite you to join the National Latino Children's Institute Network. We work to make the lives of Latino children, families and all children better. We begin by focusing our nation's attention on young Latinos and advocate for positive change so the entire family and our nation are stronger.
Expand your knowledge of issues impacting the success of young Latinos and their communities.
Establish important links to other organizations whose interest are young Latinos and children.
Become an informed advocate that takes action on issues that impact the lives of children in our nation.
Be on the inside of the news.
Join with other children's advocacy organizations and help support public policy initiatives and advocacy activities.
Share your opinions on children's issues with a national audience.
Receive a quarterly NLCI newsletter, updates and alerts.
I understand that all of my information will be kept confidential and used for the benefit of the National Latino Children's Institute only. I can unsubscribe at any time.


text:
David Livingstone Statue
Glasgow, United Kingdom (GB)
David Livingstone (19 March 1813–1 May 1873) was a Scottish Congregationalist pioneer medical missionary with the London Missionary Society and explorer in Africa. He was the first recorded European to see the Victoria Falls, to which he gave the English name in honour of his monarch, Queen Victoria. His meeting with H. M. Stanley gave rise to the popular quotation, "Dr Livingstone, I presume?".
Art - Sculpture
Links: en.wikipedia.org
By: adrbr
"Life After Iraq"
Provand's Lordship
Glasgow Royal Infirmary
Glasgow Cathedral and Necropolis
'Callandish' by Gerald Laing
Giant Diaper Pin
Wellpark Brewery
Glasgow City Chambers
Tollbooth Steeple
George Square
Britannia Panopticon Music Hall


My wife has a paid off 2011 Hyundai Elantra. We got a note from a dealer that says a customer wants it to replace a totaled one. Looking at their used inventory online, they do not have a single Elantra. My question is... is this legit? Does this happen often?
One of the ways dealerships get new customers is to reach out to existing customers to see whether or not they would be interested in trading their current car for a new one. The dealers also know that flooding your inbox with messages promising "HUGE SAVINGS" is an exercise in futility. Most of these messages end up in a spam box or get deleted without ever being read.
So now the dealers are looking to go old school in the way of traditional mail and "personalized letters" to entice people into thinking that maybe they would consider a new car. It plants the seed that a buyer could get "top dollar" for their trade, and pairs that with "rock bottom prices on all new models." Many times they will spin some story about how they are "desperate" for pre-owned inventory; sometimes they will even go a step further and concoct some story about how your particular car is needed for a specific customer.
Sure, they may have a shortage of pre-owned cars because they had a big sale month and blew out their inventory and of course they are always looking for nice trades that would make for an easy sale. But your car is not that special, if they really wanted your 2011 Elantra (or whatever other car they are begging you for) the dealer would go to an auction and find one.
These letters are not about your trade, the primary objective is to get you physically at the dealership so you can be enticed by a newer car, with updated styling and cooler features. Chances are the dealership will even promise you to, "keep your payments exactly the same." Of course if your car is paid off, I would try to take them up on that offer.
Are these letters a "scam?" No, but they are pretty effective marketing to get folks that don't really need a car into the showroom. So if you are not in the market and you get one of these notices, don't over think it, just toss it in the trash.


The 2015 Multicultural New Haven Block Party is a community event promoting and embracing New Haven's diversity. Geared especially towards residents, business owners and newly arrived students and scholars, the block party typically draws several hundred attendees to enjoy performances, food, activities, and fellowship.
For Chinese Culture demonstration table, our group honorary member Dawei Sun and Group leader Xiaoxing Cao give painting and calligraphy demonstration there during the event.
Thanks Micheal, Shraddha, Yujie Han and Lirong Wang for photography, Antje, Irina, Lindan Hu, Xiaohui Gao and all the members for table setting, answer questions and clean up. People love our works and all of us enjoy this special events.


1 for 1: You Save, We Give.
This December, Brazi Bites has teamed up with The Food Trust to improve access to healthy food and nutrition education across America. For many, wholesome food can be difficult to find, expensive, or both — help us make sure everyone has a seat at the table!
Learn more about our partnership below, or get right to the saving!
Sign up to receive a "1 FOR 1" $1 coupon.
Redeem your $1 coupon at your favorite store before 12/31.
For every coupon redeemed, Brazi Bites will donate $1 to The Food Trust.
Why Partner with The Food Trust?
Quick recap: You save $1, we donate $1 to The Food Trust. How simple is that?


text:
Q: How JQuery Ajax works? I am trying to send data from JQuery Ajax to a Generic Handler that calculates something and returns a result. The Ajax request is made inside a for loop at JQuery end. The code looks something like this:
function send(Handler, ids) {
    var URL = "http://" + window.location.host + "/Handlers/" + Handler + ".ashx";
    var i;
    for (i = 0; i < 10; i++) {
        var cur = $('.' + ids[i]);
        $.ajax({
            url: URL,
            type: "POST",
            data: JSON.stringify({
                Data: cur
            }),
            dataType: "json",
            cache: false,
            beforeSend: //my before send code,
            success: //my success code,
            error: //my error code
        });
    }
    alert('Done!');
}

I placed 3 breakpoint in Visual Studio 2012 at line:
$.ajax({

this
alert('Done!');

And third breakpoint at first line in the Generic Handler.
Now, when I try to execute this code, the Ajax works nicely in async way. But, when it reaches the first breakpoint, stops there and then I resume it, instead of reaching the Generic Handler's breakpoint, it continues the loop and goes back to first breakpoint. Then after it reaches the second breakpoint it then stops at generic handler's break point again and again for each value.
So, does that mean that Ajax first collects all the ajax requests and then after the for loop it executes them together?

A: If I am understanding you correctly, you just answered your own question. Ajax is working asynchronously meaning the for loop starts and fires out ajax requests, and continues the loop (the ajax request DOES NOT block)
Therefore it is very likely that the js is performing a loop of code before the request reaches your url (as this has to create a network call)
That said, what are you doing in your beforeSend method? maybe this is making it take enough time that it can perform all iterations of the loop before sending the first request?
To answer your question, no it shouldn't be waiting for the for loop to finish in order to send off the requests, it should be initiating the process as soon as you have made the call

A: Javascript is single threaded and non-blocking. This means that in the first iteration won't wait for the ajax call to be completed, it will go back and start the second iteration, and so on.
So, no it doesn't executes them all together. It definately starts the ajax calls in the order of the loop but there is no way to tell what will end first. It might make all the ajax calls and then get an answer (doesn't mean it is the answer of the first iteration), or in the middle of a loop it might be getting answers.

redpajama_set_name: RedPajamaStackExchange


King has a new 'On-Track Plant' development and test facility in Market Harborough. The test facility is intended to reduce the cost and turnaround of rail conversion projects for our customers. This is all part of a major investment which includes strengthening our sales and engineering team in the 'On-Track Plant' sector. With a growing order book of machine conversions and upgrades KingRail is investing in the rail industry.


Waste Accumulation Storage Tracking electronically or WASTe is a web based system that facilitates regulatory compliant labeling, tracking, collection and shipping of hazardous chemical, radioactive, and biohazardous wastes. The UC Risk & Safety Solutions application provides an interface between Principal Investigators, lab staff and other generators of regulated wastes and the UCR waste management staff.
This quick and easy-to-use system allows users to create tags for hazardous chemical, radioactive, or biohazardous waste in less than one minute. WASTe supports creation of tags for five types of waste: Chemical, Mixed, Radiological, Biological, and Universal. Lab & Facility staff can share created tag profiles, and administrators can create profiles for a single lab or for all users on campus. The system notifies Environmental Health and Safety staff when waste is ready for pickup. WASTe will enable researches and staff to dispose of hazardous waste in accordance with all state and federal regulations.
Anyone with a UCR NetID will be able to access the system at https://ehs.ucop.edu/waste/#/.
For more information please contact Richard Watson at (951) 827-4248, or Travis Underwood at (951) 827-5518.
Login to WASTe to create radioactive waste labels and request pickup. There is no need to complete any other forms.
Biohazardous and Medical Waste Pickup Request has been added to WASTe.
Login to WASTe to create biohazardous and sharps container waste labels and to request pickup.
We are pleased to offer WASTe online waste pickup request services through our integrated Waste Management program. To use these services it is important that you properly determine the type of waste and review waste disposal requirements.
For information on proper containers, labeling, and storage of biohazardous and medical waste, including how to request pickup of biohazardous and sharps waste containers, and how to request new and replacement sharps containers, please refer to this Biohazardous and Medical Waste Disposal Requirements poster.


text:
South Sudan : Rebel leader to talk to President after Ethiopian invitation
14th June 2018 Africa News 0 Comments Addis Ababa, Ethiopia, Juba, President, Riek Machar, Salva Kiir, South Sudan, Talk
South Sudan's rebel leader Riek Machar has accepted an invitation by the Ethiopian government to hold talks with South Sudanese President Salva Kiir in Addis Ababa next week.
"The … invitation … will go a long way in building confidence in the peace process," a spokesman for Machar said in an emailed statement on Wednesday.
The spokesman further said Ethiopian Prime Minister Abiy Ahmed had invited Machar, who is under house arrest in South Africa, to attend the talks on June 20.
According to the statement, the negotiations will be led by the eight-nation East African bloc of Intergovernmental Authority on Development (IGAD).
There was no immediate comment from Juba's government or from IGAD on the matter.
The bloody civil war in South Sudan, the youngest country in Africa, began in December 2013, when the incumbent president accused his former deputy, Machar, of plotting a coup. The two sides have been involved in a cycle of retaliatory killings that have split the impoverished country along the ethnic lines. Tens of thousands have been killed and millions displaced in the conflict.
The United Nations and other bodies accuse all sides in the war of committing atrocities against civilians.
In December 2017, South Sudan's government and rebel groups inked a ceasefire agreement after talks in the Ethiopian capital, Addis Ababa. The deal, however, has been violated repeatedly with both sides blaming each other.
The meeting would be the first time that Kiir and Machar have met since August 2016.
By Africafrique and agencies
← South Africa : 2 men killed by stabbing in Malmesbury mosque
Russia : 2018 FIFA World Cup schedule,Dates, fixtures, groups, venues and kick-off times →
Egypt : President appoints close military ally as transport minister
Nigeria:President Buhari extends medical leave in UK
Kenya:Named World's Least Toxic country
Somalia:14 killed as soldiers clash over food aid
Somalia-Six dead as plane carrying coronavirus aid crashes
DRC:World Diamond Council has a New CEO, Marie-Chantal Kaninda
DR Congo :Explores oil drilling allowed in wildlife parks
South Africa:President lauds Muslims for aiding economy
Egypt:President endorses new legislation for more powers


Please find out more about our nominees by clicking on their names listed above. Please also read more about the gifts we seek in potential board members.
Those elected will begin their service just after the Annual Meeting this Ocotober. You may participate in the election process as a member in good standing. All members will receive an electronic ballot and voting instructions in an email soon. Please continue to pray for wisdom and discernment in our communitand discernment in our community.


text: Q: Is there a mac and a linux version of GetLogicalDriveStrings() from win32api? The title basically explains the problem. I don't know a way to use GetLogicalDriveStrings() for os's besides windows, so it would be very helpful if someone could tell me what to use, and how. I need this so I can search all drives on a computer for a file, on the 3 main os's. There was a similar question to this one, but it had only one answer, which had a check mark, and it basically restated the problem. Plus, I couldn't comment cause I didn't have enough reputation since I mostly only read from stack exchange. Edit: I want my program to search the entire computer for a file, and run it, on the 3 main os's. That's all I want. I know windows has partitions that are treated as singular drives, and usb drives, so I want to find out how to search it all, and all of the partitions.


This recipe, originally inspired by a recipe of one of my favorite blogs, Smitten Kitchen, is an amped-up version of the grain salad posted in the most recent CSA newsletter. The recipe requires a fair amount of chopping work up front, but your hard work will pay off in the end.
Your reward? A huge bowl of pleasantly chewy barley (if you have never cooked with barley before, you are in for a treat) and tender cubes of eggplant and zucchini, punctuated by juicy sweet tomatoes, the bite of paper-thin slices of raw red onion, and a sprinkle of chopped capers. The whole delicious mess is bound together by a simple lemon vinaigrette and a healthy dose of fresh parsley and mint.
We ate this topped with a bit of feta cheese as a hearty vegetarian meal, and it lasted us through two dinners and one day of packed lunches. You could certainly serve it as a a side dish with some grilled chicken, fish, or lamb. You can also make this ahead of time and serve at room temperature for the ideal picnic take-along!
Put oven racks in upper and lower thirds of oven and preheat oven to 425°F for eggplant and zucchini.
While oven preheats, begin cooking barley: Heat 2 tablespoons oil in a 3- to 4-quart heavy pot over moderately high heat until hot but not smoking, then cook scallion, cumin, coriander, and cayenne, stirring, until fragrant, about 1 minute. Add barley and cook, stirring until well coated with oil, 2 minutes more. Add broth and water and bring to a boil. Reduce heat and simmer, covered, about 30 to 40 minutes.
While barley simmers, roast eggplant and zucchini. Toss eggplant and zucchini with 5 tablespoons oil, 3/4 teaspoon salt, and 3/4 teaspoon pepper in a bowl, then spread in 2 oiled large shallow (1-inch-deep) baking pans. Roast vegetables in oven, stirring occasionally and switching position of pans halfway through baking, until vegetables are golden brown and tender, 20 to 25 minutes total. Combine vegetables in 1 pan and cool, reserving other pan for cooling barley.
Cool barley: When all of liquid in barley pot is absorbed and barley is tender, remove from heat and let stand, covered, 5 minutes. Transfer to reserved shallow baking pan and spread to quickly cool, uncovered, to room temperature, about 20 minutes.
While barley and roasted vegetables cool, make dressing and assemble salad: Whisk together lemon juice, garlic, sugar, and remaining 1/4 teaspoon salt, 1/4 teaspoon pepper, and 3 tablespoons oil in a large bowl. Add barley, roasted vegetables, and remaining ingredients to bowl with dressing and toss until combined well. Serve topped with crumbled feta cheese (optional).


Coming Forward With a Voluntary Disclosure: On June 2, 2011, the IRS issued an updated list of frequently asked questions on the 2011 Offshore Voluntary Disclosure Initiative.
Opting Out of Such a Disclosure: On June 1, 2011, the IRS issued more guidance on how a U.S. taxpayer should proceed both before and after choosing to opt out of the current voluntary offshore account disclosure initiative, or if the individual is removed from the civil settlement structure of the 2009 offshore voluntary disclosure program and the 2011 offshore voluntary disclosure initiative. This Guidance includes several attachments to assist those preparing a taxpayer's documents and discusses the applicable Statute of Limitations.
Both of these IRS publications are attached and should be carefully read. One is a pdf file and the other is a Word document.
Open a Bottle of Good Scotch: You might want start sipping a stiff glass of Scotch, however, while you dive into these shocking new revelations.
Penalties for Ignorant Americans: Each is stunning in the extent to which it elaborates on the very nasty penalties that will be applied to Americans living overseas who may not have ever known they had to be filing these reports. Such ignorance seems to be, by definition, their inexcusable fault and they will now be appropriately punished.
Penalties for Those Who Didn't Even Know They Were Americans: An even bigger howler is that there are also penalties for individuals who never even knew they were American citizens until some administrative procedure brought this fact to their attention! They too obviously deserve all the punishment that awaits them now.
An Enduring Mystery: It nonetheless remains a great and enduring mystery why any government, especially that of the United States, which sees itself as the ideal model for everyone else to emulate, would want to treat its overseas citizens in such a nasty, brutal and unfair way. No other country has ever done this before, and, most interestingly, none seems to be in a hurry to copy our behaviour here either.
Future Nasty Nuggets: Also, one can only imagine how many other toxic nuggets have also been skilfully hidden in the more than 71,000 pages of the current U.S. tax code. Undoubtedly, inspired by this current example, some future zealots will trot these other similar gems and launch yet more fiscal catastrophes.
How is This Helping Our Trade Performance: And, just for fun, while we try to put some balm on these wounds, how is this supposed to be helping improve the trade situation of a country that now has an unbroken 36 year record of trade deficits that aggregate to more than $7 trillion? This too remains an enduring mystery.
A Manifestation of Insanity? Then again, by making life ever more miserable, expensive and legally dangerous for its own citizens, who have already been long denied the chance to compete on a level playing field in world markets, this does rather clearly seem to be just another shocking manifestation of national institutional insanity on a planetary scale.
But Why? What in the world could be driving such bizarre behaviour, other than profound self-hatred at the highest levels?
What Do You Think About This? Your thoughts and suggestions would be most appreciated.


text:
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What is pelvic inflammatory disease?
Pelvic inflammatory disease (PID) is an inflammation and infection of a woman's upper reproductive organs. It occurs when bacteria move from the vagina, upward through a layer of mucus that protects the opening of the cervix, and into the uterus, fallopian tubes, and ovaries. PID can cause an acute illness, chronic pain, or it may be present as an almost silent chronic infection. Left untreated, it can cause scarring and irreversible damage to one or more of the reproductive organs. Scarring in the fallopian tubes can cause infertility and significantly increase the risk of an ectopic pregnancy.
Pelvic inflammatory disease is primarily associated with Chlamydia trachomatis and Neisseria gonorrhoeae, two common microorganisms that cause the sexually transmitted diseases (STDs) chlamydia and gonorrhoea. PID may also be caused by bacteria typically found in the vagina or gastrointestinal tract - such as Gardnerella vaginalis or Escherichia coli, and by a variety of other pathogens.
Although PID can affect any woman at any age, with or without sexual transmission, most cases occur in sexually active women of childbearing age. Those at increased risk include women who began having sex at a young age, those with multiple sexual partners, and those who have had a previous STD or PID. A woman with an intrauterine device (IUD) may be at an increased risk for a few months after its insertion. Douching may increase the risk of PID because the practice can change the vaginal normal flora and can potentially flush bacteria into the uterus. Since the uterus is sealed off during gestation, an occurrence of PID during pregnancy is rare.
About 10,000 women in Australia are treated for PID in hospital each year. About 10 to 30 times that number are treated as outpatients.
Last Review Date: April 10, 2020
Conditions / diseases
Screening tests for newborns
Screening tests for infants
Screening tests for children
Screening tests for young adults
Screening tests for adults
Screening tests for adults: 50 and over
Pregnancy & prenatal testing
Point-of-Care Testing (PoCT)


text:
NHS Seacole: Creating a COVID-19 rehab centre from scratch
NHS Seacole
Creating a COVID-19 rehab centre from scratch
Our healthcare, public services, and people experts had experience of helping hospitals run efficiently – from using data to cut surgery waiting times at Leeds Teaching Hospitals NHS Trust to improving COVID-19 patient care by matching doctors to hospitals in New York. We're also supporting the UK Government to build seven new hospitals as part of the health infrastructure plan (HIP). From developing the business case and investment plans to providing the hands-on operational expertise, our broad range of capabilities ensures we can move at pace. Now we had to apply our experience to a very different challenge: creating a new kind of facility in a fraction of the time it would normally take.
Working closely with the UK National Health Service (NHS), our team were at the heart of creating the NHS Seacole Centre for COVID-19 rehabilitation in just 35 days. It's meant everything from co-ordinating building work and local volunteers to recruiting well-trained staff and getting hold of medical equipment. Together we also set out how the Centre would operate, so everyone from doctors to porters knew what to do, and how. By mobilising quickly and adapting as the pandemic unfolded, we've created a place that helps patients recover fully from the effects of the illness, and frees capacity in local hospitals to deal with new COVID-19 cases.
recruited a range of staff for the Centre, from nurses and therapists to healthcare assistants and office workers, for 90 whole-time-equivalent (WTE) posts
co-ordinated hundreds of people from more than 20 organisations, including volunteers to get the new rehabilitation centre up and running
opened on time on May 4, 2020, 35 days after work commenced, with the first patients received on May 31
Facing up to a five-week deadline
As the COVID-19 pandemic hit the UK, the NHS knew hospitals would come under extreme pressure. In March 2020, the UK Government announced that NHS hospitals needed to expand critical care capacity to the maximum, which included freeing up 30,000 (or more) of the English NHS's 100,000 general and acute beds. The NHS needed to act fast. They decided to create the NHS Seacole Centre, which would provide all-important ward capacity with a facility dedicated to patients recovering from effects of the illness.
But for the plan to work, the new Centre would need to be ready in only five weeks to coincide with the expected peak of the virus in May 2020. The Seacole Centre would care for 130 patients, with room for up to 300 later if needed. They're either recovering COVID-19 patients who no longer need acute hospital care, or people with symptoms who can't cope at home. Many are dealing with the virus's physical aftermath including lung, heart and muscle damage that can make it hard to even turn in bed. Others are suffering the psychological effects of the illness.
The chosen site was an ex-military rehabilitation facility in Surrey, England. But the buildings had stood empty for two years and some had been flooded. The site would need a complete transformation, including new wiring and drains, restored electricity and water, and redecoration. Wards would need converting to single rooms and the Centre kitted out with fire alarms, IT and nurse-call systems. It also needed separate accommodation for staff who wanted to stay on site rather than travel long distances or risk infecting their families by going home.
Inventing a new kind of facility
We worked side by side with the Epsom & St Helier University Hospitals NHS Trust team – under whose jurisdiction the Centre would operate – including the Chief Executive, Medical Director, Chief Operating Officer and Director of Nursing. We also worked closely with other partner organisations including Surrey County Council, Corrigenda and Surrey Heartlands Health and Care Partnership. Together we engaged closely with over 15 clinical and support service leads partners to create a comprehensive hospital operations manual. It defined scores of processes, from referring and admitting patients to transferring and discharging them and everything in between, whether it was what meals they got, how post came and went from the Centre or how medical check-ups, cleaning and portering happened. We also managed 50 local volunteers who helped assemble 300 pieces of furniture and ferry supplies around the 83-acre site.
We worked with Epsom & St Helier's HR department, plus NHS partner organisations to agree workforce loans. And we filled nursing and therapist posts with retired, student and trainee practitioners, as well as staff new to the NHS. We also needed healthcare assistants to feed and dress patients, help them get about and keep them company. To find them, we looked in industries like air travel, where crews had first aid experience and, because of the pandemic, weren't currently working.
Collaboration was vital when it came to procuring equipment, with suppliers facing more demand just when their own supply chains were disrupted. To avoid being left short of the essentials needed to open on time, we used our partners' networks to borrow items like portable x-ray machines and beds able to accommodate patients at different stages of recovery.
From start to finish, we kept everyone focused on the critical things needed to open the Centre safely and on time. And we made sure they carried on talking to each other, so they could anticipate obstacles as much as possible.
Opening on time
Working in close collaboration with Epsom & St Helier NHS Trust, we hit the opening deadline of May 4, 2020. Health Secretary Matt Hancock joined the Trust's Chief Executive, Daniel Elkeles, in hosting a virtual ceremony. Patients began arriving shortly afterwards. Now that Seacole is up and running, nearby hospitals can release COVID-19 patients earlier to start what can be a long road to full recovery. It means patients are in a more pleasant environment that aids their recovery, while other hospitals have extra capacity for new COVID-19 cases.
The ability of the PA team to turn their hand to anything and everything that was needed, from writing the operating manual for the new hospital, to scenario testing all the clinical processes to even unpacking furniture, was invaluable in enabling us to deliver this project on time. It certainly made my job much easier!
Hilary Floyd, Medical Director,
NHS Seacole, Headley Court
The PA team were the glue between a lot of different organisations – their expertise in programme management of healthcare projects plus their ability to bring a diverse group of stakeholders together was critical to the project's success. We absolutely could not have delivered at the pace we did without them.
Daniel Elkeles, Chief Executive,
Epsom & St Helier University Hospitals NHS Trust
It usually takes years to plan, design and build a hospital. We had five weeks. That made us very single-minded about focusing on the things that were going to stop us opening on time and dealing with those.
Christian Norris, Economics expert at PA
We help you develop innovative services that improve outcomes, take care closer to home and deliver a great patient experience.
Hampshire County Council: Helping people live independently for longer using consumer technology
Argenti: Helping vulnerable people stay independent during the COVID-19 outbreak
NHS Seacole Centre provided essential care for patients recovering from #Covid-19, and was operationalised in just 35 days. Discover the hospitals journey #Healthcare #NHS
Christian Norris
PA Head economist
Gareth Fitzgerald
PA healthcare expert
@gfitzgerald81


text:
\section{Introduction}

This paper is focused on the regularity of the unique solution to the Dirichlet problem
\begin{equation}\label{f:dirich}
	\begin{cases}
		-\Delta^N_{\infty} u = 1 &\text{in}\ \Omega\\
		u = 0 &\text{on}\ \partial\Omega\,,
	\end{cases}
\end{equation}

where $\Omega$ is an open bounded convex subset of $\mathbb{R}^N$ and $\Delta^N_{\infty} u$ is the normalized infinity Laplacian.
The symbolic definition of $\Delta^N_{\infty}  \varphi$ for a smooth function $\varphi$ is
\begin{equation}\label{f:newdef}
\Delta^N_{\infty} \varphi(x) := \begin{cases}
\frac{1}{|\nabla \varphi(x)|^2}\, \pscal{\nabla^2 \varphi (x)\, \nabla \varphi (x)}{\nabla \varphi (x)} & \text{ if } \nabla \varphi (x) \neq 0
\\
[\lambda_{\text{min}} (\nabla ^ 2 \varphi(x)), \lambda_{\text{max}} (\nabla ^ 2 \varphi(x))] & \text{ if } \nabla \varphi (x)=  0\, ,
\end{cases}
\end{equation}
being $\lambda_{\text{min}}(\nabla^2 \varphi (x))$ and $\lambda_{\text{max}}(\nabla^2 \varphi (x))$
respectively the minimum and the maximum eigenvalue of the Hessian matrix $\nabla^2 \varphi (x)$;
for the detailed interpretation of the pde $-\Delta^N_{\infty} u = 1$ in the viscosity sense, we refer to Section \ref{secprel} below.

The normalized infinity Laplace operator has recently attracted an increasing interest for its applications and connections with different areas, such as mass transportation \cite{EGan}, shape metamorphism \cite{CEPB}, and especially differential games \cite{KoSe, PSSW}. In fact, according to Kohn and Serfaty \cite{KoSe}, and Peres et al. \cite{PSSW}, the equation $- \Delta^N_{\infty} u = 1$ is satisfied by the continuum value of a differential game called ``tug of war'' (a description can be found  for instance in \cite{ArmSm2012}).

The development of the existence, uniqueness, and regularity theory for boundary value problems involving the normalized infinity Laplace operator is still at its early stages.

The well-posedness of the Dirichlet problem \eqref{f:dirich} (possibly with a more general source term $f$) has been proved independently in
\cite{PSSW} with probabilistic methods,
in \cite{LuWang2} with pde methods (see also \cite{LuWang3}),
and in \cite{ArmSm2012} using a finite difference approach.
Let us mention that the existence  and uniqueness questions have been attacked also in the parabolic framework, see the interesting paper \cite{JuKa} on the evolution governed by $\Delta^N_{\infty} u$.

About regularity theory, so far no result seems to be available.  As it is well-known,
the fundamental contributions on regularity  for pde's involving the infinity Laplace operator
are the celebrated works by Evans-Smart \cite{EvSm} and Evans-Savin \cite{EvSav}, which concern infinity harmonic function and establish they are differentiable in any space dimension and $C ^{1, \alpha}$ in dimension two.
Recently, some of these results have been extended to a class of inhomogeneous Dirichlet problems for the not-normalized infinity Laplacian  in any dimension:
the everywhere differentiability property  has been obtained by Lindgren  \cite{Lind},
and the $C^1$ regularity in case of a constant source term on a convex domain
has been proved in our previous paper \cite{CFd}.

In case of the normalized operator, its
definition via a dichotomy involving the maximum and minimum directions of the Hessian necessarily augments the difficulty of enforcing  regularity techniques, and in fact no result beyond Lipschitz regularity is currently known.

Aim of this paper is to present two new regularity results for the unique solution to problem~\eqref{f:dirich} on convex domains.

The first result establishes that the solution is power-concave, precisely, $1/2$-concave, see Theorem \ref{t:concave}. To prove such result, we apply the convex envelope method by Alvarez-Lasry-Lions we already exploited in our previous paper \cite{CFd}, but in the current case this requires a more delicate  procedure
which is outlined for the benefit of the reader at the beginning of Section~\ref{secconc}. In particular, this procedure exploits as a crucial tool  a comparison principle proved in \cite{ArmSm2012}.

The second result is obtained as a consequence of the first one, and states that the solution is of class $C ^1(\Omega)$, see Theorem \ref{t:diff}.
The proof relies on the local semiconcavity of the solution, combined with an estimate for semiconcave functions near singular points proved in \cite{CFd}.

The paper is organized as follows. In Section \ref{secprel} we provide and discuss the definitions and properties of viscosity solution to  problem \eqref{f:dirich} and to more general second order equations that we need to consider in the proofs. In Sections \ref{secconc} and \ref{secdiffe} we state and prove respectively  the power-concavity and the $C^1$ regularity of the solution.

\section {Preliminaries} \label{secprel}

In this section we review the definition of normalized infinity Laplace operator, and that of viscosity sub- and super-solutions of \eqref{f:dirich}, as well as of more general second order equations (that  we shall need to use). Afterwards, we give some remarks to enlighten some basic features of solutions.

For a $C^2$ function $\varphi$ defined in a neighborhood of $x\in\mathbb{R}^n$,
we define the
{\it (not normalized) infinity Laplace operator}
\[
\Delta_{\infty} \varphi(x) := \pscal{\nabla^2\varphi(x)\, \nabla\varphi(x)}{\nabla\varphi(x)}
\]
and the operators
\[
\begin{split}
\Delta^+_{\infty}\varphi(x) & :=
\begin{cases}
|\nabla\varphi(x)|^{-2}\, \Delta_{\infty} \varphi(x),
&\text{if}\ \nabla\varphi(x)\neq 0,\\
\lambda_{\text{max}}(\nabla^2\varphi(x)),
&\text{if}\ \nabla\varphi(x)= 0,
\end{cases}
\\
\Delta^-_{\infty}\varphi(x) & :=
\begin{cases}
|\nabla\varphi(x)|^{-2}\, \Delta_{\infty} \varphi(x),
&\text{if}\ \nabla\varphi(x)\neq 0,\\
\lambda_{\text{min}}(\nabla^2\varphi(x)),
&\text{if}\ \nabla\varphi(x)= 0,
\end{cases}
\end{split}
\]
where, for a symmetric matrix $A\in\Sym(n)$, $\lambda_{\text{min}}(A)$ and $\lambda_{\text{max}}(A)$
denote respectively the minimum and the maximum eigenvalue of $A$.

In the following, if $u,v\colon\Omega\to\mathbb{R}$ are two functions and
$x\in\Omega$, by
\[
u \prec_x v
\]
we mean that $u(x) = v(x)$ and $u(y) \leq v(y)$ for every $y\in\Omega$.

Moreover we recall that {\it second order sub-jet} (resp.\ {\it super-jet}),
$J^{2, -}_{\Omega} u (x)$ (resp. $J^{2, +}_{\Omega} u (x)$), of a function $u\in C(\overline{\Omega})$
at a point $x\in \overline{\Omega}$,  is by definition the set of pairs
$(p, A) \in \mathbb{R} ^n \times \mathbb{R} ^ { n \times n }_{{\rm sym}}$ such that, as $y \to x,\ y\in \overline{\Omega}$, it holds
\begin{equation}\label{d:jet}
 u (y) \geq  (\leq) \ u ( x) + \pscal{ p}{y- x}
+ \frac{1}{2} \pscal {A (y- x_0)}{y- x_0} + o ( |y - x|^2)
\,.
\end{equation}

\begin{definition}\label{d:visc}
Let $f\colon\Omega\to\mathbb{R}$ be a continuous function and consider the normalized infinity Laplace equation
\begin{equation}
\label{f:norm}
-\Delta^N_{\infty} u = f(x)\qquad \text{in}\ \Omega\,.
\end{equation}

(i)  An upper semicontinuous function $u\colon\Omega\to\mathbb{R}$
is a {\it viscosity sub-solution of \eqref{f:norm}} if, for every $x\in\Omega$,
\[
-\Delta^+_{\infty} \varphi(x) \leq f(x)\qquad \forall\varphi\in C^2(\Omega)\ \text{s.t.}\ u\prec_x \varphi.
\]
The explicit formulation reads
$$
\begin{cases}
-\Delta_{\infty} \varphi(x) \leq f(x) |\nabla\varphi(x)|^2 & \forall\varphi\in C^2(\Omega)\ \text{s.t.}\ u\prec_x \varphi,\text{ if } \nabla \varphi  (x) \neq 0
\\
- \lambda_{\text{max}}(\nabla^2\varphi(x)) \leq f(x) &  \forall\varphi\in C^2(\Omega)\ \text{s.t.}\ u\prec_x \varphi,\text{ if } \nabla \varphi  (x) =0\,,
\end{cases}
$$
or in terms of super-jets
$$
\begin{cases}
- \langle X p,  p  \rangle \leq f(x) |\nabla\varphi(x)|^2 & \forall(p, X) \in J ^ {2, +} _\Omega u (x), \text{ if } p \neq 0
\\
-\lambda _{\max} (X) \leq f(x) & \forall(p, X) \in J ^ {2, +} _\Omega u (x), \text{ if }  p =0\,.
\end{cases}
$$

(ii) A lower semicontinuous function $u\colon\Omega\to\mathbb{R}$
is a {\it viscosity super-solution of~\eqref{f:norm}}
if, for every $x\in\Omega$,
\[
-\Delta^-_{\infty} \varphi(x) \geq f(x)\qquad \forall\varphi\in C^2(\Omega)\ \text{s.t.}\ \varphi\prec_x u\,.
\]
The explicit formulation reads
$$
\begin{cases}
-\Delta_{\infty} \varphi(x) \geq f(x) |\nabla\varphi(x)|^2 & \forall\varphi\in C^2(\Omega)\ \text{s.t.}\ \varphi \prec_x u,\text{ if } \nabla \varphi  (x) \neq 0
\\
- \lambda_{\text{min}}(\nabla^2\varphi(x)) \geq f(x) &  \forall\varphi\in C^2(\Omega)\ \text{s.t.}\ \varphi \prec_x u,\text{ if } \nabla \varphi  (x) =0\,,
\end{cases}
$$
or in terms of super-jets
$$
\begin{cases}
- \langle X p,  p  \rangle \geq f(x) |\nabla\varphi(x)|^2 & \forall(p, X) \in J ^ {2, -} _\Omega u (x), \text{ if } p \neq 0
\\
-\lambda _{\min} (X) \geq f(x) & \forall(p, X) \in J ^ {2, -} _\Omega u (x), \text{ if }  p =0\,.
\end{cases}
$$

(iii) A function $u\in C(\Omega)$ is a {\it viscosity solution of~\eqref{f:norm}}
if $u$ is both a viscosity sub-solution and a viscosity super-solution
of~\eqref{f:norm}.

(iv)  A function $u\in C(\overline{\Omega})$ is a {\it viscosity solution
of~\eqref{f:dirich}} if $u=0$ on $\partial\Omega$ and
$u$ is a viscosity solution of~\eqref{f:norm} in $\Omega$.

\end{definition}

\begin{definition}\label{d:general}
Let $I\subset\mathbb{R}$ be an open interval,
 let $H\colon \Omega \times I\times\mathbb{R}^n\times\Sym(n)\to\mathbb{R}$, and consider the equation
\begin{equation}
\label{f:F}
H(x,u, \nabla u, \nabla^2 u) = 0 \qquad \text{in}\ \Omega\,.
\end{equation}
(i) An upper semicontinuous function $u\colon\Omega\to I$
is a {\it viscosity sub-solution of  \eqref{f:F}}
if, for every $x\in\Omega$,
\[
H_*(x, u(x), \nabla\varphi(x), \nabla^2\varphi(x)) \leq 0
\qquad \forall\varphi\in C^2(\Omega)\ \text{s.t.}\ u\prec_x \varphi\, ,
\]
where $H_* $ is the lower semicontinuous envelope of $H$.

(ii) A lower semicontinuous function $u\colon\Omega\to I$
is a {\it viscosity super-solution of~\eqref{f:F}}
if, for every $x\in\Omega$,
\[
H^*(x, u(x), \nabla\varphi(x), \nabla^2\varphi(x)) \geq 0
\qquad \forall\varphi\in C^2(\Omega)\ \text{s.t.}\ u\prec_x \varphi\,,
\]
where $H^*$ is the upper semicontinuous envelope of $H$.

(iii) A function $u\in C(\Omega, I)$ is a {\it viscosity solution of~\eqref{f:F}}
if $u$ is both a viscosity sub-solution and a viscosity super-solution
of~\eqref{f:norm}.

(iv) A function $u\in C(\overline{\Omega}, I)$ is a {\it viscosity solution
of the homogeneous Dirichlet problem for equation \eqref{f:F}} if $u=0$ on $\partial\Omega$ and
$u$ is a viscosity solution of~\eqref{f:F} in $\Omega$.

\end{definition}

\begin{remark}
By taking the function
$$H (p, A) := - \Big \langle{A \frac{p}{|p|}}{\frac{p}{|p|}}  \Big \rangle\,, \qquad \forall (p, A) \in (\mathbb{R}^n\setminus \{ 0 \} )\times\Sym(n)\,, $$
and computing its upper and lower semicontinuous envelopes,
we see that Definition  \ref{d:general} gives back Definition  \ref{d:visc}.  (Note that $H _* = - (- H)^*$ and $H ^* = - ( - H ) _*$).
This argument justifies the apparently  strange
notions of  the operators $\Delta^+_{\infty}$ and $\Delta^-_{\infty}$. \end{remark}
\begin{remark}
It is clear that the viscosity solution of the
Dirichlet problem~\eqref{f:dirich} is also a
viscosity solution of
\begin{equation}
\label{f:dirich2}
\begin{cases}
-\Delta_{\infty} u = |\nabla u|^2 & \text{in}\ \Omega,\\
u = 0 & \text{on}\ \partial\Omega.
\end{cases}
\end{equation}
On the other hand, the converse is not true,
and in fact the Dirichlet problem~\eqref{f:dirich2} has not,
in general, a unique solution.
To shed some light on this feature, it is enough to look at the one-dimensional  case.
If $\Omega$ is the interval $(-R, R)$, problems \eqref{f:dirich} and \eqref{f:dirich2}  read respectively
\begin{equation}
\label{f:1dim}
\begin{cases}
- u ''  = 1 & \text{in}\ (-R, R),\\
u (\pm R)= 0 \,.
\end{cases}
\qquad \qquad
\begin{cases}
- u '' (u') ^2 = (u')^2 & \text{in}\ (-R, R),\\
u (\pm R)= 0 \,.
\end{cases}
\end{equation}
It is immediate to check that  the function
$$u _ r (x) := \begin{cases}
\frac{R ^ 2 - r ^ 2}{2} & \text{ if  } |x| \leq r
\\  \noalign{\medskip}
\frac{R ^ 2 - x ^ 2}{2} & \text{ if  }r \leq  |x| \leq R
\end{cases}
$$
is a solution to the second problem in \eqref{f:1dim} for every $r \in [0, R]$, whereas it is a solution to the first problem in~\eqref{f:1dim} only for $r= 0$.
Indeed, if $r \in (0, R]$,  for every  $x$ with  $|x| < r$ there exist smooth functions $\varphi$ such that $\varphi \prec _ x u$ but  violate the condition $-\varphi '' (x) \geq 1$, so that $u$ is not a super-solution  to $- u '' = 1$.
\end{remark}

\begin{remark}\label{r:pos}
The viscosity solution $u$ to problem~\eqref{f:dirich} is strictly positive in $\Omega$.
Indeed, it is nonnegative by the comparison result proved in
\cite[Thm.~2.18]{ArmSm2012}.
Assume by contradiction that $u (x_0) = 0$ at some point $x _0 \in \Omega$. Then
 the function $\varphi \equiv 0$ touches $u$ from below at $x_0$, and hence
$u$ cannot be a viscosity supersolution to the equation
$- \Delta^N_{\infty}  u =1$ at $x_0$.
\end{remark}

\section{Power-concavity of solutions on convex domains}\label{secconc}

In this section we prove:
\begin{theorem}\label{t:concave}
Assume that $\Omega$ is an open bounded convex subset of $\mathbb{R}^n$,
and let $u$ be the solution to problem~\eqref{f:dirich}.
Then $u^{1/2}$ is concave in $\Omega$.
\end{theorem}
Our proof strategy is the following.
\begin{itemize}
\item[{Step 1.}]
We show that
the map $u \mapsto w := -u^{1/2}$
establishes a one-to-one correspondence between
positive viscosity sub- and super-solution of $-\Delta^N_{\infty} u = 1$ in $\Omega$
and a ``restricted class'' of, respectively, negative viscosity super- and sub-solution of the equation
\begin{equation}
\label{f:modif}
F(w, \nabla w, \nabla^2 w) = 0
\qquad\text{in}\ \Omega,
\end{equation}
where the function $F\colon (-\infty, 0) \times\mathbb{R}^n\times\Sym(n)\to\mathbb{R}$
is defined by
\begin{equation}\label{f:defF}
F(w, p, A) := -\pscal{A p}{p} - \frac{1}{w} \left( |p|^4 + \frac{1}{2} |p|^2\right)\,.
\end{equation}

\item[{Step 2.}]
Under the additional assumption that
\begin{equation}
	\text{the convex set $\Omega$ satisfies an interior sphere condition, }
	\label{f:ipo}
\end{equation}
we can adapt the convex envelope method of
Alvarez, Lasry and Lions (see \cite{ALL}),
proving that
the convex envelope $w_{**}$ of
a (restricted) super-solution $w$ of~\eqref{f:modif}
is still a (restricted) super-solution.

\item[{Step 3.}]
Using the comparison principle
proved in~\cite[Thm.~2.18]{ArmSm2012},
we conclude that, if~\eqref{f:ipo} is fulfilled, then
$w$ is convex, namely $u^{1/2}$ is concave.

\item[{Step 4.}]
By approximating $\Omega$ with outer parallel sets, we finally show that
the assumption~\eqref{f:ipo} can be removed.
\end{itemize}

 \begin{remark}
 The assumption~\eqref{f:ipo} is used in a crucial way to prove
 Lemma~\ref{l:emptyjet} below,
 so that we do not need to impose
 \textsl{state constraints boundary conditions} on $\partial \Omega$
 (see \cite[Definition~2]{ALL}).
 In this respect, we mention that the power concavity of solutions
 to~\eqref{f:modif} has been discussed by Juutinen in \cite{Juu};
 nevertheless, as kindly pointed out by the Author himself,
 his proof is flawed precisely in the argument
 used to show the validity of these
 state constraints boundary conditions.
 (See Lemma~{4.1} in \cite{Juu} where, at boundary points,
 the emptyness
 of $J^{2, -}_{\overline\Omega}$ instead of $\overline{J}^{2, -}_{\overline\Omega}$ is proved.)
 \end{remark}

\begin{remark}
The comparison principle
proved in~\cite[Thm.~2.18]{ArmSm2012}, that we use as a crucial tool in Step 3, holds true for solutions to problem \eqref{f:dirich}.
In view of the one-to one correspondence mentioned in Step 1, it is therefore irremissible to deal with ``restricted'' solutions to the equation \eqref{f:modif}, according to Definition \ref{d:restricted} below.
 \end{remark}

\textbf{Step 1.} We set the following
\begin{definition}\label{d:restricted}
We say that
$w$ is a {\it restricted viscosity super-solution of~\eqref{f:modif}}, if it satisfies
\begin{equation}
\label{f:Rsuper}
\forall x\in\Omega, \ \forall \psi\prec_x w
\ \Longrightarrow \
\begin{cases}
F(w(x), \nabla\psi(x), \nabla^2\psi(x)) \geq 0,
\\
\lambda_{\text{min}}(\nabla^2\psi(x)) \leq -\dfrac{1}{2 w(x)}\quad \text{if}\ \nabla \psi (x) = 0.
\end{cases}
\end{equation}
We say that
$w$ is a {\it restricted viscosity sub-solution of~\eqref{f:modif}}, if it satisfies
\begin{equation}
\label{f:Rsub}
\forall x\in\Omega, \ \forall w\prec_x \psi
\ \Longrightarrow \
\begin{cases}
F(w(x), \nabla\psi(x), \nabla^2\psi(x)) \leq 0,
\\
\lambda_{\text{max}}(\nabla^2\psi(x)) \geq -\dfrac{1}{2 w(x)}\quad \text{if}\ \nabla \psi (x) = 0.
\end{cases}
\end{equation}
\end{definition}

\begin{lemma}
\label{l:equiv}
A positive upper semicontinuous function $u\colon\Omega\to\mathbb{R}^+$
is a viscosity sub-solution of $-\Delta^N_{\infty} u = 1$ in $\Omega$
if and only if the function $w = - u^{1/2}$ is a restricted viscosity super-solution of \eqref{f:modif}.
Likewise, a positive lower semicontinuous function $u\colon\Omega\to\mathbb{R}^+$
is a viscosity super-solution of $-\Delta^N_{\infty} u = 1$ in $\Omega$
if and only if the function $w = - u^{1/2}$ is a restricted viscosity sub-solution of \eqref{f:modif}.
\end{lemma}

\begin{proof}
We are going to prove only the first part of the statement.
To that aim it is enough to observe that
\[
u \prec_x \varphi
\quad\Longleftrightarrow\quad
\psi := -\varphi^{1/2} \prec_x w,
\]
and the test functions $\varphi$ and $\psi$ satisfy
\[
\nabla\psi(x) = - \frac{\nabla\varphi(x)}{2 \varphi (x) ^ {1/2}}\,,
\qquad
\nabla^2\psi(x) = \frac{1}{4 \varphi(x)^{3/2}}\, \nabla\varphi(x) \otimes \nabla\varphi(x)
- \frac{1}{2\varphi(x)^{1/2}}\, \nabla^2\varphi(x).
\]
In particular, $\nabla\varphi(x) = 0$ if and only if $\nabla\psi(x) = 0$ and,
if this is the case,
$\lambda_{\text{max}}(\nabla^2\varphi(x)) \geq -1$ if and only if
$\lambda_{\text{min}}(\nabla^2\psi(x)) \leq -1/(2w(x))$.
\end{proof}

\textbf{Step 2.}
Let us
show that, if \eqref{f:ipo} is satisfied and $w$ is a restricted viscosity solution to
\begin{equation}
\label{f:probmod}
\begin{cases}
-\Delta_{\infty} w - \frac{1}{w}\left( |\nabla w|^4 + \frac{1}{2} |\nabla w|^2\right) = 0
& \text{in}\ \Omega,\\
w = 0 & \text{on}\ \partial\Omega,
\end{cases}
\end{equation}
then $w$ is convex.
We denote by $w_{**}$  the largest convex function below $w$.
We first establish that,
under the assumption~\eqref{f:ipo},
for every $x \in \Omega$, in the characterization
\[
w_{**} (x)  = \inf \left\{
 \sum _ {i = 1} ^ k \lambda _i w (x_i) \ :\ x = \sum _{i=1} ^k \lambda _ i x _i \, ,\
 x_i \in \overline \Omega\, , \ \lambda _i >0\, ,\ \sum  _ {i = 1} ^ k \lambda _i = 1\, , \ k \leq n+1 \right\}\,
\]
the infimum can be attained only at interior points $x_i \in \Omega$:

\begin{lemma}\label{l:emptyjet}
Assume \eqref{f:ipo}, and let $u$ be the  solution to problem~\eqref{f:dirich}.
Set $w := - u^{1/2}$.
For a fixed $x\in\Omega$, let $x_1,\ldots,x_k\in\overline{\Omega}$,
$\lambda_1,\ldots,\lambda_k > 0$, with  $\sum_{i=1} ^ k \lambda _ i = 1$, be such that
\[
x= \sum_{i=1}^k \lambda_i x_i\, , \quad
w_{**}(x) = \sum_{i=1}^k \lambda_i w(x_i).
\]
Then $x_1, \ldots, x_k\in\Omega$.
\end{lemma}

\begin{proof}
Assume by contradiction that at least one of the $x_i$'s, say $x_1$,
belongs to $\partial\Omega$.
Let $B _ R (y)\subset \Omega$ be a ball such that $\partial B _ R (y) \cap \partial \Omega = \{ x_1 \}$.
Since $-\Delta_\infty u = |\nabla u|^2$, by Lemma~2.2 in \cite{CEG} the function $\tilde u := -u$ enjoys the property of comparison with cones from above according to Definition 2.3 in the same paper.
Then, by Lemma 2.4 in \cite{CEG}, the function
\[
r \mapsto \max _{x \in \partial B _ r (y)} \frac{\tilde u (x) -  \tilde u  (y)}{r} = - \min  _{x \in \partial B _ r (y)}  \frac{ u (x) -  u  (y)}{r}
\]
is monotone nondecreasing on the interval $(0, R)$.
Namely, for all $r \in (0, R)$, there holds
\begin{equation}
\label{f:cones2}
\min_{x \in \partial B _ r (y)} \frac{ u (x) -  u  (y)}{|x-y|}  \geq \min_{x \in \partial B _ R (y)} \frac{ u (x) -  u  (y)}{|x-y|} = -  \frac{u (y)} {R}\ ,
\end{equation}
where the last equality comes from the fact that
$u$ is non-negative in $\Omega$ (cf.\ Remark \ref{r:pos}).
By \eqref{f:cones2}, we have
\[
u (x) \geq u (y ) \Big ( 1 - \frac{|x-y|}{R} \Big ) \qquad \forall x \in B _R (y) \, ,
\]
and hence
\begin{equation}
\label{f:upbd}
w (x) \leq w (y ) \Big ( 1 - \frac{|x-y|}{R} \Big ) ^ {1/2} \qquad \forall x \in B _R (y) \, .
\end{equation}
Let us define the unit vector $\zeta := (x-x_1) / |x-x_1|$
and let $\nu = (y-x_1) / |y-x_1|$ denote the inner normal of
$\partial\Omega$ at $x_1$.
Since $\Omega$ is a convex set and $x\in\Omega$, we have
that $\pscal{\zeta}{\nu} > 0$
and $x_1 + t\zeta\in B_R(y)$ for $t>0$ small enough.
Moreover, $w_{**}$ is affine on $[x_1, x]$: indeed, since the epigraph of $w_{**}$ is the convex envelope of the epigraph of $w$, it is readily seen that $w_{**}$ is affine on the whole set of convex combinations of the points $\{x_1, \dots, x_k \}$.
Taking into account that $w_{**}(x_1) = w(x_1) = 0$, we infer that
there exists $\mu > 0$ such that
\[
w(x_1 + t\zeta) \geq w_{**}(x_1+t \zeta) = - \mu t
\qquad\forall t\in [0,1].
\]
{}From \eqref{f:upbd} we obtain
\[
-\mu t \leq w(y) \left( 1 - \frac{|t\zeta - R\nu|}{R}\right)^{1/2}
= w(y) \left(\pscal{\zeta}{\nu} \frac{t}{R} + o(t)\right)^{1/2},
\qquad t\to 0^+,
\]
and, recalling that $w(y) < 0$,
\[
\mu t^{1/2} \geq K + o(1),\qquad t\to 0^+
\]
with $K>0$, a contradiction.
\end{proof}

On the basis of the lemma just proved, we obtain:

\begin{lemma}\label{p:All}
Assume \eqref{f:ipo}, and let $w$ be a restricted viscosity super-solution to~\eqref{f:probmod}.
Then also $w_{**}$ is a restricted viscosity
super-solution to the same problem.
\end{lemma}

\begin{proof}

Let $w$ be a restricted viscosity super-solution to~\eqref{f:probmod}.
In order to show that $w _{**}$ is still a restricted viscosity super-solution to the same problem, we begin by observing that that $w _{**}$ agrees with $w$ on $\partial \Omega$, namely $w_{**} = 0$ on $\partial \Omega$  (since \cite[Lemma 4.1]{ALL} applies).

Now let us check that $w_{**}$ satisfies ~\eqref{f:Rsuper}. In terms
of sub-jets, such property can be rephrased as
\begin{equation}
\label{f:Rsuper2}
\forall x\in\Omega,\
\forall (p,A)\inJ^{2, -}_{\overline\Omega} w_{**}(x)
\ \Longrightarrow\
\begin{cases}
F(w_{**}(x), p, A) \geq 0  \quad \text{if}\ p \neq 0,\\
\lambda_{\text{min}}(A) \leq -\dfrac{1}{2 w_{**}(x)} \quad \text{if}\ p = 0.
\end{cases}
\end{equation}

Let $x\in\Omega$  and consider first the case when $(p,A)\inJ^{2, -}_{\overline\Omega} w_{**}(x)$, with $p \neq 0$.

For every $\epsilon>0$ small enough,
applying Proposition~1 in \cite{ALL}
and Lemma~\ref{l:emptyjet},
we obtain
points $x_1,\ldots, x_k \in \Omega$,
positive numbers $\lambda_1,\ldots,\lambda_k$ satisfying
$\sum_{i=1}^k \lambda_i = 1$,
and elements
$(p, A_i) \in \overline{J}^{2, -}_{\overline\Omega} w(x_i)$, with
$A_i$ positive semidefinite,
such that
\[
\sum_{i=1}^k \lambda_i x_i = x,\quad
\sum_{i=1}^k \lambda_i w(x_i) = w_{**}(x),\quad
A-\epsilon A^2 \leq
\left(\sum_{i=1}^k \lambda_i A_i^{-1}\right)^{-1} =: B.
\]
We recall that, here and in the sequel,
it is not restrictive to assume that
the matrices $A$, $A_1,\ldots, A_k$ are positive definite,
since the case of degenerate matrices can be handled
as in \cite{ALL}, p.~273.
Moreover we recall that the ``closure'' $J^{2, -}_{\overline\Omega} v (x_0)$
is the set of $(p,A)\in \mathbb{R} ^n \times \mathbb{R} ^ { n \times n }_{{\rm sym}}$
for which there is a sequence $(p_j, A_j)\in J^{2, -}_{\overline\Omega} v(x_j)$ (according to \eqref{d:jet})
such that
$(x_j, v(x_j), p_j, A_j) \to (x_0, v(x_0), p, A)$.

Then, since by assumption $w$ is a super-solution to~\eqref{f:probmod}, we have $F(w(x_i), p, A_i) \geq 0$, i.e.
\[
-w(x_i) \leq \frac{1}{\pscal{A_i p}{p}}\left(|p|^4 + \frac{1}{2} |p|^2 \right)\, ,
\]
so that
\[
- \frac{1}{\sum_{i=1}^k \lambda_i w(x_i)}
\left(|p|^4 + \frac{1}{2} |p|^2 \right)
\geq
\left(\sum_{i=1}^k \lambda_i \frac{1}{\pscal{A_i p}{p}}\right)^{-1}.
\]

\smallskip
Then, using the degenerate ellipticity of $F$ and the concavity of the map
$Q\mapsto 1/ \text{tr} \big ( (p \otimes p )  Q^{-1} \big )$
(see \cite{ALL}, p. 286), we obtain
\[
\begin{split}
F(w_{**}(x), p, A-\epsilon A^2) & \geq
-\pscal{B p}{p}
- \frac{1}{\sum_{i=1}^k \lambda_i w(x_i)}
\left(|p|^4 + \frac{1}{2} |p|^2 \right)
\\ & \geq
-\pscal{B p}{p}
+ \left(\sum_{i=1}^k \lambda_i \frac{1}{\pscal{A_i p}{p}}\right)^{-1}
\\ & \geq 0\,.
\end{split}
\]

\medskip
It remains to consider the case when $(0,A)\inJ^{2, -}_{\overline\Omega} w_{**}(x)$. We have to show that
$$\lambda_{\text{min}}(A) \leq -\frac{1}{2 w_{**}(x)} \,.$$
In terms of test functions, this is equivalent to prove that
\[
\psi \prec_x w_{**},\
\nabla\psi(x) = 0
\quad\Longrightarrow\quad
\lambda_{\text{min}}(\nabla^2\psi(x)) \leq - \frac{1}{2 w_{**}(x)}\,.
\]

Since $w_{**}$ is a convex function,
the conditions
$\psi \prec_x w_{**}$ and $\nabla\psi(x) = 0$
imply that $x$ is a minimum point of $w_{**}$.
In particular, we must have
$w(x_1) = \cdots = w(x_k) = w_{**}(x)$.

If $k=1$, then $w_{**}(x) = w(x)$, $B = A_1$ and
$\lambda_{\text{min}}(A-\epsilon A^2)\leq \lambda_{\text{min}}(B) = \lambda_{\text{min}}(A_1) \leq -1/(2 w(x))$,
so that the required inequality follows.

Assume now that $k>1$,
so that $x$ is not a strict minimum point of $w_{**}$.
Since $x$ belongs to the relative interior of the convex polyhedron with
vertices $x_1, \ldots, x_k$, if we choose
$q := (x_1 - x) / |x_1 - x|$ we get that
$w_{**}(x + t q)$ is constant for $|t|$ small enough,
so that $\psi(x + tq) \leq w_{**}(x) = \psi(x)$ for $|t|$ small.
Hence
\[
\lambda_{\text{min}}(\nabla^2\psi(x)) \leq \pscal{\nabla^2\psi(x)\, q}{q} \leq 0
< - \frac{1}{2 w_{**}(x)}
\]
completing the proof.
\end{proof}

\medskip
\noindent
\textbf{Step 3.}
Let us prove that, under the additional assumption~\eqref{f:ipo},
the unique solution $u$ to~\eqref{f:dirich}
is $1/2$-power concave.

Let  $w= - u^{1/2}$.
By Lemma~\ref{l:equiv}, $w$ is a restricted super-solution to~\eqref{f:probmod}
hence, by Lemma~\ref{p:All},
also  $w_{**}$ is a restricted super-solution to~\eqref{f:probmod}.
Invoking again Lemma~\ref{l:equiv},
the function $v := (w_{**})^2$ is a viscosity
sub-solution to~\eqref{f:dirich}.
By the comparison principle proved in~\cite[Thm.~2.18]{ArmSm2012}
we deduce that $v \leq u$,
i.e.\ $(w_{**})^2 \leq w^2$.
On the other hand,
since $w_{**}\leq w$ (by definition of convex envelope) and $w\leq 0$,
we have that
$(w_{**})^2 \geq w^2$,
so that $w = w_{**}$, namely $w$ is a convex function.

\medskip
\noindent
\textbf{Step 4.}
Let us finally show that the conclusions of Step 3 (i.e.\ the power concavity of $u$)
remains true if $\Omega$ is any bounded convex domain.

For $\varepsilon\in (0,1]$ let $\Omega_\varepsilon$ denote
the outer parallel body of $\Omega$ defined by
\[
\Omega_\varepsilon := \{x\in\mathbb{R}^n:\ \text{dist}(x, \Omega) < \varepsilon\}\,,
\]
and let $u_\varepsilon$ denote the solution to
\[
	\begin{cases}
	-\Delta^N_{\infty} u_\varepsilon = 1 &\text{in}\ \Omega_\varepsilon\\
	u_\varepsilon = 0 &\text{on}\ \partial\Omega_\varepsilon\,.
	\end{cases}
\]
Since $\Omega_\varepsilon$ satisfies an interior sphere condition
(of radius $\varepsilon$), by Step~2 the function
$u_\varepsilon^{1/2}$ is concave in $\Omega_\varepsilon$.
Therefore, to show that $u^{1/2}$ is concave in $\overline{\Omega}$,
it is enough to show that, as $\varepsilon \to 0$, $u_\varepsilon \to u$
uniformly in $\overline{\Omega}$.
In turn, by Theorem~5.3 in \cite{LuWang2}, this convergence
holds true provided that $\left.{u_\varepsilon}\right|_{\partial\Omega}$
converges uniformly to $0$.

Let $y\in\partial\Omega$, and let $x_\varepsilon\in\partial\Omega_\varepsilon$
be such that $|x_\varepsilon - y| = \varepsilon$.
Let us consider the polar quadratic polynomial
\[
\eta(x) := \frac{1}{2}\, \text{diam} (\Omega_\varepsilon) \, |x-x_\varepsilon|
- \frac{1}{2}\, |x-x_\varepsilon|^2.
\]
Since $u_\varepsilon \leq \eta$ on $\partial\Omega_\varepsilon$,
and $u_\varepsilon$ enjoys the comparison with quadratic cones
(see \cite[Theorem~2.2]{LuWang2} or \cite[Lemma~5.1]{ArmSm2012}),
we have $u_\varepsilon \leq \eta$ on $\Omega_\varepsilon$,
and, in particular,
\[
u_\varepsilon(y) \leq \frac{\varepsilon}{2}\, (\text{diam} (\Omega) + 1)
- \frac{1}{2}\, \varepsilon^2.
\]
Hence  $\left.{u_\varepsilon}\right|_{\partial\Omega}$
converges uniformly to $0$ in $\partial\Omega_\varepsilon$.

\section{Local semiconcavity and $C^1$-regularity  of solutions}\label{secdiffe}

In this section we show the local semiconcavity and the $C ^1$-regularity of the unique solution to problem \eqref{f:dirich}.

We recall that  $u\colon \Omega \to \mathbb{R}$ is called {\it semiconcave (with constant $C$) in $\Omega$} if
\[
u(\lambda x + (1-\lambda)y) \geq \lambda u(x) + (1-\lambda) u(y)
-C\frac{\lambda(1-\lambda)}{2}\, |x-y|^2
\qquad \forall [x,y] \subset \Omega \ \text { and } \ \forall \lambda \in [0,1]\,.
\]
We say that $u$ is {\it locally semiconcave in $\Omega$} if it is semiconcave on compact subsets of $\Omega$.

\begin{proposition}\label{c:locsemiconc}
Assume that $\Omega$ is an open bounded convex subset of $\mathbb{R}^n$,
and let $u$ be the solution to problem~\eqref{f:dirich}.
Then $u$ is locally semiconcave in $\Omega$.
\end{proposition}

\begin{proof}
Let $K$ be a compact convex subset of $\Omega$,
and let $M$ be the Lipschitz constant of $v:= u ^ {1/2}$ in $K$.
We claim that $u$ is semiconcave in $K$ with semiconcavity constant
$C = 2 M^2$.
Namely, given $x,y\in K$ and $\lambda\in [0,1]$
and using the concavity of $v$ established in Theorem \ref{t:concave},  we get
\[
\begin{split}
& u(\lambda x + (1-\lambda)y) - \lambda u(x) - (1-\lambda) u(y) + \frac{C}{2}\lambda(1-\lambda)|x-y|^2
\\ & \qquad \geq [\lambda v(x) + (1-\lambda) v(y)]^2 - \lambda v(x)^2 - (1-\lambda) v(y)^2 +
M^2 \lambda(1-\lambda)|x-y|^2
\\ & \qquad
= \lambda(1-\lambda) \left[M^2 |x-y|^2 - |v(x) - v(y)|^2\right]
\geq 0\,.
\qedhere
\end{split}
\]
\end{proof}
\begin{remark}
In the above result  the semiconcavity property of $u$  is stated  just {\it locally} in $\Omega$. The reason can understood by inspection of the proof, and analyzing the behaviour of the constant $M$ appearing therein as the compact set $K \uparrow \Omega$. Indeed, recalling that the function $w= -v = -u ^ {1/2}$ satisfies  \eqref{f:upbd},
choosing $x = x_1 + \lambda \nu$,
and taking into account that $w (x_1) = 0$,
it is readily seen that
\[
\lim _{\lambda \to 0 ^ +}
\frac{w ( x_1 + \lambda \nu) - w (x_1)}{ \lambda}
\leq  \lim _{\lambda \to 0 ^ +} \frac{w ( y)}{ \lambda} \Big ( \frac{\lambda}{R} \Big ) ^ {1/2} = - \infty\,.
\]
This means that the normal derivative of $w$ with respect to
the external normal is $+\infty$ at every boundary point of $\Omega$ (so that $M \to + \infty$ as $K \uparrow \Omega$).

\end{remark}

Next,   let us quote an estimate for locally semiconcave functions near singular points
that we are going to exploit in order to arrive at the $C ^1$-regularity; such result
was proved in our previous paper~\cite[Thm.~8]{CFd} (see also~ \cite[Thm.~5]{CFc}).

Given a function  $u\in C(\Omega)$, we denote by $\Sigma (u)$ the singular set of $u$, namely the set of points where $u$ is not differentiable.
For every $x_0\in\Sigma(u)$,
 the {\it super-differential} of
$u$ at $x_0$, which is defined by
\[
D^+ u (x_0) :=
\left\{ p \in \mathbb{R} ^n \ :\ \limsup _{ x \to x_0}
\frac{u(x) - u(x_0) -\langle p, x-x_0 \rangle }{|x-x_0| }
\leq 0  \right\} \,,
\]
turns out to be a
nonempty compact convex set different from a singleton. In particular,  $D^+ u(x_0)\setminus \extr D^+ u(x_0)$
is not empty and contains non-zero elements.

\begin{theorem}\label{t:estid} {\rm (\cite[Theorem 8]{CFd})}
Let $u\colon\Omega\to\mathbb{R}$ be a locally semiconcave function,
let $x_0 \in \Sigma(u)$, and let $p\in D^+ u(x_0)\setminus \extr D^+ u(x_0)$.
Let
$R>0$ be such that
$\overline{B}_R(x_0)\subset\Omega$, and let $C$ denote
the semiconcavity constant of $u$ on
$\overline{B}_R(x_0)$.
Then
there exist
a constant $K>0$ and a unit vector
$\zeta\in\mathbb{R}^n$
satisfying the following property:
\begin{equation}\label{f:estidist0}
u(x) \leq u(x_0) + \pscal{p}{x-x_0}
- K\, |\pscal{\zeta}{x-x_0}| +\frac{C}{2}
|x-x_0|^2
\quad \forall x\in\overline{B}_R(x_0)\,.
\end{equation}
In particular, for every $c>0$, setting  $\delta := \min\{K/c, R\}$,
it holds
\begin{equation}\label{f:estidist}
u(x) \leq u(x_0) + \pscal{p}{x-x_0}
- c \pscal{\zeta}{x-x_0}^2 +\frac{C}{2}
|x-x_0|^2 \quad  \forall x \in  \overline B_{\delta}(x_0)\,.
\end{equation}
Furthermore, if $p\neq 0$ then the vector $\zeta$ can
be chosen so that $\pscal{\zeta}{p}\neq 0$.
\end{theorem}

\bigskip
We are now ready to give our  $C^1$-regularity result:

\begin{theorem}\label{t:diff}
Let $u\in C(\Omega)$ be a viscosity solution to
$-\Delta_{\infty}u = f(x,u)$ in $\Omega$.
If $u$ is locally semiconcave in $\Omega$,
then $u$ is everywhere differentiable (hence of class $C^1$)
in $\Omega$.

In particular, if $\Omega$ is an open bounded convex subset of $\mathbb{R}^n$,
and  $u$ is the unique solution to problem~\eqref{f:dirich}, then $u \in C ^ 1 (\Omega)$.
\end{theorem}

\begin{proof} Let $u\in C(\Omega)$ be a locally semiconcave viscosity solution to
$-\Delta_{\infty}u = f(x,u)$ in $\Omega$.
Assume by contradiction that $\Sigma(u)\neq \emptyset$.
Without loss of generality we can assume that $0\in \Sigma(u)$.
Let $p\in D^+u(0)\setminus \extr D^+u(0)$, $p\neq 0$.
By Theorem~\ref{t:estid}, there exists a unit vector
$\zeta\in\mathbb{R}^n$, with $\pscal{\zeta}{p}\neq 0$, such that,  for every $c>0$, the inequality
\[
u(x) \leq
u(0) + \pscal{p}{x} - c \pscal{\zeta}{x}^2 +
\frac{C}{2}\, |x|^2
\]
holds true for all $x\in B_{\delta}(0)$, with $\delta$ depending on $c$.
Thus, setting $\varphi (x):= u(0) + \pscal{p}{x} - c \pscal{\zeta}{x}^2 +
\frac{C}{2}\, |x|^2$, it holds $u \prec_0\varphi $.
Since $\nabla \varphi (0) = p \neq 0$, we have
\[
\Delta^+_{\infty} \varphi(0) =
-2 c  \pscal{\zeta}{p}^2 + C|p|^2,
\]
choosing $c>0$ large enough we get $-\Delta^+_{\infty} \varphi(0) > f(0, u(0))$,
a contradiction.

Since $u$ is differentiable everywhere in $\Omega$,
by \cite[Prop.~3.3.4]{CaSi}
we conclude that $u\in C^1(\Omega)$.

Finally, if $\Omega$ is an open bounded convex subset of $\mathbb{R}^n$,
and  $u$ is the unique solution to problem~\eqref{f:dirich}, the first part of the statement just proved applies (and hence $u \in C ^ 1 (\Omega)$) because we know from Proposition \ref{c:locsemiconc} that $u$ is locally semiconcave.

\end{proof}

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\providecommand{\MR}{\relax\ifhmode\unskip\space\fi MR }
\providecommand{\MRhref}[2]{%
	\href{http://www.ams.org/mathscinet-getitem?mr=#1}{#2}
}
\providecommand{\href}[2]{#2}

redpajama_set_name: RedPajamaArXiv


His head was usually full of ah ha!, a luminescence that folded around obstacles like smoke. Sometimes the ah ha! conformed to the world around him, sometimes it rippled across the flat sea of induction, threatening to alter the way the world waddles on its hips. The sad thing is that he couldn't always stop the ah ha! from fizzing away. You know how sheep disappear on a green meadow when you've been watching them all day? They're there, of course, but for all intents and purposes they've sneaked away like a group of night singers in the storybooks about Merry Old England. The ah ha! drifted away much the same way. There isn't much difference in the world. The ah ha! was a product of his heart and his mind, and although it can be argued there is some difference between the heart and the mind--if the heart could think it would stop beating, for example--his mind and his heart always traveled in the same direction, filling each other with treasures as they traveled along. But, nevertheless, more and more he had to watch the ah ha! fizzle away. She, on the other hand, is too sly and leonine to drift away. She keeps her good stuff in a locker, determined to find the right time to put it to use. I don't know how the two met and to convince you I did I'd have to invent a story and worry about it disappearing before I wrote it down. But they did meet and it's more important to know that she stopped his fizz and he unlocked her locker, and now they are as indivisible as the white sheep are from the green meadow.
Pointed and effervescent as broken crystal.
"They're there, of course, but for all intents and purposes they've sneaked away like a group of night singers in the storybooks about Merry Old England. The ah ha! drifted away much the same way."
Thanks, everyone for the nice comments. I'm delighted it appeals to each of you.
"But they did meet and it's more important to know that she stopped his fizz and he unlocked her locker, and now they are as indivisible as the white sheep are from the green meadow."
Strange, nonchalant, unruly love story. The best kind. Fav.
Thanks, Mia. Your praise means a lot to me.
"But, nevertheless, more and more he had to watch the ah ha! fizzle away. She, on the other hand, is too sly and leonine to drift away."
Thanks, Sally. Glad you enjoyed it.
This made me think of Zefrank's video, The Sweetness, but with twist at the end where they grab a scrap of sweetness together.
Lovely piece and very sweet, which is something we need. Fav'd.
"she stopped his fizz and he unlocked her locker"
The key to happiness! Great!
Thanks for reading the nice comments Laura, Kyle, Gary, Bill and Beate. Delighted you enjoyed it.
Wonderful, sad and damn those fading ah-ha's! Thank you.
Thanks, Kari, Patti and David.


text:
Five-Man Band/Live Action TV-N-S
Trope, Narrative Devices, Religion Tropes, Religious Russian Roulette
Religious Russian Roulette
Quotes •
Playing With • Useful Notes •
Guidance Counsellor: Dwicky, you really believe in aliens?
Dwicky: Ha ha, not any more! All the childlike wonder was ripped from my heart the day my foot was stuck in an escalator, and aliens didn't come rescue me.
—Invader Zim: Vindicated
Bob is in a crisis, so he turns to God, gods, Crystal Dragon Jesus, aliens, or Trope-tan for aid but laces it with an ultimatum. Bob will be the bestest follower and prophet of his/her/its/their greatness if they'll just come through with this one teeny-tiny miracle. If they don't? Well then he'll have empirical proof of the absence or jerkassness of God, and will convert to another religion or become a Hollywood Atheist.
This gets very thorny very quickly as Bob's miracle fails to manifest. Did he not pray hard enough? Was he ignored? Is God even there? Bob just pulled the trigger on the one loaded chamber in his game of Religious Russian Roulette. After blowing his theological brains out, Bob will embrace atheism with all the Fundamentalist fervor he can muster: if the Powers That Be didn't come through for him, why would they do so for anyone else? This nice little angry depression will last right up until a left field miracle kicks in. It is Maybe Magic, Maybe Mundane, but Bob now has something spiritual to chew on.
Typically, any non-answer from the Powers That Be is chalked up to Bob asking for the wrong thing or for the wrong reason, such as selfish or hurtful wishes or expecting God to out and out prove His existence on demand. If it's anything like a Chick Tract, the appearance of a true miracle (on demand, no less!) will instantly convert Bob to whatever religion the author wants. In works where the existence of God and the Devil are established facts, this situation is often caused because either God Is Evil, a Jerkass, inept, or simply flawed. At the very least, it's a case for God and Satan Are Both Jerks.
See also: Smite Me Oh Mighty Smiter.
Examples of Religious Russian Roulette include:
In Major League, Pedro Cerrano is a practitioner of voodoo, worshiping the spirit Jabu. In his last at bat, Pedro tells Jabu that if he can't hit this curve, he won't believe in him anymore. After two strikes, he decides to abandon him anyway. "I say 'Fuck you.' I do it myself." He hits it, it's a homer, and by the sequel he's converted to Buddhism. It's also implied that this is how he came to worship Jabu in the first place; when a teammate talks to him about Christianity, Pedro remarks that he likes Jesus very much -- but he never helped him hit breaking balls.
Conan the Barbarian, rather than giving Crom an ultimatum, simply says, "To Hell with you! I'll do it myself." And it pays off; when Conan is about to be struck down, a vision of Valeria appears so he can finish the battle.
Probably the most accurate part of the movie: Crom, being the war-god of a Proud Warrior Race, has no time to aid weaklings who cannot overcome their challenges alone.
In the books Crom explicitly can't be prayed to; those who believe in him say that the only response to such audacity would be death. According to them Crom has already given them everything worth having by giving them life and the ability to do battle. It's up to them to do the rest.
An unusual example in Mistborn: After Sazed crosses the Despair Event Horizon, he loses his sense of generalized faith and starts going through the 200+ religions he knows about looking for one that he feels can offer a suitable explanation for everything wrong with the world. He winds up throwing out all of them for having logical flaws before eventually realizing it doesn't really work like that.
In The Facts of Life, Blair describes how this applies to her: she once prayed to God to stop her parents' divorce; He didn't, and she stopped believing in Him.
Sergeant Major Williams does this in an episode of It Ain't Half Hot Mum where he tried praying to Hare Krishna.
Calvin in Calvin and Hobbes as he unsuccsessfully begs for it to snow:
Calvin: Do you want me to become an atheist?!
Played with in another Calvin and Hobbes strip where Calvin, just before beaning Susie with a snowball, prays for a sign that this would be wrong. No sign; he beans her and she beats him up. "How come the universe always gives you the sign after you do the thing?" In this case, he's hoping for a negative result, and the object of the experiment is objective morality rather than God, but otherwise the structure is the same.
This can sometimes be the impetus for real life turns to atheism. Something doesn't happen the way their religious belief suggests it would, and they start to doubt. However, rather than leave the person a bitter, god-hating shell, it's more likely to make the person more perceptive to atheists reasons for disbelief. Sometimes it sticks. Other times, the religious person will end up renewing their faith. Either way, the common end of this trope, being angry at god, is at most a temporary state, if it happens at all. The person either returns to their faith, or stops believing, and why be angry at a non-existent deity?
Christianity calls this trope the mortal sin of "Tempting God". Demanding an action from God in order to justify faith or prove himself real. A Real Life example is Abusive Parents refusing medical treatment for their child because they expect God to heal them. The most common, but extreme example to explain this to children would be jumping off a cliff and expecting God to catch you.
Notably, this is exactly what Satan tries to convince Jesus to do during the second temptation. Jesus replies, "[Scripture] also says, 'You shall not put the Lord your God to the test.'"
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