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Language-Based Learning in Children - Unlocking Clues
University of Missouri-Columbia - missouri.edu
Synopsis : University of Missouri study that focuses on typically learning children, has found a link between working memory and how children learn.
According to the National Center for Learning Disabilities (NCLD), one in five individuals are impacted by language-based learning disabilities - one of the most common being dyslexia, which involves difficulty in reading or interpreting words, letters and other symbols.
Working memory is defined as a cognitive system with a limited capacity responsible for the transient holding, processing, and manipulation of information. Working memory is a short-term memory buffer that allows for the manipulation of stored information, while short-term memory is only involved in the short-term storage of information and does not entail the manipulation or organization of material held in memory. Working memory is an important process for reasoning and the guidance of decision making and behavior, and is essential for navigating complex cognitive tasks such as comprehension, learning and reasoning.
Now, a new study from the University of Missouri that focuses on typically learning children, has found a link between "working memory" and how children learn. Researchers feel this discovery may later help educators uncover new ways to teach children with learning disabilities.
"Working memory, or the retention of a small amount of information that is readily available, is an integral aspect to learning," said Nelson Cowan, a professor psychological sciences in the MU College of Arts and Science. "Our goal was to understand the structure of working memory and how it relates to language use and intelligence in children."
Working memory is the small amount of information an individual holds in his or her mind at one time, Cowan says. For example, if an individual is listening to a story, he or she holds the beginning sentence or paragraph in his or her mind until the story continues to the point where the listener can make sense of it. Cowan believes that people with language disorders such as dyslexia may have a working memory deficit.
In the study, Cowan and his team used three different memory models to assess the relationship between working memory and learning in 168 typically developing second grade children. The children participated in an array of tests that challenged the visual-spatial, phonetic and auditory aspects of their working memory.
Results showed that an aspect of working memory referred to as the focus of attention was found to have a strong correlation with learning and intelligence. In his previous work, Cowan found that attention is a key component to working memory.
"Attention is vital to understanding phonetic sounds, reading words and solving mathematical problems," Cowan said. "To help children with language learning disabilities, educators can reduce the amount of information that is given to a child at one time, allowing them to channel their attention onto one task instead of three or four."
The study, "The Structure of Working Memory in Young Children and its Relation to Intelligence," was published in the Journal of Memory and Language and was funded by a grant from the National Institutes of Health (NIDCD #R01 DC010784). Shelley Gray, a professor of speech and hearing science in the Mary Lou Fulton Teachers College at Arizona State University was first author on the study. The content is solely the responsibility of the authors and does not necessarily represent the official views of the funding agency.
Editor's Note: For more on the story, please see: coas.missouri.edu/news/professor-explores-relationship-between-working-memory-and-intelligence
Journal: Disabled World. Language: English. Author: University of Missouri-Columbia. Electronic Publication Date: 2016/12/19. Last Revised Date: 2016/12/19. Reference Title: "Language-Based Learning in Children - Unlocking Clues", Source: Language-Based Learning in Children - Unlocking Clues. Abstract: University of Missouri study that focuses on typically learning children, has found a link between working memory and how children learn. Retrieved 2019-07-19, from https://www.disabled-world.com/disability/education/special/lbl.php - Reference Category Number: DW#275-12568.


These common cracks are caused by excessive lateral earth pressure on block basements. Southeastern Wisconsin's clay soil expands significantly when it gets wet, putting lateral pressure on the basement wall that can cause it to crack. The size of the crack is insignificant – the fact of the crack indicates the structure has failed and needs immediate attention.
All horizontal cracks are serious enough that profession evaluation is warranted. By no means do all horizontal cracks require repair, but all horizontal cracking should be monitored for any movement over time.
"Mike did a good job explaining the problem to me. Professional & Courteous. "


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Chairman: Alfred H. Moses
Executive Director: Hillel Neuer
Founder: Morris B. Abram
UN Watch Database
The Pillay Commission
Focus on China
Focus on Iran
Durban 4
ILO on Auto-Pilot against Israel
Hamas, ILO, Palestinians, UN Ethics, UN-Israel Controversies
The ILO issued a deeply flawed, misleading and misrepresentative report on the Palestinian territories.
Last month, at the 103rd session of the International Labour Conference, the International Labour Organization (ILO), the UN agency dealing with labor issues, presented their 2014 report on "The situation of workers of the occupied Arab territories." The report, however, is deeply flawed, misleading and misrepresentative of realities on the ground.
The report puts sole responsibility for the economic hardships, poor working conditions, stalled growth and deteriorating markets of the Palestinian's economy on the Israeli "occupation." The report echoes the President of the Palestinian Authority Mahmoud Abbas, who proclaims, with simplicity, "The economic situation is really difficult, and the main reason is the Israeli occupation."
The ILO report holds only one party responsible for "Barriers to economic development in the West Bank" — Israel.
The report blames Israel for the "Deepening of Gaza's six-year siege [which] paralyses [the] economy."
In a section titled "Rising tensions and violence," the report discusses in depth violence against Palestinians throughout the West bank. This includes discussions of:
"Violent acts and intimidation by Israeli settlers against Palestinians and their property."
"Serious concerns are being expressed that occupation and settler activity is leading to a dangerous culture of impunity."
"In addition to settler violence, there is a worrying trend of violence linked to confrontations between Israeli armed forces and Palestinians."
"A recent United Nations report notes that Israeli settlement-related activities and settler violence are at the core of most of the violations of human rights in the Occupied Palestinian Territory, including the right to freedom of movement, work, health, education, housing and an adequate standard of living (UN, 2014b)."
However, the ILO does not present the whole and honest truth and instead submits a careless and sloppy work that is seemingly offered as part of the habitual castigation of the State of Israel.
IGNORING CORRUPTION OF THE PALESTINIAN AUTHORITY IN THE WEST BANK: While the report places the blame on Israel for Palestinian's economic woes, outside observers find that corruption plays a pivotal role in Palestinian struggles. Last year, Britain's Sunday Times reported that "Billions of euros in European aid to the Palestinians may have been misspent, squandered or lost to corruption, according to a damning report by the European Court of Auditors, the Luxembourg-based watchdog." The auditors found that the Palestinian Authority wasted almost €2 billion ($2.7 billion) in European aid. Additionally, the Palestinian Anti-Corruption Commission (PACC), has received 65 corruption cases regarding Palestinian misconduct and notes that the commission is currently investigating several senior officials in the Palestinian Authority. There are also disturbing reports of misuses of international aid to the Palestinian Authority.
IGNORING CORRUPTION OF HAMAS IN GAZA: The ILO, which places blame on Israel for the faltering economy in Gaza, makes no mention of Hamas throughout their entire report. Instead, what solution did the report provide for fixing the Palestinian economy? "Ultimately, the blockade must be lifted." Hamas — a group many governments have designated as a terrorist organization — remains a corrupt institution which has a chokehold on the Palestinian economy. Reports of Hamas corruption include accusations of graft and other illicit activities, including embezzlement of funds, purchases of exorbitant sports cars, looting of public resources and outright theft. This includes accusations from the Palestinian Authority Minister for Religious Endowment Mahmoud Habbash that Hamas has stolen "800 million dollars from the poor, through front companies for investments." Even more disturbing are United Nation's reports that Hamas has directly stolen UN aid. Hamas irked the ire of the UN after it raided 100 aid trucks entering the Gaza strip and sold the contents to "the highest bidders," and then, on two separate days, it stole truckloads of goods from the United Nations Relief and Works Agency (UNRWA) in the Gaza Strip. The UN was thus prompted to halt all aid to the Gaza Strip, before reinstating it a few days later. A few weeks prior, in a separate incident, Jordanian authorities confirmed that Hamas seized 12 trucks full of Jordanian humanitarian aid.
IGNORING PALESTINIAN TERROR: Although the report does not shy away from asserting that there is "increased settler violence" in the West Bank, it refuses to explicitly mention any violence committed by Palestinians against Israeli civilians. The report additionally does not mention Palestinian extremism or terrorism. While the report is highly critical of the Israeli blockade, the report neglects to mention that the reason Israel maintains a blockade on the Gaza Strip is because thousands of rockets and mortars have been fired into Israel directly targeting, injuring and killing Israeli civilians.
IGNORING ISRAELI EFFORTS TO IMPROVE PALESTINIAN ECONOMY: The report remains silent on Israeli efforts to aid and assist in the improvement of the Palestinian economy. These efforts range from small scale efforts to large scale initiatives. For example, in order to ameliorate the Palestinian economy, Israel has lifted restrictions on importing most building materials to Gaza, increased the gas and fuel imports into Gaza, expanded the number of work permits issued to West Bank residents and opened several border crossings — including the opening of the Kerem Shalom crossing in Gaza. There has been international praise for the larger efforts of the Israeli government in reaching out and working with Palestinian authorities including from the World Bank and the EU. Currently, commerce between Israelis and Palestinians is estimated at $20 billion USD annually and the International Chamber of Commerce and the Quartet have recently taken initiatives to strengthen economic relations between the two. Most notably is the Initative for the Palestinian Economy (IPE), which is led by Quartet Representative Tony Blair. The Quartet states that the ultimate goal of the initiative is to continue to lead Palestinians on a path of growth and development — something the current Palestinian leadership has failed to do.
Ultimately, the facts show that the ILO presents a slanted and partial view of the economic situation of the Palestinians and disregards substantial truths, data and information. While the ILO is quick and thorough in scrutinizing, criticizing and exaggerating Israeli actions and inactions, it remains quiet on the Palestinian responsibility for their own failures and lack of success.
Syria in denial on denying women's rights
ICJ Advisory Opinion Relied on Founder of "Gaddafi Human Rights Prize"
Qatar's envoy loses bid for human rights post, called Jews 'enemies' & gays 'disgusting'
Watchdog Urges UN to Reject Bid by Antisemitic Qatar Envoy to Chair Human Rights Forum
The Compendium of World Condemnations of Antisemitic UN Investigator Miloon Kothari
HRC 44 Written Statement: Antisemitism and Terrorist Incitement in Palestinian Education
Watchdog calls out UN rights council's hypocrisy, anti-Israel bias
How the UK Funds the Targeting of Israel Under the UNHRC's Agenda Item 7
Your Voice at the United Nations
UN Watch is a non-profit organization dedicated to holding the United Nations accountable to its founding principles. UN Watch is funded entirely by charitable donations
Copyright © 2014–2022. UN Watch | Your voice at the UN.
UN Watch

redpajama_set_name: RedPajamaCommonCrawl


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Q: What files make one Wordpress site different from another Migrating wordpress sites between hosts can take a lot of time, especially when the hosting platforms are different.
I have been trying to migrate my sites from Cpanel to Mediatemple, but it seems like im just not getting it right.
There is various options

*

*Use the guide they provide
https://kb.mediatemple.net/questions/1556/Migrating+your+websites+to+the+Grid#gs

*

*When moving the files in this way the permissions of the files are not set properly and I would have to got back through them and figure out which ones need to change.

*The database export from PHPMyAdmin also does not look the same it looks in the screenshot

*Using InfiniteWP

*

*To use InfiniteWP you must provide the url of the site and since I dont want to change the DNS until the site is moved this option does not seems to be ideal.

*This option might work if its ok for the sites to be unavailable for a day or so while the DNS resolves...

*But I don't want the sites to be unavailable

*Using Mediatemples "one click apps" to install wordpress and then moving only the files that are unique to the site from the old host to the new host.

*

*I would like to use this option

*I think that the content of the WP-Content folder needs to be moved that the database needs to be moved.

My question is
- what folders and files in a standard wordpress install typically hardly ever changes from one site to the other.
- can I use the wordpress database export and import function to move the database from one site to the other.
Any help will be appreciated
Thank you

A: With InfiniteWP, you can use the clone an existing site command (which can be found in "Tool"->"Install / Clone WP") to migrate a site to a new server.
You have to use a temporary (sub)domain pointing to the new server.
To answer your questions :

*

*/wp-content/ stores all your files and sometimes plugin files, wp-config.php is where your configuration is stored (e.g. credentials to access the database). Depending on your servers, the .htaccess files may be different.

*I would recommend to create a dump file of your entire database using phpMyAdmin.

redpajama_set_name: RedPajamaStackExchange


This ornately shaped headboard features an antique white finish that provides the perfect mix of traditional and cottage design elements. The headboard's inset upholstered cushion with button tufting offers the perfect finishing touch, adding a welcoming comfort to the overall feel of this piece.
The Realyn Twin Upholstered Panel Headboard with Button Tufting by Signature Design by Ashley at Pilgrim Furniture City in the Hartford, Bridgeport, Connecticut area. Product availability may vary. Contact us for the most current availability on this product.


We understand that for people living with OCD, family members, and professionals who treat OCD and related disorders, it can sometimes be a challenge to find comprehensive information that addresses your specific needs. In addition, with all of the information out there, it can be difficult to know what sources are vetted and trustworthy.
At the International OCD Foundation (IOCDF) we understand these challenges. That is why we collaborate with leading experts to bring you informative and accurate information about the latest therapy, research, and treatment in OCD and related disorders. In 2018, it is our goal to ensure that more of this information is available to you on our website. That's why we have recently revamped our "Expert Opinions" section to now be called "From the Experts", and include even more relevant and informative articles.
All of the articles that we share have been firmly vetted by the IOCDF, so you can trust that what you are reading is presented accurately and honestly.
You can find our "From the Experts" section located under the "Find Resources" header on our homepage or by visiting iocdf.org/from-the-experts. We will be adding new articles on a quarterly basis to ensure that you have access to fresh content that covers new areas of interest.
We strive to deliver information that offers value to the community and welcome your suggestions about specific subjects or areas that you would like to see covered by our experts. If you have suggestions please feel free to email them to membership@iocdf.org. In addition, you can become an IOCDF member in order to gain access to our quarterly OCD newsletter, which features new expert articles in each issue. For more information, please visit www.iocdf.org/membership.
We hope that you will find this information helpful!
Can we expect gene therapy for OCD soon??
Can stem cells help to manage OCD? I heard , it may cure OCD.


text:
Gut bacterium supports growth in infants with severe acute malnutrition
By the SOURCE/Medicine & Health • April 15, 2022
A new study led by Washington University School of Medicine in St. Louis and the International Centre for Diarrhoeal Disease Research in Dhaka, Bangladesh, shows that a standard milk-based therapy plus treatment with a specific strain of gut bacteria known as Bifidobacterium infantis (B. infantis) reduces gut inflammation and promotes weight gain in infants with severe acute malnutrition. Pictured is a mother feeding her infant, who has severe acute malnutrition. (Photo: Ridwan Islam, International Centre for Diarrhoeal Disease Research, Bangladesh)
About 18 million children under age 5 suffer from severe acute malnutrition, and over 3 million children die from it each year. Treatment with high-calorie supplemental foods and antibiotics can prevent deaths, but these interventions often have limited impact on the long-term effects of severe acute malnutrition, such as persistent stunted growth, disrupted immune function and impaired brain development. Even when treated with standard therapeutic foods, many children continue to have moderate forms of the disease and are at risk of falling back into severe acute malnutrition.
A new study, published April 13 in the journal Science Translational Medicine, from Washington University School of Medicine in St. Louis and the International Centre for Diarrhoeal Disease Research in Dhaka, Bangladesh (icddr,b), shows that a standard milk-based therapy plus treatment with a specific strain of gut bacteria known as Bifidobacterium infantis (B. infantis) for four weeks promotes weight gain in infants with severe acute malnutrition, with accompanying reductions in gut inflammation. The clinical trial was conducted in Dhaka.
The B. infantis strain was chosen for the trial because it has been shown to be safe to give to infants as a probiotic and is known to thrive on specific carbohydrates present in human breast milk. The strain, which had been isolated from the gut microbes of a healthy infant in the U.S., is a commercially available probiotic that meets the Food and Drug Administration's good manufacturing practice guidelines and was provided by Evolve BioSystems.
Importantly, the investigators found that B. infantis was either undetectable or present in markedly reduced amounts in infants with severe acute malnutrition compared to those with healthy growth.
Although treatment with the U.S. strain of B. infantis produced significant improvements in growth in Bangladeshi infants with severe acute malnutrition, its level of colonization was still tenfold to a hundredfold lower than levels documented in healthy Bangladeshi infants. To identify strains that might colonize to a greater degree and potentially produce superior clinical responses, the research team cultured B. infantis strains from healthy children living in the same community as those with severe acute malnutrition who were enrolled in the clinical trial.
The scientists found one strain that contained genes that allow it to utilize complex carbohydrates present in the local diet more efficiently than the strain investigated in the clinical trial. They went on to study germ-free mice colonized with gut microbial communities — called the microbiota — obtained from an infant with severe acute malnutrition before treatment to show that this Bangladeshi strain was a more efficient colonizer than the U.S. B. infantis strain. These findings set the stage for future clinical studies to test the effects of this newly discovered strain on restoring growth in children with severe acute malnutrition.
"Bacterial strain selection is a critical element in designing future therapies for repairing dysfunctional gut microbial communities that lead to malnutrition," said co-senior author Jeffrey I. Gordon, MD, the Dr. Robert J. Glaser Distinguished University Professor and director of the Edison Family Center for Genome Sciences and Systems Biology at Washington University School of Medicine. "This pilot study not only shows promising results in treating children with severe acute malnutrition but also underscores how proper development of the gut microbiota is linked to healthy growth of infants."
The clinical trial was led by co-senior author Tahmeed Ahmed, MBBS, PhD, executive director of the icddr,b, and Sharika Nuzhat, MBBS, a pediatrician in the Nutrition and Clinical Services Division of icddr,b. The trial involved 60 infants ages 2 months to 6 months with severe acute malnutrition, meaning their weight-for-height growth curves were at least three standard deviations below the average for healthy children.
Jeffrey Gordon


text:
Captain plunges overboard with cop to evade arrest for "ganja" on boat
-three others nabbed on vessel, policeman rescued from waters
One man is on the run while three others are in police custody after they were nabbed inside of a boat witg 103 kilogrammes (kgs) of "ganja" concealed in several parcels.
The boat captain, whose alias is "Royo", along with the other three suspects had been apprehended on the vessel but while in police custody he managed to plunge overboard and escape.
Reports are that on Tuesday, ranks of the Guyana Police Force (GPF) were patrolling in the Canje Creek when they intercepted a wooden vessel in the vicinity of Sandaka with the four males.
A search of the boat revealed nine (9) taped and compressed parcels. Upon investigating their contents, each revealed a quantity of cannabis.
According to the police, the parcels weighed 103 kgs of "ganja".
The four suspects were arrested and being escorted to New Amsterdam along with their boat which is said to be powered by a 60HP outboard engine.
However, while in the vicinity of New Forest, West Canje, one of the suspects (the Captain) "suddenly held on to one of the ranks and plunged overboard with him."
The policeman was quickly rescued but the boat Captain swan ashore and escaped the clutches of the lawmen.
Meanwhile, the other three suspects were taken to the police station and are presently in custody as investigations continue.
The three men hail from Charlestown and East Ruimveldt, Georgetown.
Meanwhile, the police are on the hunt for the boat Captain who is said to hail from Berbice.
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text:
Navitas Systems Expands Michigan Manufacturing Plants
Research, Engineering, and Design: The Talent Cluster in a Talent-Focused World
Navitas Systems, a leader in integrated design, technology development, and manufacturing of innovative energy-enabled system solutions and energy storage products, was awarded a $1 million Michigan Business Development Program grant for the creation of 125 new jobs and a $9.2 million investment at its facility in Pittsfield Charter Township, Michigan.
The company is located in Ann Arbor, with both a 48,000-square-foot R&D, engineering, and manufacturing center, as well as a 100,000-square-foot lithium battery systems manufacturing center.
According to the Michigan Economic Development Corporation, Navitas is experiencing growth in its commercial segment and plans to expand into a second facility in Scio Township, where it will house its commercial battery manufacturing, additional engineering resources and battery cell production, and office space. The project is expected to generate a total private investment of $3.36 million and create 111 high-paying new jobs, resulting in a $750,000 MBDP performance-based grant. Michigan was chosen over competing sites in Illinois and Pennsylvania.
"We appreciate the opportunity to continue our development path in the state of Michigan. The job incentive grant is an important part of our justification for making the additional investments in Ann Arbor as compared to other sites," said Alan Elshafei, founder and CEO of Navitas Systems. "MEDC has been a strategic partner for Navitas since 2013 and we appreciate the confidence MEDC has placed in our company as well as the resources to help turn our plans into new jobs and opportunities."
The project aligns with MEDC's focus industries of automotive manufacturing, engineering, design, and development as well as advanced manufacturing, state officials said. Navitas is focused on promotion from within and provides a complete benefits package and tuition reimbursement to its employees. Ann Arbor SPARK has worked to help further the company's success since 2013, through a range of activities including support in finding key talent, staff time, and other resources.
In August 2019, East Penn Manufacturing partnered with Navitas Systems and acquired majority interest in the company. East Penn is a privately-owned company operating the largest single-site battery manufacturing facility in the world.


text:
Thirst (Spanish: La sed) aka Hijo de Hombre, aka Choferes del Chaco is a 1960 Argentine-Spanish war film directed by Lucas Demare. It is set during the Chaco War (1932–35) between Bolivia and Paraguay, sometimes known as the "War of the Thirst". The script is based on a chapter of Augusto Roa Bastos novel Hijo de Hombre. It was made as a co-production between Argentina Sono Film and the Spanish company Suevia Films.

Cast
 Vicente Ariño
 Susy Castell
 Lucas Demare
 Carlos Dorrego
 Carlos Estrada
 Dora Ferreiro
 Francisco Rabal
 Manuel Rosón
 José María Salort
 Sergio Vandes
 Jorge Villalba
 Olga Zubarry

References

Bibliography
 Helene C. Weldt-Basson. Postmodernism's Role in Latin American Literature: The Life and Work of Augusto Roa Bastos. Springer, 2010.

External links

1960 films
1960s Spanish-language films
Films directed by Lucas Demare
Films set in the 1930s
Films set in Paraguay
Chaco War films
Spanish war films
Argentine war films
1960s war films
1960s Argentine films

redpajama_set_name: RedPajamaWikipedia


text:
(Mark) Mercado, Christina
(Mark) Mercado, Christina - New York, NY
1 Penn PlazaSuite 5315New York, NY 10119
Block O'Toole & Murphy
Christina (Mark) Mercado graduated from New York University with a degree in journalism and earned her juris doctor from Notre Dame Law School. She joined Block O'Toole & Murphy, LLP as an associate, after demonstrating exceptionally strong legal research and writing skills, an industrious work ethic, and a heartfelt dedication to fighting for clients' rights as a summer intern for the firm. Prior to joining the firm, Ms. Mercado worked in the Appeals Bureau of the...
Christina (Mark) Mercado graduated from New York University with a degree in journalism and earned her juris doctor from Notre Dame Law School. She joined Block O'Toole & Murphy, LLP as an associate, after demonstrating exceptionally strong legal research and writing skills, an industrious work ethic, and a heartfelt dedication to fighting for clients' rights as a summer intern for the firm.
Prior to joining the firm, Ms. Mercado worked in the Appeals Bureau of the Nassau County District Attorney and in the chambers of the Honorable Joan O'Dwyer of Queens County Supreme Court. At Notre Dame, she worked as a devoted advocate for clients at the Notre Dame Legal Aid Clinic and served as the Executive Articles Editor of the law school's prestigious Journal of Law, Ethics & Public Policy, after receiving accolades for her brief-writing skills. Ms. Mercado also previously worked as a professional writer and reporter in New York City.
Since joining Block O'Toole & Murphy, LLP, Ms. Mercado has handled a wide range of cases including wrongful death, motor vehicle accidents, premises liability, labor law and construction accidents. Ms. Mercado is known for her superior writing and analytical skills. She is the lead associate in the firm's appellate and complex motion practices, and regularly writes briefs to the Appellate Division and the Court of Appeals. She is also heavily relied upon to research and brief critical legal issues in substantial cases across all of the firm's practice areas.
Recent Results:
$8,800,000 settlement for victim in Queens pedestrian accident who suffered a brain injury*
$7,525,000 for family devastated by fatal Suffolk County auto accident
$3,925,000 settlement for injured "Good Samaritan" in locomotive, tractor-trailer collision at railroad crossing
$2,650,000 awarded to Local 3 Electrician after a back injury caused by a trip and fall*
$2,250,000 construction worker recovers for lower back and knee injuries
$2,250,000 settlement awarded to window washer who fell from Manhattan building*
$2,150,000 settlement for woman injured in Nassau County auto collision
$1,850,000 settlement for victim of a delivery truck collision in the Bronx
$1,750,000 settlement for city sanitation worker and wife who suffered neck injuries in collision with truck*
$1,500,000 settlement for a woman injured in slip and fall on snow on front landing and steps
$1,500,000 settlement for a Bronx driver who was struck by a New York City street sweeper
$1,500,000 for a pedestrian who injured her neck and shoulder*
$1,475,000 for injured Brooklyn product delivery man*
$1,410,000 for a married couple from Brooklyn who injured their necks in a rear-end motor vehicle accident*
$1,325,000 settlement for a worker whose forearm was lacerated by a shard of falling glass
$1,000,000 settlement for a 35-year-old truck loader who sustained neck and shoulder injuries in crash
Outside of her work for the firm, Ms. Mercado feels compelled to serve and help those in need. Ms. Mercado regularly contributes her time, legal skills and counseling to homeless and low-income New Yorkers as a volunteer for Open Hands Legal Services (www.openhandslegalservices.org). Ms. Mercado also volunteers annually as a Running Buddy for Girls on the Run (www.gotrnyc.org).
In 2017 and 2018, Ms. Mercado was selected as a 'Rising Star' by Super Lawyers for Personal Injury - Plaintiff. Only 2.5% of attorneys are designated 'Rising Stars' in their practice area in each state.
*Ms. Mercado contributed to the results as a non-handling attorney/junior associate on the case.
https://www.blockotoole.com
www.blockotoole.com


ROUTLEDGE, Mrs. Scoresby [Katherine]. The Mystery of Easter Island. The Story of an Expedition.
ROUTLEDGE, Mrs. Scoresby [Katherine]. The Mystery of Easter Island. The Story of an Expedition. London, privately printed for Sifton, Praed & Co., February, 1919.
8vo. Original green cloth, decorated and lettered in gilt, top edge gilt; pp. xxi, 404, , frontispiece qand portrait in photogravure, with tissue guards, plates, maps and planse after drawings and photographs inluding one folding panorama; hinges a little rubbed, internally here and there a few light spots, a very good copy with ownership inscription, dated 1919 top front fly-leaf.
Scarce first edition of anm important standard work on the history, art and archaeology of the Easter Islands. The Routledge couple sailed before the outbreak of WWI to the Easter Island and was forced to remain there and investigated culture and artifacts on the archipelago, the outcome of which is this work, the fist and most important on the subject.


A selection of NZ souvenir cycling accessories can be purchased online. If you do not find what you are looking for please do not hesitate to contact us.
Great looking New Zealand replica/souvenir shorts. Made in 240gsm lycra and featuring Luxury Blue chamois and silicone leg gripper.
Traditional cycling cap in a smart New Zealand silver fern design.
QuickDry fabric to keep you cool.
New Zealand cycling gloves. Based on the popular Tineli Aero Glove this is a souvenir glove with all the function of a quality road glove.
New Zealand socks with cycling specific shape and cooling mesh section on the top of the foot.


with your personal spin on romance and fun.
Over a hundred year old chapel and romantic garden is available for wedding ceremonies.
Weddings up to 70 guests can be catered for, with exceptional food and services.
Have more freedom to customise your wedding to your own tastes.
A small wedding gives you the opportunity to get your creative juices flowing and make your unique wedding a reflection of the two of you.
Create a one-of-a-kind wedding in a historic chapel that captures elegance and classic romance. The stunning views over the hinterland hills and valleys from the chapel window make your moment unforgettable.
Let mother nature bring the drama for your special day. You and your guests will be swept away by the natural beauty of 26 acres of bushland that has been untouched to support native species of plants and animals.
Get away from the town and celebrate your wedding day in a relaxed vibe – the quick change in scenery will feel like another world. Take advantage of the on-site accommodation available for you and your guests.
sharing with you on Instagram.
Hold your ceremony in the greenery. Indoor Chapel Weddings and Outdoor Garden Weddings available.
Let your hair down and celebrate the day with stunning views over the Hinterland hills and valleys.
Shine on your precious day with a little something extra. Display your love stories on the wall like an art gallery.
Get a FREE 'Save the Date' Card Design Template when you book a wedding with us. High-quality and full colour design templates are ready to go. Choose the one suitable to your wedding theme among 3 different designs, and then we will finalise the template with your personal information such as Bride/Groom's name and date.
*We provide you with a high quality print ready PDF file to supply to your preferred printer for this event.
We welcome the opportunity to meet with you to show you around our wedding venue. We open from Wednesday to Sunday by appointment, so please fill out the contact form to organise an appointment.
Do not hesitate to get in touch at any time to discuss your wedding plans. Simply fill out the enquiry form with your questions. One of our friendly staff will be in contact with you as soon as possible.


Compriband. The Original since 1963. Only genuine with "C"!
The production of compressed foams started in Traun/Austria 1963. At this time the company Compriband was a pioneer and introduced this new development of sealants to Austria. Compriband was immediately successful, hence, an international conglomerate was established with affiliated companies and production sites on all continents.
Today the original product Compriband together with many innovative products is used not only by the civil-/structural-engineering, but also by all industries. The product range comprises a huge variety of sealants.
all kinds and materials of sealants are supplied, whereas we are very flexible in responding to the customer's needs on an individual basis.
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Sydney, 18 May 2016- A drop in sublease availability in Melbourne has levelled out slight increases in Sydney and Brisbane, according to the latest Sublease Barometer report from CBRE.
In Q1, the total volume of sublease space in Australia's sublease markets was down by 4,200sqm to 320,600sqm. Melbourne recorded the largest drop, decreasing by 21% over the quarter to reach its lowest level in over two years. Perth also decreased, while Sydney and Brisbane both recorded increases.
CBRE Regional Director, Office Services, Andrew Tracey said that Melbourne was experiencing rising rents and lowering vacancy level, as well as a general increase in business confidence levels.
"Melbourne CBD has been seeing a solid take up, particularly in better quality space with fitout in situ. Good quality sublease space is moving well. This is resulting in positive net absorption off the back of growth in white collar employment. The take up of better opportunities is leading to solid rental growth in the CBD proper with the market moving quickly back towards equilibrium rather than sitting firmly in the favour of the tenant as is has for the past seven or eight years," Mr Tracey said.
The report states that nationally, the resource sector is the major source of sublease availability, concentrated mostly in Perth and Brisbane.
Sydney – After a large decline in Q4 2015, sublease space in the Sydney CBD rose by 3,200sqm in Q1 2016 to total 52,400sqm. Availability is likely to ease again over the remainder of the year, however, as tenants relocate from secondary stock being withdrawn for conversion to alternate uses. Space offered by the Financial & Insurance Services sector continues to trend down with just 3,800sqm now available, down from 18,600sqm one year ago.
Melbourne – Sublease availability totalled 61,300sqm in March 2016, declining 21% (or by 15,900sqm) over the quarter to its lowest level in over two years. The Finance and Insurance sector remained the largest source of availability, despite recording a decline of 30% (or 5,980sqm) over the quarter. Despite contraction remaining the primary driver of availability, with 68% of space offered for this reason, the quarter recorded a substantial drop in the actual amount of space offered as a result of contraction, down 21,750sqm from Q4 2015. This represents the lowest contraction driven availability seen in over two years.
Brisbane – As expected, the impact of the completion of 480 Queen Street and 180 Brisbane has driven a renewed spike in sublease availability to 72,500sqm, a level last seen in mid-2014. Contraction remains the dominant reason for subleasing space, accounting for 65% of availability (or 46,800sqm).
Perth – Sublease availability dropped by just under 9,500sqm to 88,800sqm during Q1. The decline, however, was driven predominantly by space converting to direct vacancy as leases end their initial terms. The Mining sector remains the major source of availability, although the amount of sublease space offered by the sector did decline by just under 10,000sqm over the quarter to 40,900sqm (46% of total availability).
Adelaide – Sublease space totalled 29,180sqm at the end of March 2016, an increase of 6,970sqm (or 31%) over the quarter. The Property and Business Services sector continues to provide the most sublease space, totalling 9,620sqm, unchanged over the quarter but up by 3,000sqm since the start of 2015. Other industries driving availability were Mining, Manufacturing & Construction (7,260sqm) and Finance & Insurance (4,100sqm).
Canberra – Sublease availability measured 16,400sqm in the March quarter, a slight improvement from the 16,900sqm recorded in December and continuing the downward trend evident over the past year. Sublease vacancy in the City Centre is becoming increasingly scarce with just 6,400sqm available down from 48,900sqm one- year ago.


text:
\subsection{Unimodularity}\label{subsec: unimodularity}
In this subsection, we provide a short exposition of unimodular groups;~see~\cite{GR2015[Folland]} for a textbook exposition.

Let $ \ab{G,\lambda} $ be a second-countable locally-compact Hausdorff group endowed with a left Haar measure.
Let $ \Bs $ be the collection of all Borel subsets of $ G \:$
Define, for $ g\in G\, \Sc\in\Bs \;$ $\lambda^g(\Sc)\defeq\lambda(\Sc g)\: $
Note, for $ g\in G \;$ $ \lambda^g $ is a left Haar measure and there exists $ c(g)=c>0 $ such that $ \lambda^g=c\lambda \:$
Define
\[\ts D:G\to (0,+\infty):g\mapsto c(g) \: \]
Here $ D $ is called the \textit{modular function} of $ G \;$ and it can be shown that $ D $ is a continuous \mbox{homomorphism}.
Say $ \ab{G,\lambda} $ is \textit{unimodular} if $ D\equiv 1 \;$ or, equivalently, $ \lambda^g=\lambda $ for every $ g\in G \:$
Define, for $ \Sc\in\Bs \;$ $ \lambda^*\p{\Sc}\defeq \lambda(\inv{\Sc})\: $
It can be shown that if $ \ab{G,\lambda} $ is unimodular, then $ \lambda^*=\lambda \:$
Note, if $ G $ is discrete or abelian, then $ \ab{G,\lambda} $ is unimodular.
Observe if $ G/[G,G] $ is compact, then $ \ab{G,\lambda} $ is unimodular.
Indeed, the image of $ [G,G]\defeq \s{[x,y]:x,y\in G} $ under $ D $ is trivial since $ D $ is a homomorphism and $ D(G)=D(G/[G,G]) $ is compact since $ D $ is continuous and the only compact subgroup of $ (0,+\infty ) $ is $ \s{1} \:$
As a result, if $ G $ is compact, then $ \ab{G,\lambda} $ is unimodular.
Other examples can be found within \cite{GR2015[Folland]} such as connected nilpotent Lie groups, e.g., the Heisenberg group.

Henceforth, if $ \ab{G,\lambda} $ is unimodular, then we write $ dh=d\lambda(h) $ for every $ h\in G \:$
\subsection{Amenability}\label{subsec: amenability}
In this subsection, we provide a short exposition of amenable groups; see \cite{AG} for a textbook exposition.

Let $ \ab{G,\lambda} $ be a second-countable locally-compact Hausdorff group endowed with a left Haar measure.
Say $ \ab{G,\lambda} $ is \textit{amenable} if for each $ \eta>0 $ and for each compact subset $\Fc$ of $ G \;$ there exists a compactly-supported continuous nonzero $ \chi \in L^2(G) $ such that for each $ h\in \Fc\;$
\[\ts \vv{\cramped{\chi^h-\chi}}\le \eta\vv{\chi}\: \]
Here $ \cramped{\chi^h}:G\to \C:g\mapsto\chi(gh) \:$
Note, if $ G $ is compact, then $ \ab{G,\lambda} $ is amenable.
It can be shown that if $ G $ is solvable, then $ \ab{G,\lambda} $ is amenable.
As a result, if $ G $ is abelian, then $ \ab{G,\lambda} $ is amenable.
The free group $ \F_2 $ with two generators is an example of a group that is not amenable.

Because $G$ is second-countable, locally-compact, and Hausdorff, there exists a proper continuous metric $d$ on $G$ that is left invariant, i.e., for all $j,g,h\in G\;$ $ d(jg,jh) = d(g,h)\: $
Say $ \ab{G,\lambda} $ has \textit{strict polynomial growth} if there exist $ b\ge 1\, c_1>c_0>0 $ such that for each $ r\ge 1 \;$
\[\ts c_0r^b\le\lambda(B(r))\le c_1r^b \:\]
Here $ B(r) $ is the open ball centered at the identity element of $ G $ with radius $ r \:$
We remark that this notion of growth can be found within \cite{DS2006[Nevo]}, and it can be shown that it implies $ \ab{G,\lambda} $ is amenable.

Henceforth, if $ G $ has a proper continuous left invariant metric, then we write $ \v{\inv{g}h}=d(g,h) $ for every $ g,h\in G \:$
\subsection{Hausdorff Distance}\label{subsec: hausdorff distance}
Let $ Z $ be a metric space with metric $ d \:$
Let $ S_{\ttm 1} $ and $ S_{\ttm 2} $ be subsets of $ Z \:$
The \textit{Hausdorff distance} between $ S_{\ttm 1} $ and $ S_{\ttm 2} $ induced by $ d $ is
\[\ts \max\s{\sup_{s_{\ttm 1}\in S_{\ttm 1}}d(s_{\ttm 1},S_{\ttm 2}),\sup_{\smash{s_{\ttm 2}}\in \smash{S_{\ttm 2}}}d(s_{\ttm 2},S_{\ttm 1})}\eqdef d_{\mathsf{H}}(S_{\ttm 1},S_{\ttm 2}) \:\]
If $ (Z,G) $ is a topological dynamical system and $ Z $ is a compact metric space, then it can be shown that the collection $ \Ss $ of all invariant nonempty closed subsets of $ Z $ endowed with the Hausdorff metric $ d_{\ssm H} $ is a compact metric space; this result can be found within \cite{MP2018SpecrtalContinuityI}.

\subsection{Dynamically-Defined Operator Families}\label{subsec: DDOF}
In this subsection, basic facts are recovered.
Specifically, we prove a covariance relation, a uniform boundedness, and a strong continuity property for dynamically-defined operator families and their adjoints.

Let $ \ab{G,\lambda} $ be a second-countable locally-compact Hausdorff group endowed with a left Haar measure, where $ \ab{G,\lambda} $ is unimodular.
Let $ \ab{Z,G} $ be a topological space endowed with a left \mbox{continuous} action.
Let $ k $ be a bounded continuous (on $ Z $) complex-valued function on $ Z\times G \;$ where $ \sup_{x\in Z}\int \sup_{g\in G}\v{k(\inv{g}x,h)} dh<+\infty \:$
Define $ A_x:L^2(G)\to L^2(G) \;$ for $ x\in Z \;$ by
\[\ts
(A_x\psi)(g)=\int k(\inv{g}x,\inv{g}h)\psi(h) dh
\]
for every $ \psi\in L^2(G)\, g\in G \:$
\begin{lem}\label{lem: (1)}
Define $ \smash{U_j}:L^2(G)\to L^2(G) \;$ for $ j\in G \;$ by
\[\ts (\smash{U_j}\psi)(g)=\psi(\inv{j}g) \]
for every $ \psi\in L^2(G)\, g\in G \:$
Fix $ x\in Z \:$
Then $ A_x $ is linear and for each $ j\in G \;$
\[\ts \smash{U_{j}}A_x\smash{U_{\inv{j}}}=A_{jx} \]
and for each $ \vp\in L^2(G) \;$
\[\ts \vv{A_x\vp}\le \p{\int \sup_{g\in \supp\p{\vp}\inv{h}}\v{k(\inv{g}x,h)} dh}\vv{\vp} \]
and for each $ y\in Z \;$
\[\ts \vv{(A_y-A_x)\vp}\le \p{\int \sup_{g\in \supp\p{\vp}\inv{h}}\v{k(\inv{g}y,h)-k(\inv{g}x,h)} dh}\vv{\vp}\: \]
\end{lem}
\begin{proof}
\allowdisplaybreaks%
Observe $ A_x $ is linear.
Also, for $ j\in G\, \psi\in L^2(G)\, g\in G \;$
\begin{align*}
&\ts({{U}_{j}A_x{U}_{\inv{j}}}\psi)(g)\\
&\ts=\int k(\inv{(\inv{j}g)}x,\inv{(\inv{j}g)}h)\psi(jh) dh\\
&\ts=\int k(\inv{g}jx,\inv{g}h)\psi(h) dh\\
&\ts=(A_{jx}\psi)(g)\:
\end{align*}
Also, for $ \vp\in L^2(G) \;$
\begin{align*}
&\ts\vv{A_x\vp}^2\\
&\ts=\int\v{\int k(\inv{g}x,h)\vp(gh) dh}^2dg\\
&\ts\le\int\p{\int \v{k(\inv{g}x,h)}\v{\vp(gh)} dh}^2dg\\
&\ts\le\int\p{\int \v{k(\inv{g}x,h)}\1_{\supp\p{\vp}}(gh) dh}\p{\int \v{k(\inv{g}x,h)}\v{\vp(gh)}^2 dh}dg
\tag*{\scalebox{1}{\(\mathllap{\smash{\p{\lsubstack{\textrm{Hölder's}\\\textrm{inequality}}}}}\)}}\\
&\ts\le\int \p{\int \sup_{\gch\in \supp\p{\vp}\inv{h}}\v{k(\inv{\gch}x,h)} dh}\p{\int \sup_{\check{g}\in \supp\p{\vp}\inv{h}}\v{k(\inv{\check{g}}x,h)}\v{\vp(gh)}^2 dh}dg\\
&\ts= \p{\int \sup_{g\in \supp\p{\vp}\inv{h}}\v{k(\inv{g}x,h)} dh}^2\vv{\vp}^2\:
\end{align*}
Also, for $ \vp\in L^2(G)\, y\in Z \;$
\begin{align*}
&\ts\vv{(A_{y}-A_{x})\vp}^2\\
&\ts=\int\v{\int (k(\inv{g}y,h)-k(\inv{g}x,h))\vp(gh) dh}^2dg\\
&\ts\le\int\p{\int \v{k(\inv{g}y,h)-k(\inv{g}x,h)}\v{\vp(gh)} dh}^2dg\\
&\ts\le\int\p{\int \v{k(\inv{g}y,h)-k(\inv{g}x,h)}\1_{\supp\p{\vp}}(gh) dh}\p{\int \v{k(\inv{g}y,h)-k(\inv{g}x,h)}\v{\vp(gh)}^2 dh}dg\\
&\ts\le\int
\!\begin{array}[t]{l}
\p{\int \sup_{\check{g}\in \supp\p{\vp}\cramped{\inv{h}}}\v{k(\inv{\check{g}}y,h)-k(\inv{\check{g}}x,h)} dh}\\
{}\cdot\p{\int \sup_{\check{g}\in\supp\p{\vp}\cramped{\inv{h}}}\v{k(\inv{\check{g}}y,h)-k(\inv{\check{g}}x,h)}\v{\vp(gh)}^2 dh}dg
\end{array}\\
&\ts=\p{\int \sup_{g\in \supp\p{\vp}\inv{h}}\v{k(\inv{g}y,h)-k(\inv{g}x,h)} dh}^2\vv{\vp}^2\:\qedhere
\end{align*}
\end{proof}
\begin{lem}
Define $ a_x:G\times G\to \C \;$ for $ x\in Z \;$ by
\[\ts a_x(g,h)=k(\inv{g}x,\inv{g}h) \]
for every $ g,h\in G \:$
Assume $ k $ is continuous on $ Z\times G \:$
Fix $ x\in Z \:$
Then $ a_x $ is continuous and
\begin{align*}
\ts A_x=A_x\mspace{-3mu}{}^* &\ts\iff \p{\f g,h\in G}\b{a_x(g,h)=\overline{a_x(h,g)}\mspace{1.5mu}}\;\\
\ts A_xA_x\mspace{-3mu}{}^*=A_x\mspace{-3mu}{}^*A_x &\ts\iff \p{\f g,h\in G}\b{\int a_x(g,j)\overline{a_x(h,j)} dj=\int \overline{a_x(j,g)}a_x(j,h) dj}\:
\end{align*}
\end{lem}
\begin{proof}
\allowdisplaybreaks%
Observe $ a_x $ is continuous.
Also, for $ \vp,\psi\in L^2(G) \;$
\begin{align*}
\ab{\vp,A_x\psi}&\ts=\iint\overline{\vp(g)}a_x(g,h)\psi(h)dhdg\;\\
\ab{A_x\vp,\psi}&\ts=\iint\overline{a_x(h,g)}\mspace{1.5mu}\overline{\vp(g)}\psi(h)dgdh\:
\end{align*}
Also, for $ \psi\in L^2(G)\,  g\in G \;$
\begin{align*}\SwapAboveDisplaySkip
(A_xA_x\mspace{-3mu}{}^{*}\psi)(g)&\ts=\iint a_x(g,j)\overline{a_x(h,j)}\psi(h)dhdj\;\\
(A_x\mspace{-3mu}{}^{*}A_x\psi)(g)&\ts=\iint \overline{a_x(j,g)}a_x(j,h)\psi(h)dhdj\:\qedhere
\end{align*}
\end{proof}
\begin{lem}\label{lem: (3)}
Fix $ x\in Z \:$
Then for each $ \vp\in L^2(G) \;$
\[\ts \vv{A_x\mspace{-3mu}{}^{*}\vp}\le \p{\int \sup_{g\in \supp\p{\vp}}\v{k(\inv{g}x,h)} dh}\vv{\vp} \]
and for each $ y\in Z \;$
\[\ts \vv{(A_y-A_x)^*\vp}\le \p{\int \sup_{g\in \supp\p{\vp}}\v{k(\inv{g}y,h)-k(\inv{g}x,h)} dh}\vv{\vp}\: \]
\end{lem}
\begin{proof}
\allowdisplaybreaks%
Observe, for $ \vp\in L^2(G) \;$
\begin{align*}
&\ts\vv{A_x\mspace{-3mu}{}^{*}\vp}^2\\
&\ts=\int\v{\int \overline{k(\inv{h}\inv{g}x,\inv{h})}\vp(gh) dh}^2dg\\
&\ts\le\int\p{\int \v{\overline{k(\inv{h}\inv{g}x,\inv{h})}}\v{\vp(gh)} dh}^2dg\\
&\ts\le\int\p{\int \v{k(\inv{(gh)}x,\inv{h})}\1_{\supp\p{\vp}}(gh) dh}\p{\int \v{k(\inv{(gh)}x,\inv{h})}\v{\vp(gh)}^2 dh}dg\\
&\ts\le\int \p{\int \sup_{\gch\in \supp\p{\vp}\inv{h}}\v{k(\inv{(\gch h)}x,\inv{h})} dh}\p{\int \sup_{\check{g}\in \supp\p{\vp}\inv{h}}\v{k(\inv{(\check{g}h)}x,\inv{h})}\v{\vp(gh)}^2 dh}dg\\
&\ts= \p{\int \sup_{g\in \supp\p{\vp}}\v{k(\inv{g}x,\inv{h})} dh}^2\vv{\vp}^2\:
\end{align*}
Also, for $ \vp\in L^2(G)\, y\in Z \;$
\begin{align*}
&\ts\vv{(A_{y}-A_{x})^*\vp}^2\\
&\ts=\int\v{\int (\overline{k(\inv{h}\inv{g}y,\inv{h})}-\overline{k(\inv{h}\inv{g}x,\inv{h})})\vp(gh) dh}^2dg\\
&\ts\le\int\p{\int \v{\overline{k(\inv{h}\inv{g}y,\inv{h})}-\overline{k(\inv{h}\inv{g}x,\inv{h})}}\v{\vp(gh)} dh}^2dg\\
&\ts\le\int
\!\begin{array}[t]{l}
\p{\int \v{{k((gh)^{-1}y,\inv{h})}-{k((gh)^{-1}x,\inv{h})}}\1_{\supp\p{\vp}}(gh) dh}\\
{}\cdot\p{\int \v{{k((gh)^{-1}y,\inv{h})}-{k((gh)^{-1}x,\inv{h})}}\v{\vp(gh)}^2 dh}dg
\end{array}\\
&\ts\le\int
\!\begin{array}[t]{l}
\p{\int \sup_{\check{g}\in \supp\p{\vp}\cramped{\inv{h}}}\v{{k((\check{g}h)^{-1}y,\inv{h})}-{k((\check{g}h)^{-1}x,\inv{h})}} dh}\\
{}\cdot\p{\int \sup_{\check{g}\in\supp\p{\vp}\cramped{\inv{h}}}\v{{k((\check{g}h)^{-1}y,\inv{h})}-{k((\check{g}h)^{-1}x,\inv{h})}}\v{\vp(gh)}^2 dh}dg
\end{array}\\
&\ts=\p{\int \sup_{g\in \supp\p{\vp}}\v{{k({g}^{-1}y,\inv{h})}-{k({g}^{-1}x,\inv{h})}} dh}^2\vv{\vp}^2\:\qedhere
\end{align*}
\end{proof}
\subsection{Continuity of Spectrum Maps}\label{subsec: CSM}
In this subsection, we establish sufficient conditions, e.g., amenability, implying semi-continuity of the spectrum $ \sigma(A_x) $ and implying continuity of the spectrum map $ X\mapsto \overline{\bigcup_{x\in X}\sigma(A_x)} $ with respect to a dynamically-defined operator family.

Let $ \ab{G,\lambda} $ be a second-countable locally-compact Hausdorff group endowed with a left Haar measure, where $ \ab{G,\lambda} $ is unimodular.
Let $ e $ be the identity element of $ G \:$
Let $ \ab{Z,G} $ be a topological space endowed with a left continuous action.
Let $ k $ be a bounded continuous (on $ Z $) complex-valued function on $ Z\times G \;$ where $ \sup_{x\in Z}\vv{k_x}_{\infty,1}<+\infty \:$
Here
\[\ts \vv{k_x}_{\infty,1}\defeq\int\sup_{g\in G}\v{k(\inv{g}x,h)} dh \:\]
Define $ A_x:L^2(G)\to L^2(G) \;$ for $ x\in Z \;$ by
\[\ts
(A_x\psi)(g)=\int k(\inv{g}x,\inv{g}h)\psi(h) dh
\]
for every $ \psi\in L^2(G)\, g\in G \:$
Let $ \Rc_x \;$ for $ x\in Z \;$ be the support of the map $ h\mapsto \sup_{g\in G}\v{k(\inv{g}x,h)} \:$
\begin{thm}\label{thm: (4)}
Fix $ x\in Z $ with $ A_xA_x\mspace{-3mu}{}^*=A_x\mspace{-3mu}{}^*A_x \:$
Fix $ E\in\sigma(A_x) \:$
Fix $ \ep>0 \:$
Let $ \vp:G\to\C $ be a nonzero $ L^2 $-vector satisfying $ \vv{(A_{x}-E)\vp}\le \ep\vv{\vp} \:$
Let $ \chi:G\to\C $ be a compactly-supported continuous nonzero $ L^2 $-vector.
Define, for $ j\in G \;$
\begin{align*}
\ts\chi_{j}&\ts:G\to\C:g\mapsto\chi(\inv{j}g)\;\\
\ts\cramped{\chi^{j}}&\ts:G\to\C:g\mapsto\chi(gj)\:
\end{align*}
Then there exists $ j\in G $ such that $ \vv{\vp_j}\ne 0 $ and for each $ y\in Z \;$
\[\ts
\vv{(A_y-E)\vp_j}\le
\Bigp{
\vv{(A_y-A_x)\smash{\frac{\vp_j}{\vv{\vp_j}}}}
+\vv{k_x}_{\infty,1}\sup_{h\in\Rc_x\cup\s{e}}\smash{\frac{\vv{\cramped{\chi^h-\chi}}}{\vv{\chi}}}
+2\ep
}\vv{\vp_j}\;
\]
where $ \vp_j:G\to \C:g\mapsto \chi_j(g)\vp(g) \:$
\end{thm}
\begin{proof}
\allowdisplaybreaks%
Define
\[\ts
\de\defeq
\begin{cases}
\ep\p{\vv{k_x}_{\infty,1}\sup_{h\in\Rc_x\cup\s{e}}\smash{\frac{\vv{\cramped{\chi^h-\chi}}}{\vv{\chi}}}}^{-1}&\iph\p{\vv{k_x}_{\infty,1}\sup_{h\in\Rc_x\cup\s{e}}\smash{\frac{\vv{\cramped{\chi^h-\chi}}}{\vv{\chi}}}}\ne 0\;\\
3^{-1}&\elss\:
\end{cases}
\]
Let $ \Sc $ be the support of the map $ j\mapsto \vv{\vp_{j}}^2 \:$
Observe, for $ j\in G \;$
\begin{align*}
&\ts\v{\vv{\vp_j}^2-\vv{\vp_e}^2}\\
&\ts\le \int\v{\v{\vp_j(g)}^2-\v{\vp_e\p{g}}^2}dg\\
&\ts\le 2\vv{\chi}_{\infty}\vv{\chi_j-\chi}_{\infty}\vv{\vp}^2\:
\end{align*}
Because $ \chi\in C_{\ssm c}(G) \;$ the map $ j\mapsto \vv{\vp_{j}}^2 $ is continuous and $ 0<\lambda(\Sc) \:$
Define $ \bigb{A,B}:L^2(G)\to L^2(G) \;$ for $ A,B\in \Ls\p{L^2(G)} \;$ by
\[\ts
(\bigb{A,B}\psi)(g)=(AB\psi)(g)-(BA\psi)(g)
\]
for every $ \psi\in L^2(G)\, g\in G \:$
Define $ \bigb{\chi_j(\cdot)}:L^2(G)\to L^2(G) \;$ for $ j\in G \;$ by
\[\ts
(\bigb{\chi_j(\cdot)}\psi)(g)=\chi_j(g)\psi(g)
\]
for every $ \psi\in L^2(G)\, g\in G \:$
Define, for $ j\in G \;$
\begin{align*}
\vp_{\ep}&\defeq (A_{x}-E)\vp\;\\
\vp_{j,\ep}&\defeq (A_{x}-E)\bigb{\chi_{j}(\cdot)}\vp\;\\
\vp_{\ep,j}&\defeq \bigb{\chi_{j}(\cdot)}(A_{x}-E)\vp\:
\end{align*}
Observe
\begin{align*}\SwapAboveDisplaySkip
&\ts\int \vv{\vp_{\ep,j}}^2dj\\
&\ts=\iint\v{\chi_j(g)}^2\v{\vp_{\ep}(g)}^2dgdj\\
&\ts=\iint\v{\chi_j(g)}^2\v{\vp_{\ep}(g)}^2djdg\\
&\ts=\vv{\vp_{\ep}}^2\vv{\chi}^2\\
&\ts\le\ep^2\vv{\vp}^2\vv{\chi}^2\\
&\ts=\iint\ep^2\v{\chi_j(g)}^2\v{\vp(g)}^2djdg\\
&\ts=\iint\ep^2\v{\chi_j(g)}^2\v{\vp(g)}^2dgdj\\
&\ts=\int \ep^2\vv{\vp_{j}}^2dj
\end{align*}
and
\begin{align*}\SwapAboveDisplaySkip
&\ts\int\vv{\bigb{A_x-E,\bigb{\chi_j\p{\cdot}}}\vp}^2dj\\
&\ts=\iint\v{\int k(\inv{g}x,h)(\chi_j(gh)-\chi_j(g))\vp(gh)dh}^2dgdj\\
&\ts\le\iint\p{\int \v{k(\inv{g}x,h)}\v{\chi_j(gh)-\chi_j(g)}\v{\vp(gh)}dh}^2dgdj\\
&\ts\le\iint\p{\int \v{k(\inv{g}x,h)}dh}\p{\int \v{k(\inv{g}x,h)}\v{\chi_j(gh)-\chi_j(g)}^2\v{\vp(gh)}^2dh}dgdj\\
&\ts\le\iint
\!\begin{array}[t]{l}
\p{\int \sup_{\gch\in G}\v{k(\inv{\gch}x,h)}dh}\\
{}\cdot\p{\int \sup_{\gch\in G}\v{k(\inv{\gch}x,h)}\v{\chi_j(gh)-\chi_j(g)}^2\v{\vp(gh)}^2dh}djdg
\end{array}\\
&\ts=\p{\int \sup_{g \in G}\v{k(\inv{g}x,h)}dh}\p{\int \sup_{g \in G}\v{k(\inv{g}x,h)}\vv{\cramped{\chi^h}-\chi}^2\vv{\vp}^2dh}\\
&\ts=\p{\int \sup_{g \in G}\v{k(\inv{g}x,h)}dh}\p{\int \sup_{g \in G}\v{k(\inv{g}x,h)}\vv{\cramped{\chi^h}-\chi}^2\frac{1}{\vv{\chi}^2}dh}\vv{\chi}^2\vv{\vp}^2
\\
&\ts=\p{\int \sup_{g\in G}\v{k(\inv{g}x,h)}dh}\p{\int \sup_{g\in G}\v{k(\inv{g}x,h)}\smash{\frac{\vv{\cramped{\chi^h-\chi}}^2}{\vv{\chi}^2}}dh}\int\vv{\vp_j}^2dj\\
&\ts\le\int(\vv{k_x}_{\infty,1}\sup_{h\in \Rc_x\cup\s{e}}\smash{\frac{\vv{\cramped{\chi^h-\chi}}}{\vv{\chi}}})^2\vv{\vp_j}^2dj\:
\end{align*}
Therefore
\begin{align*}\SwapAboveDisplaySkip
&\ts\int\vv{\vp_{j,\ep}}^2dj\\
&\ts=\int\vv{\bigb{A_{x}-E,\bigb{\chi_j\p{\cdot}}}\vp+\vp_{\ep,j}}^2dj\\
&\le\ts \int (1+\de)\vv{\bigb{A_{x}-E,\bigb{\chi_j\p{\cdot}}}\vp}^2+(1+\inv{\de})\vv{\vp_{\ep,j}}^2dj
\tag*{\scalebox{1}{\(\mathllap{\smash{\p{\lsubstack{\textrm{Peter--Paul's}\\\textrm{inequality}}}}}\)}}\\
&\le\ts \int (1+\de)(\vv{k_x}_{\infty,1}\sup_{h\in \Rc_x\cup\s{e}}\smash{\frac{\vv{\cramped{\chi^h-\chi}}}{\vv{\chi}}})^2\vv{\vp_{j}}^2+(1+\inv{\de})\ep^2\vv{\vp_{j}}^2dj\:
\end{align*}
Because $ 0<\lambda(\Sc) \;$ there exists $ j\in \Sc $ such that $ \vv{\vp_j}\ne 0 $ and
\[\ts \vv{\vp_{j,\ep}}^2\le\bigp{(1+\de)(\vv{k_x}_{\infty,1}\sup_{h\in \Rc_x\cup\s{e}}\smash{\frac{\vv{\cramped{\chi^h-\chi}}}{\vv{\chi}}})^2+(1+\inv{\de})\ep^2}\vv{\vp_{j}}^2\: \]
Therefore
\[\ts
\begin{cases}
\vv{\vp_{j,\ep}}\le \bigp{(\vv{k_x}_{\infty,1}\sup_{h\in \Rc_x\cup\s{e}}\smash{\frac{\vv{\cramped{\chi^h-\chi}}}{\vv{\chi}}})+\ep}\vv{\vp_j}&\iph\p{\vv{k_x}_{\infty,1}\sup_{h\in\Rc_x\cup\s{e}}\smash{\frac{\vv{\cramped{\chi^h-\chi}}}{\vv{\chi}}}}\ne 0\;\\
\vv{\vp_{j,\ep}}\le 2\ep\vv{\vp_j}&\elss\:
\end{cases}
\]
Also, for $ y\in Z \;$
\begin{align*}
&\ts\vv{(A_{y}-E)\vp_{j}}\\
&\ts=\vv{(A_{y}-A_{x}+A_{x}-E)\vp_{j}}\\
&\ts\le\vv{(A_{y}-A_{x})\vp_{j}}+\vv{\vp_{j,\ep}}\\
&\ts\le\vv{(A_{y}-A_{x})\vp_{j}}+\bigp{(\vv{k_x}_{\infty,1}\sup_{h\in \Rc_x\cup\s{e}}\smash{\frac{\vv{\cramped{\chi^h-\chi}}}{\vv{\chi}}})+2\ep}\vv{\vp_{j}}\:\qedhere
\end{align*}
\end{proof}
\begin{cor}\label{cor: 4.1}
Fix $ x\in Z \:$
Assume $ \ab{G,\lambda} $ is amenable and $ \Rc_x $ is compact and $ A_yA_y\mspace{-3mu}{}^*=A_y\mspace{-3mu}{}^*A_y $ for every $ y $ in some neighborhood of $ x \:$
Then
\[\ts \lim_{y\xrightarrow[]{\smash{Z}}x}\sup_{E\in\sigma(A_x)}\dist(E,\sigma(A_y))=0\: \]
\end{cor}
\begin{proof}
\allowdisplaybreaks%
Fix $ \eta>0 \:$
Because $ \ab{G,\lambda} $ is amenable and $ \Rc_x $ is compact, there exists a compactly-supported continuous nonzero $ L^2 $-vector $ \chi:G\to\C $ satisfying
\[\ts \sup_{h\in\Rc_x\cup\s{e}}\vv{\cramped{\chi^h-\chi}}\le \eta\vv{\chi}\: \]
Here $ \cramped{\chi^h}:G\to \C:g\mapsto\chi(gh) \:$
Define, for $ j\in G \;$ $ \chi_j:G\to \C:g\mapsto\chi(\inv{j}g) \:$
Fix $ \ep>0 \:$
Because $ A_x $ is a bounded operator, there exist elements $ E_1,\ldots,E_N $ of $ \sigma(A_x) $ satisfying
\[\ts \sigma(A_x)\subseteq \bigcup_{n=1}^N B(E_n,\ep) \:\]
Here $ B(E_n,\ep) $ is the open ball centered at $ E_n $ with radius $ \ep \:$
Fix $ n\in\s{1,\ldots,N} \:$
Because $ A_x $ is a normal operator, there exists a nonzero $ L^2 $-vector $ \vp:G\to\C $ satisfying
\[\ts \vv{(A_{x}-E_n)\vp}\le \ep\vv{\vp} \:\]
Here $ \vp $ is an approximate eigenfunction.
By Theorem~\ref{thm: (4)}, there exists $ j\in G $ such that $ \vv{\vp_j}\ne 0 $ and for each $ y\in Z \;$
\[\ts
\vv{(A_y-E_n)\vp_j}\le
\Bigp{
\vv{(A_y-A_x)\smash{\frac{\vp_j}{\vv{\vp_j}}}}
+\vv{k_x}_{\infty,1}\eta
+2\ep
}\vv{\vp_j}\;
\]
where $ \vp_j:G\to \C:g\mapsto \chi_{j}(g)\vp(g) \:$
Let $ y $ be an element of $ Z $ satisfying
\[\ts A_yA_y\mspace{-3mu}{}^*=A_y\mspace{-3mu}{}^*A_y\: \]
By properties of normal operators,
\begin{align*}
&\ts \dist(E_n,\sigma(A_y))\\
&\ts =\vv{(A_y-E_n)^{-1}}^{-1}\\
&\ts\le {
\vv{(A_y-A_x)\smash{\frac{\vp_j}{\vv{\vp_j}}}}
+\vv{k_x}_{\infty,1}\eta
+2\ep
}\\
&\ts =
\vv{(A_y-A_x)^*\smash{\frac{\vp_j}{\vv{\vp_j}}}}
+\vv{k_x}_{\infty,1}\eta
+2\ep\:
\end{align*}
Observe
\[\ts
\supp\p{\vp_j}\subseteq \supp\p{\chi_j}=j\supp\p{\chi}\:
\]
By Lemma~\ref{lem: (3)},
\begin{align*}\SwapAboveDisplaySkip
&\ts \vv{(A_y-A_x)^*\smash[b]{\frac{\vp_j}{\vv{\vp_j}}}}\\
&\ts \le \int \sup_{g\in \supp\p{\vp_j}}\v{k(\inv{g}y,h)-k(\inv{g}x,h)} dh\\
&\ts \le \int \sup_{g\in \supp\p{\chi}}\v{k(\inv{g}\inv{j}y,h)-k(\inv{g}\inv{j}x,h)} dh\:
\end{align*}
Therefore
\begin{align*}\SwapAboveDisplaySkip
&\ts\sup_{E\in \sigma(A_x)\cap B(E_n,\ep)}\dist(E,\sigma(A_y))\\
&\ts\le \int \sup_{g\in \supp\p{\chi}}\v{k(\inv{g}\inv{j}y,h)-k(\inv{g}\inv{j}x,h)} dh
+\vv{k_x}_{\infty,1}\eta
+3\ep\:
\intertext{Because $ k $ is continuous on $ Z $ and because $ \supp\p{\chi} $ is compact,}
&\ts \lim_{y\xrightarrow[]{\smash{Z}}x}\sup_{E\in\sigma(A_x)}\dist(E,\sigma(A_y))=0\:\qedhere
\end{align*}
\end{proof}
\begin{cor}\label{cor: 4.2}
Let $ d $ be a continuous metric on $ Z \:$
Let $ \Ss $ be the collection of all invariant nonempty closed subsets of $ Z \:$
Let $ \Ks $ be the collection of all nonempty compact subsets of $ \C \:$
Define
\[\ts
\bar{\sigma}:(\Ss,d_{\ssm H})\to (\Ks,\dist_{\ssm H}):X\mapsto\overline{\bigcup_{x\in X}\sigma(A_x)}\:
\]
Assume $ \ab{G,\lambda} $ is amenable and $ \overline{\bigcup_{x\in Z}\Rc_x} $ is compact and $ A_yA_y\mspace{-3mu}{}^*=A_y\mspace{-3mu}{}^*A_y $ for every $ y\in Z \:$
Then
\[\ts
\lim_{\de\to 0^+}\sup_{X,Y\in\Ss,d_{\ssm{H}}(X,Y)\le \de}\dist_{\ssm{H}}(\bar{\sigma}(X),\bar{\sigma}(Y))=0\:
\]
Here $ d_{\ssm H}(X,Y) $ and $ \dist_{\ssm H}(\bar{\sigma}(X),\bar{\sigma}(Y)) $ are Hausdorff distances between subsets of metric spaces.
\end{cor}
\begin{proof}
\allowdisplaybreaks%
Fix $ \eta>0 \:$
Define $ \Rc\defeq \overline{\bigcup_{x\in Z}\Rc_x} \:$
Because $ \ab{G,\lambda} $ is amenable and $ \Rc $ is compact, there exists a compactly-supported continuous nonzero $ L^2 $-vector $ \chi:G\to\C $ satisfying
\[\ts \sup_{h\in\Rc\cup\s{e}}\vv{\cramped{\chi^h-\chi}}\le \eta\vv{\chi}\: \]
Here $ \cramped{\chi^h}:G\to \C:g\mapsto\chi(gh) \:$
Define, for $ j\in G \;$ $ \chi_j:G\to \C:g\mapsto\chi(\inv{j}g) \:$
Fix $ \ep>0 \:$
Fix $ X,Y\in\Ss \:$
Define $ \de_{\ssm H}\defeq d_{\ssm H}(X,Y) \:$
Fix $ E\in\bigcup_{x\in X}\sigma(A_x) \:$
Let $ x $ be an element of $ X $ satisfying
\[\ts E\in\sigma(A_x) \:\]
Because $ A_x $ is a normal operator, there exists a nonzero $ L^2 $-vector $ \vp:G\to \C $ satisfying
\[\ts \vv{(A_x-E)\vp}\le \ep\vv{\vp} \:\]
Here $ \vp $ is an approximate eigenfunction.
By Theorem~\ref{thm: (4)}, there exists $ j\in G $ such that $ \vv{\vp_j}\ne 0 $ and for each $ y\in Z \;$
\[\ts
\vv{(A_y-E)\vp_j}\le
\Bigp{
\vv{(A_y-A_x)\smash{\frac{\vp_j}{\vv{\vp_j}}}}
+\vv{k_x}_{\infty,1}\eta+2\ep
}\vv{\vp_j}\;
\]
where $ {\vp}_{j}:G\to \C:g\mapsto \chi_{j}(g){\vp}(g) \:$
Let $ y $ be an element of $ Y $ satisfying
\[\ts d(\inv{j}{x},\inv{j}{y})= d(\inv{j}{x},Y)\le \de_{\ssm H} \:\]
By properties of normal operators,
\begin{align*}
&\ts \dist(E,\sigma(A_y))\\
&\ts =\vv{(A_y-E)^{-1}}^{-1}\\
&\ts\le {
\vv{(A_y-A_x)\smash{\frac{\vp_j}{\vv{\vp_j}}}}
+\vv{k_x}_{\infty,1}\eta
+2\ep
}\\
&\ts ={
\vv{(A_y-A_x)^*\smash{\frac{\vp_j}{\vv{\vp_j}}}}
+\vv{k_x}_{\infty,1}\eta
+2\ep
}\:
\end{align*}
Observe
\[\ts
\supp\p{\vp_j}\subseteq \supp\p{\chi_j}=j\supp\p{\chi}\:
\]
By Lemma~\ref{lem: (3)},
\begin{align*}\SwapAboveDisplaySkip
&\ts \vv{(A_y-A_x)^*\smash{\frac{\vp_j}{\vv{\vp_j}}}}\\
&\ts \le \int \sup_{g\in \supp\p{\vp_j}}\v{k(\inv{g}y,h)-k(\inv{g}x,h)} dh\\
&\ts \le \int \sup_{g\in \supp\p{\chi}}\v{k(\inv{g}\inv{j}y,h)-k(\inv{g}\inv{j}x,h)} dh\:
\end{align*}
Therefore
\begin{align*}
&\ts\sup_{E\in \bigcup_{x\in X}\sigma(A_x)}\dist(E,\bigcup_{y\in Y}\sigma(A_y))\\
&\ts\le\sup_{\lsubstack{\check{x}\in X,\check{y}\in Y\\\mathrlap{d(\check{x},\check{y})\le \de_{\ssm H}}}}\int \sup_{g\in \supp\p{\chi}}\v{k(\inv{g}\check{y},h)-k(\inv{g}\check{x},h)} dh
+\p{\sup_{\check{z}\in Z}\vv{k_{\check{z}}}_{\infty,1}}\eta+2\ep\:
\intertext{Because $ k $ is continuous on $ Z $ and because $ \supp\p{\chi} $ is compact,}
&\ts\lim_{\de\to 0^+}\sup_{X,Y\in\Ss,d_{\ssm{H}}(X,Y)\le \de}\dist_{\ssm{H}}(\bar{\sigma}(X),\bar{\sigma}(Y))=0\:\qedhere
\end{align*}
\end{proof}
\subsection{Quantitative Continuity of Spectrum Maps}\label{subsec: QCSM}
In this subsection, we establish sufficient conditions, e.g., strict polynomial growth and Lipschitz continuity on $ Z $ of both the action $ G\curvearrowright Z $ and the kernel $ k:Z\times G\to\C \;$ implying $ \frac{1}{2} $-Hölder continuity of the spectrum map $ X\mapsto \overline{\bigcup_{x\in X}\sigma(A_x)} $ with respect to a dynamically-defined operator family.
Furthermore, if the kernel is instead locally-constant on $ Z $, then the spectrum map is instead Lipschitz continuous.

Let $ \ab{G,\lambda} $ be a second-countable locally-compact Hausdorff group endowed with a left Haar measure, where $ \ab{G,\lambda} $ is unimodular.
Let $ e $ be the identity element of $ G \:$
Let $ d $ be a proper continuous left invariant metric on $ G \;$ where $ \ab{G,\lambda} $ has strict polynomial growth, i.e., there exist $ b\ge 1\, c_1>c_0>0 $ such that for each $ r\ge 1 \;$
\[\ts c_0r^b\le\lambda(B(r))\le c_1r^b \:\]
Here $ B(r) $ is the open ball centered at $ e $ with radius $ r \:$
Define, for $ j\in G \;$ $ \v{j}\defeq d(e,j) $ so that $ d(g,h) \;$ for $ g,h\in G \;$ is instead now $ \v{\inv{g}h} \:$
Let $ \ab{Z,G} $ be a topological space endowed with a left continuous action.
Let $ k $ be a bounded continuous (on $ Z $) complex-valued function on $ Z\times G \;$ where $ \sup_{x\in Z}\vv{k_x}_{S}<+\infty \:$
Here
\[\ts \vv{k_x}_{S}\defeq\int\sup_{g\in G}\v{k(\inv{g}x,h)}\v{h} dh \:\]
Define $ A_x:L^2(G)\to L^2(G) \;$ for $ x\in Z \;$ by
\[\ts
(A_x\psi)(g)=\int k(\inv{g}x,\inv{g}h)\psi(h) dh
\]
for every $ \psi\in L^2(G)\, g\in G \:$
\begin{lem}\label{lem: (5)}
Define $ t\defeq\p{{\frac{(2+b)^{2+b}c_1}{2b^bc_0}}}^{1/2} \:$
Fix $ r\ge 1 \:$
Define $ \chi:G\to \C:g\mapsto \1_{B(r)}(g)\p{\smash{\frac{r-\v{g}}{r}}} \:$
Define, for $ h\in G \;$ $ \cramped{\chi^h}:G\to\C:g\mapsto\chi(gh) \:$
Then $ \chi $ is a compactly-supported continuous nonzero $ L^2 $-vector and for each $ h\in G \;$
\[\ts
\vv{\cramped{\chi^h-\chi}}\le\smash{\frac{t}{r}}\v{h}\vv{\chi}\:
\]
\end{lem}
\begin{rem}
Lemma~\ref{lem: (5)} establishes the implication that strict polynomial growth implies amenability.
\end{rem}
\begin{proof}
\allowdisplaybreaks%
Define $ \smash{\de\defeq \p{\frac{2}{2+b}}r} \:$
Observe $ \chi $ is a compactly-supported continuous nonzero $ L^2 $-vector.
Also, for $ h\in G \;$
\begin{align*}
&\ts\vv{\cramped{\chi^h-\chi}}^2\\
&\ts=\int_{B(r)\inv{h}\setminus B(r)}(\frac{r-\v{gh}}{r})^2dg
+\int_{B(r)\setminus B(r)\inv{h}}(\frac{r-\v{g}}{r})^2dg
+\int_{B(r)\cap B(r)\inv{h}}(\frac{\v{g}-\v{gh}}{r})^2dg
\\
&\ts\le\int_{B(r)\inv{h}\setminus B(r)}(\frac{\v{g}-\v{gh}}{r})^2dg
+\int_{B(r)\setminus B(r)\inv{h}}(\frac{\v{gh}-\v{g}}{r})^2dg
+\int_{B(r)\cap B(r)\inv{h}}(\frac{\v{g}-\v{gh}}{r})^2dg
\\
&\ts\le\int_{B(r)\inv{h}\setminus B(r)}r^{-2}\v{h}^2dg
+\int_{B(r)\setminus B(r)\inv{h}}r^{-2}\v{h}^2dg
+\int_{B(r)\cap B(r)\inv{h}}r^{-2}\v{h}^2dg
\\
&\ts\le r^{-2}\v{h}^22(c_1r^b)
\shortintertext{and}
&\ts\vv{\chi}^2\\
&\ts=\int_{B(r)\setminus B(r-\de)}(\frac{r-\v{g}}{r})^2dg
+\int_{B(r-\de)}(\frac{r-\v{g}}{r})^2dg
\\
&\ts\ge\int_{B(r-\de)}(\frac{r-\v{g}}{r})^2dg\\
&\ts\ge\int_{B(r-\de)}r^{-2}\de^2dg\\
&\ts\ge r^{-2}\de^2(c_0(r-\de)^b)\:\qedhere
\end{align*}
\end{proof}
\begin{thm}\label{thm: (6)}
Fix $ x\in Z $ with $ A_xA_x\mspace{-3mu}{}^*=A_x\mspace{-3mu}{}^*A_x \:$
Fix $ E\in\sigma(A_x) \:$
Fix $ \ep>0 \:$
Let $ \vp:G\to\C $ be a nonzero $ L^2 $-vector satisfying $ \vv{(A_{x}-E)\vp}\le \ep\vv{\vp} \:$
Let $ \chi:G\to\C $ be a compactly-supported continuous nonzero $ L^2 $-vector.
Define, for $ j\in G \;$
\begin{align*}
\ts\chi_{j}&\ts:G\to\C:g\mapsto\chi(\inv{j}g)\;\\
\ts\cramped{\chi^{j}}&\ts:G\to\C:g\mapsto\chi(gj)\:
\end{align*}
Assume there exists $ \eta>0 $ such that for each $ h\in G \;$ $ \vv{\cramped{\chi^h-\chi}}\le \eta\v{h}\vv{\chi} \:$
Then there exists $ j\in G $ such that $ \vv{\vp_j}\ne 0 $ and for each $ y\in Z \;$
\[\ts
\vv{(A_y-E)\vp_j}\le
\Bigp{
\vv{(A_y-A_x)\smash{\frac{\vp_j}{\vv{\vp_j}}}}
+\vv{k_x}_{S}\eta
+2\ep
}\vv{\vp_j}\;
\]
where $ \vp_j:G\to \C:g\mapsto \chi_j(g)\vp(g) \:$
\end{thm}
\begin{rem}
Theorem~\ref{thm: (6)} does not need that $ \ab{G,\lambda} $ has strict polynomial growth.
\end{rem}
\begin{proof}
\allowdisplaybreaks%
Define
\[\ts
\de\defeq
\begin{cases}
\ep\p{\vv{k_x}_{S}\eta}^{-1}&\iph\vv{k_x}_{S}\ne 0\;\\
3^{-1}&\elss\:
\end{cases}
\]
Let $ \Sc $ be the support of the map $ j\mapsto \vv{\vp_{j}}^2 \:$
Observe, for $ j\in G \;$
\begin{align*}
&\ts\v{\vv{\vp_j}^2-\vv{\vp_e}^2}\\
&\ts\le \int\v{\v{\vp_j(g)}^2-\v{\vp_e\p{g}}^2}dg\\
&\ts\le 2\vv{\chi}_{\infty}\vv{\chi_j-\chi}_{\infty}\vv{\vp}^2\:
\end{align*}
Because $ \chi\in C_{\ssm c}(G) \;$ the map $ j\mapsto \vv{\vp_{j}}^2 $ is continuous and $ 0<\lambda(\Sc) \:$
Define $ \bigb{A,B}:L^2(G)\to L^2(G) \;$ for $ A,B\in \Ls\p{L^2(G)} \;$ by
\[\ts
(\bigb{A,B}\psi)(g)=(AB\psi)(g)-(BA\psi)(g)
\]
for every $ \psi\in L^2(G)\, g\in G \:$
Define $ \bigb{\chi_j(\cdot)}:L^2(G)\to L^2(G) \;$ for $ j\in G \;$ by
\[\ts
(\bigb{\chi_j(\cdot)}\psi)(g)=\chi_j(g)\psi(g)
\]
for every $ \psi\in L^2(G)\, g\in G \:$
Define, for $ j\in G \;$
\begin{align*}
\vp_{\ep}&\defeq (A_{x}-E)\vp\;\\
\vp_{j,\ep}&\defeq (A_{x}-E)\bigb{\chi_{j}(\cdot)}\vp\;\\
\vp_{\ep,j}&\defeq \bigb{\chi_{j}(\cdot)}(A_{x}-E)\vp\:
\end{align*}
Observe
\begin{align*}\SwapAboveDisplaySkip
&\ts\int \vv{\vp_{\ep,j}}^2dj\\
&\ts=\vv{\vp_{\ep}}^2\vv{\chi}^2\\
&\ts\le\ep^2\vv{\vp}^2\vv{\chi}^2\\
&\ts=\int \ep^2\vv{\vp_{j}}^2dj
\end{align*}
and
\begin{align*}\SwapAboveDisplaySkip
&\ts\int\vv{\bigb{A_x-E,\bigb{\chi_j\p{\cdot}}}\vp}^2dj\\
&\ts=\iint\v{\int k(\inv{g}x,h)(\chi_j(gh)-\chi_j(g))\vp(gh)dh}^2dgdj\\
&\ts\le\iint\p{\int \v{k(\inv{g}x,h)}\v{\chi_j(gh)-\chi_j(g)}\v{\vp(gh)}dh}^2dgdj\\
&\ts\le\iint\p{\int \v{k(\inv{g}x,h)}\v{h}dh}\p{\int \v{k(\inv{g}x,h)}\v{h}^{-1}\v{\chi_j(gh)-\chi_j(g)}^2\v{\vp(gh)}^2dh}dgdj\\
&\ts\le\iint
\!\begin{array}[t]{l}
\p{\int \sup_{\gch\in G}\v{k(\inv{\gch}x,h)}\v{h}dh}\\
{}\cdot\p{\int \sup_{\gch\in G}\v{k(\inv{\gch}x,h)}\v{h}^{-1}\v{\chi_j(gh)-\chi_j(g)}^2\v{\vp(gh)}^2dh}djdg
\end{array}\\
&\ts=\p{\int \sup_{g\in G}\v{k(\inv{g}x,h)}\v{h}dh}\p{\int \sup_{g\in G}\v{k(\inv{g}x,h)}\v{h}^{-1}\vv{\cramped{\chi^h}-\chi}^2\vv{\vp}^2dh}\\
&\ts=\p{\int \sup_{g\in G}\v{k(\inv{g}x,h)}\v{h}dh}\p{\int \sup_{g\in G}\v{k(\inv{g}x,h)}\v{h}^{-1}\vv{\cramped{\chi^h}-\chi}^2\frac{1}{\vv{\chi}^2}dh}\vv{\chi}^2\vv{\vp}^2\\
&\ts=\p{\int \sup_{g\in G}\v{k(\inv{g}x,h)}\v{h}dh}\p{\int \sup_{g\in G}\v{k(\inv{g}x,h)}\v{h}^{-1}\smash{\frac{\cramped{\vv{\chi^h-\chi}^2}}{\vv{\chi}^2}}dh}\int\vv{\vp_j}^2dj\\
&\ts\le\int(\vv{k_x}_{S}\eta)^2\vv{\vp_j}^2dj\:
\end{align*}
Therefore
\begin{align*}\SwapAboveDisplaySkip
&\ts\int\vv{\vp_{j,\ep}}^2dj\\
&\ts=\int\vv{\bigb{A_{x}-E,\bigb{\chi_j\p{\cdot}}}\vp+\vp_{\ep,j}}^2dj\\
&\le\ts \int (1+\de)\vv{\bigb{A_{x}-E,\bigb{\chi_j\p{\cdot}}}\vp}^2+(1+\inv{\de})\vv{\vp_{\ep,j}}^2dj\\
&\le\ts \int (1+\de)(\vv{k_x}_{S}\eta)^2\vv{\vp_{j}}^2+(1+\inv{\de})\ep^2\vv{\vp_{j}}^2dj\:
\end{align*}
Because $ 0<\lambda(\Sc) \;$ there exists $ j\in G $ such that $ \vv{\vp_j}\ne 0 $ and
\[\ts \vv{\vp_{j,\ep}}^2\le\bigp{(1+\de)(\vv{k_x}_{S}\eta)^2+(1+\inv{\de})\ep^2}\vv{\vp_{j}}^2\: \]
Therefore
\[\ts
\begin{cases}
\vv{\vp_{j,\ep}}\le \bigp{(\vv{k_x}_{S}\eta)+\ep}\vv{\vp_j}&\iph\vv{k_x}_{S}\ne 0\;\\
\vv{\vp_{j,\ep}}\le 2\ep\vv{\vp_j}&\elss\:
\end{cases}
\]
Also, for $ y\in Z \;$
\begin{align*}\SwapAboveDisplaySkip
&\ts\vv{(A_{y}-E)\vp_{j}}\\
&\ts=\vv{(A_{y}-A_{x}+A_{x}-E)\vp_{j}}\\
&\ts\le\vv{(A_{y}-A_{x})\vp_{j}}+\vv{\vp_{j,\ep}}\\
&\ts\le\vv{(A_{y}-A_{x})\vp_{j}}+\bigp{(\vv{k_x}_{S}\eta)+2\ep}\vv{\vp_{j}}\:\qedhere
\end{align*}
\end{proof}
\begin{cor}\label{cor: 6.1}
Let $ d $ be a continuous metric on $ Z \;$ where the group action is Lipschitz continuous on $ Z $ and the kernel is Lipschitz continuous on $ Z \:$
Specifically, there exists $ m>0 $ such that for each $ g\in G $ and for all $ x,y\in Z \;$
\[\ts d(gx,gy)\le (m\v{g}+1)d(x,y)\]
and there exists $ M\in L^1(G) $ such that for $ \lambda $-a.e.\ $ h\in G \;$ $ M(h)\ge 0 $ and for all $ x,y\in Z \;$
\[\ts\v{k(x,h)-k(y,h)}\le M(h)d(x,y)\:\]
Define $ t\defeq \smash[t]{\p{\frac{(2+b)^{2+b}c_1}{2b^bc_0}}^{1/2}} \:$
Define $ \vv{k_Z}_S\defeq \smash[t]{\sup_{x\in Z}\vv{k_x}_{S}} \:$
Let $ \Ss $ be the collection of all invariant nonempty closed subsets of $ Z \:$
Let $ \Ks $ be the collection of all nonempty compact subsets of $ \C \:$
Define
\[\ts
\bar{\sigma}:(\Ss,d_{\ssm H})\to (\Ks,\dist_{\ssm H}):X\mapsto\overline{\bigcup_{x\in X}\sigma(A_x)}\:
\]
Assume $ \vv{k_Z}_S>0 $ and $ \vv{M}_1>0 $ and $ A_yA_y\mspace{-3mu}{}^*=A_y\mspace{-3mu}{}^*A_y $ for every $ y\in Z \:$
Fix $ X,Y\in\Ss \:$
Then
\[\ts
d_{\ssm H}(X,Y)\le{\min\s{\frac{t \vv{k_Z}_S}{m\vv{M}_1},1}}
\lRightarrow
\dist_{\ssm H}(\bar{\sigma}(X),\bar{\sigma}(Y))\le {\bigp{2(t\vv{k_Z}_Sm\vv{M}_1)^{1/2}+\vv{M}_1}} d_{\ssm H}(X,Y)^{1/2}\:
\]
Here $ d_{\ssm H}(X,Y) $ and $ \dist_{\ssm H}(\bar{\sigma}(X),\bar{\sigma}(Y)) $ are Hausdorff distances between subsets of metric spaces.
\end{cor}
\begin{proof}
\allowdisplaybreaks%
Define $ \de_{\ssm H}\defeq d_{\ssm H}(X,Y) \;$ $ r\defeq (\frac{t\vv{k_Z}_S}{\de_{\ssm H}m\vv{M}_1})^{1/2} \;$ $ \chi:G\to \C:g\mapsto\1_{B(r)}(g)(\frac{r-\v{g}}{r}) \:$
Fix $ \ep>0 \:$
Fix $ E\in \bigcup_{x\in X}\sigma(A_x) \:$
Let $ x $ be an element of $ X $ satisfying $ E\in \sigma(A_x)\:$
Because $ A_x $ is a normal operator, there exists a nonzero $ L^2 $-vector $ \vp:G\to \C $ satisfying
\[\ts \vv{(A_x-E)\vp}\le \ep\vv{\vp} \:\]
Here $ \vp $ is an approximate eigenfunction.
By Lemma~\ref{lem: (5)} and Theorem~\ref{thm: (6)}, there exists $ j\in G $ such that $ \vv{\vp_j}\ne 0 $ and for each $ y\in Z \;$
\[\ts
\vv{(A_y-E)\vp_j}\le
{\Bigp{
\vv{(A_y-A_x)\smash{\frac{\vp_j}{\vv{\vp_j}}}}
+\vv{k_x}_{S}\frac{t}{r}
+2\ep
}}\vv{\vp_j}\;
\]
where $ \vp_j:G\to \C:g\mapsto \chi_{j}(g)\vp(g) $ and $ \chi_{j}:G\to \C:g\mapsto\chi(\inv{j}g) \:$
Indeed, $ r\ge 1 \:$
Let $ {y} $ be an element of $ Y $ satisfying $ d(\inv{j}{x},\inv{j}{y})= d(\inv{j}{x},Y)\le\de_{\ssm H} \:$
By properties of normal operators,
\begin{align*}
&\ts \dist(E,\sigma(A_y))\\
&\ts =\vv{(A_y-E)^{-1}}^{-1}\\
&\ts\le {
\vv{(A_y-A_x)\smash{\frac{\vp_j}{\vv{\vp_j}}}}
+\vv{k_x}_{S}\frac{t}{r}
+2\ep
}\\
&\ts ={
\vv{(A_y-A_x)^*\smash{\frac{\vp_j}{\vv{\vp_j}}}}
+\vv{k_x}_{S}\frac{t}{r}
+2\ep
}\:
\end{align*}
Observe
\[\ts
\supp\p{\vp_j}\subseteq \supp\p{\chi_j}=j\supp\p{\chi}\:
\]
By Lemma~\ref{lem: (3)},
\begin{align*}\SwapAboveDisplaySkip
&\ts \vv{(A_y-A_x)^*\smash{\frac{\vp_j}{\vv{\vp_j}}}}\\
&\ts \le \int \sup_{g\in \supp\p{\vp_j}}\v{k(\inv{g}y,h)-k(\inv{g}x,h)} dh\\
&\ts \le \int \sup_{g\in \supp\p{\chi}}\v{k(\inv{g}\inv{j}y,h)-k(\inv{g}\inv{j}x,h)} dh\:
\end{align*}
Therefore
\begin{align*}
&\ts\sup_{E\in \bigcup_{x\in X}\sigma(A_x)}\dist(E,\bigcup_{y\in Y}\sigma(A_y))\\
&\ts\le\sup_{\lsubstack{\check{x}\in X,\check{y}\in Y\\\mathrlap{d(\check{x},\check{y})\le \de_{\ssm H}}}}\int \sup_{g\in \supp\p{\chi}}\v{k(\inv{g}\check{y},h)-k(\inv{g}\check{x},h)} dh
+\p{\sup_{\check{z}\in Z}\vv{k_{\check{z}}}_{S}}\frac{t}{r}+2\ep\\
&\ts\le \vv{M}_1(m r+1)\de_{\ssm H}
+\vv{k_Z}_{S}\smash[b]{\frac{t}{r}}+2\ep\:\qedhere
\end{align*}
\end{proof}
\begin{cor}\label{cor: 6.2}
Let $ d $ be a continuous metric on $ Z \;$ where the group action is Lipschitz continuous on $ Z $ and the kernel is locally-constant on $ Z \:$
Specifically, there exists $ m>0 $ such that for each $ g\in G $ and for all $ x,y\in Z \;$
\[\ts d(gx,gy)\le (m\v{g}+1)d(x,y)\]
and there exists $ M\in L^\infty(G) $ such that for $ \lambda $-a.e.\ $ h\in G \;$ $ M(h)\ge 1 $ and for all $ x,y\in Z \;$
\[\ts d(x,y)\le \frac{1}{M(h)}\lRightarrow k(x,h)=k(y,h)\:\]
Define $ t\defeq \smash[t]{\p{\frac{(2+b)^{2+b}c_1}{2b^bc_0}}^{1/2}} \:$
Define $ \vv{k_Z}_S\defeq \smash[t]{\sup_{x\in Z}\vv{k_x}_{S}} \:$
Let $ \Ss $ be the collection of all invariant nonempty closed subsets of $ Z \:$
Let $ \Ks $ be the collection of all nonempty compact subsets of $ \C \:$
Define
\[\ts
\bar{\sigma}:(\Ss,d_{\ssm H})\to (\Ks,\dist_{\ssm H}):X\mapsto\overline{\bigcup_{x\in X}\sigma(A_x)}\:
\]
Assume $ \vv{k_Z}_S>0 $ and $ A_yA_y\mspace{-3mu}{}^*=A_y\mspace{-3mu}{}^*A_y $ for every $ y\in Z \:$
Fix $ X,Y\in\Ss \:$
Then
\[\ts
d_{\ssm H}(X,Y)\le{\frac{1}{(m+1)\vv{M}_{\infty}}}
\lRightarrow
\dist_{\ssm H}(\bar{\sigma}(X),\bar{\sigma}(Y))\le {\bigp{t\vv{k_Z}_S(m+1)\vv{M}_{\infty}}} d_{\ssm H}(X,Y)\:
\]
Here $ d_{\ssm H}(X,Y) $ and $ \dist_{\ssm H}(\bar{\sigma}(X),\bar{\sigma}(Y)) $ are Hausdorff distances between subsets of metric spaces.
\end{cor}
\begin{proof}
\allowdisplaybreaks%
Define $ \de_{\ssm H}\defeq d_{\ssm H}(X,Y) \;$ $ r\defeq \frac{1-\de_{\ssm H}\vv{M}_\infty}{\de_{\ssm H}m\vv{M}_\infty} \;$ $ \chi:G\to \C:g\mapsto\1_{B(r)}(g)(\frac{r-\v{g}}{r}) \:$
Fix $ \ep>0 \:$
Fix $ E\in \bigcup_{x\in X}\sigma(A_x) \:$
Let $ x $ be an element of $ X $ satisfying $ E\in \sigma(A_x)\:$
Because $ A_x $ is a normal operator, there exists a nonzero $ L^2 $-vector $ \vp:G\to \C $ satisfying
\[\ts \vv{(A_x-E)\vp}\le \ep\vv{\vp} \:\]
Here $ \vp $ is an approximate eigenfunction.
By Lemma~\ref{lem: (5)} and Theorem~\ref{thm: (6)}, there exists $ j\in G $ such that $ \vv{\vp_j}\ne 0 $ and for each $ y\in Z \;$
\[\ts
\vv{(A_y-E)\vp_j}\le
{\Bigp{
\vv{(A_y-A_x)\smash{\frac{\vp_j}{\vv{\vp_j}}}}
+\vv{k_x}_{S}\frac{t}{r}
+2\ep
}}\vv{\vp_j}\;
\]
where $ \vp_j:G\to \C:g\mapsto \chi_{j}(g)\vp(g) $ and $ \chi_{j}:G\to \C:g\mapsto\chi(\inv{j}g) \:$
Indeed, $ r\ge 1 \:$
Let $ {y} $ be an element of $ Y $ satisfying $ d(\inv{j}{x},\inv{j}{y})= d(\inv{j}{x},Y)\le\de_{\ssm H} \:$
By properties of normal operators,
\begin{align*}
&\ts \dist(E,\sigma(A_y))\\
&\ts =\vv{(A_y-E)^{-1}}^{-1}\\
&\ts\le {
\vv{(A_y-A_x)\smash{\frac{\vp_j}{\vv{\vp_j}}}}
+\vv{k_x}_{S}\frac{t}{r}
+2\ep
}\\
&\ts ={
\vv{(A_y-A_x)^*\smash{\frac{\vp_j}{\vv{\vp_j}}}}
+\vv{k_x}_{S}\frac{t}{r}
+2\ep
}\:
\end{align*}
Observe
\[\ts
\supp\p{\vp_j}\subseteq \supp\p{\chi_j}=j\supp\p{\chi}\:
\]
By Lemma~\ref{lem: (3)},
\begin{align*}\SwapAboveDisplaySkip
&\ts \vv{(A_y-A_x)^*\smash{\frac{\vp_j}{\vv{\vp_j}}}}\\
&\ts \le \int \sup_{g\in \supp\p{\vp_j}}\v{k(\inv{g}y,h)-k(\inv{g}x,h)} dh\\
&\ts \le \int \sup_{g\in \supp\p{\chi}}\v{k(\inv{g}\inv{j}y,h)-k(\inv{g}\inv{j}x,h)} dh\:
\end{align*}
Therefore
\begin{align*}
&\ts\sup_{E\in \bigcup_{x\in X}\sigma(A_x)}\dist(E,\bigcup_{y\in Y}\sigma(A_y))\\
&\ts\le\sup_{\lsubstack{\check{x}\in X,\check{y}\in Y\\\mathrlap{d(\check{x},\check{y})\le \de_{\ssm H}}}}\int \sup_{g\in \supp\p{\chi}}\v{k(\inv{g}\check{y},h)-k(\inv{g}\check{x},h)} dh
+\p{\sup_{\check{z}\in Z}\vv{k_{\check{z}}}_{S}}\frac{t}{r}+2\ep\\
&\ts\le 0
+\vv{k_Z}_{S}\smash[b]{\frac{t}{r}}+2\ep\:
\end{align*}
Indeed, for each $ \xch\in X $ and for each $ \ych\in Y \;$ if $ d(\xch,\ych)\le \de_{\ssm H} \;$ then for each $ g\in\supp\p{\chi} \;$
\[\ts
d(\inv{g}\ych,\inv{g}\xch)
\le(mr+1)\de_{\ssm H}
\le \frac{1}{\vv{M}_{\infty}}\:\qedhere
\]
\end{proof}
\begin{comment}
\begin{lem}
Let $ \ab{(\G,d),\#} $ be a countable discrete left-metric group endowed~with the counting measure, where for each $ m\ge 0 \;$
\[\ts \#(B(m))<+\infty\:\]
Let $ \beta:[0,+\infty)\to [1,+\infty) $ be an order-preserving embedding, where for all $ m_{\ttm 1},m_{\ttm 2}\in [0,+\infty) \;$ $\beta(m_{\ttm 1}+m_{\ttm 2})\le \beta(m_{\ttm 1})\beta(m_{\ttm 2})$ and $ \beta(0)=1 $ and $ \sup_{m\ge 0}  \beta(m)  = +\infty \:$
Let $ l $ be an integer, where $ l\ge 2 \:$
Define $ \Ac\defeq \s{0,\ldots,l-1} \:$
Define $ Z\defeq \Ac^\G \:$
Define, for $ x,y\in{Z} \;$
\[\ts d(x,y)\defeq 1\meet \inf\s{\smash[b]{\frac{1}{\beta(m)}}:m\in[0,+\infty)\, \smash[b]{x|_{B(m)}}=\smash[b]{y|_{B(m)}}}\:\]
Define, for $ g\in \G\, x\in{Z} \;$
\[\ts g x\defeq x\circ\inv{g}\:\]
Let $ e $ be the identity element of $ \G \:$
Then
\begin{enumerate}[(i)]
\item
$ d $ is an ultrametric on $ {Z} \:$
\item
$ \p{{Z},d} $ is compact and totally-disconnected.
\item
$ \p{{Z},d} $ has no isolated points $ \impliedby $ $ \G $ is infinite.
\item
$ ex=x $ for every $ x\in {Z} \:$
\item
$ h(gx)=(hg)x $ for every $ h,g\in\G\, x\in {Z} \:$
\item
$ d(g x,g y)\le \beta(\v{g})d(x,y) $ for every $ g\in \G\, x,y\in {Z} \:$
\end{enumerate}
\end{lem}
\begin{proof}
\allowdisplaybreaks%
\textit{(i)}
Fix $ x,y,z\in {Z} \:$
Observe $ d(x,y)=0 $ if and only if $ x=y \:$
Observe $ d(x,y)=d(y,x) \:$
Suppose $ d(x,y)\join d(y,z)<d(x,z) \:$
Observe $ 0<d(x,y)\meet d(y,z)\meet d(x,z) $ and $ x(e)=z(e) $ and $ d(x,y)\join d(y,z)\join  d(x,z)<1 \:$
Let $ m_{xy},m_{yz},m_{xz} $ be elements of $ [0,+\infty) $ satisfying $ d(x,y)=\frac{1}{\beta(m_{xy})} $ and $ d(y,z)=\frac{1}{\beta(m_{yz})} $ and $ d(x,z)=\frac{1}{\beta(m_{xz})} \:$
Observe $ \beta(m_{xz})<\beta(m_{xy})\meet \beta(m_{yz}) $ and $ m_{xz}<m_{xy}\meet m_{yz} $ and $ x|_{B(m_{xy}\meet m_{yz})}=y|_{B(m_{xy}\meet m_{yz})}=z|_{B(m_{xy}\meet m_{yz})} $ and $ m_{xy}\meet m_{yz}\le m_{xz} $---a contradiction, and $ d(x,z)\le d(x,y)\join d(y,z) \:$
\smallskip\\
\textit{(ii)}
Fix $ \s{x^{(n)}}_{n\in\N}\subseteq {Z} \:$
Suppose for each $ \ep>0 \;$ there exists $ n_\ep\in\N $ such that for all $ i,j\ge n_\ep \;$ $ d(x^{(i)},x^{(j)})<\ep \:$
Define $ x^{(\infty)}\defeq (g\mapsto x^{(n_{(1/\beta(\v{g}))})}(g)) \:$
Observe $ \lim_{n\to +\infty}x^{(n)}=x^{(\infty)} \;$ and $ ({Z},d) $ is complete.
Observe for each $ \ep>0 \;$ there exists $ m_\ep\in [0,+\infty) $ such that $ 1/\beta(m_\ep)<\ep $ and $ \Ac^{B(m_\ep)} $ is finite, and $ ({Z},d) $ is totally-bounded.
Fix $ y,z\in {Z} \:$
Fix $ r>0 \:$
Suppose $ d(x^{(n)},y)<r $ for every $ n\in\N $ and $ \lim_{n\to+\infty}x^{(n)}=z \:$
Observe $ d(z,y)\le d(z,x^{(n)})\join d(x^{(n)},y) $ for every $ n\in\N $ and $ d(z,y)<r $ and $ \s{x\in {Z}:d(x,y)<r} $ is clopen, and $ \p{{Z},d} $ is totally-disconnected.
\smallskip\\
\textit{(iii)}
Fix $ x\in {Z} \:$
Fix $ u,v\in\Ac $ with $ u\ne v \:$
Define, for $ n\in\N \;$
\[\ts
x^{(n)}:G\to\Ac:g\mapsto
\begin{cases}
x(g)&\iph g\in B(n)\\
u&\iph g\notin B(n)\, x(g) \ne u\\
v&\iph g\notin B(n)\, x(g) = u\:
\end{cases}
\]
Observe $ x^{(n)}\ne x $ for every $ n\in\N $ and $ \lim_{n\to +\infty}x^{(n)}=x \:$
\smallskip\\
\textit{(iv)}
Fix $ x\in {Z} \:$
Observe $ ex=x\circ\inv{e}=x\circ e=x\: $
\smallskip\\
\textit{(v)}
Fix $ h,g\in\G \:$
Fix $ x\in {Z} \:$
Observe $ h(gx)=h(x\circ \inv{g})=x\circ \inv{g}\circ \inv{h}=x\circ (hg)^{-1}=(hg)x\:$
\smallskip\\
\textit{(vi)}
Fix $ g\in \G \:$
Fix $ x,y\in{Z} \:$
If $ 0=d(gx,gy)\meet d(x,y) $ or $ x(e)\ne y(e) $ or $ (gx)(e)\ne (gy)(e) \;$ then $ d(g x,g y)\le \beta(\v{g})d(x,y)\:$
Suppose $ 0<d(gx,gy)\meet d(x,y) $ and $ x(e)=y(e) $ and $ (gx)(e)= (gy)(e) \:$
Let $ m_g,m_e $ be elements of $ [0,+\infty) $ satisfying $ d(gx,gy)=\frac{1}{\beta(m_g)} $ and $ d(x,y)=\frac{1}{\beta(m_e)} \:$
Observe $ m_e\le \v{g}+m_g $ and $ \beta\p{m_e}\le \beta\p{\v{g}+m_g} $ and $d(g x,g y)\le \beta(\v{g})d(x,y)\:$
\end{proof}
\begin{thm}
Let $ \ab{(\G,d),\#} $ be a countable discrete left-metric group endowed~with the counting measure, where there exist $ b\ge 1\, a_1>a_0>0 $ such that for each $ r\ge 1 \;$
\[\ts a_0r^b\le\#(B(r))\le a_1r^b \:\]
Let $ \beta:[0,+\infty)\to [1,+\infty) $ be an order-preserving embedding, where for all $ m_{\ttm 1},m_{\ttm 2}\in [0,+\infty) \;$ $\beta(m_{\ttm 1}+m_{\ttm 2})\le \beta(m_{\ttm 1})\beta(m_{\ttm 2})$ and $ \beta(0)=1 $ and $ \sup_{m\ge 0}  \beta(m)  = +\infty \:$
Let $ l $ be an integer, where $ l\ge 2 \:$
Define $ \Ac\defeq \s{0,\ldots,l-1} \:$
Define $ Z\defeq \Ac^\G \:$
Define, for $ x,y\in Z \;$
\[\ts d(x,y)\defeq 1\meet \inf\s{\smash[b]{\frac{1}{\beta(m)}}:m\in[0,+\infty)\, \smash[b]{x|_{B(m)}}=\smash[b]{y|_{B(m)}}}\:\]
Define, for $ g\in \G\, x\in Z \;$
\[\ts g x\defeq x\circ\inv{g}\:\]
Let $ k:Z\times \G\to \C $ be a bounded continuous map, where $ \sup_{x\in Z}\sum_{h\in \G} \sup_{g\in \G}\v{k(\inv{g}x,h)}\v{h} <+\infty $ and there exists $ D\in L^1(\G) $ such that $ \vv{D}_1>0 $ and for each $ h\in \G \;$ $ D(h)\ge 0 $ and for all $ x,y\in Z \;$
\[\ts\v{k(x,h)-k(y,h)}\le D(h)\p{\inv{\beta}\p{d(x,y)^{-1}}}^{-1}\:\]
Define, for $ x\in Z \;$
\[\ts \smash[b]{\vv{k_x}_{\infty,1_+}}\defeq\sum_{h\in\G}\sup_{g\in \G}\v{k(\inv{g}x,h)}\v{h}\: \]
Define $ A_x:L^2(\G)\to L^2(\G) \;$ for $ x\in Z \;$ by
\[\ts
(A_x\psi)(g)=\sum_{h\in\G} k(\inv{g}x,\inv{g}h)\psi(h)
\]
for every $ \psi\in L^2(\G)\, g\in \G \:$
Assume for each $ x\in Z \;$ $ \smash[b]{\vv{k_x}_{\infty,1_+}}\ne 0 $ and $ A_xA_x\mspace{-3mu}{}^*=A_x\mspace{-3mu}{}^*A_x \:$
Let $ \Ss $ be the collection of all invariant nonempty closed subsets of $ Z \:$
Define, for $ X\in\Ss \;$ $\bar{\sigma}(X)\defeq \overline{{\bigcup_{x\in X}\sigma(A_{x})}}\:$
Fix $ X,Y\in\Ss \:$
Then
\[\ts
d_{\ssm H}(X,Y)\le\smash{\frac{1}{\beta(\frac{c+1}{c})}}
\lRightarrow
\dist_{\ssm H}(\bar{\sigma}(X),\bar{\sigma}(Y))\le \smash{\bigp{\p{\sqrt{\vv{D}_1}+\sqrt{st}}^2}} \p{\inv{\beta}\p{d_{\ssm H}(X,Y)^{-1}}}^{-1}\;
\]
where $ s\defeq\sup_{x\in Z}\smash[b]{\vv{k_x}_{\infty,1_+}} $ and $ t\defeq \smash[b]{\p{\frac{(2+b)^{2+b}a_1}{2b^ba_0}}^{1/2}} $ and $ c\defeq\smash[b]{(\frac{st}{\vv{D}_1})^{1/2}} \:$
\end{thm}
\begin{proof}
\allowdisplaybreaks%
Define $ \de_{\ssm H}\defeq d_{\ssm H}(X,Y) \;$ $ r\defeq \frac{\p{\inv{\beta}\p{\bp{\de}{\ssm H}{\smash{-1}}}}c}{c+1} \;$ $ \chi:\G\to \C:g\mapsto\1_{B(r)}(g)(\frac{r-\v{g}}{r}) \:$
Fix $ \ep>0 \:$
Fix~$ E\in \bigcup_{x\in X}\sigma(A_x) \:$
Let $ x $ be an element of $ X $ satisfying $E\in \sigma(A_x)\:$
Let $ \vp:\G\to\C $ be a nonzero $ L^2 $-vector satisfying $ \vv{(A_{x}-E)\vp}\le \ep\vv{\vp} \:$
By Lemma~\ref{lem: (5)} and Theorem~\ref{thm: (6)}, there exists $ j\in \G $ such that $ \vv{\vp_j}\ne 0 $ and for each $ y\in Z \;$
\[\ts
\vv{(A_y-E)\vp_j}\le
{\Bigp{
\vv{(A_y-A_x)^*\smash{\frac{\vp_j}{\vv{\vp_j}}}}
+\smash[b]{\vv{k_x}_{\infty,1_+}}\frac{t}{r}
+\ep
}}\vv{\vp_j}\;
\]
where $ \vp_j:\G\to \C:g\mapsto \chi_{j}(g)\vp(g) $ and $ \chi_{j}:\G\to \C:g\mapsto\chi(\inv{j}g) \:$
Let $ {y} $ be an element of $ Y $ satisfying $ d(\inv{j}{x},\inv{j}{y})= d(\inv{j}{x},Y)\le\de_{\ssm H} \:$
Observe $\supp\p{\vp_j}\subseteq \supp\p{\chi_j}=j\supp\p{\chi}=jB(r)$ and
$ \smash[b]{\sup_{g\in B(r)}}d(\inv{g}\inv{j}y,\inv{g}\inv{j}x)\le\smash[b]{\frac{1}{\vphantom{\de^{-1}}\beta(\inv{\beta}\p{\bp{\de}{\ssm H}{\smash{-1}}}-r)}} $ and
\begin{align*}
&\ts\dist(E,\sigma(A_y))\\
&\ts=\vv{(A_y-E)^{-1}}^{-1}\\
&\ts\le {\Bigp{
\p{\sum_{h\in \G} \sup_{g\in B(r)}\v{k(\inv{g}\inv{j}y,\inv{h})-k(\inv{g}\inv{j}x,\inv{h})}}
+\smash[b]{\vv{k_x}_{\infty,1_+}}\frac{t}{r}
+\ep
}}
\shortintertext{and}
&\ts\sup_{E\in\bigcup_{x\in X}\sigma(A_x)}\dist(E,\bigcup_{y\in Y}\sigma(A_y))\\
&\ts\le \sup_{\lsubstack{\check{x}\in X,\check{y}\in Y\\\mathrlap{d(\check{x},\check{y})\le \de_{\ssm H}}}}\p{\sum_{h\in \G} \sup_{g\in B(r)}\v{k(\inv{g}\check{y},\inv{h})-k(\inv{g}\check{x},\inv{h})}}
+\p{\sup_{\check{z}\in Z}\smash[b]{\vv{k_{\check{z}}}_{\infty,1_+}}}\frac{t}{r}
+\ep\\
&\ts\le \vv{D}_1\smash[b]{\frac{1}{\vphantom{\de^{-1}}\inv{\beta}\p{\bp{\de}{\ssm H}{\smash{-1}}}-r}}
+s\smash[b]{\frac{t}{r}}+\ep\:\qedhere
\end{align*}
\end{proof}
\end{comment}

\subsection*{Acknowledgements}
\NEWvbadness{10000}
\let\originalurl\url
\renewcommand{\url}[1]{\nobreak\hfil\allowbreak\hfilneg\originalurl{#1}}
\renewcommand{\refname}{References}
\renewcommand{\bibname}{\refname}

\section{Introduction}\label{sec: Introduction}
\input{_introduction}
\clearpage
\section{Preliminaries}\label{sec: Preliminaries}
\input{_preliminaries}
\section{Proof of Main Results}\label{sec: Proof of Main Results}
\input{_proof_of_main_results}
\bookmarksetup{startatroot}
\input{_references}
\end{document}

redpajama_set_name: RedPajamaArXiv


text:
Q: How to add else condition to takeIf in kotlin? I know that for checking
 if(storesList!=null && storesList.isNotEmpty()){
    // do this
    } else {
  //else do this
    }

we can write like this,
storesList?.takeIf { it.isNotEmpty() }.apply {//do this
  }

How can i add an else condition to this, I'm not taking about takeUnless

A: You can add an Elvis Operator..
storesList?.takeIf { it.isNotEmpty() }?.apply {
    //it.isNotEmpty() is true
} ?: //it.isNotEmpty() is false

So if it.isNotEmpty() is true, takeIf returns a non-null value and the apply block will be called.
If false, the expression is null and the elvis operator will execute the expression behind it. The elvis operator is a bit like if (expression before == null) -> execute statement after elvis operator.
For more information, have a look at the docs: https://kotlinlang.org/docs/reference/null-safety.html#elvis-operator

A:
Use Elvis Operator

eg.)   a ?: b // here ifais not null value =a else value=b
in your case -
storesList?.takeIf { it.isNotEmpty() }?.apply {
    //it.isNotEmpty() is true
} ?: run{} //`else condition goes here`

for more check here - https://en.wikipedia.org/wiki/Elvis_operator

A: In Kotlin way, you can use safe call operator ('?') to do stuffs on nullable objects without crash or even making it inside if (obj!=null) block.
So, some object like
if (someObj != null )
    someObj.someOperation() //just doing some operation

is the same thing by calling like : someObj?.someOperation().

So, if you want to check emptiness of list without if-else condition, you can use like below (Which you've already done).
storesList?.takeIf { it.isNotEmpty() }?.apply {
    // Provides you list if not empty
}

But what about else condition here?
For that, you can use elvis operator to satisfy condition. What this operator do is if left hand-side of operation is null or doesn't satisfy specific condition then take right hand-side of operand.
So, final code should look like:
storesList?.takeIf { it.isNotEmpty() }?.apply {
// Provides you list if not empty
} ?: run {
// Else condition here
}

Explanation: if storesList is empty or null it goes to else part (that is after elvis) otherwise goes to apply block.


text:
import Oauth2 from 'torii/providers/oauth2-code';
import {configurable} from 'torii/configuration';

var computed = Ember.computed;

/**
 * This class implements authentication against AzureAD
 * using the OAuth2 authorization flow in a popup window.
 * @class
 */
var AzureAdOauth2 = Oauth2.extend({
  name: 'azure-ad-oauth2',

  baseUrl: computed(function() {
    return 'https://login.windows.net/' + this.get('tennantId') + '/oauth2/authorize';
  }),

  tennantId: configurable('tennantId', 'common'),

  // additional url params that this provider requires
  requiredUrlParams: ['state', 'api-version', 'client_id'],

  optionalUrlParams: ['scope', 'nonce', 'response_mode'],

  responseMode: configurable('responseMode', null),

  responseParams: computed(function () {
    return [ this.get('responseType') ];
  }),

  state: 'STATE',

  apiVersion: '1.0',

  responseType: configurable('responseType', 'code'),

  redirectUri: configurable('redirectUri', function(){
    // A hack that allows redirectUri to be configurable
    // but default to the superclass
    return this._super();
  })
});

export default AzureAdOauth2;

redpajama_set_name: RedPajamaGithub


Our medical information is perhaps considered the most personal and valuable of all. Its integrity and safety is critical and recent data breaches have shown how vulnerable access to medical information can be. Regulatory pressures and compliance guidelines naturally require health care providers to handle patient information with the highest degree of privacy and care.
But how can we ensure a seamless patient experience in receiving care and in communicating with for instance physicians and insurance providers, while protecting patient privacy and at the same time safeguarding health care providers against fraud? With health care related fraud reaching an estimate of 56 billion in the EU alone, finding the right balance is crucial.
This is where BioTrust can be of help!
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Our solutions ensure patient confidentiality while minimizing the risk of fraud.
Prevent medical identity theft and safeguard the privacy of healthcare data with a secure and convenient authentication experience for physicians, partners and patients.
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Go beyond usernames, passwords and insecure questions. Apply biometric authentication for patients and health care providers to eliminate the risk of fraud.
Use voice biometrics to allow elderly and sick people to order prescription medication without the need to leave the house.


text:
Bruno Mars – The Other Side feat. Cee Lo Green & B.o.B
Saturday, 28 January 2023, 17:13 pm Lalas Hip Hop 0
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Bruno Mars - Finesse (Remix) feat. Cardi B
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Bruno Mars – Count on Me
Lyrics: Pheelz – Finesse
Lyrics: Ayo & Teo – Lit Right Now
Lyrics: Erigga – Head Pan
Lyrics: Lil Nas X – Rodeo (Remix)
Lyrics: POP SMOKE – DIOR
Lyrics: Lil Pump – Gucci Gang
Lyrics: CKay – Emiliana
Lyrics: BNXN & Zinoleesky – Kilometer
Lyrics: A$AP Rocky – D.M.B.
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2023: Buhari's aide reacts as Osinbajo declares for Presidency
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E-mentoring, also known as online mentoring or virtual mentoring, is changing the way that traditional mentor and protégé dyad members interact with each other. Mentoring has been widely known for its ability to enhance the career development, and to provide psychosocial support, for more junior organizational members. Through the use of computer-mediated communication technology, e-mentoring may allow individuals to bridge geographic and time differences. However, there is still much we do not know about e-mentoring and its social effects. This chapter focuses on whether or not computer-mediated communication (CMC) technology will allow for true mentoring relationships to develop, as well as what personal characteristics may be necessary to grow these virtual relationships. A model and proposition for future research are offered.