text:
Fined for parking outside own home
A HUDDERSFIELD couple have attacked Kirklees over the council's refusal to overturn their parking fine.
A HUDDERSFIELD couple have attacked Kirklees over the council's refusal to overturn their parking fine.
The council is insisting on charging Gavin and Sharlene Noble for being illegally parked in a residential zone.
The problem, say the couple, is that they were parked outside their Lindley home at the time!
Mrs Noble, 37, had her hands full when she returned to her home on Union Street after a shopping trip.
Her arms were loaded with carrier bags and she had to attend to a screaming seven-week-old baby.
She rushed into her house, forgetting to prop up the parking display permit in her car.
When she returned 30 minutes later, a Penalty Charge Notice had been slapped on the vehicle.
Mr Noble, 32, appealed against the fine to the council.
The appeal was rejected on the grounds that the car was left in a residents' permit parking area and a valid permit was not on display.
Mr Noble said: "I explained the situation, but the council said I had not given sufficient reason.
"I understand that the warden did not know we were residents.
"But as after the event we proved that we were residents I don't think it is unreasonable to say we should be granted an appeal.
"When I spoke to someone at the council they said if it had been in the town I wouldn't be able to appeal - but I don't live in the car park!
"It's a bit silly if you can't appeal, even though you are a resident.
"If I had parked illegally outside the scheme I wouldn't appeal.
"It is annoying as when you go out on to our street there are so many non-residents parked there. None of them seem to get parking tickets.
"We stood outside one day and counted at least a dozen non-residents' vehicles. Not one had been given a ticket.
"It is just common sense that if you are a resident involved in a residential scheme you should be granted an appeal."
The couple do have a choice of either paying the £30 fine or appealing again. They are likely to pay the fine as if they wait to contest it again they run the risk of the fine being increased for late payment.
Mr Noble said it was more about the principle than the money.
He said: "I really just want to highlight the stupidity of the system.
"I want the council to take a look at it and apply common sense in the future.
"There should be a system in place to reverse the charge if something like
this happens."
A Kirklees spokesperson said: "It is in the interests of all the residents with such permits that the zone be enforced.
"There is no discretion when individuals fail to display their permits as parking attendants can only tell if the vehicle is in a scheme if it has a valid permit on display.
"The couple may now submit a formal appeal."
UK & World NewsFamily who spend £160 on takeaways each month are evicted from council houseThe couple had fallen into rent arrears and have been booted from their home


Building upon a strong foundation and a rich industrial heritage, 21c Museum Hotel Oklahoma City is a contemporary art museum, a 135-room boutique hotel with spacious guest rooms reminiscent of a loft apartment in New York's Soho neighborhood, a cultural civic center and the home of Mary Eddy's Kitchen x Lounge, led by executive chef Jason Campbell. Stay with us and the iconic Purple Penguin.
Contemporary art is the heart of the 21c experience. Look for the sixteen-foot-high perforated steel tree, Woozy Blossom, which produces a continuous fog along Main Street; then explore the current exhibition; dine in a sculpturally expressive space at Mary Eddy's; or discover the art of time with the site-specific installation River of Time.
Situated in the former home of a Ford Motor Company assembly plant, which dates back more than a century and is listed on the National Register of Historic Places, 21c Oklahoma City takes full advantage of the scale and openness of its historical setting.
The New York City-based architecture firm Deborah Berke Partners sought to create a contemporary interpretation of the building's industrial heritage. Inspired by the assembly line and technological advances of the automotive industry, the design team restored historical features and complemented them with contemporary finishes and furnishings.
Features two (incredibly comfortable) queen beds and seating area with table, plus all the usual 21c luxury amenities – 43″ HDTV, Malin+Goetz bath amenities, free Wi-Fi, Nespresso coffee machine and original artwork on display in the room.
King-sized bed (featuring incredibly comfortable and luxurious bedding), seating area with table and standing shower. Includes all the usual 21c amenities – 43″ HDTV, Malin+Goetz bath amenities, free Wi-Fi, Nespresso coffee machine and original artwork on display in the room.
King-sized bed (of the most comfortable sort) with an exclusive view into a sculptural lightwell that captures sunlight during the day – through a full wall of glass – and features distinctive architectural lighting. Bathroom includes standing shower and Malin+Goetz bath amenities. Plus all the usual 21c amenities – 43″ HDTV, free Wi-Fi, Nespresso coffee machine and original artwork on display in the room.
One bedroom suite with a king bed, separate living area, dressing table and standing shower. Most feature wet room with soaking tub as well. 49″ HDTV in living area and 43″ HDTV in bedroom. Plus the usual 21c amenities – Malin+Goetz bath amenities, free Wi-Fi, Nespresso coffee machine and original artwork on display in the room.
Two separate bedrooms (the most comfortably of sort) complete with two en-suite bathrooms, dressing table, and separate living area. One bedroom has a spacious bathroom featuring a wet room with soaking tub and standing shower. Plus the usual 21c amenities – Malin+Goetz bath amenities, free Wi-Fi, three HDTV's, Nespresso coffee machine and original artwork on display in the room.
Availability and pricing upon request. Call 405.982.6900 or email OKC.Reservations@21cHotels.com.
A most comfortable one bedroom suite complete with a king-sized bed, dressing table, separate living area, and wet room with soaking tub and standing shower. Features a 49″ HDTV in the living area and 43″ HDTV in bedroom. Plus the usual 21c amenities – Malin+Goetz bath amenities, free Wi-Fi, Nespresso coffee machine and original artwork on display in the room.
One room studio-style suite with (an incredibly comfortable) king-sized bed. Features exclusive view into a sculptural lightwell that captures sunlight during the day – through a full wall of glass – and features distinctive architectural lighting. Includes extra living space, 49″ HDTV in living area, 43″ HDTV in bed area and standing shower in bathroom. Plus the usual 21c amenities – Malin+Goetz bath amenities, free Wi-Fi, Nespresso coffee machine and original artwork on display in the room.
One bedroom suite with a king bed and extra private space with a daybed. Features separate living area with desk, dressing table, and wet room with soaking tub and standing shower. 49″ HDTV in living area and 43″ HDTV in bedroom. Plus the usual 21c amenities – Malin+Goetz bath amenities, free Wi-Fi, Nespresso coffee machine, luxurious bedding and original artwork on display in the room.
Our penthouse-level suite features 2 bedrooms, each with a king bed, and a large terrace that includes outdoor living and dining areas, with sweeping views of downtown OKC. The open concept living and dining area includes a folding glass wall to the terrace. Kitchen includes a full-size refrigerator, ice machine and wine cooler, and the dining table seats 12. Living room includes 2 80″ HDTVs and each bedroom includes a 43″ HDTV. En suite bathrooms with wet room feature a soaking tub, standing shower with dual showerhead and natural light. Controlled access. Plus the usual 21c amenities –Malin+Goetz bath amenities, free Wi-Fi, Nespresso coffee machine, luxurious bedding and original artwork on display in the room.
Mary Eddy's Kitchen x Lounge showcases executive chef Jason Campbell's bold and eclectic take on New American cuisine in an art-filled, communal setting that honors the building's industrial roots. Menus playfully combine unexpected flavors and celebrate regional, seasonal products.
Mary Eddy's dedicated entrance on Main Street welcomes guests into the original historic Model T showroom space where historical elements mingle with contemporary finishes and custom furnishings. The design takes full advantage of the scale and openness of the original building, resulting in a light-filled and expressive space.
Feeling stressed? 21c has your back. (Literally).
Let the skilled hands of our massage therapists use a variety of techniques to ease those aching muscles, reawaken your senses and soothe away your stress.
All spa treatments are by appointment only and must be made at least 24 hours in advance. Simply call or email us and we'll happily ensure your appointment is arranged.
21c Oklahoma City brings more than 14,000 square feet of art-filled exhibition and event space to the West End of downtown Oklahoma City.
The Oklahoma City Thunder, cowboy culture and Midwestern charm – and that's just what you already know about Oklahoma City.
Oklahoma City, an incubator for vibrant arts, music and culinary communities possesses small-town Midwestern charm, big-city amenities (it's Oklahoma's largest!) and a rich heritage. You won't have to venture far from 21c to experience the city's unique flavor; just use our guide to Oklahoma City to explore.
Nothing compares to experiencing art in person. We say the same thing about experiencing 21c. From highly anticipated annual events to weekly docent tours, there's always something going on at 21c Oklahoma City.
Docent Tours at 21c Oklahoma City.
Stay connected. Sign up for the latest on events, development updates, special offers and Oklahoma City travel packages.


text:
Q: Grandstream GXP2000 unresponsive I have two Grandstream GXP2000 hardphones that power on (the LEDs all come on green), but the display is blank, and the switch doesn't recognize the phones as being connected.
In the past, this has happened, except the phones still responded to the network, so I was able to reboot them using the web interface, and they worked again.  Simply unplugging it, and plugging it back in doesn't help.  I can't reset it to the factory settings, because the display is blank.
I am fairly certain that the phone is the problem, since I have plugged known good phones into the same power and network connections, and they worked.  Likewise, the bad phones connected to known-good power and network don't work.
From what I have read online, it sounds like this might be a common problem, but I thought I'd ask here to see if anyone knows any tricks.

A: From my experience, the GXP handsets were the worst I've ever used. I found much better results using their ATA converter and a cheap £5 handset.
I had problems with the shocking web interface not loading, phones not registering, NAT issues, STUN issues all sorts.
May seem slightly ostentatious but I'd replace them all for SPA941 or 942. You'll have much more fun using them than trying to keep them online :)

redpajama_set_name: RedPajamaStackExchange


Even if it says Mobile in title of this chipset, it is not mobile. Use older version You can download it here, http: We are going to use Game Booster and edit Graphics Properties for now. Have tried countless times to get this working on Win 8 Bit, have uninstalled old drivers, cleared registry, disable integrity checks and turned of driver signing checks. Add Poll to this thread.
Amit-The Learner April 27, at I'll have to give these a go. Didn't realise the guy making that utility still handn't released a app for X users, in that case I'll whip up a quick INF mod file for people to use.
Full readme for rev 2. They probably all do the exact same thing or have some minor variation to them.
Then it load default driver and should be warp. I have tried everything for months.
Maybe try running a 3D Benchmarking program while alternating settings and see if there is any change to the scores.
Would want to test it but have P and HD on my laptop.
One thing I would recommend is to try setting it in "Software Mode" – that should make the card emulate Vertex shaders on the CPU rather than hardware. Figured more people would have laptops here than it seems.
It does give you a slight boost. Skill Information and Ggm45 G. I'm not sure, you may try coz I didn't test Windows 10 Technical Preview yet already got the ISO, no time to install and play with it yet.
I did install intl RoyalBNA more than 10 times on 2 different laptops and it worked great every time. I followed this tutorial to play Portal 2. Anonymous November 11, at 8: Also if I have problems with deleting drivers, I sometimes use Driver cleaner pro after deleting drivers.
I used them more than year, year and half, maybe more without problems, and also many people use them. That's the first mobile version of the gma Anonymous June 3, at It should be capable of whatever resolution the device can support – and maybe a bit further – I can set mine as high as x, but it won't work with my monitor. Anonymous June 1, at 7: Perfetto, aggiornato col driver Royal v3 Scheda video prestazioni Aero: The problem is that none of them seem to do anything vm45 my ThinkPad x Here are the download links: Just one word for u: Maybe this has to do with a setting in the original IntelliModder32 driver.
Yeah, I noticed it's still broken. Thanks for your reply anyway. Each new driver development series hm45 somehow different. And you need to have cleaned computer, defragmented and disable all programs when playing. I noticed improvements in call of duty 4 too.
They say only W7, vista and W8 works for inel drivers. The function allowing me to go higher than native res is not there. I will test out some games tomorrow.
It's present in Windows 7, but not enabled by default and you could download it for Vista.


Lily Allen's Keane Cover To Return To No.1?
Lily Allen's Somewhere Only We Know could return to the top of the Official UK Singles Chart this weekend if midweek figures are to be believed.
However, in today's Official Chart Update, One Direction look like they're hot on her heels with Story Of My Life steadily climbing up to No.2.
This week no new entries seem to be making a dent in the Top 10 as current chart-toppers Calvin Harris & Alesso fall to No.3 with Under Control, Bastille climb one place to No.4 with Of The Night and Gary Barlow falls to No.5 with Let Me Go.
Eminem & Rihanna climb to No.6 with The Monster, Ellie Goulding climbs to No.7 with How Long Will I Love You and Avicii could have his fourth Top 10 single of the year with Hey Brother curenlty looking to climb to No.8 this week.
Former No.1 Martin Garrix climbs to No.9 with Animals while Lady Gaga & R Kelly round off the Top 10 with Do What U Want.
Find out who gets the gold by at 7pm on Sunday by following @MTVMusicUK on Twitter.


text:
Nicholas Bostick
Please read web version for embedded content
Post Malone's "Posty Fest" Will Include a Number of Underground Names
Nicholas Bostick | October 3, 2018 | 4:00am
Post Malone, Dallas' favorite New Yorker, has finally unveiled the rest of the lineup for his upcoming day-long festival. The event dubbed Posty Fest, which was announced to take place Oct. 28 earlier this year, is already being headlined by Malone himself as well as Travis Scott. However the rest of the lineup is a relatively obscure group of up-and-coming hip-hop artists for the most part.
Lil'Skies, the "Catch Me Outside" rapper Ski Mask and the Slump God are among a few that will also join. The bill, which almost reads like a playlist straight out of Malone's own phone, mostly features an eclectic mix of bleeding edge acts. Odd Future founder, Tyler, the Creator, will also round out the marquee acts for the festival.
Tyler made a surprise comeback last year, racking up a Grammy nomination for Flower Boy and putting out some of the best bars he has since 2013's Wolf. Getting Grammy nods is a big change for the 20-something rapper, especially after being banned from Britain and accused of inciting riots. Though he's far from the most notable act scheduled for Malone's festival.
A North Philadelphia native, artist/rapper Tierra Whack made more than a splash earlier this year after the release of Whack World back in March. The album is a 15-minute look into the mind of its creator, an audio/visual experience composed of 15 one-minute songs of varying styles with accompanying videos. With skull-cracking force, Whack's album shows the listener exactly what's on her mind, and her inclusion on the bill is perhaps the most intriguing so far announced.
Posty Fest is so far shaping up to be a pretty decent show, so long as your musical tastes align with Malone's. Some of the lesser known names like Saint JHN and Tyla Yaweh have a few decent tracks apiece that definitely take modern conventions to new places. The former's "3 Below," released last year, is a croon-ish slow burner and sleeper SoundCloud hit. Los Angeles-based Yaweh's "Drugs and Pain" has an impressive flow, despite his green lyricism and honey-bee high voice.
So it seems Posty Fest is all that it's been cracked up to be thus far, on paper at least. Rumors of special guests and surprise appearances still hang heavily over the event, so who's to say who will show up. With the likes of Tyler and Travis Scott anchoring the event, and the deep cuts in between, there's enough to be excited about just on face value.
Tickets start at $135.

redpajama_set_name: RedPajamaCommonCrawl


Greatest warrior of Mahabharat – Vinothbabu's Desk!
Mahabharata is one of the greatest literature's in the world. The beauty of Mahabharata is such it reflects our emotions and questions our wisdom every time you read. There is no equivalent work in history which brings out the complexity of human problems in such a profound and entertaining way. It shows us the application of philosophy when it comes to deciding between the right and wrong in such cases where the answer is not straightforward.
It always puzzles on who is the greatest warrior in Mahabharata whether its Bhisma, Drona, Arjuna or Karna. I would narrow down to these two as Arjun and Drona were greatest warriors but unmatchable against Bhisma or Karna.
Its almost tough decide on who is greatest between these two as both were disciples of parsuram and favorite too. Bhisma was invinsible, Had boon of Ichyamrityu-no chance of being killed. Truly invincible. Bhishma Pitamaha is the one who first called Krishna by name of "Vasudev". He is Ganga Putra and he is the only person ever who had the right choose his own death.
On the other hand we have Karna who could only be defeated only by a combination of curses that dwindled his prowess and also foul play by others. He was truely invincible through his skills. Its merely impossible to defeat had he not given his kavach and kundal to Indra.
3. Krishna's idea of using Ghatotkach, to complete his powerful vajra weapon.
4. Indra seeked Kavach and Kundal from him.
5. Arjun puts aside the rules of warfare.
6. His vachan to his mother- Kunthi.
1. His mastery in weaponry and warfare.
3. ICCHA maran varadan from his father.
But both were killed because of the side which they stood for, they bide dharma but stood on the side of adharma. so they were conquered through illegal means.
Bhishma being Bhishma was just too powerful. He was completely dominating the battlefield and the Pandavs were losing. Regardless of whether Arjuna had the will or not, he most certainly lacked the strength to back it in face of Bhishma so Krishna, partly due to frustration (since Pandavas winning is his only reason) and partly to hasten Arjuna to arrive at "defeat Bhishma by 'any means'", picks up a chariot wheel and charges at Bhishma.
Krishna wants to break that pride of Bhishma. He wants to tell him that "Dharma" is not just about following "Rules". If following a "vow" is causing wrong repercussions then one must not hesitate to break it. Therefore, Krishna in-spite of his vow to not fight in the war lifts the cart wheel. He tells Bheesma that he doesn't even need a weapon to kill him and only a cart wheel is enough for him. It is now when Krishna tells Bheesma about his mistaken sense of "Dharma". Bhishma lays down his weapon and asks Krishna to end his life.
This incident really shows us that if you act against the welfare of society, dharma, truth and morality, you're screwed.
The brahmin said, "Your life has been one of great generosity and goodness. I ask that you give me the effects of all that goodness (punya)." Karna signalled to the brahmin that he take out the arrow that pierced his heart. As Krishna did that, the blood flowed out of Karna and he breathed his last. So finally, the great warrior met his end at the hands of Krishna and attained salvation.
This incident proves the the superiority of Karna. Throughout his life Karna indeed did limitless dhana, and during his end times instead of being roused to uncontrollable anger , Karna was happy that even at death he could donate something. He pulled over an arrow from his body and , with the gushing blood , gave away all his punya ( merits ) earned through his charity.The whole battle field was amazed.Krishna assumed Visvarupa,showed His grace on Karna and granted him moksha. Though Karna was killed,he attained immortality through his charity and loyalty.
That's why people say there is nothing superior than giving dhaan/helping others.
If you belive that krishna is a god then you have to accept that karna was the greatest warrior. He was the unsung hero, the deserving winner, the struggler, the fighter, in courage more powerful than others. His only fault was he took the side of Duryodhana in Mahabharata, but alas- even then he was following his Dharma of friendship as it was Duryodhana who was Karna's one and all.
What do we learn from the great karna?


The joke was posted on Twitter on March 2, 2011.
I wanted to sue Ryanair because they damaged my luggage. I showed the damaged remains to my lawyer. He said "You don't have much of a case".
Wanted to sue the airline bcos they damaged my luggage.Showed the badly damaged remains to lawyer.He said "You don't have much of a case".
I showed the damaged remains of my luggage to my lawyer.
He replied 'You don't have much of a case'.


text:
Roger Federer and Rafael Nadal on course for Wimbledon semi-finals clash
Rafael Nadal at Wimbledon
Press Association · Columnist
15:00 · July 07, 2019 · 2 min read
Roger Federer and Rafael Nadal remain on course for a blockbuster semi-final showdown and their first Wimbledon meeting since their classic final in 2008.
Incredibly, the two most successful male players in history have not faced each other on the grass at SW19 since Nadal claimed his first title, 9-7 in the fifth, 11 years ago in what is widely regarded as the greatest match in the history of the Championships.
The next obstacles standing in their way, in the fourth round, are big-serving Italian Matteo Berrettini and gritty Portuguese 30-year-old Joao Sousa.
Berrettini is a lively outsider enjoying a decent grass-court season who saved four match points in beating Diego Schwartzman in five sets.
Federer will be facing the 23-year-old for the first time, and he said: "I don't know him very well, so that makes it a bit more tricky.
"I saw him play a little bit in Halle and I saw his run in Stuttgart. Now he's backing it up here again. That's not easy to do, especially when you're sort of newer on the tour.
"That's the way to go - start winning, going deep in those. You gain momentum, you start rolling at the bigger events. He did exactly that. I almost played him in the Halle finals when he ran into (David) Goffin, who I thought played a great match.
"I'm expecting a tough one. I hope he has no energy left. I'm sure he'll recover. He's young. I'm sure we'll see a tough match on Monday."
Berrettini was six when Federer lifted his first Wimbledon trophy.
"When I was a child playing tennis I was looking at him, the best player. I was cheering for him," he said.
"Now I kind of stopped cheering for him when I saw - I don't remember which tournament, but I saw his name and then my name in the draw.
"So it was, like, now I'm playing same tournament, I cannot cheer for him. But I grew up watching him. So for sure it's gonna be one of dreams."
Nadal is facing less of a step into the unknown than Federer, against Sousa, the man who ended Dan Evans' run.
"We know each other very well, we've practised plenty of times together," he said.
"He's a player who when he's winning matches, he's a super dangerous opponent against everybody. He is very quick, very good physical performance. He's aggressive with his shots, he's going for the shots all the time.
"He's a player that, sometimes he loses matches during the whole year. But when he starts to have a positive dynamic, is very, very dangerous."
Elsewhere, top seed Novak Djokovic continues the defence of his title against Ugo Humbert, Kei Nishikori plays Mikhail Kukushkin, David Goffin faces Fernando Verdasco and Guido Pella takes on Milos Raonic.
There is an all-American affair between Sam Querrey and Tennys Sandgren while Roberto Bautista Agut comes up against Benoit Paire.
Help your friends Know It All by sharing this article to your social media.
The Open: Thomas first for value
Justin Thomas can make another fast start in the Open Championship as Ben Coley previews the first round.
Last updated 9hGolf
The Open: Strike while irons are hot
Ben Coley expects precision approach play to be the key to conquering Royal Portrush in this week's Open Championship. Don't miss his in-depth betting preview.
Last updated 19hGolf
Last updated 12hRacing
Dettori in Killarney: Verdict
David Ord assesses Frankie Dettori's four mounts as the jockey visits Killarney for the first time on Wednesday evening.
Sporting Life Tipping Record
A look at how the Sporting Life recommended bets have performed in July 2019.
Last updated 9h
Saturday's Bets of the Day
We have selections from our racing team to help boost those Royal Ascot coffers, plus Richard Mann's cricket advice.
Last updated 1M
Our best bets
Check out Sporting Life's best bets across racing, football and more.
All Football TipsTips & Previews


The Memorial Day parade was canceled because of rain, but the rest of Monday's activities went off without much of a hitch.
Town officials had moved the speaking program for the town's traditional holiday observance into one of the large display buildings at the Brooklyn Fairgrounds when the steady rain, punctuated by persistent flashes of lightning and the rumble of thunder, convinced First Selectman Austin Tanner to call off the parade.
Tanner said ceremonies would be held at the town's veterans monuments and at South Cemetery later Monday if the weather allowed.
Principal speaker Robert Kelleher reminded the crowd in the exhibition hall that Memorial Day is set aside to honor those who have defended America's way of life, especially those who gave their lives fighting to preserve the freedoms we enjoy today.
Kelleher said the holiday now had a personal connection for his family because his older son's sister-in-law left last month for a year's deployment in Iraq with her Connecticut Army National Guard unit.
In traditional parts of the ceremonies, Brooklyn Middle School students Angelise Martin and Erin Bagdoian, respectively, recited, "Cover Them Over With Beautiful Flowers" and Lincoln's Gettysburg Address; and Drew Dionne read Gen. Logan's orders that established Memorial Day.


MELBOURNE, AUSTRALIA - Pinnacle Health Group, a fast-growing provider of physiotherapy, other Allied Health and GP services in Australia, today announced the launch of a national Telehealth initiative in collaboration with patient engagement and Telehealth solution Physitrack. Thanks to this initiative, Pinnacle Health Group patients can access consultations with their GP, physiotherapist, dietitian or psychologist from the comfort of their home or workplace via Physitrack's secure PhysiApp patient app.
Physitrack is one of a select number of healthcare companies, which closely collaborates with Apple (NASDAQ: AAPL) as part of their ground-breaking Apple Mobility Partner Program.
Pinnacle Health Group has been an early adopter of Physitrack's patient engagement and Telehealth solution, extending the scope of their services to offer smartphone, tablet and computer-based video consultations before, during or after in-patient treatments, while at the same time providing patients with clinical home exercise programs, educational content or clinical outcomes tracking, all uniquely integrated into the video stream.
"We place a significant focus on continually improving the patient experience – from better facilities, customer service and accessibility, to better use of technology and adding value to health consultations through ongoing follow-ups.
"Physitrack and its PhysiApp patient app have enabled our patients to access our prescribed home exercise programs and educational content outside of the treatment room, and communicate with our practitioners via the built-in messaging and Telehealth features – which all leads to better adherence to treatment plans and overall improved health outcomes".
Pinnacle Health Group operates from eight clinics across Melbourne and Sydney, including their Corporate Wellness Centres at large companies such as ANZ Bank and Westpac, and specialises in delivering physiotherapy, other Allied Health and GP services.
Physitrack is a privately held UK company with offices in Melbourne, London, Houston, Santa Monica, Amsterdam and New Delhi. The company provides healthcare companies with an easy-to-use patient engagement solution for clinical home exercise prescription and patient education, patient outcomes tracking and secure Telehealth video consultations.


Tiny Betty Christmas is a joy to behold in a red skirt and dark green felt jacket that has red trim worn over a white ribbed blouse. The skirt has a checked waistband that matches the hair band in her dark curly hair. White undies, painted white socks and black shoes completes Tiny Betty Christmas ensemble and goes beautifully with her blue eyes. Retired item - available for limited time.


Thin fries topped with beer cheese, smoked bacon, and green onion.
Thin fries topped with shredded cheddar, smoked bacon, beer onions, and pub sauce.
Three beef sliders topped with a trio of cheeses: pepper jack, Swiss, and American.
Breaded onion petals served with house made ranch.
Breaded cheese curds served with house made ranch.
Breaded pickle chips served with house made ranch.
Tortilla chips served with fire roasted salsa.
Shredded pork, shredded cheddar, onion, green chives, and sour cream, drizzled in our house-made BBQ sauce on a bed of waffle fries.
Beef or chicken, tortilla chips, nacho cheese, lettuce, onion, tomato, jalapenos, black olives, fire roasted salsa, and sour cream.
Buffalo, BBQ, sweet chili, mango habanero, stinging honey garlic.
Spring greens, tomato, onion, cheese, and croutons.
Grilled chicken tossed in Buffalo sauce, spring greens, tomato, onion, shredded cheese, and blue cheese crumble.
Grilled chicken breast, romaine, Parmesan cheese, croutons, and Caesar dressing.
Chipotle black bean patty, romaine, avocado, tomato, jalapenos, and shredded cheese, topped with a cilantro-lime creme and fire roasted salsa.
Served with choice of French fries, waffle fries, or kettle chips (sub salad for extra charge). All burgers can be subbed with a black bean patty.
Applewood smoked bacon, American cheese, lettuce, tomato, and onion.
Bleu cheese crumbles, Swiss cheese, lettuce, tomato, and onion.
Pepper jack cheese, avocado, lettuce, tomato, and onion.
Shredded cheddar, applewood smoked bacon, and fried egg.
Swiss cheese, sauteed onions, and mushrooms, on a marble rye bun.
Shredded cheddar, applewood smoked bacon, onion strings, and BBQ.
American cheese, beer infused onions, pub sauce, lettuce, tomato, and pickle.
Shredded mozzarella, bacon jam, topped with a garlic aioli.
Pepper jack cheese, avocado, topped with a cilantro-lime creme, on sourdough.
Two 1/4 pound patties, Swiss cheese, American cheese, lettuce, tomato, and onion.
Swiss cheese, sauerkraut, beer onions, and honey Dijon mustard.
Two half pound patties, four slices of bacon, American cheese, Swiss cheese, beer onions, and beer cheese, topped with an egg.
Wraps & PitasToggle visibility of menu section "Wraps & Pitas"
Served with choice of French fries, waffle fries, or kettle chips (sub salad for extra charge).
Fried chicken, applewood smoked bacon, lettuce, tomato, shredded cheese, and ranch dressing.
Fried chicken tossed in Buffalo sauce, lettuce, tomato, shredded cheese, blue cheese crumble, and ranch dressing.
Grilled chicken tossed in spicy honey garlic sauce, lettuce, tomato, crunchy noodles, and ranch dressing.
Grilled chicken, lettuce, applewood smoked bacon, Parmesan cheese, and Caesar dressing.
Chipotle black bean patty, shredded cheddar, lettuce, tomato, avocado, and cilantro cream.
Grilled chicken, feta cheese, tomato, onion, black olives, and greens tossed in a Greek vinaigrette.
Grilled chicken breast, lettuce, tomato onion, and mayonnaise.
Applewood smoked bacon, lettuce, tomato, and mayonnaise on sourdough bread.
Grilled chicken breast blackened, pepper jack cheese, and spicy mayonnaise.
Grilled chicken breast, American cheese, applewood smoked bacon, and avocado.
Seasoned pork topped with our house made whiskey BBQ sauce.
Homemade meatloaf topped with our gold sauce, served on sourdough.
Fried haddock, shredded lettuce, tomato, tartar, and pickles.
Seasoned pork, ham, Swiss cheese, and pickles, drizzled in gold sauce.
Grilled chicken, pepper jack cheese, onions, peppers, and jalapenos.
Shaved prime rib, Swiss cheese, sauteed onions and peppers, served with a side of jus.
Battered haddock served with fries and sourdough toast.
Fried top sirloin seasoned served with fries and sourdough toast.
Breaded tenders served with fries and sourdough roast.
Shrimp served with fries and sourdough roast.
All kids meals include choice of drink.
Vanilla ice cream, fried banana foster, caramel, chocolate, topped with whipped cream and cherries.
Served with tortilla chips and toasted pita bread.
Seasoned pork topped with mango salsa and chipotle sour cream.
Beef or chicken, tortilla chips, nacho cheese, lettuce, onion, tomato, jalapenos, black olives, fire roasted salsa and sour cream.
Buffalo, BBQ, sweet chili, mango habanero, string honey garlic.
Spring greens, tomato, onion, cheese, croutons.
Fried chicken tossed in Buffalo sauce, spring greens, tomato, onion, shredded cheese, and bleu cheese crumble.
Chipotle black bean patty, romaine, avocado, tomato, jalapeno, shredded, cheese, topped with a cilantro-lime creme and fire roasted salsa.
Cajun top sirloin, spring greens, onions, tomato, green pepper, bleu cheese crumbles, tossed in Cajun vinaigrette.
Served with choice of French fries, waffle fries, or kettle chips.
Apple wood smoked bacon, American cheese, lettuce, tomato, and onion.
Swiss cheese, sauteed onions and mushrooms, on a marble rye bun.
Pepper jack cheese, sauteed jalapenos, and spicy mustard.
Grilled chicken tossed in honey garlic sauce, lettuce, tomato, shredded cheese, crunchy noodles, and ranch dressing.
Grilled chicken, lettuce, tomato, applewood smoked bacon, Parmesan cheese, Caesar dressing.
Chipotle black bean patty, shredded cheddar, lettuce, tomato, avocado, chipotle sour cream.
Grilled chicken breast, lettuce, tomato onion, and mayo.
Apple wood smoked bacon, lettuce, tomato, and mayo on sourdough bread.
Grilled chicken breast, American cheese, applewood smoked bacon, avocado.
Seasoned pork, house made whiskey BBQ sauce, topped with fried pickles.
Fried Cajun shrimp, lettuce, tomato, drizzled in remoulade sauce.
Seasoned pork, ham, Swiss cheese, pickles, drizzled in gold sauce.
Season chicken, pepper jack cheese, sauteed onions, peppers, and jalapenos.
Seasoned grilled top sirloin served with fries and sourdough toast.


Eddie Byrom, Zimbabwean born with an Irish passport, made his Championship debut for Somerset in their first-ever floodlit first-class match, against Hampshire at the Ageas Bowl in 2017.
A dependable left-handed opener of great potential, he then built on the experience of eight Championship matches with an attritional maiden first-class hundred in Harare on his debut for Rising Stars (a collection or young talent from different franchises) against Matabeleland Tuskers in the Logan Cup.
Byrom, the son of a former Zimbabwean sports journalist, had moved to Kings College in taunton two years earlier and was signed by Somerset on a scholarship contract. Before his move, he attended St John's College and played regularly for Zimbabwe's age group teams between 2011 and 2015.


text:
Tag: Trinidad and Tobago
General Grants Says, "I Want My Flowers While I'm Still Here."
The name 'General Grant' is one that most people in Trinidad and Tobago and across the region, know and appreciate.
Alison Hinds, Nessa Preppy Heat Up BLOCKO.
The ladies did not come to play.
Carnival Is Here! This Exec Producer Aims to Highlight 'Everybody.'
A new riddim that pushes pure talent and performance skill has hit the airwaves ahead of Carnival 2023 and the artistes are all ready to make magic in the season of revelry.
Watch Matara 'Ride!' This Carnival The Social Media Influencer's A Soca Jockey!
Well look at this now!
"Hold on To Soca. It's Yours." The Revelry Has Begun.
Like clockwork, the music's on time.
Leonce, Rayzor Take on T&T with New Music Ahead of C2K23.
Two of Trinidad and Tobago's foreign based artistes are back and revving to go.
Debit Card Holders Can Now Buy Event Tickets Online at Island E Tickets.
Recent confirmation that internationally sought-after megastar, Burna Boy, would be performing live in Trinidad this December, has been met with even bigger news for Trinidad and Tobago.
Destra Brings The Heat With Spicy New Video.
The heat is here!
Maloney Native Preedy Keeps Showing Up for His Community and We're Here for It!
Maloney is being highlighted by one of its own for something positive.
Blackie's Back with 'Do Not Disturb' for Carnival 2023.
Never before has the anticipation for Trinidad and Tobago's carnival – the mecca of them all, been this high!


Behind The Blogger: Chatting With Ashley From She Is Blank Space + Giveaway!
Behind The Blogger: Chatting with Ashley from She Is Blank Space!
Today we would like to introduce Ashley, owner of the lifestyle blog, She Is Blank Space! Check out Ashley's latest blog post, Renovating Her Office Space featuring stationery and decor from Graphique! And make sure you enter her giveaway on Instagram to win the items she featured! The giveaway runs from today until Friday, February 24th! All you have to do is head over to She Is Blank Space to read Ashley's blog post, follow us on Instagram @graphiquedefrance, and then enter to win at @sheisblankspace. Now, let's get to know a little more about Ashley!
My name is Ashley Solberg. My blog is She Is Blank Space and I'm from Tampa, FL.
After having my three kids, I felt a little lost as a person. The last two were twins, and I got so immersed in changing diapers and keeping one of them from jumping off the dining room table, I forgot about me. I started our blog to regain focus on some of the things I love: encouraging other women, fashion, food and motherhood (in no particular order:)).
How did you go about finding the other 17 contributors for your blog?
There are some women that you just connect with a little extra, and with our writers, it just felt right. They are all at different ages and stages in life, so it's perfect for our varied audience so everyone has something that speaks to them.
Oh boy, not sure you're ready for all of it, but I'll summarize! Here goes nothing: Wake up before the sun, pack lunch for my oldest, get her off to school, get the twins dressed, respond or send lots of emails, play time with twins, take photographs for blog or social media, format our blog entries to share, coordinate with our writers, dream a little, try to conquer mountains of laundry (emphasis on try), call my hubby to say hello, pick up oldest from school, snack time, answer a few more emails, finalize blog entries, dinner time, bed time for littles, quality chat time with hubby with electronics off, back to work until the wee hours, catch some Zzz's, then start all over again. I'm sure there was a tantrum or two I managed in there somewhere too!
How do you manage running a blog while being a mom to 3 kids?
I keep them important firstly. I don't ever want them to feel less important than my phone, computer or anything else I'm working on. So, when we are spending time together, I'm all in. Tea parties, dinosaur stomps, dress up–the focus is all on them. I look into their eyes when I talk to them, hold them close, tell them words of affirmation. I've found when they feel fully attended to in those moments, they are ok when mommy takes a few minutes to work alongside them.
What are some of your favorite products from Graphique and why?
I am a words girl, I mean as a blogger words are super important, but they are to me directly too. I love the wall art, pens, and notebooks with fun, but positive, sayings on them. The world has enough negativity, so I choose to surround myself with positive people and a work space that also provides the same.
Do you have any advice for others that are looking to start a blog?
Be Y-O-U! We are a very unique set up with so many writers. I used to wonder how we would fit in the blogging world, but then realized I didn't want to be limited by what may be the standard as we should be ourselves. I hoped people will feel our genuine personalities shining through and appreciate who we are.
What is the biggest lesson you've learned since starting She Is Blank Space?
Trying not to get choked up here, but I've realized to go after your dreams, even if they seem out of reach or different from what you imagined. Doing this was not something I imagined in any way, but so glad I said yes to my heart. So many women are reached and encouraged as we support each other in life and it feels like we are still just beginning.
Love and grace. They are so hard sometimes, but that's why they are important to me. It takes a lot to love people with grace where they are sometimes, as we are human. But, we all make mistakes and should always treat others the way we would want to be if the shoe were on the other foot. So even in misunderstandings, wrong doings or just flat out hard days, I still choose love and grace as it's much better than the alternative.
Thanks Ashley! You can visit She Is Blank Space here, and don't forget to check out the giveaway to enter for your chance to win some new accessories for your desk!


The fatigue crack growth rate behavior of a SAE1045 steel in the as received condition and four different quenched and tempered conditions was studied as a function of stress ratio and peak compressive overload. The threshold stress intensity behavior of the quenched and tempered conditions was not sensitive to changes of monotonic mechanical properties. The threshold decreased linearly with increasing positive stress ratio and compressive peak load level. As received ferritic-pearlitic SAE1045 steel was much more sensitive to stress ratio and compressive peak load than any of the quenched and tempered conditions studied.
Yu MT, Topper TH. The Effects of Material Strength, Stress Ratio, and Compressive Overload on the Threshold Behavior of a SAE1045 Steel. ASME. J. Eng. Mater. Technol. 1985;107(1):19-25. doi:10.1115/1.3225765.


From its opening in 1992, the Eye Institute has grown in terms of the breadth of expertise in vision related research. Due to the efforts of six basic scientists, well-established research programs in cellular and molecular biology, biomedical instrumentation, electrophysiology, optics, psychophysics, and visual performance have been developed.
In addition, there is a strong clinical science expertise in many areas including epidemiology, ocular pathology, ocular genetics, neuro-ophthalmology, cornea, uveitis, and glaucoma.
Development of instrumentation that will give physicians new, non-invasive ways to measure the health of the eye in patients.
All of these new discoveries and technologies have been accomplished due to the collaborative efforts of both clinicians and basic researchers working closely together.
New developments have been made in the creation of an artificial cornea for transplantation, which potentially can provide visually impaired or blind individuals with useful vision. Also, cell-regeneration studies are underway to discover methods of preventing or repairing ocular damage that results in loss of vision, from such diseases as glaucoma, macular degeneration and diabetes.
The Eye Institute, from its opening, has embraced the investigation of new state-of-the-art technology as a primary focus. As a result of these studies, the Ottawa community has been able to receive the latest diagnostic tests, and be treated with new therapies for ocular inflammation and lasers for glaucoma, macular degeneration and corrective eye surgery.
Part of the success of the research effort at the Eye Institute lies in its collaborative and multidisciplinary approach to problem-solving. Through collaboration, scientists and clinicians have developed successful research programs in a number of areas, including refractive surgery, tissue engineering, biomedical instrumentation, and ocular genetics, amongst others. Few eye care facilities around the world have been able to achieve the same level of cooperation and interaction.


text:
Feature Wreck: M24 Midget Submarine
The fascinating story of the M24 Japanese Midget Submarine
Credit: Liam Allen
For more than 60 years the whereabouts of the M24, a Japanese midget submarine, was one of the greatest Australian wartime and maritime mysteries. Three midget submarines invaded Sydney Harbour on the evening of 31 May 1942. That night, the harbour was full of Allied naval vessels and the midget submarines were on a mission to inflict maximum damage.
Two of the midget submarines (M22 and M27) were destroyed almost immediately and recovered from Sydney Harbour within a week, but the third (M24) could not be found.
The M24 was the only submarine of the three able to launch its torpedoes, and with terrible effect. It sank a naval vessel, HMAS Kuttabul, which before the war had operated as a ferry. Twenty-one men on board were killed and another 10 injured.
There were many theories about what might have happened to the missing submarine and many so-called 'discoveries'.
It was not until November 2006 that a group of weekend divers called 'No Frills Divers' located the still-intact Japanese midget submarine M24 off Bungan Head, Newport (Sydney, Australia). The submarine was on the seabed and entangled in nets 54 metres below the surface.
Like all shipwrecks, the M24 has a fascinating story to tell. It played a major part in the events in Sydney Harbour on the night of 31 May and the early morning of 1 June 1942 and its discovery recalls the role of Japanese midget submarines and the Japanese submarine campaign along Australia's eastern seaboard during World War II.
Heritage NSW continues to actively manage the site and undertakes an ongoing program of archaeological surveys.
Following the discovery of the M24 submarine, the NSW and Australian governments jointly protected the site under the NSW Heritage Act 1977 and the Commonwealth Historic Shipwrecks Act 1976. The NSW Government continues to work collaboratively with the Commonwealth and Japanese governments to protect this highly significant site.
M24 public trial Diver Open Day
For over 10 years, a wide variety of detailed archaeological surveys, research, and conservation efforts have been undertaken on the M24. To help this valuable work take place, a moratorium on diving was placed on the wreck with only archaeological and scientific diving permitted.
To coincide with the 75th anniversary of the 1942 raid Heritage NSW, in collaboration with the Commonwealth Historic Shipwrecks Program, decided that the M24 site could be opened for limited and controlled public access on a trial basis. Twelve permits were allowed and to ensure a fair and equitable allocation of permits a public ballot was held. More than 300 applications were received for the 12 permits.
In November 2017, two groups of six divers were allowed access to the M24 Protected Zone. This trial Diver Open Day showed that controlled and respectful public access to the M24 site was possible. OEH is currently working with the Commonwealth and Japanese governments to find ways of opening up the site to recreational divers on a regular basis.
M24 3D model
For the first time a high-resolution 3D model of the M24 has been produced. This model provides an unprecedented level of detail of the site and accurately shows what the submarine looks like 54 metres below the surface. The model will provide a baseline recording that will allow OEH to monitor changes to the wreck over time and it will also allow the public to 'visit' the site without having to get wet.
The 3D recording project was initiated by Office of Environment and Heritage with help from the Australian and New Zealand Chapter of the Explorers Club led by maritime archaeologist Matt Carter who was granted a special-access permit to undertake the diving survey. The 3D model was produced in partnership with ARCHAEOTecnic and Tempus Archaeology.
Credit: 3D Animation of M24 Midget Submarine Courtesy of ARCHAEOTecnic and Tempus Archaeology


The new edition of the London Reader is Drama & Dragons: Stories about games and growing up. It's a collection of fiction, poetry, interviews, and art that reflects on the effect that RPGs and other fantasy games have had on our lives, stories, and culture.
I wrote one of the introductions, and I think this issue is something that many of you will enjoy. There's an interview with Margaret Weis and – in a great nod to the old-school game magazines – no less than three one-page RPGs. "Complete in this issue!" as we used to say.
To follow the magazine on social media . . .
To subscribe now . . .


This lovely Orgonite® pendant contains aluminium,bronze, brass and copper shavings. The main stone is natural Citrine which you can see in the front around thethird chakra symbol. Rhodizite (to amplify the other crystals), Pyrite (to enhance), black Tourmaline (to ground excess energy and for protection), Selenite (to clean and charge the other crystals) and clear quartz (for piezo electric effect) in a durable resin setting.
It consists of lots of metal shavings which is required for the orgonite® to properly remove stagnant energy according to William Reich's research.
It is the center of your personal power, ambition, emotion and is where feelings and sense of touch are processed. This area is where our "gut" feeling comes from and your instinctively yes/no response. Stress has a huge impact on this Chakra and when it is not balanced we often feel drained. The idea of trying to do too much, running around (like a headless chicken), multitasking, trying to be in too many places at one time makes a person unfocused and exhausted. This also has the same effect when we have ideas/thoughts but never put these into action. A balanced third Chakra allows us to feel centered, have self esteem, self respect, be less sensitive to criticism and therefore, can care for yourself and others. Also, a balanced third chakra allows you to have control over your thoughts and emotional responses, set healthy boundaries, and be at peace with yourself.
Its physical association is with digestive function.
Because Citrin can clear negative energy and influences from the aura, it is useful for meditation, psychic awareness, and spiritual development. By removing unwanted energies it paves the way for this spiritual and psychic growth. It is also excellent for dream recall and dream work.
In the mental arena, Citrin enhances mental clarity, confidence, and will power. The mental clarity and confidence support bringing increased creativity and honesty.
Since Citrin brings self-confidence as well as positive energy, it can also help eliminate fears of being judged or of others' ideas. It is also used to dissipate fear of being alone or unworthy of love.
Citrin is a happy stone. It can bring happiness to the person who carries or wears it. Because of its removal of negative energy and its own bring positivity, it can relieve depression, self-doubt, anger, and irrational mood swings.
Citrine is used to reduce self-destructive tendencies. Citrin can help overcome emotional traumas and grief that may lead to these emotionally self-destructive issues.
Citrin also does not absorb any negative energies from its surroundings, and thus never needs energetic clearing.


text:
\section{Introduction}
\label{sec:intro}

The stars of the Milky Way and other galaxies are embedded in the interstellar medium (ISM), a mixture of gas, cosmic rays and magnetic fields.   The ISM is an important and active ingredient in the Galactic system despite comprising only about $10\%$ of the total baryonic mass of the Galaxy  \citep{ferriere01}.
New stars form from cold, dense, parts of the ISM, while stellar evolution driven by energy release from thermonuclear reactions returns some of the stellar mass to the ISM via stellar winds and supernova explosions.   This injection of energy generates turbulent motions and shocks in the ISM, producing a highly heterogeneous random structure.
Accurate knowledge of the spatial distribution of the ISM is required to understand the properties and evolution of galaxies.

One way to probe the ISM is to observe
neutral atomic hydrogen (H\,{\sc i}), as about  $90\%$  of atoms in the interstellar gas are hydrogen \citep{kalberla09}. H\,{\sc i} emits and absorbs radio waves at the frequency of $1420$\,MHz  and large data sets are now available for detailed analysis.  Figure \ref{fig:gass} shows H\,{\sc i} distribution in a section of the southern sky.  These data were obtained by the Galactic All-Sky Survey (GASS) using the Parkes $64\,{\rm m}$ radio telescope \citep{mclure09, kalberla10}.  The second and third releases of the data are available at {\tt http://www.astro.uni-bonn.de/hisurvey}. The figure shows the antenna-temperature distribution  $T(l, b)$, which is related to the gas density, as a function of position on the sky, using the coordinates of Galactic longitude, $l$, and latitude, $b$. The distance to the gas cannot be measured directly, but the Doppler-shift of the emission, dominated by the differential rotation of the Galaxy, produces a line-of-sight velocity $v$ that can in principle be used to determine the location of the gas in three dimensions. The transformation is complicated, however, and not necessary for our purposes. Instead, we obtained the two-dimensional data in Figure \ref{fig:gass}  by integrating   $T$ over  velocities from $v=20.6$ to $v=40.4\,{\rm km/s}$.

\begin{figure}[t]
\begin{center}
\includegraphics[width=\textwidth]{gass.pdf}
\end{center}
\caption{\it Emission of neutral atomic hydrogen \textup{H}\,{\sc i} in a region of the Southern sky \citep{mclure09}. The coordinates are Galactic longitude $l$ and latitude $b$, centered on the Sun, where the mid-plane of the Galactic disc is at $b=0^{\circ}$ and the Galactic centre is in the direction $l=0^{\circ}$.  The brightness temperature shown is proportional to the total mass of  \textup{H}\,{\sc i} along the line of sight.  Dashed squares from left to right identify the Regions 1--3 used in Section 6.  } \label{fig:gass}
\end{figure}

The GASS data and other surveys are rich enough to allow subtle comparisons between observations and the results of sophisticated  magneto-hydrodynamic (MHD) simulations of the ISM. The data are represented as random fields with large-scale gradients and a complex topology widely believed to be related to turbulence and outflows from the Galactic disk. At a more local level, Gaussian random fields (GRF), or simple transformations thereof, have underpinned the modelling of small-scale variations in H\,{\sc i}.  In this paper we investigate whether these models are sufficient to describe the small-scale properties of $T(l, b)$.

We consider the three regions marked by dashed lines in Figure \ref{fig:gass}, which we refer to as Regions 1--3, moving from left to right. Each consists of a $256 \times 256$ array of temperature values.  The selection of the regions was arbitrary: we did not consider any astronomical information about the locations when drawing their boundaries.  In order to concentrate on small-scale variation we removed the trend from the plots by fitting to each region a polynomial surface of order four in each of $l$ and $b$. The residuals were then marginally transformed to N(0,1).  If this transformation results in a realisation of a GRF then all information would be captured by the correlation function.  We therefore consider two questions.

\begin{enumerate}
\item[]  Q1. Are the transformed data sets consistent with stationary isotropic Gaussian random fields?
\item[]  Q2. Are there differences between the three data sets, to which the correlation function is insensitive?
\end{enumerate}

We will address both questions using techniques  in topological data analysis, which is becoming  a popular approach
to the analysis of random fields and more generally \citep{ ABBSW2010, AdTay2011, bubenik15, CarGun2009, Ed2014, fasy14, YA2012}. Topological invariants such as Betti numbers, the Euler characteristic, persistence diagrams and persistence barcodes, rank functions and landscapes, have been used in areas such as astrophysics \citep{Li2016arXiv}, cosmology \citep{Gay2010, Sousbie2011part1, Sousbie2011part2,  Pranav2015}, fluid dynamics \citep{Mischaikow2015arXiv, Li2016arXiv} and medicine
\citep{Davis2008, Chung2009, Chung2014arXiv}. A difference in our case compared with most previous work however is
that we have just a single observation for each region, so that inferential techniques based on sampling and asymptotics are not appropriate.

In Section \ref{sec:sums} we describe several topological summaries that are appropriate for data on two-dimensional lattices.
In   Section \ref{sec:grf} we study characteristics when a GRF is appropriate.
In Section \ref{sec:fields} we demonstrate that non-Gaussian random fields can sometimes be distinguished by topological features even though first and second order properties (marginal distribution and correlation function) are the same. In Section \ref{sec:tests} we propose a simple  procedure for comparing two single realisations of random fields and in Section \ref{sec:app} we describe our analysis of the GASS data.

\section{Topological descriptors of a random field}
\label{sec:sums}

Here we describe a number of topological measures that are suitable for analysing data that are distributed on a two-dimensional rectangular lattice.  For more general definitions and interpretations and further information see, for instance, \cite{adler10,bubenik15} or \cite{fasy14}.

\subsection{Level sets, persistent homology and persistence diagrams}

Let $z(x)$ be the value of a random field at location $x$ on a two-dimensional lattice. For any real $t$, the {\it lower-level set} is defined as the locations that have field values below $t$,
\[  \mathcal F_t= \{x: z(x)\leq t \}.\]
Increasing $t$ from below defines a filtration which is used in {\it persistent homology} to describe the evolution of  topological structures in the field. In our context, there are two \emph{topological features} of interest, namely \emph{components} and \emph{holes}, whose counts in a level set determine
  the {\it Betti numbers} of order zero and one, $\beta_0$ and $\beta_1$, respectively \citep{CarGun2009}.  Figure \ref{fig:grid} shows four lower-level sets for a simulated field on a $10 \times 10$ lattice and will be used to illustrate some basic concepts.

\begin{figure}[h]
\begin{center}
\includegraphics[width=4.5in,height=4.4in]{gridfig.pdf}
\end{center}
\caption{{\it Four level sets (black) for a field on a $10 \times 10$ lattice.} \label{fig:grid}}
\end{figure}

A {\it component} is a group of one or more  pixels in a lower level set that are connected to each other, where for now we define neighbouring pixels to be connected if they have a common edge, and non-neighbouring pixels
to be connected if there is a path of connected neighbours between them. The Betti number of order zero, $\beta_0$,  for $\mathcal F_t$ is  the number of components in the level set.  In Figure \ref{fig:grid} for instance, in panel (a) at $t=-2$ we have $\beta_0=3$, as we have assumed that pixels that share only a vertex are not connected.  By $t=-1$ and panel (b) we have $\beta_0=7$, and then $\beta_0=5$ and $\beta_0=1$ in panels (c) and (d) respectively. The emergence of a new component is described as a {\it birth} and the merger of two components is interpreted as the continuation of the component with the earlier birth time and the {\it death} of the other.  In a two-dimensional field each local minimum is associated with the birth of a component.

A {\it hole} is a group of one or more pixels that are not in the level set, are connected to each other but are isolated from other pixels that are also outside the level set.  The Betti number of order one, $\beta_1$, for $\mathcal F_t$ is  the number of holes in the level set.  Thus in  panel (d) of Figure  \ref{fig:grid} we have $\beta_1=9$ as we ignore common vertices.  In panel  (c) we have $\beta_1$=5,   in (b) $\beta_1=2$ and in (a) we have $\beta_1=1$.  New holes are created when existing ones split, and again we can define their birth and death levels $t$. The death of a hole is associated with a local maximum. Holes could alternatively be defined by symmetry as components in a similarly defined {\it upper} level set, with the filtration now running from high to low $t$.  We will always refer to lower level sets so as to  avoid confusion.

A {\it persistence diagram} is a scatterplot of birth levels against death levels  for features of interest, in our case either components or holes.  The left plot of Figure
\ref{fig:pers} shows a persistence diagram for components in Region 1 of the transformed GASS data.  The first component to be born is by construction the last to die, producing the single point in the top left.   Otherwise, the points are clustered in a loose oval. Points near the diagonal represent less persistent structures mostly associated with noise, whereas more significant features are usually associated with points away from the diagonal.

\begin{figure}
\begin{center}
\includegraphics[width=6in,height=3.5in]{persfig.pdf}
\end{center}
\caption{{\it Component persistence diagram (left) and 90\% convex peel (right, as dots) for Region 1 in the GASS data. Grey lines in the right panel represent the peeled convex hulls.} \label{fig:pers}}
\end{figure}

\subsection{Convex peels and summary statistics}

It is sometimes difficult to interpret or compare persistence diagrams, either because of the large number of points or the bunching of many points along the diagonal.  We propose peeling successive convex hulls \citep{barnett76} until only a prescribed proportion of points remain, as illustrated in the right panel of Figure \ref{fig:pers}. In this way we extract the general shape of the persistence diagram without undue influence of either outliers or the mass of points near the boundary.  We summarise the shape of the final convex hull by the following five statistics:

\begin{enumerate}
\item the two centroid coordinates $(C_b,C_d)$;
\item the  perimeter $P$;
\item the area $A$;
\item the filamentarity, defined as \citep{bharad00,makarenko15}
\[ F = \frac{P^2-4\pi A}{P^2+4\pi A},\;\;\;\;\;\;\;0 \leq F \leq 1. \]
Thus defined, $F=0$ for a circle and $F=1$ for a line.
\end{enumerate}

\subsection{Bottleneck distance}

The bottleneck distance provides a measure in the space of persistence diagrams and can be used as a quantitative
summary of the difference between two persistence diagrams, $A$ and $B$ say. A readable description, and further information, is given by  \cite{fasy14}.

First, the points in $A$ are matched to the points in $B$. This means that each point in $A$ is mapped  either to a unique point in $B$, or to its projection onto the diagonal line of birth-death equality.  The same is true for $B$.  Use of the diagonal is necessary because $A$ and $B$ can have different numbers of points.

The {\it cost} of a mapping from  a point $(a_{ix},a_{iy})$ in $A$ to a point  $(b_{jx},b_{jy})$ in $B$ is the norm
\[ c_{ij} = \max\{ \mid a_{ix}-b_{jx} \mid, \; \mid a_{iy} - b_{jy} \mid \}. \]
The total cost  of a matching $P$ between $A$ and $B$ is $C(P)=\sum_{i,j}c_{ij}$.  The bottleneck distance is
then
\[ w(A,B) = \min_P C(P), \]
where the minimum is taken over all possible selections of the linked pairs.
This, the bottleneck distance,  is essentially the smallest cost that could be incurred in mapping all points of $A$ into all points of $B$. Calculation is  numerically expensive, though efficient algorithms are available. For our application, we used the {\tt tda} package in R.

\section{Topology of a Gaussian random field on a lattice}
\label{sec:grf}

The persistent homology of a two-dimensional random field depends upon the number and distribution of local maxima and minima. In this section, we investigate the number of local extrema in a Gaussian random field on a  $d\times d$ lattice.  The results will be used in Section 6 to benchmark an analysis of the H\,{\sc i} data.  We assume stationarity, isotropy and N(0,1) margins throughout.

 The density of  critical points of a Gaussian random field in continuous space is a well-studied problem
\citep{bard86}. It depends upon the joint Gaussian distribution of function values and first and second derivatives.  We are not aware of published  results for local extrema of a field on a discrete lattice however, although there has been work on the distribution of the global maximum over a region \citep{taylor07}.

\subsection{Expected number of local extrema}

Given a field $z(x)$, let $N_0$ and $N_1$ be the numbers of points in persistence diagrams for components and holes respectively.
As previously stated, each point in a persistence diagram for components  corresponds to a local minimum of the field, and each point in a persistence diagram for holes corresponds to a local maximum. Due to  the symmetry of the Gaussian distribution, the number of local minima has the same statistical properties as the number of local maxima.  Because it is slightly tidier notationally, we will consider local maxima in this section.

Let $z_1=z(x_1)$ be the (scalar) field value at some location $x_1$ and let $z_2=z(x_2)$ be the $k$-dimensional vector of field values at the  immediate neighbours $x_2$ of $x_1$, with the neighbourhood to be defined later. Then, because we have standardised,  $(z_1,z_2)$ is Gaussian, with zero mean and  variance matrix
\[  \left(\begin{array}{cc}
 1 & r^T \\
  r & R
\end{array}
\right).\]
Here the $k$-vector of correlations $r$ between $z_1$ and $z_2$, and the $k \times k$ correlation matrix $R$ of $z_2$, each depend on the locations $x_1$ and $x_2$, though this is suppressed in the notation.

We have a local maximum at $x_1$ if $z_2 < z_1\mathbf{1}_k$ where $\mathbf{1}_k$ is a $k-$vector of ones. We can calculate this from the conditional distribution of $z_2$ given $z_1$ and the marginal distribution of $z_1$. First let $\phi^{(p)} (x; \mu, \Sigma)$ be the $p$-dimensional Gaussian probability density with mean vector $\mu$ and covariance matrix $\Sigma$, with $\Phi^{(p)} (x; \mu, \Sigma)$ being the corresponding cumulative distribution function. So $z_1$ has density  $\phi^{(1)} (z_1; 0, 1)$, $z_2$ has density
 $\phi^{(k)} (z_2; \mathbf{0}_k, R)$, where $\mathbf{0}_k$ is a $k-$vector of zeros, and the conditional density for $z_2$ given $z_1$ is
\[ \phi^{(k)} (z_2; rz_1, R-rr^T).\]

Then

\begin{eqnarray*}
p_1=P(\mbox{local maximum at } x_1) & = & P(z_2 < z_1\mathbf{1}_k) \\
\\
   & = & \int_{z_1} \left\{ \int_{z_2 < z_1\mathbf{1}_k}  \phi^{(k)} (z_2; rz_1, R-rr^T)\,dz_2 \right\}\phi^{(1)} (z_1; 0, 1)\,  dz_1 \\
\\
 & = & \int_{z_1}   \Phi^{(k)} (z_1\mathbf{1}_k; rz_1, R-rr^T)     \phi^{(1)} (z_1; 0, 1)\,  dz_1 \\
\\
 & = & \int_{z_1}   \Phi^{(k)} ((\mathbf{1}_k-r)z_1; \mathbf{0}_k, R-rr^T)     \phi^{(1)} (z_1; 0, 1) \, dz_1. \\
\end{eqnarray*}

A general skew-Normal distribution described by
\citet{gupta04, azzalini05, arnold09} and \cite{barrett14}
can be used to show that
\begin{equation}
\Phi^{(q)}(D\mu; \nu, \Delta+D\Sigma D^T)  = \int_y  \Phi^{(q)}(Dy; \nu, \Delta)  \phi^{(p)} (y ; \mu,\Sigma) \, dy,
\label{eqn:fx}
\end{equation}
where $\mu$ is of dimension $p$, $\nu$ is of dimension $q$, $\Sigma$ and $\Delta$ are $p \times p$ and $q \times q$ covariance matrices respectively, and $D$ is an arbitrary $q \times p$ matrix. Hence, if we set
\[  p=1,\;\;\;q=k,\;\;\;\mu=0,\;\;\;\Sigma=1, \;\;\;\nu=\mathbf{0}_k,\;\;\; \Delta=R-rr^T,\;\;\;y=z_1 \mbox{ and } D=\mathbf{1}_k-r, \]
then
\begin{eqnarray}
 p_1 &  = &  \Phi^{(k)}(\mathbf{0}_k; \mathbf{0}_k,R-rr^T+(\mathbf{1}_k-r)(\mathbf{1}_k-r)^T) \nonumber \\
 & = &  \Phi^{(k)}(\mathbf{0}_k; \mathbf{0}_k, R+\mathbf{1}_k\mathbf{1}_k^T-\mathbf{1}_k r^T-r \mathbf{1}_k^T).
\label{eqn:expn}
\end{eqnarray}

Summing over all locations  yields $E[N_1]=E[N_0]=\sum_j p_j$, where $p_j$ is obtained from the equivalent of (\ref{eqn:expn}) for location $x_j$ instead of $x_1$. To calculate, we next need to define the neighbourhood of a lattice node.  Thinking of a lattice as an array of square pixels, those with common boundaries are clearly neighbours.  With this choice, an  interior pixel thus has $k=4$ neighbours, with differences in location of the form $(\pm 1, 0)$ or $(0, \pm 1)$.  We will call this the {\it cross} neighbourhood, which is what was used in the discussion of connected locations in Section 2.1.  We may alternatively  consider in addition pixels that share at least one vertex as neighbours. An interior pixel then has $k=8$ neighbours, consisting of the previous four and the four corners $(\pm 1, \pm 1)$. We will call this the {\it square} neighbourhood.

Table \ref{tab:expn} shows $E[N_1]$ for three different grid sizes, assuming exponential correlation function with correlation length (the integral of the correlation function) equal to 20, which is close to that  in the H\,{\sc i} data.  We used the {\tt pmvnorm} routine in the {\tt mvtnorm} package within R, with the Miwa algorithm to calculate $\Phi^{(k)}$, and we adjusted neighbourhoods appropriately for pixels on boundaries.  The table also gives the means and associated standard errors of counts from 1000 simulations in each case.  The simulation results match the analytical results well. For larger  correlation lengths of $z(x)$ (not shown), there are fewer maxima or minima for a fixed size lattice, and the opposite for shorter correlation lengths. At zero correlation length, the number of minima (or maxima) is about 1/5 of the total number of points for the cross neighbourhood, and 1/9 for the square, as expected.

\begin{table}
\caption{{\it Expected number of local maxima for a discrete Gaussian random field on a $d \times d$ lattice, with exponential correlation function of correlation length 20. The simulation results are the  means (standard errors) from 1000 repetitions.} \label{tab:expn}}
\begin{center}
\begin{tabular}{lcccc}
              & \multicolumn{4}{c}{Neighbourhood} \\
              &\multicolumn{2}{c}{Cross, $k=4$} & \multicolumn{2}{c}{Square, $k=8$}\\
              & Simulation & Eq.(\ref{eqn:expn}) & Simulation & Eq.(\ref{eqn:expn})\\
$d=32$        &   128.3  & 128.4 & 68.8 & 68.9\\
              &  (0.3) & & (0.2)\\
$d=64$        & 498.3 & 497.9 & 258.3 & 258.3    \\
              & (0.5)  & & (0.4)\\
$d=256$       & 7786.7 & 7786.5 & 3927.4 & 3929.5\\
              & (2.2) & & (1.6)\\
\end{tabular}
\end{center}
\end{table}

\subsection{Variance of number of local extrema}

Let $I_j$ be an indicator of a local maximum at location $x_j$.  In order to determine the variance of $N_1=\sum I_j$ we need $E[I_iI_j]$ for all pairs of locations $x_i$ and $x_j$.  First we introduce some notation.
Let $z_1=(z_{11}(x_{11}), z_{12}(x_{12}))$ be the (bivariate) field value at any two locations
 $x_{11}$ and $x_{12}$. Let $z_{21}(x_{21})$ be the $k_1$-dimensional vector of field values over the neighbours $x_{21}$ of $x_{11}$, and let $z_{22}(x_{22})$ be the $k_2$-dimensional vector of field values over the neighbours $x_{22}$ of $x_{12}$.  Finally, let $z_2$ be the $(k_1+k_2)$-vector made up of  $z_{21}(x_{21})$ and $z_{22}(x_{22})$.

We need to consider separately the three possible arrangements of the neighbours:  first, when $x_{11}$ and $x_{12}$ have no common neighbours; second, when a neighbour is shared between $x_{11}$ and $x_{12}$;  and third,  when one of $x_{11}$ and $x_{12}$ is itself in the neighbourhood of the other.
In the first case, of no common neighbours,  $x_{21}$ and $x_{22}$ are distinct and $(z_1,z_2)$ is Gaussian with zero mean and  variance matrix
\[  \left(\begin{array}{cc}
 R_{11} & R_{12}^T \\
  R_{12} & R_{22}
\end{array}
\right),\]
where the sub-matrices once more depend on the locations, though this is still suppressed in the notation.
Similarly to the previous section, we can use (\ref{eqn:fx})  to show that
\begin{equation}
 E[I_1 I_2]= \Phi^{(k)} (\mathbf{0}_k; \mathbf{0}_k; R_{22}-R_{12}  R_{11}^{-1}  R_{12}^T +DR_{11}D^T),
\label{eqn:i1i2}
\end{equation}
where $k=k_1+k_2$, $D= J -  R_{12}R_{11}^{-1}$ and $J$ is the $k\times 2$ matrix
\[ J  = \left(\begin{array}{cc}
 \mathbf{1}_{k_1} & \mathbf{0}_{k_1} \\
\\
\mathbf{0}_{k_2} & \mathbf{1}_{k_2}
\end{array}\right).   \]
From this we get the covariance between any pair of indicators $I_1$ and $I_2$ provided
there is no neighbour in common.

The second arrangement is when  $x_{21}$ and $x_{22}$ share one or more points so that $x_{11}$ and $x_{12}$  have at least one common neighbour. In that case $R_{22}$ is singular and we might anticipate difficulties. However, the variance matrix $R_{22}-R_{12}  R_{11}^{-1}  R_{12}^T +DR_{11}D^T$ in Equation (\ref{eqn:i1i2}) is {\it not} singular, at least in general, and  (\ref{eqn:i1i2}) still applies.

The final arrangement is simple. When either $x_{11} \in x_{22}$ or $x_{12} \in x_{21}$, then clearly there cannot be a local maximum at each of $x_{11}$ and $x_{12}$, so $E[I_1I_2]=0$ and Cov($I_1,I_2)=-E[I_1]E[I_2]$.

In principle, the variance of $N_1=\sum I_i$ can now be calculated.  However, the number of terms involved in the $d^2 \times d^2$ covariance matrices is unmanageable for lattices of large dimension $d$. Nonetheless, if the separation between pixels $x_i$ and $x_j$ is not small,  the covariance between $I_i$ and $I_j$ is negligible.    A working assumption of ignoring covariances between locations which are  separated by at least some threshold $\delta_0$ seems reasonable. Hence, our proposed estimator is
\begin{equation}
 \widehat{\mbox{Var}}(N_1)  = \sum_i \left\{E[I_i](1-E[I_i])+\sum_{j \in \mathcal N_i}\mbox{Cov}(I_i, I_j) \right\},
\label{eqn:varhat}
\end{equation}
where $\mathcal N_i=\left\{j: \mid x_i-x_j \mid \leq \delta_0\right\}$. In the following, we took $\delta_0=3$ lattice distance units.  The approximation is used together with Equation (\ref{eqn:i1i2}) in Table \ref{tab:sdn}. We used the GenzBretz algorithm  within the {\tt pmvnorm} routine for the covariances,  because the dimension of $R_{22}$ is too large for the Miwa routine.  The GenzBretz method involves Monte Carlo evaluations and hence leads to some uncertainty.   Nonetheless the agreement between the Monte Carlo results and our approximation is good.  The variance of $N_1$  is a little overestimated at $d=32$ and underestimated at $d=256$. More refined approximations with wider neighbourhoods and more careful treatment of boundary effects might give improvements, but the above seems adequate.

\begin{table}
\caption{{\it Standard deviation of the number of local maxima of a discrete Gaussian random field on a $d \times d$ lattice, with an exponential correlation function of correlation length 20. Simulation results are standard deviations from 1000 simulations, while the approximation is based on equations (\ref{eqn:expn}), (\ref{eqn:i1i2})  and (\ref{eqn:varhat}). }\label{tab:sdn}}
\begin{center}
\begin{tabular}{lcccc}
              & \multicolumn{4}{c}{Neighbourhood} \\
              &\multicolumn{2}{c}{Cross, $k=4$} & \multicolumn{2}{c}{Square, $k=8$}\\
              & Simulation &  Approximation & Simulation &  Approximation\\
$d=32$        & 8.6  & 9.0 & 5.9 & 6.6 \\

\\
$d=64$        & 17.2  & 17.2  & 12.2 & 12.2   \\
\\
$d=256$       & 70.7  & 66.0 & 49.1 & 45.2 \\

\end{tabular}
\end{center}
\end{table}

\section{Topology of non-Gaussian random fields}
\label{sec:fields}

We are interested in assessing whether a topological approach can  be used to distinguish random fields when traditional methods fail.  For this purpose,  we explore properties of five different distributions on lattices, each stationary and isotropic, with N(0, 1) marginal distributions at the individual pixel level, and with correlation functions that are indistinguishable given  a single realisation. Thus, their quite different higher-order characteristics  would not be identified through analysis of first- and second-order properties.

The first model is a Gaussian random field (GRF) which we use as a reference. The other models are easily generated functions of one or more GRFs,  but are not themselves either GRFs or back-transformable to GRFs.
  We stress that the four non-Gaussian random fields are used for illustration only, and we do not claim any to be appropriate for the H\,{\sc i} data.

\subsection{Simulated random fields}

The five distributions are constructed as follows.

\begin{description}
\item Model 1:  GRF $z$, with N(0,1) margins and exponential correlation function of correlation length $\eta$:
\[ \mbox{Corr}\left(z(x),z(x+d)\right)= \exp\left(-|d|/\eta\right). \]
Our default is  $\eta=20$.

\item Model 2:  $\chi^2_1$.  We start with a GRF $z_1$ with Matern correlation structure
\[ \mbox{Corr}\left(z_1(x),z_1(x+d)\right)=\frac{2^{1-\nu}}{\Gamma(\nu)}\left(\sqrt{2\nu}|d|/\eta\right)^\nu K_\nu\left(\sqrt{2\nu}|d|/\eta\right), \]
where $K_\nu(.)$ is a modified Bessel function of the third kind.  For  $\nu=0.5$ the Matern correlation function reduces to exponential. We construct a $\chi^2_1$ field as $z^\ast=z_1^2$ and then marginally transform using $z=\Phi^{-1}\left(F_{\chi^2_1}(z^\ast)\right)$, where $\Phi(.)$ and $F_{\chi^2_1}(.)$ are the N(0,1) and $\chi^2_1$ cumulative distribution functions, respectively.

\item Model 3:  $\chi^2_3$.  We begin with three independent GRFs $z_1, z_2, z_3$, each with the same Matern correlation structure.  We transform as
\[ z^\ast = \sum_{i=1}^{3} z_i^2, \hspace{1cm} z= \Phi^{-1}\left(F_{\chi^2_3}(z^\ast)\right). \]

\item Model 4: $T_3$.  We use four GRF with Matern correlation and our transformation is
\[ z^\ast= \frac{z_1}{\left(\sum_{i=2}^4 z_i^2/3\right)^{1/2}},
\hspace{1cm} z= \Phi^{-1}\left(F_{T_3}(z^\ast)\right). \]

\item Model 5: $F_{3,3}$.  This time we have six GRF and take
\[ z^\ast= \frac{\sum_{i=1}^3 z_i^2/3}{\sum_{i=4}^6 z_i^2/3},
\hspace{1cm} z= \Phi^{-1}\left(F_{F_{3,3}}(z^\ast)\right). \]
\end{description}

By construction, the marginal distributions are all N(0,1).  We estimated the Matern parameters $\nu$ and $\eta$ for Models 2--5 numerically, so as to match as closely as possible the computed correlation functions of the final fields $z$ to that of the reference Model 1.
Parameter estimates and simulation results for correlations at various distances are given in Table \ref{tab:cors}.  The mean results for Models 2--5 match Model 1 closely, well within sampling variation.  We contend that, as intended, these distributions cannot be distinguished through sample correlations on a single $256 \times 256$ lattice.

\begin{table}
\caption{{\it Parameter estimates and mean (standard deviation) correlations in the distance range 1--50
for five types of random field.  Correlations results for Model 1 (Gauss) are exact, the others are empirical estimates based on 50 simulated $256 \times 256$ random fields with the given parameters.  }\label{tab:cors}}
\begin{center}
\begin{tabular}{lrrccccccc}
 & & & \multicolumn{7}{c}{Distance}\\
Model &   $\nu$ & $\eta$ &    1&     2&     3&     5&    10&    25&    50\\
1: Gauss      &0.50 & 20 &0.950& 0.905& 0.861& 0.779& 0.607& 0.287& 0.082\\
\\
2: $\chi^2_1$ &0.74& 41  &0.946& 0.894& 0.846& 0.755& 0.565& 0.220& 0.029\\
              &    &     &(0.012)& (0.022)& (0.031)& (0.046)& (0.076)& (0.106)& (0.083)\\
3: $\chi^2_3$ &0.54& 42  &0.952& 0.903& 0.857& 0.770& 0.590& 0.264& 0.033\\
              &     &    &(0.009)& (0.017)& (0.025)& (0.040)& (0.066)& (0.095)& (0.068)\\
4: $T_3$      &0.58 & 22 &0.950& 0.900& 0.851& 0.763& 0.585& 0.260& 0.049\\
              &     &    &(0.006)& (0.013& (0.018)& (0.028)& (0.047)& (0.076)& (0.082)\\
5: $F_{3,3}$  &0.54 & 50 &0.948& 0.897& 0.850& 0.762& 0.584& 0.272& 0.063\\
              &     &    &(0.009)& (0.017)& (0.024)& (0.037)& (0.061)& (0.080)& (0.061)\\
\end{tabular}
\end{center}
\end{table}

\subsection{Comparison of topological summaries}

Figure \ref{fig:tda} shows topological summaries of persistence diagrams for both components and holes,  for 50 simulated realisations of each model on  $256 \times 256$ lattices. We plot the total  number of points in the persistence diagram, either $N_0$ or $N_1$, together with the five simple statistics described in  Section \ref{sec:sums}, based on  90\% convex peels: the centroids, area, perimeter and filamentarity.  The top row is for the persistence diagrams defined by components, and the second row for holes. We used the cross neighbourhood: results for the square are similar.

The most obvious differences between the models is in the numbers of points, whether  $N_0$ or $N_1$. The solid vertical line in the right-most panels shows the expected value obtained in Section \ref{sec:grf}, with the broken lines marking two standard deviations on either side, again using the results of Section \ref{sec:grf}.  The theoretical values match the simulated data well and it is clear that of the models considered, the Gaussian random field has the largest numbers of both components and holes.  Model 2, based on $\chi^2_1$, has fewest structures of both types.    In most panels there is separation between at least some of the models, and there are some interesting contrasts between the rows.  For example, the $\chi^2_1$ model has the highest area and perimeter when components are considered, but lowest for holes. Also, there is evidence from the lower row  that Models 3 ($\chi^2_3$) and 5 ($F_{3,3}$) have similar counts of local maxima
but differing values on the other summaries.  This suggests that there can certainly be information in the shape of a persistence diagram over and above the number of structures.

\begin{figure}
\begin{center}
\includegraphics[width=7in,height=6in]{geomfig.pdf}
\end{center}
\caption{{\it Number of structures and persistence diagram summaries for Models 1--5. Results from 50 realisations of $256\times 256$ fields, with the cross neighbourhood and 90\% convex peels.  The top row refers to components defined by local minima and the bottom row refers to holes  defined by local maxima. The solid vertical lines mark the expected count under a Gaussian random field, with the dashed lines indicating two standard deviations to each side.} \label{fig:tda}}
\end{figure}

\section{Testing for differences between fields}
\label{sec:tests}

We can test an observed field against a Gaussian random field assumption by comparing the observed number of topological features with an interval based on the GRF properties. Figure \ref{fig:tda} suggests this may work well.
More generally, we might want to distinguish one field from another without assuming a particular distribution.  The previous section shows that differences in underlying structure might be detected through topological features but, without theoretical values or an underlying model, we need to develop a data-based approach.

Our proposal is to split the data from each field into subsets which are sufficiently separated for between--subset dependence to be weak, calculate appropriate summary statistics, and then use standard non-parametric statistical tests to compare the summaries from one field with another. Concentrating on $256\times256$ grids and correlation length around 20 as in the GASS data, we suggest splitting  each field  into nine $64\times64$ subsets with buffers of size 32  between them.  This gives a reasonable balance between the number of subsets (9) and correlation between them.  At the correlation length of 20 that we consider, correlation is 0.20 at the shortest distance between blocks (32), and 0.01 at the separation of block centres (96). Thus although the separate blocks are not independent, association is weak.

Our experience is that the most useful approach is to base
tests  on two of the summaries:   counts $N_0$ of components and  filamentarity $F$ of persistence diagrams for holes. We suggest a 90\% convex peel  and use Wilcoxon tests to compare the 9 counts in one field with the 9 in another, and similarly the 9 filamentarities. A Bonferonni correction is used to adjust for the use of two tests.  The tests are of course not necessarily independent, and, as stated, we know the subsets are not independent.  Nonetheless the simulation results in Table \ref{tab:sims} suggest that performance is good. The standard deviation on test size is 0.01 and all estimated test sizes are within noise of the nominal 5\%.  It is clearly very easy to detect difference between $\chi^2_1$ fields and any of the others,  or between  Gauss and $T_3$.  There is good power for the other comparisons, except for $\chi^2_3$ against $F_{3,3}$, which are evidently hard to distinguish.  Power in this case can be increased by focussing if required only on filamentarity and not using the Bonfernonni correction.

\begin{table}
\caption{{\it Test size and power for nonparametric comparison between pairs of $256 \times 256$ random fields.  Each value is obtained from 500 simulated pairs, using 5\% overall tests after Bonferonni adjustment.  }\label{tab:sims}}
\begin{center}
\begin{tabular}{lrrrrr}
 & Gauss & $\chi^2_1$ & $\chi^2_3$ & $T_3$ & $F_{3,3}$\\

Gauss & 0.050  &1.000 & 0.782 & 1.000 & 0.776\\
$\chi^2_1$ &      & 0.062 & 1.000 & 1.000 & 1.000\\
$\chi^2_3$ &        &     &0.056 & 0.698 & 0.160\\
$T_3$ &             &     &      & 0.038  & 0.602\\
$F_{3,3}$   &       &     &      &        &  0.040\\
\end{tabular}
\end{center}
\end{table}

\section{Neutral Hydrogen Distribution in the Milky Way}
\label{sec:app}

We now return to the astronomical data, to understand whether the small-scale variation in the regions marked in Figure \ref{fig:gass} can be described by simple transfomations of a Gaussian random field, and whether the topologies of this variation can be considered to be the same in each region.

Figure \ref{fig:app} shows, in the top row, 90\% convex peels of persistence diagrams for components and holes: it seems that in each case  the persistence diagram is more filamentary for Region 3 than the other regions. The bottom row of the figure shows cumulative counts of components and holes as the filtration $\mathcal F_t$ increases. They include for Region 3 broken lines at plus and minus two estimated standard deviations around the count, obtained from the estimated predictable variation of
the counting process \citep{and93}.  We have omitted similar intervals for Regions 1 and 2 as they confuse the plot, but we remark that their widths are very close to those for Region 3. Evidently, the cumulative counts are very similar for Regions 1 and 2 but very different for Region 3. This is confirmed by the final counts, which are given in Table \ref{tab:gasscounts}.  The table also gives the expected counts for a Gaussian random field with Matern correlation matching fits to the data.  Clearly, the observed counts are much lower than would be expected for a Gaussian random field, and the difference is stronger for Region 3.

\begin{figure}[p]
\begin{center}
\includegraphics[width=\textwidth,height=5.5in]{appfig.pdf}
\end{center}
\caption{{\it Topological summaries for the GASS data.  Regions 1, 2 and 3 as solid, dashed and dotted lines respectively.  The broken lines in the bottom row indicate two estimated standard deviations on either side of the Region 3 counts.  Standard deviations for the other regions are similar.} \label{fig:app}}
\end{figure}

Table \ref{tab:gasspvals} shows the $p$-values obtained from the data-splitting procedure of Section \ref{sec:tests}, based on filamentarity and counts.  Conclusions are as above, and our answers to both original questions are thus negative.  None of the regions has small-scale variation consistent with a simple transformation of a Gaussian random field, and Region 3 has markedly different structure from the other regions.

\begin{table}
\caption{{\it Component and hole counts for the GASS data in the regions shown in Figure \ref{fig:gass}.  The first two numerical columns give estimated parameters for fitted  Matern correlation functions. Values in brackets are standard deviations obtained from estimated predictable variation.}\label{tab:gasscounts}}
\begin{center}
\begin{tabular}{lcccccc}
  & $\nu$  & $\eta$ & Components  & Holes & Gauss expected & Gauss SD\\
Region 1 &0.55 & 12.98 & 5201  & 5100  &   7896.1  & 65.9 \\
         &     &       & (63.1)& (63.6)\\
Region 2 &0.70 & 13.50 & 5103  &  5095 &  7884.2 & 65.9 \\
         &     &       & (62.2)& (66.4)\\
Region 3 & 1.08& 14.90 & 3831  & 3661 & 7856.1 & 65.9\\
         &     &       & (57.9)& (60.5)\\
\end{tabular}
\end{center}
\end{table}

\begin{table}
\caption{{\it $P$-values for comparing the GASS Regions 1--3, based on data splitting into 9 sub-regions and Wilcoxon tests.}\label{tab:gasspvals}}
\begin{center}
\begin{tabular}{llcc}
              & Feature & Filamentarity & Number \\
Region 1 v Region 2 &  Components & 0.387 & 0.825\\
              &  Holes & 0.863 & 0.825\\
\\
Region 1 v Region 3 & Components & 0.077 & 0.040\\
              & Holes & 0.019 & 0.040\\
\\
Region 2 v Region 3 & Components & 0.222 & 0.027\\
              & Holes & 0.031 & 0.027\\
\end{tabular}
\end{center}
\end{table}

Finally for this section we note that the well-known bottleneck distance has no discriminatory power for these data.  Table \ref{tab:gassbotts} summarises bottleneck distances between all pairs of sub-regions, both within and between regions.  Mean intra-region distances can sometimes be larger than mean inter-region values, and any differences in means are small compared to the standard deviations.

\begin{table}
\caption{{\it Mean and standard deviation of bottleneck distances between persistence diagrams of sub-regions.  The column $M$ shows the number of pairs for each comparison.}\label{tab:gassbotts}}
\begin{center}
\begin{tabular}{llccccc}
      &    &     & \multicolumn{2}{c}{Components} & \multicolumn{2}{c}{Holes}\\
  \multicolumn{2}{c}{Comparison} & $M$ & Mean & SD & Mean & SD\\
   Region 1 & Region 1&   36& 1.130 &0.574&    0.542& 0.144\\
   Region 1 & Region 2&   81& 1.237 &0.528&    0.671& 0.280\\
   Region 1 & Region 3&   81& 1.314 &0.558&    0.596& 0.131\\
   Region 2 & Region 2&   36& 1.486 &0.575&    0.799& 0.406\\
   Region 2 & Region 3&   81& 1.528 &0.620&    0.682& 0.365\\
   Region 3 & Region 3&   36& 1.287 &0.539&    0.533& 0.171\\
\end{tabular}
\end{center}
\end{table}

\section{Discussion}
\label{sec:disx}

Our analyses of the map of interstellar hydrogen (Fig.~\ref{fig:gass}), in terms of the shape of the persistence diagrams and the cumulative counts of topological features, identified Region 3 as being topologically distinct from the other two regions. The regions were deliberately selected without any prior knowledge of the properties of the interstellar medium in these parts of the sky. After carrying out the topological analysis we looked for an astronomical explanation for the distinctive nature of the gas in Region 3 and found that about half of this patch of the sky overlaps a nearby region of recent star formation known as the Orion-Eridanus superbubble \citep{naranan76, burrows93, pon16}. Stellar winds and/or supernova explosions have produced a bubble of hot gas, a few hundred light years across, whose boundaries can be traced in many different observational windows, including the H\,{\sc i} emission.  No structures due to a distinct astronomical object are present in the fields of Regions 1 and 2; in these parts of the sky, the distribution of the H\,{\sc i} is most likely the result of pervasive turbulent flows in the interstellar medium. Whilst recognising our identification of the Orion-Eridanus superbubble as retrospective, we speculate this as an explanation for the topological differences between Region 3 and either Regions 1 or 2.

The vast majority of analyses of the GASS or similar astronomical data rely on an assumption of an underlying  Gaussian random field, whether for the observational data with or without a simple transformation such as log, or as residuals around some large-scale structure. We have shown that all three of the regions of H\,{\sc i} that we examined contain strongly non-Gaussian fields, after trend removal. We suggest that statistical analysis of topological properties is an attractive alternative to existing approaches based on higher-order moments.  So far we have considered only two-dimensional data. Extension to three-dimensions may bring further power and is an important problem to address.

\bibliographystyle{agsm}

redpajama_set_name: RedPajamaArXiv