Since I was little I started to collect stuff. You can say that I'm a born collector and that I have the heart of a collector. I have collected so many different things over the years and I think most people who know me don't even know all my collections. Time to reveal it! I'm absolutely NOT a horder. I can put away things that I don't want to have anymore really easily. But when it comes to serious collections, I still have most of them somewhere!
I believe it all started off with stickers, since I have to stick them into a map after injecting growing hormones everyday. I still have a lot of stickers left and I use them for my letters and that kind of stuff. Then I had this obsession with the Teletubbies. It was my first favorite series I guess haha! It sounds lame, but I had a few different things of the Teletubbies. Don't judge, I was a toddler back then haha!
After that, I came into my K3 fandom phase. Yep, music became a real thing in my life and I started to collect EVERYTHING of that grou[. I have a box filled with K3 stuff in the attic. Not long after that I discovered the world of stamps. And that is something I actually still collect! It might sound boring and geeky, but there are so many different stamps! I even own really old ones and I have a lot of different stamps from all over the world. My most special ones are my Hello Kitty stamps from Taiwan. I got them recently in the mail, thanks to someone that contacted me, because of my Hello Kitty collection.
Diddle became a really popular thing during my last few years on primary school. You could literally get everything you could think about printed with Diddle. But the most popular things were the notebooks, writing papers and paper bags. Whenever you could switch your doubles for a new one with others at school, you did that. I remember one of my friends back then, organized a Diddle themed party for her birthday and she send in a picture of us to the magazine Diddle's Kaasblad and we got in it! So awesome!
I also had a "horses are my favorite animals" phase. So I collected magazines, newspapers, stickers and all that kind of stuff about horses. I even had several pc games about horses. Besides The Sims, I played those horses games the most. Yep pretty obsessed. I still love horses though, and I still don't dare to ride them haha! I'm more obsessed with cats now.
Speaking of cats, here we go to my biggest collection: Hello Kitty. Okay yeah actually she only looks like a cat but she is a girl. (Background information blabla). I don't even remember when I exactly started to collect Hello Kitty stuff. But what I do know is that I'm doing it for years now and my collection has insanely grown over the years! Everyone keeps an eye out for me when it comes to Hello Kitty stuff, which is really really sweet! Fun fact: most of the stuff were gifted to me, I only bought a small part myself haha, oops! I keep discovering new Hello Kitty stuff and I want to visit Sanrio Puroland in Tokyo so bad! I hope I will ever go there. I still need to make a Hello Kitty collection update on video. Stay tuned!
And last but not least, I love to collect memories. I always keep entree tickets when I go somewhere, I take pictures etc. Today I've started to make a concert book. All of my concert memories are going to be in that book. I'm thinking of making a video about it when most of it is in the book. Let me know if you would like to see a video about it! I want to collect cd's and vinyls as well. Music is such a big part of my life!
Jaaa Diddl. En nog veeeeel meer ander spul. Zo leuk om aan terug te denken, en sommige spullen heb ik zelfs nog op zolder staan.


We are continuing with our player season reviews with Sergio Romero. The Argentine goalkeeper had a season that could be called poor. Not because he did not play much – that was always expected when David De Gea is his rival for the starting spot, but because the season finished in the worst possible way for him. He got injured in training ahead of the World Cup and the regular first-team goalkeeper of Argentina is now forced to miss the tournament.
Sergio Romero really did not play much. He had the opportunity to start one Premier League match, on the last day of season against Watford. De Gea played every minute of the previous 37 games and it was the opportunity for Romero to get the taste of the competition this season. In Champions League he got the chance to be in goal twice – against Basel and CSKA Moscow. Those were the two final group stage matches for the Devils and he conceded once to both sides.
In FA Cup, he was in goal four times, while he made three appearances in the League Cup. Looking overall, he conceded just five goals in those 10 matches, which is obviously a good record. What is more impressive is that he did not concede a single goal in 2018, keeping five clean sheets in five appearances. It is too bad he will not have the chance to try and prove his quality in Argentina's jersey this summer.
Numerically, it is always harder to measure the performances of goalkeepers, especially when you do not have the access to all of those nice analytical tools. What we can derive from Romero's season is that he has been reliable when given the chance and that chance was never going to come often.
Was he an important player for the club?
No, obviously not. Manchester United are lucky they had such an experienced goalkeeper as their backup option for the past two seasons. Romero has been really solid for Argentina for many years now and there is no doubt that he would do well if he had been given more chances. However, next to De Gea that was never going to happen. More importantly, De Gea raised the bar so high that Romero would probably never reach his levels, but that does not mean that Romero is not a very good goalkeeper.
Obviously, De Gea's performances meant that United conceded much less goals than it would be expected on average and Romero would probably be amongst those keepers that would struggle with so much work to do during matches. However, there are almost no goalkeepers in the world that would do the sam job De Gea did, after all.
It is expected for Sergio Romero to leave the club this summer. He has not been playing for two years and it is normal to get a bit agitated and wanting to play more. Also, there were already rumours of potential suitors, so it will not be a surprise if we do not see him in Manchester come next season.


text:
SwitchArcade Round-Up: '10 Second Ninja X', 'Apple Slash', 'Xenogunner', and Today's Other New Releases and Sales
Posted on July 30, 2021 July 30, 2021 by Shaun Musgrave
Hello gentle readers, and welcome to the SwitchArcade Round-Up for July 30th, 2021. In today's article, we look at the rest of the new releases for the week. Some decent games in here, but after the week we've had I'll forgive you if you want to take a little breather. We've got summaries of all the new games anyway, along with the usual lists of incoming and outgoing sales for you to have a good gander at. Let's dig in!
10 Second Ninja X ($11.99)
It's ultra-challenging platformer time. Each level gives you exactly ten seconds to clear it, so you'd better find your feet fast. There are one hundred levels in total, with tons of different obstacles and hazards to overcome. Are you fast enough to get a three-star rating on every level? There are different character costumes to unlock, hidden minigames, online leaderboards for every level, marathon modes, and even some accessibility options for those who want to play their way. If you enjoyed stuff like League of Evil, N++, or Super Meat Boy, you may want to look into this.
Alfonzo's Arctic Adventure ($4.99)
Oh hey, I recognize this. I think this one came out a couple years back on Steam and as an NES cartridge. It's from the same developer as Jay & Silent Bob: Mall Brawl, and it's not too bad. You play as a handful of different characters in a side-scrolling platformer where you need to collect fish. There are more than ninety levels to play across six different worlds, and there are some secrets, branching paths, and cool bosses to battle. There's also a two player battle mode via local multiplayer. Not the finest 8-bit platformer around, but fun enough for a fiver.
Apple Slash ($4.99)
Some good slashing fun, but gosh it is short. I don't need my games to be long but it feels like this one barely gets going before it's done. Oh well. You're the Apple Knight and you basically just swing your blade around like wild, destroying enemies as you go. You can get new abilities, meet and make friends with various creatures, and travel around an ancient swamp full of foes to fillet. The price is fair and everything, I just was having such a good time with this game that I wanted it to continue. If you too want to have a good time that ends a bit too soon, pick up Apple Slash.
Xenogunner ($14.99)
This is a rather enjoyable run-and-gun side-scroller that focuses on intense boss battles. You know, kind of like how Treasure used to do with games like Gunstar Heroes or Alien Soldier. I'm not making that comparison to suggest that you should expect this game to be on that level, because it's not quite there. But what is, really? The point is that this is a slice of old-school fun that 2D action fans will likely have a good time with. I'll have a review of this one for you soon, but I liked it.
Horror Tales: The Wine ($14.99)
Horror tales! The wine did not match the meal! SEND IT BACK NOW BEFORE IT IS TOO LATE. That's how I imagine fancy people spend their birthdays. Probably not with Dominos Pizza and an ice cream cake, anyway. Anyway, this game sees you exploring a Mediterranean island after a deadly pandemic swept through and caused it to be abandoned. While it may look beautiful enough on the surface, the island is full of danger. Chief among them is a mysterious stalker that hunts you as you explore. Solve the puzzles, try to sort out the island's riddles, and most importantly stay alive. Not my bag at all, but it may be yours.
Dungeon of Crawl ($4.99)
A low-cost dungeon crawling RPG seems like a thing people would want, yes. This one has fifteen levels to play through, set across five different regions. Each of those regions has its own enemies, weapons, and boss to encounter. You can pick from four different character classes, which will naturally affect how you need to approach combat situations. There's also apparently an arena mode if you just feel like jumping in and beating up on some baddies. It looks like it costs, in my opinion, and while you might have a bit of fun with it I think this is very much a case of getting what you pay for versus something more robust like Vaporum or Operencia.
Papa's Quiz ($9.99)
Friends, I hate to come off as a scold at times in these summaries, but I have to point out: you can't just go ahead and use pictures of stuff you don't own the rights to in your game, even if it seems perfectly reasonable. Oh well. This is a quiz game for up to eight players via local multiplayer, and you can use mobile devices as extra controllers if you need them. Answer over 3000 questions in 185 categories, battle over which category to play, and do a little dance when things go your way. You can customize your avatar with all kinds of odd accessories and clothing items and give them whichever name you like. It's fairly hard to mess up a quiz game, at the very least. Just maybe replace some of those category photos before the lawyers come knocking, yeah?
Cuccchi ($7.99)
There are some weird games in today's list, aren't there? In this one, you'll be diving into the paintings of Enzo Cucchi, an Italian painter with a distinctive impressionist style. Here, his paintings have been transformed into spaces you can move through and in some cases interact with. There are seven different areas in total, and you need to try to find fifty-one collectibles hidden across them. If you can pull it off, you'll unlock an archive gallery that shows off the original paintings. I would say this is more for fans of Mr. Cucchi's art than people looking for an enjoyable game, but it's a unique effort that I can't help but respect anyway.
Alone With You ($9.99)
Your space colony has completely failed, and you are the last survivor. In this lonely aftermath, you wander the remains of the colony by day and revisit its past via holographic companions at night. Solve puzzles as you piece together what happened, and learn about the lives and ultimate fates of the colonists. Converse with the holographic companions and build relationships with them by making the right dialogue choices, eventually leading to one of multiple endings. I don't know that it necessarily succeeds as an adventure game, but it is a pretty neat story to experience. Worth a shot if you like the premise.
Bustafellows ($39.99)
This is a noir otome visual novel where you meet and attempt to romance five different and rather dangerous men. The heroine is a journalist who has the unique power to leap a short ways into the past, occupying someone's body in that time. Useful! She'll meet a handsome sketchy lawyer, a handsome hitman who kills other hitmen, a handsome plastic surgeon, a handsome autopsy specialist, and a handsome underground criminal boss. Wow, so much handsome, and only so many smooches to go around! Bustafellows is well-written and has some gorgeous art, so if you're on the hunt for a high-quality otome game today this is a good call.
Super Squidlit ($9.99)
Just as Squidlit attempted to create a game within Game Boy specs, Super Squidlit limits itself to what the Game Boy Color could pull off. As such, it has a very authentic look to it that I rather like. For the most part, it's a side-scrolling platformer. There are also some wild first-person RPG sections that use ray-casting to rather impressive effect. If you liked the original game, you'll probably enjoy this one. If you haven't played that game, it's only two bucks and is worth checking out. Then you'll be all set to enjoy this one!
Dream House Days DX ($14.00)
This is where I usually talk about how you can get this game cheaper and in a form that plays just as well on mobile, but this particular Kairosoft release was an odd one. Kairosoft briefly flirted with free-to-play versions of their games that had ads and IAPs out the wazoo, and this was one of them. Now, this is probably similar to the version you would get by buying the $4.99 IAP for removing the ads, but I wouldn't rule out some balance adjustments. The idea behind this one is that you are building a sort of apartment complex, creating special rooms and collecting rent while helping tenants achieve their life goals. It's a fun one, provided you're not tired of the formula yet.
City Match – A Block Pop Puzzle Game ($6.99)
Basically just a block-bursting puzzle game like Toon Blast or Pet Rescue Saga. This one is going with a building demolition theme, but it doesn't really run all that far with it so don't spend too much time thinking about it. No button controls, so this one is for undocked play only. Very high chance it's a hastily scrabbled-together Android template. You could buy this game for seven dollars, or you could download one of the thousand or so free ones on your mobile device and probably get a better experience. As a bonus, you would also keep your seven dollars.
(North American eShop, US Prices)
Yes, it's another one of those sales lists. Some good games in there, but nothing all that rare. Chasm, Hand of Fate 2, Titan Quest, Kingdoms of Amalur, and the Picross games would be my picks, but you had best check the list yourself to see what your eye picks up on. The weekend outbox doesn't have a whole lot going on either, at least in my opinion, but do give that a once-over while you're at it too.
Select New Games on Sale
Castle of No Escape ($1.99 from $4.99 until 8/1)
Chasm ($5.99 from $19.99 until 8/4)
Night Call ($7.99 from $19.99 until 8/5)
West of Dead ($11.99 from $19.99 until 8/5)
Dandara: Trials of Fear ($5.24 from $14.99 until 8/5)
Star Renegades ($14.99 from $24.99 until 8/5)
GoNNER ($1.99 from $9.99 until 8/5)
Out There: Omega The Alliance ($3.74 from $14.99 until 8/5)
B.ARK ($7.99 from $9.99 until 8/5)
Atomicrops ($7.49 from $14.99 until 8/5)
Uurnog Uurnlimited ($1.99 from $9.99 until 8/5)
Kingdom: New Lands ($3.74 from $14.99 until 8/5)
Mosaic ($9.99 from $19.99 until 8/5)
El Hijo: A Wild West Tale ($13.99 from $19.99 until 8/6)
Hand of Fate 2 ($9.89 from $29.99 until 8/6)
Absolute Drift ($5.39 from $11.99 until 8/6)
Little Big Workshop ($11.99 from $19.99 until 8/6)
OS Omega ($11.19 from $13.99 until 8/8)
Piczle Puzzle & Watch ($6.99 from $7.99 until 8/9)
Super Dungeon Tactics ($4.99 from $9.99 until 8/10)
Sentinels of Freedom ($12.49 from $24.99 until 8/10)
Bite the Bullet ($5.99 from $14.99 until 8/12)
Tower of Time ($16.24 from $24.99 until 8/12)
REZ PLZ ($5.99 from $14.99 until 8/12)
Titan Quest ($13.19 from $39.99 until 8/12)
1979 Revolution: Black Friday ($2.99 from $11.99 until 8/12)
Ancestors Legacy ($15.99 from $39.99 until 8/12)
Bleed ($3.59 from $11.99 until 8/12)
Bleed 2 ($4.49 from $14.99 until 8/12)
Vertical Drop Heroes HD ($2.49 from $9.99 until 8/12)
Spectrum ($2.39 from $11.99 until 8/12)
Adventures of Chris ($5.99 from $14.99 until 8/12)
I.F.O ($2.79 from $3.99 until 8/12)
MX vs ATV All Out ($26.79 from $39.99 until 8/12)
Pipe Push Paradise ($2.74 from $10.99 until 8/12)
Omega Strike ($4.49 from $14.99 until 8/12)
Glass Masquerade ($3.59 from $11.99 until 8/12)
Glass Masquerade 2 ($5.99 from $11.99 until 8/12)
The King's Bird ($4.99 from $19.99 until 8/12)
Uncanny Valley ($2.49 from $9.99 until 8/12)
Aquatic Adv. of the Last Human ($3.89 from $12.99 until 8/12)
Joggernauts ($3.74 from $14.99 until 8/12)
Mable & The Wood ($3.74 from $14.99 until 8/12)
Hacky Zack ($2.49 from $9.99 until 8/12)
Shikhondo: Soul Eater ($4.19 from $13.99 until 8/12)
Underhero ($10.19 from $16.99 until 8/12)
Don't Die, Mr Robot! ($2.24 from $8.99 until 8/12)
Tamashii ($5.99 from $11.99 until 8/12)
Signs of the Sojourner ($14.99 from $19.99 until 8/12)
Unto the End ($17.49 from $24.99 until 8/12)
Three Fourths Home: Extended ($2.24 from $8.99 until 8/12)
Double Cross ($4.99 from $19.99 until 8/12)
Sunless Sea: Zubmariner Edition ($11.99 from $19.99 until 8/12)
Kingdoms of Amalur: Re-Reckoning ($29.99 from $39.99 until 8/12)
SpongeBob SquarePants: BfBB ($20.09 from $29.99 until 8/12)
Battle Chasers: Night War ($11.99 from $29.99 until 8/12)
Darksiders Warmastered Edition ($13.49 from $29.99 until 8/12)
Darksiders II Deathinitive ($13.49 from $29.99 until 8/12)
Darksiders Genesis ($17.99 from $39.99 until 8/12)
Ink ($2.24 from $8.99 until 8/12)
Paranautical Activity ($2.39 from $7.99 until 8/12)
Blue Fire ($11.99 from $19.99 until 8/12)
Discolored ($7.99 from $9.99 until 8/12)
Lost Artifacts ($2.99 from $9.99 until 8/13)
Legend of Keepers: CoaDM ($14.99 until $19.99 until 8/13)
SmileBASIC 4 ($16.74 from $24.99 until 8/17)
Water Balloon Mania ($1.99 from $4.99 until 8/17)
Black Book ($21.24 from $24.99 until 8/17)
Birds and Blocks ($1.99 from $4.99 until 8/17)
Croc's World ($1.99 from $4.99 until 8/17)
Croc's World Run ($1.99 from $5.99 until 8/17)
Pacific Wings ($1.99 from $4.99 until 8/17)
Airfield Mania ($1.99 from $5.99 until 8/17)
Jewel Fever 2 ($1.99 from $4.99 until 8/17)
Jewel Rotation ($1.99 from $5.99 until 8/17)
Yellow Fins ($1.99 from $4.99 until 8/17)
16-Bit Soccer ($1.99 from $4.99 until 8/17)
Picross S ($6.39 from $7.99 until 8/18)
Picross S2 ($7.19 from $8.99 until 8/18)
Kemono Friends Picross ($7.99 from $9.99 until 8/18)
Picross Lord of the Nazarick ($7.99 from $9.99 until 8/18)
The Explorer of Night ($1.99 from $4.99 until 8/19)
Evil Inside ($10.99 from $12.99 until 8/19)
ReactorX ($2.49 from $4.99 until 8/19)
Exodemon ($10.99 from $12.99 until 8/19)
Undead Battle Royale ($8.10 from $9.00 until 8/19)
Eight Dragons ($6.79 from $7.99 until 8/19)
The Lost Cube ($8.49 from $9.99 until 8/19)
AER Memories of Old ($1.99 from $19.99 until 8/19)
Bullet Beat ($1.99 from $4.99 until 8/19)
Who Wants to Be A Millionaire? ($19.49 from $29.99 until 8/19)
Nexoria: Dungeon Rogue Heroes ($1.99 from $4.99 until 8/19)
The Long Journey Home ($1.99 from $19.99 until 8/19)
Creature in the Well ($7.49 from $14.99 until 8/19)
Cybxus Hearts ($1.99 from $4.99 until 8/19)
Robothorium ($10.04 from $14.99 until 8/19)
Bot Vice ($1.99 from $9.99 until 8/19)
Silence ($1.99 from $19.99 until 8/19)
Our Flick Erasers ($6.48 from $12.96 until 8/19)
Rev Up! RC Grand Prix ($9.28 from $13.26 until 8/19)
Dark Grim Mariupolis ($1.99 from $4.99 until 8/19)
Edna & Harvey The Breakout ($4.99 from $19.99 until 8/19)
Edna & Harvey: Harvey's New Eyes ($4.99 from $19.99 until 8/19)
Super Star Path ($1.99 from $4.99 until 8/19)
Zombie Scrapper ($1.99 from $2.99 until 8/19)
Kuukiyomi: Consider It! ($3.74 from $4.99 until 8/19)
State of Mind ($1.99 from $19.99 until 8/19)
Felix the Reaper ($1.99 from $14.99 until 8/19)
Smart Moves ($2.39 from $5.99 until 8/19)
Void Source ($2.49 from $5.99 until 8/19)
Virtual Battle ($5.38 from $7.69 until 8/19)
The Dark Eye: Memoria ($9.99 from $19.99 until 8/19)
The Dark Eye: Chains of Satinav ($9.99 from $19.99 until 8/19)
Shift Happens ($1.99 from $14.99 until 8/19)
Voxel Shot ($1.99 from $8.00 until 8/19)
Voxel Sword ($1.99 from $7.00 until 8/19)
Voxel Galaxy ($1.99 from $7.43 until 8/19)
Voxel Pirates ($1.99 from $7.28 until 8/19)
Tetragon ($11.99 from $14.99 until 8/19)
Chalk Dash Carnival ($1.99 from $7.09 until 8/19)
Desktop Rugby ($1.99 from $7.43 until 8/19)
Desktop Soccer ($2.84 from $7.11 until 8/19)
Desktop Baseball ($2.91 from $7.29 until 8/19)
Desktop Table Tennis ($2.96 from $7.41 until 8/19)
Desktop Bowling ($2.95 from $7.39 until 8/19)
Desktop Dodgeball ($2.98 from $7.45 until 8/19)
Desktop Basketball ($2.88 from $7.20 until 8/19)
Desktop Voleyball ($2.97 from $7.43 until 8/19)
Banners of Ruin ($15.99 from $19.99 until 8/19)
Fire Tonight ($4.79 from $5.99 until 8/19)
Hoa ($13.49 from $14.99 until 8/20)
Sales Ending This Weekend
.cat ($1.99 from $9.99 until 7/31)
Ace Invaders ($4.99 from $9.99 until 7/31)
Arkham Horror: Mother's Embrace ($11.99 from $19.99 until 7/31)
AXS ($6.99 from $14.99 until 7/31)
BDSM ($9.99 from $19.99 until 7/31)
Boris The Rocket ($10.04 from $14.99 until 7/31)
Carcassonne ($9.99 from $19.99 until 7/31)
Catan ($9.99 from $19.99 until 7/31)
Dragon Pinball ($2.00 from $2.99 until 7/31)
HardCube ($2.10 from $7.00 until 7/31)
Lord of the Rings: Adv. Card Game ($9.99 from $19.99 until 7/31)
Munchkin: Quacked Quest ($12.49 from $24.99 until 7/31)
My Hidden Things ($4.68 from $6.99 until 7/31)
Mystery Mine ($1.99 from $3.99 until 7/31)
One Dog Story ($5.99 from $14.99 until 7/31)
Save Koch ($3.00 from $20.00 until 7/31)
Shinobi Blade ($4.95 from $9.90 until 7/31)
Speedway Racing ($2.99 from $9.99 until 7/31)
Titans Pinball ($2.00 from $2.99 until 7/31)
88 Heroes: 98 Heroes Edition ($5.99 from $29.99 until 8/1)
Anthill ($3.99 from $9.99 until 8/1)
Conga Master Party! ($1.99 from $9.99 until 8/1)
Crashnauts ($7.99 from $14.99 until 8/1)
Crayon Shinchan Kasukabe Runner ($5.99 from $9.99 until 8/1)
Curious Expedition ($7.49 from $14.99 until 8/1)
Decay of Logos ($7.99 from $19.99 until 8/1)
Farm Builder ($7.99 from $9.99 until 8/1)
Flipping Death ($3.99 from $19.99 until 8/1)
Fly Punch Boom! ($7.49 from $14.99 until 8/1)
Giga Wrecker Alt. ($9.99 from $24.99 until 8/1)
Happy Words ($7.99 from $9.99 until 8/1)
Holy Potatoes! A Weapon Shop?! ($4.49 from $14.99 until 8/1)
Holy Potatoes! We're In Space?! ($4.49 from $14.99 until 8/1)
Holy Potatoes! What The Hell?! ($7.50 from $15.00 until 8/1)
Lonely Mountains: Downhill ($13.39 from $19.99 until 8/1)
Ninja Showdown ($2.99 from $14.99 until 8/1)
Pony World 3 ($3.49 from $4.99 until 8/1)
Postal Redux ($4.49 from $9.99 until 8/1)
Promesa ($3.49 from $4.99 until 8/1)
RICO ($7.99 from $19.99 until 8/1)
Say No! More ($10.04 from $14.99 until 8/1)
Stick It to The Man ($2.39 from $11.99 until 8/1)
Super Soccer Blast ($3.99 from $7.99 until 8/1)
Super Sports Blast ($12.49 from $24.99 until 8/1)
Super Tennis Blast ($7.49 from $14.99 until 8/1)
Super Volley Blast ($4.99 from $9.99 until 8/1)
Super Soccer Blast: America vs Europe ($5.99 from $9.99 until 8/1)
Trailblazers ($5.99 from $29.99 until 8/1)
Wuppo: Definitive Edition ($3.74 from $14.99 until 8/1)
That's all for today, this week, this month, and hey, this year of my life, friends. I'm definitely getting up there now, and all I really hope is that I can keep getting up there. This last year has been a hard one for me, and I still don't even know what my condition is and how serious it will turn out to be. Still, for today I will have pizza, cake, and play some good games. That's good enough for now. I hope you all have a great weekend, and as always, thanks for reading!
SwitchArcade
Previous PostPrevious 'Final Fantasy II' Pixel Remaster Review – The Emperor Strikes Back
Next PostNext TouchArcade Game of the Week: 'Infinity Island'

redpajama_set_name: RedPajamaCommonCrawl


Utilities and power plants use metal hose and expansion joints for steam piping systems as well as fuel delivery lines in power generation equipment. Included in this industry are steam turbine manufacturers who also use metal hose for high pressure lubrication line applications and metal expansion joints found within the various piping systems. Similarly, nuclear power plants use metal hose and expansion joints for many of their critical service applications.


CORAL GABLES, Fla. (Dec. 30, 2017) – Lily May Humphreys watched her first putt Saturday spin out so hard it came to rest six feet from the cup. A birdie chance wound up turning into a bogey.
One hole later, Humphreys' birdie attempt did a hard horseshoe back at her.There's stroking it with authority, but this was a little much.
"I was just trying to hit everything with confidence – but it needed to go in the middle for it to fall in," said the English teen, who found herself 2-over par through her first four holes on the final day of the 54th Junior Orange Bowl International.
Fortunately for Humphreys, she had plenty of cushion to work with. And once she regained her touch, there was nothing to stand in the way of a record performance – finishing nine shots clear of her nearest pursuers at the historic Biltmore Golf Course.
Jerry Ji captured the boys' title in similarly comfortable fashion, holding steady as France's Pierre Pineau waged a brief challenge before falling away for an eventual seven-stroke margin of victory.
The European duo became the newest additions to to a champions list that includes Tiger Woods, LPGA Hall of Famer Inbee Park and such other future major champions as Mark Calcavecchia, Cristie Kerr, Lexi Thompson and Brooke Henderson.
"I just hope I can follow in their footsteps," said Humphreys, 15, who already has built an impressive junior portfolio in Europe but was making her first competitive venture to the United States.
For Ji, the Junior Orange Bowl International marks his first international victory after a series of close calls.
Intermittent clouds and northerly breezes created chilly conditions for golfers on Saturday, a stark contrast to the first three days when temperatures rose into the 80s. Ji's 70 made him one of just 12 boys to break par in the final round; none of the girls managed that.
"The score wasn't that bad, all things considered," said Humphreys, whose 2-over-par 73 was four shots higher than anything else she recorded on the week.
Had her chip shot at No.18 managed to fall in the hole for eagle – it stopped less than a foot away – Humphreys still would have been the first girls' champion to finish at double digits under par.
As it was, her total of 9-under 275 was two shots better than Somi Lee's previous standard established just last year. For Humphreys, it's her first tournament record.
She'll add the JOBI title to five others she's won in 2017, including the English Women's Amateur, European Young Masters, Henry Cooper Masters and Girls' British Open. She also notched four victories in 2016.
The frustration came to a head after another power lipout at No.8, her par attempt rolling away 8 feet from the cup. "I said to myself, 'You're hitting everything way too hard. Just slow down,' " she recalled.
She made the comebacker to save bogey, then didn't card another bogey the rest of the way.
Japan's Miyu Goto (72) claimed second at even-par 284, one stroke ahead of Mexico's Cory Lopez (72). France's Pauline Roussin-Bouchard (71) was another shot back.
Ji took a five-shot lead into the final round, but saw it quickly cut to three when Pineau eagled the par-5 opening hole. The lead fluctuated between three and five for the rest of the front nine, narrowing to three again when Ji bogeyed No.9.
It remained that way for two more holes before Pineau blinked with a bogey at No.12, while Ji stroked home a 6-foot par save. "From then on, it just felt comfortable," Ji said.
Three more Pineau bogeys followed, sealing the outcome.
Pineau wound up carding a 72, leaving him at 9-under 275. Belgium's Adrien Dumont de Chassart was another stroke back after a 71 – his second top-3 finish in three Junior Orange Bowl visits.
Defending champion Karl Vilips of Australia tied for seventh, recovering from a tough start to produce a 69-69 finish. The JOBI still has yet to see a multiple winner on the boys' side.


You got here right here since you wish to download the guide referred to as The Temple of Dawn and it is written by Yukio Mishima. You can download it in PDF or EPUB format, however you too can read it on-line. It belongs to the style . We hope you get pleasure from studying.


text:
\section{Introduction}

The Fundamental Measure Theory (FMT) approach to building approximate free
energy density functionals has proven very successful in describing the
properties of inhomogeneous hard-sphere fluids including the hard-sphere
FCC\ solid\cite{Rosenfeld1,Rosenfeld2,Rosenfeld_1997_1,tarazona_2000_1,tarazona_2002_1,WhiteBear}. The original form
of FMT\ proposed by Rosenfeld\cite{Rosenfeld1,Rosenfeld2} was based
on a generalization of the ideas underlying scaled particle theory. The
Rosenfeld functional gave a useful description of an inhomogeneous fluid but
did not stabilize the FCC solid. One important formal property of the
functional was that its second functional derivative with respect to the
density reduced to the Percus-Yevik direct correlation function in the
uniform liquid limit. Subsequent refinements were motivated in particular by
the requirement that the free energy functional reduce to the known exact
form when the density was restricted to zero- and one-dimensional systems%
\cite{Rosenfeld_1997_1}. The resulting theories, discussed in more detail
below, retains all of the useful properties of the original Rosenfeld
functional while also stabilizing the FCC solid and predicting the
liquid-solid coexistence reasonably accurately\cite{Rosenfeld_1997_1,tarazona_2000_1,tarazona_2002_1,WhiteBear}. The theory also
gives a good description of the mean-squared displacement of the atoms in
the solid phase including the nontrivial prediction that the mean-squared
displacement goes to zero as the density approaches close packing.

The purpose of this paper is to present the results of the application of
the FMT to other crystalline structures of the hard-sphere solid. There are
two good reasons for carrying out such a study. First, from a practical
point of view, one would like to be able to use the FMT to study solid-solid
phase transitions as well as a basis for the study, via thermodynamic
perturbation theory, of non-FCC solids along the lines discussed in
refs.\cite{CurtinAschroftPert,Lutsko_1991_3,Lutsko_1992_1, Song_Perturbation_Theory}. To do so with any confidence first requires that the predictions
of the FMT for non-FCC hard-sphere solids be well understood. The second
reason is that more loosely-packed crystal structures may provide a more
demanding test of the theory than does the FCC phase since the structure of
the FCC phase is not that dissimilar to that of the liquid:\ other structures
differ more from the liquid structure, e.g. in terms of the nearest-neighbor
coordination. As shown below, this expectation is borne out and the
predictions of the properties of the BCC\ phase in particular are not as
satisfactory as for the FCC phase.

The remainder of this paper is organized as follows. Section \ref{sec2} reviews
the elements of Density Functional Theory (DFT) and FMT and discusses the difference between the three
FMTs considered in this work. The results of the calculations are presented
in Section \ref{sec3} where it is shown that the theories give very different
results for different lattice structures. The final Section summarizes
the results of these calculations and discusses obstacles that prevent
any simple modification of these theories to give a better description of the
BCC phase.

\section{Fundamental Measure Theory}
\label{sec2}

As is usual in DFT, the Helmholtz free energy $F$ is written as a
sum of an ideal contribution $F_{id}$ and an excess contribution, $F_{ex}$.
The former is given by%
\begin{equation} \label{fid}
\beta F_{id}\left[ \rho \right] =\int \left[ \rho \left(
\overrightarrow{r}\right) \ln \Lambda ^{3}\rho \left( \overrightarrow{r}%
\right) -\rho \left( \overrightarrow{r}\right) \right] d\overrightarrow{r}
\end{equation}%
where $\rho \left( \overrightarrow{r}\right) $ is the local density, $N=\int
\rho \left( \overrightarrow{r}\right) d\overrightarrow{r}$ is the total
number of particles and $\beta =1/k_{B}T$ is the inverse temperature.
In
the FMT\ approximation, the excess free energy functional is written in
terms of a set of local functions%
\begin{equation}
\beta F_{ex}\left[ \rho \right] = \int \sum_{i=1}^{3}\beta \phi
_{i}\left( \left\{ n_{\alpha }\left( \overrightarrow{r},\left[ \rho \right]
\right) \right\} \right) d\overrightarrow{r}  \label{fex}
\end{equation}%
which, as indicated, include a functional dependence on the density through
a family of local functionals of the form%
\begin{equation}
n_{\alpha}\left( \overrightarrow{r},\left[ \rho \right]\right) =\int w_{\alpha }\left(
\overrightarrow{r}-\overrightarrow{r}^{\prime }\right) \rho \left(
\overrightarrow{r}^{\prime }\right) d\overrightarrow{r}^{\prime }.  \label{2}
\end{equation}%
The weights $w_{\alpha}\left(\overrightarrow{r}\right)$ occurring in eq.(\ref{2}) are $\Theta \left( \frac{\sigma }{2}%
-r\right) $, $\delta \left( r-\frac{\sigma }{2}%
\right) $ , $\widehat{r}\delta \left( r-\frac{%
\sigma }{2}\right) $ and $\widehat{r}\widehat{r}%
\delta \left( r-\frac{\sigma }{2}\right) $ yielding the local density functionals
 which will be denoted $\eta \left( \overrightarrow{r}\right) $, $s\left(
\overrightarrow{r}\right) $, $\overrightarrow{v}\left( \overrightarrow{r}%
\right) $, $\overleftrightarrow{T}\left( \overrightarrow{r}\right) $
respectively. For a uniform liquid, in which $\rho \left( \overrightarrow{r}%
\right) =\overline{\rho }$, one has, in three dimensions, that $\eta \left(
\overrightarrow{r}\right) =\frac{1}{6}\pi \sigma ^{3}\overline{\rho }$ which
is the usual definition of the packing fraction. The names of the other
quantities are motivated by their scalar, vector and tensor natures
respectively. The first two contributions to the excess free energy are%
\begin{eqnarray}
\beta \phi _{1} &=&-\frac{1}{\pi \sigma ^{2}}s\left( \overrightarrow{r}\right) \ln
\left( 1-\eta \left( \overrightarrow{r}\right) \right)   \label{3} \\
\beta \phi _{2} &=&\frac{1}{2\pi \sigma }\frac{s\left( \overrightarrow{r}\right)
^{2}-v^{2}\left( \overrightarrow{r}\right) }{\left( 1-\eta \left(
\overrightarrow{r}\right) \right) }  \nonumber
\end{eqnarray}%
These are the same as in the original Rosenfeld theory. The third function
has been the focus of most efforts to refine the FMT and we write it in the
form%
\begin{equation}
\beta \phi _{3}=\frac{1}{8\pi \left( 1-\eta \left( \overrightarrow{r}\right)
\right) ^{2}}f\left( \overrightarrow{r}\right)
\end{equation}%
where the only fundamental constraint on \ $f$ is that $\lim_{\rho
\rightarrow 0}f=1$ \cite{Rosenfeld_1997_1}. Three proposals in the literature, aside from the
original form $f=\frac{1}{3}s^{3}\left( \overrightarrow{r}\right) $, which
does not stabilize the bulk FCC solid, are%
\begin{eqnarray} \label{f3}
f_{RSLT} &=&\frac{1}{3}s^{3}\left( \overrightarrow{r}\right) \left( 1-\xi
^{2}\left( \overrightarrow{r}\right) \right) ^{3},\;\xi ^{2}=\frac{%
v^{2}\left( \overrightarrow{r}\right) }{s^{2}\left( \overrightarrow{r}%
\right) } \\
f_{T} &=&\frac{3}{2}\left( \overrightarrow{v}\left( \overrightarrow{r}%
\right) \cdot \overleftrightarrow{T}\left( \overrightarrow{r}\right) \cdot
\overrightarrow{v}\left( \overrightarrow{r}\right) -s\left( \overrightarrow{r%
}\right) v^{2}\left( \overrightarrow{r}\right) -Tr\left( \overleftrightarrow{%
T}^{3}\left( \overrightarrow{r}\right) \right) +s\left( \overrightarrow{r}%
\right) Tr\left( \overleftrightarrow{T}^{2}\left( \overrightarrow{r}\right)
\right) \right)   \nonumber \\
f_{E} &=&\frac{2}{3}\left( 1-\eta \left( \overrightarrow{r}\right) \right)
^{2}\left( \beta f_{ex}^{l}\left( \eta \left( \overrightarrow{r}\right)
\right) +\log \left( 1-\eta \left( \overrightarrow{r}\right) \right) -\frac{%
3\eta \left( \overrightarrow{r}\right) }{1-\eta \left( \overrightarrow{r}%
\right) }\right) f_{T}  \nonumber
\end{eqnarray}%
where $f_{RSLT}$ was proposed by Rosenfeld et al\cite{Rosenfeld_1997_1} $%
f_{T}$ is the tensor form proposed by Tarazona\cite{tarazona_2000_1} and $%
f_{E}$ is a heuristic modification of $f_{T}$ that allows for the insertion
of an empirical equation of state for the liquid, $\beta f_{ex}^{l}\left(
\eta \right) $. In the latter case, if the equation of state is chosen to be
the Carnahan-Starling equation\cite{HansenMcdonald}, the resulting free
energy functional is the single-species version of the so-called ''White
Bear'' functional of Roth et al.\cite{WhiteBear}, which was also proposed
by\ Tarazona\cite{tarazona_2002_1}. The first and second forms are
closely related. In fact,\ writing the tensor density as $%
\overleftrightarrow{T}\left( \overrightarrow{r}\right) =\overleftrightarrow{U%
}\left( \overrightarrow{r}\right) +\frac{1}{3}s\left( \overrightarrow{r}%
\right) \overleftrightarrow{1}$, where $\overleftrightarrow{U}\left(
\overrightarrow{r}\right) $ is traceless, a natural approximation for the
traceless part is to make it proportional to the only traceless tensor that
can be formed from the other densities%
\begin{equation}
\overleftrightarrow{U}\left( \overrightarrow{r}\right) \approx A\frac{%
\overrightarrow{v}\left( \overrightarrow{r}\right) \overrightarrow{v}\left(
\overrightarrow{r}\right) -\frac{1}{3}\overrightarrow{v}^{2}\left(
\overrightarrow{r}\right) \overleftrightarrow{1}}{s\left( \overrightarrow{r}%
\right) }
\end{equation}%
where $A$ is a constant and the denominator is needed so that $%
\overleftrightarrow{U}\left( \overrightarrow{r}\right) $ scales linearly
with the density in the sense that $\rho \left( \overrightarrow{r}\right)
\rightarrow \lambda \rho \left( \overrightarrow{r}\right) $ implies that $%
\overleftrightarrow{U}\left( \overrightarrow{r}\right) \rightarrow \lambda
\overleftrightarrow{U}\left( \overrightarrow{r}\right) $. Substituting this
into $f_{T}$ gives $f_{T}=\frac{1}{3}s^{3}\left( \overrightarrow{r}\right)
\left( 1-3\xi ^{2}+3A\xi ^{4}-A^{3}\xi ^{6}\right) $ which reduces to $%
f_{RSLT}$ provided that one takes $A=1$. The choice $A=0$ corresponds to the
original FMT\ theory of Rosenfeld. Note that one could also divide by $\left| \overrightarrow{v}\left( \overrightarrow{r}\right)\right|$
rather than $s\left( \overrightarrow{r}\right)$ in this ansatz, which  is equivalent
to the replacement $A \rightarrow A' \xi^{-1}\left( \overrightarrow{r}\right)$
giving $f_{T}=\frac{1}{3}s^{3}\left( \overrightarrow{r}\right)
\left( 1-3\xi ^{2}+A'(3-A'^{2})\xi ^{3} \right)$. Taking $A'=1$ gives the ``interpolation''
form for $\phi_{3}$ discussed in ref. \cite{Rosenfeld_1997_1}.

The calculations reported here were performed using  the standard approximation of the density as a
sum of Gaussians%
\begin{equation}
\rho \left( \overrightarrow{r}\right) =\sum_{n}f\left( \overrightarrow{r}-%
\overrightarrow{R}_{n}\right)
\end{equation}%
where $f\left( r\right) =x_{0}\left( \frac{\alpha }{\pi }\right) ^{3/2}\exp
\left( -\alpha r^{2}\right) $. For a perfect lattice, one has occupancy $x_{0}=1$
whereas values $x_{0}<1$ characterize a solid with vacancies. The parameter $%
\alpha $ characterizes the width of the Gaussians and $\left\{
\overrightarrow{R}_{n}\right\} $ is the set of lattice vectors of the
desired crystal structure. The density can also be written in terms of components in
reciprocal space as
\begin{equation}
\rho \left( \overrightarrow{r}\right) =\sum_{n}\exp \left( i\overrightarrow{K%
}_{n}\cdot \overrightarrow{r}\right) \rho _{n}
\end{equation}%
where $\left\{ \overrightarrow{K}_{n}\right\} $ is the corresponding set of
reciprocal lattice vectors and $\rho _{n}=x_{0}N_{cell}\rho _{cell}\exp
\left( -K_{n}^{2}/4\alpha \right) $ with $N_{cell}$ being the number of
lattice sites per unit cell and $\rho _{cell}$ being the number of unit cells per
unit volume. Thus, $N_{cell}\rho _{cell}$ is just the number density of the
perfect solid. From the fundamental theorems of DFT, the equilibrium values
of the parameters $x_{0},\rho _{cell}$ and $\alpha $ are those which
minimize the grand potential, $\beta \Omega = \beta F_{id}\left[ \rho \right] +\beta F_{ex}%
\left[ \rho \right] -\beta \mu N$
 for fixed chemical potential $\mu
$ and volume $V$. The  number of atoms is $N = \overline{\rho}V$ where the average density is $\overline{\rho} = \frac{1}{V}\int_{V} \rho(\overrightarrow{r})d\overrightarrow{r} = x_{0}N_{cell}\rho_{cell}$.

The spatial integrals in equations (\ref{fid})-(\ref{fex}) were performed by
 sampling the arguments on a grid of points covering the standard unit cell.
The primary consideration in fixing the number of points in the grid is that
the spacing be sufficiently fine so as to be able to sample the width of the
Gaussians, $\left(\Delta R\right)^{2} = \frac{3}{2\alpha}$. In the calculations discussed below,
the
 spacing of the points in each Cartesian direction was set to of $dr = a/(2q)$
 where $a$ is the lattice parameter and $q$ was initially set to 20 and then doubled until $dr/\Delta R < 0.5$. Increasing the number of points beyond this limit had no significant effect on the results. Note that in
the Gaussian approximation and in the limit of $\alpha a^2 >> 1$ , the ideal contribution to the free energy approaches
the asymptotic value
\begin{equation}
\beta f_{id} \rightarrow x_{0}\left(\frac{3}{2}\ln\left(\frac{x_{0} \alpha \sigma^{2}}{\pi}\right) -2.5 \right).
\end{equation}
This approximation was used for $\alpha a^2 > 200$. The plots of the free energy
as a function of $\alpha$ presented below were all generated based on sampling the
free energy at intervals of $\delta \alpha \sigma^2 = 1, 5, 10, 50$ and $ 250$ for
$\alpha \sigma^{2} < 20, 200, 2000, 10,000$ and $20,000$,
respectively. The free energy minima were determined using the the Simplex algorithm of Nelder
and Mead, see e.g. ref. \cite{NR}, as
implemented in the Gnu Scientific Library\cite{GSL}, which was terminated when the simplex size was smaller
than $10^{-4}$.

\section{Results}
\label{sec3}

\subsection{The FCC crystal}

\begin{figure}[tbp]
\begin{center}
\resizebox{12cm}{!}{
{\includegraphics{fig1}}}
\end{center}
\caption{Dimensionless Helmholtz free energy per atom, $\beta f = \beta F/N $,  as a function of $\alpha$ for the FCC hard-sphere solid as calculated
using (a) the RSLT theory, (b) the Tarazona theory and (c) the White Bear theory. From bottom to
top, the curves are for $\overline{\eta} = 0.56 - 0.68$ in steps of $0.02$. The arrows indicate
the positions of the secondary solid minima.}
\label{fig1}
\end{figure}

Figure \ref{fig1} shows the free energies as a function of $\alpha $ for the FCC\
solid as calculated from the three theories for a variety of average packing
fractions $\overline{\eta} =\frac{\pi }{6}\overline{\rho }\sigma ^{3}$. These
calculations are performed for $x_{0}=1$ which is known to be correct at the
solid minimum\cite{Rosenfeld_1997_1} and which we have verified is very
close to the free energy minimum at all values of $\alpha $. All three
theories show a minimum for all densities at $\alpha =0$ corresponding to
the liquid phase. This is unphysical for the higher densities which are well
above random close packing, $\overline{\eta} _{rcp}\sim 0.64$, and is a result of the
use of the Percus-Yevik and Carnahan-Starling approximate equations of state
which are only singular at $\overline{\eta} =1$. On the other hand, note the
consistency in the location of the free energy minimum at finite $\alpha $
as well as of the value of the free energy at the minimum. While the
location of the minimum in RSLT theory is somewhat higher than for the other
two theories, the differences are not large.  For all three theories, the
location of the solid minimum increases rapidly with density which is a
characteristic feature of the FMT\cite{Rosenfeld_1997_1,tarazona_2002_1}. The Tarazona and White Bear functionals differ primarily
in the small-$\alpha $ region, due to the difference in the equations of
state of the liquid, but at large $\alpha $ the differences are negligible.
The main difference between the tensor theories  and the RSLT\ theory is also in the small-$\alpha$ region where, for $\overline{\eta} > 0.64$ the RSLT\ theory predicts much larger
energy barriers between the liquid and solid minima than do the tensor
theories. Note also that for a narrow range of densities, $0.61\precsim
\overline{\eta }\precsim 0.63$, the RSLT\ theory actually shows \emph{two}
minima at non-zero values of $\alpha $. This appears to be quite unphysical
since all atoms are, by hypothesis, assumed to be identical and so it is
hard to see how there could be two metastable states with the same density but different vibrational amplitudes.

The equilibrium free energy of the solid phase as a function of average
density was determined by minimizing the free energy functional with respect
to both $\alpha $ and the lattice parameter. From this, the densities of the
coexisting liquid and solid as predicted by each theory were determined in
the usual way by finding liquid and solid states with the same pressure and
chemical potential. The RSLT theory gives the coexisting liquid and solid at
packing fractions $0.491$ and $0.534$ respectively (compared to $%
0.491$ and $0.540$ reported in \cite{Rosenfeld_1997_1} and $0.491$ and $0.534$ reported in \cite{Song_FMT}), the Tarazona theory
gives $0.466$ and $0.516$ (compared to $0.467$ and $0.516$ reported in ref.%
\cite{tarazona_2002_1}) and the White Bear theory $0.489$ and $0.535$
(compared to $0.489$ and $0.536$ reported in ref.\cite{tarazona_2002_1}).
For comparison, the values from simulation are $0.492$ and $0.545$\cite%
{Hoover_1968_1}. The reason that the Tarazona theory gives relatively poor
predictions for coexistence is, paradoxically, because it gives a rather
good description of the solid while the liquid equation of state is still
that of the  Percus-Yevik theory which is inaccurate at the high densities
that characterize coexistence\cite{tarazona_2002_1}. This is one of the main motivations for
introducing the empirical equation of state so that a free energy functional
is obtained that yields accurate values over the entire range of structures.

\begin{figure}[tbp]
\begin{center}
\resizebox{12cm}{!}{
{\includegraphics{fig2}}}
\end{center}
\caption{The Lindemann parameter for the FCC phase as a function of $\overline{\eta}$ as calculated using  the RSLT theory (lower line) , the Tarazona theory (middle curve labeled T) and the White Bear theory (upper curve labeled WB). Also shown as dotted lines are the quadratic interpolation of the curves to $L=0$  based on the data for $\overline{\eta} > 0.60$. }
\label{fig2}
\end{figure}

The predicted
Lindemann parameter, $L$ which is the ratio of the root-mean-square displacement
to the nearest neighbor distance, is
shown in Fig.\ref{fig2}. For the FCC crystal, it decreases monotonically with increasing
density and extrapolates to zero at a density near the FCC close-packing
density of $\overline{\eta}_{FCC-CP} = \pi\sqrt{2}/6 \sim 0.74$.

It is worth noting that similar calulations have been performed for the closely related HCP phases. In all cases, the
results closely mirrored those for the FCC phase shown here. In particular, the free
energies were always larger than, but within 1\% of, the FCC free energy.

\subsection{The BCC crystal}

\begin{figure}[tbp]
\begin{center}
\resizebox{12cm}{!}{
{\includegraphics{fig3}}}
\end{center}
\caption{Same as \ref{fig1} for the BCC phase. In (b) and (c), the arrows show the positions of the solid minima. From bottom to
top, the curves are for $\overline{\eta} = 0.52, 0.54, 0.56, 0.58, 0.59, 0.60, 0.61, 0.62, 0.63, 0.64, 0.65$.}
\label{fig3}
\end{figure}

Figure \ref{fig3} shows the same calculations for the BCC solid. In this case, all
three theories give multiple solid minima over some range of densities. For $%
\overline{\eta} \geq 0.56,$the RSLT theory gives two solid minima with the low-density
minimum having lower free energy than the high-density minimum and thus
being the preferred state. The Tarazona theory shows multiple minima for $%
\overline{\eta }\geq 0.61$ with energies that differ by less than one
percent. In the range $0.61\leq \overline{\eta }\leq 0.63$ the low-$\alpha $
branch has slightly lower free energy while for $0.635\leq \overline{\eta }$
the high-$\alpha $ minimum is favored. The White Bear functional only
develops a high-$\alpha $ minimum for $0.63\leq \overline{\eta }$ and in all
cases the low-$\alpha $ minimum has lower free energy.

\begin{figure}[tbp]
\begin{center}
\resizebox{12cm}{!}{
{\includegraphics{fig4}}}
\end{center}
\caption{The Lindemann parameter for the BCC phase as a function of packing fraction, $\overline{\eta}$, as calculated using  the RSLT theory, the Tarazona theory (labeled T) and the White Bear theory (labeled WB). Both the low-$\alpha$ and high-$\alpha$ branches are shown with the stable branch being drawn with full lines and the unstable branch with dashed lines. Also shown as dotted lines are the quadratic interpolation of the curves to $L=0$  based on the data for $\overline{\eta} > 0.60$. }
\label{fig4}
\end{figure}

As seen in Fig. \ref{fig4}, the behavior of the Lindemann parameter for the BCC crystal is more complex than in the case of the
FCC phase. All three theories give a ``low-$\alpha$'' branch
which is not monotonic: the Lindemann parameter decreases with increasing density
at low densities, but then increases with increasing density at high densities.
This behavior is similar to that of the effective-liquid-type DFTs\cite{Lutsko_1990_1, tarazona_2000_1}
and is considered quite unphysical since, if the trend  persisted
up to close packing, it would
imply that spheres are able to interpenetrate. Rather, one expects the mean-squared displacement to
decrease uniformly with increasing density as in the case of the FCC crystal.
All three FMTs also give  a second,
 ``high-$\alpha$'' branch which does behave physically at high densities and, in particular, the Lindemann parameter
extrapolates to zero near the BCC close packing density of $\overline{\eta}_{BCC-CP} = \pi\sqrt{3}/8 \sim 0.68$. However, for with the RSLT theory,  up
to $\overline{\eta} = 0.625$, the well-behaved branch has higher free energy than does the unphysical branch. The Tarazona theory gives similar results except that the high-$\alpha$ minima are thermodynamically favored at high density so that, as noted by Tarazona\cite{tarazona_2000_1, tarazona_2002_1}, the prediction of the theory is that the Lindemann parameter appears to vanish near close-packing, as it should.
Nevertheless, the prediction of the theory is that the Lindemann parameter increases with increasing density at intermediate densities as well as giving a discontinuous jump in the Lindemann parameter for a packing fraction near $0.60$, all of which seem somewhat unphysical. Finally, the White Bear theory also has a second branch but it only appears at very high densities and also has higher free energy than does the low-$\alpha$ branch. Comparison of Figures 2b and 2c shows that the reason for the observed behavior is that the Carnahan-Starling equation of state used in the White Bear functional lowers the liquid free energy relative to  the Percus-Yevik equation of state that comes out of the Tarazona theory. As $\alpha$ increases, the effect lessens until at high $\alpha$ the theories are almost identical. Thus, the free energy of the low-$\alpha$ minimum is somewhat lower in the White Bear theory leading to its stability persisting even up to high densities.

\subsection{The SC phase}

\begin{figure}[tbp]
\begin{center}
\resizebox{12cm}{!}{
{\includegraphics{fig5}}}
\end{center}
\caption{Same as Fig. \ref{fig1} for the SC phase. From bottom to
top, the curves are for $\overline{\eta}$ in the range $[0.43,0.52]$ in steps of $0.01$.}
\label{fig5}
\end{figure}

For a purely repulsive potential, the simple cubic phase is expected to be unstable with respect
to shear. Nevertheless, there is no reason that DFT should not give a well-defined free energy
for this structure since it should be metastable. Indeed, at least one of the effective-liquid
theories, the GELA, does give a stable structure with small values of $\alpha$\cite{Lutsko_1990_1}.

Figure \ref{fig5} shows the free energy curves calculated using the three theories. The results
from the RSLT theory are much as expected: the simple cubic phase stabilizes at quite low densities, $\overline{\eta} \sim 0.45$, and
the location of the free energy minimum increases with increasing density and appears to diverge as
the simple cubic close packing, $\overline{\eta}_{SC-CP} = \pi/6 \sim 0.524$ is approached. In this case, there is no indication
of multiple minima. In contrast, the tensor theories show no sign of a $\alpha \ne 0$ minimum except at densities very close to close packing. Extending the calculations up to $\alpha \sigma^{2} = 20,000$, no minimum was found in the $\overline{\eta} = 0.51$ curve using either of the tensor models while for $\overline{\eta} = 0.52$ the Tarazona theory stablizes the solid at $\alpha \sigma^{2} \sim 11,500$ and there is again no minimum using the White Bear theory.

\subsection{Relative Stability}
\label{rel}

\begin{table}[tbp]
\caption{The minimum packing fractions at which the various FMT models predict the
different crystal structures become stable and the approximate value of the Gaussian
parameter $\alpha$ at this density. }
\label{tab1}%
\begin{ruledtabular}
\begin{tabular}{cccc}
Structure & Theory & $\overline{\eta }_{sol}$ & $\alpha \sigma^{2}$ \\ \hline
FCC & RSLT & 0.49 & 27 \\
 & T & 0.46 & 22 \\
 & WB & 0.48 & 26 \\
BCC & RSLT & 0.51 & 14 \\
 & T & 0.48 & 20 \\
 & WB & 0.51  & 24 \\
SC & RSLT &  0.47 & 1150 \\
SC & T &  0.52 & 11,250\\
SC & WB &  --- & --- \\
\end{tabular}
\end{ruledtabular}
\end{table}

The last question we adress is the relative stability of the various structures. Table \ref{tab1} shows the minimum densities at which each structure becomes stable in the various FMT models. For the FCC and BCC structures, the RSLT and WB theories give quite similar results while the Tarazona theory generally stabilizes the solid at a lower density about 5\% below the other theories. In all cases, the FCC phase is stable at lower densities than the BCC phase.

The free energies as functions of the average density are shown in Fig. \ref{fig6}. In all models, the FCC phase has the lowest free energy of any solid phase at all densities. The free energy of the BCC phase is very close to that of the FCC at low densities and steadily diverges at increasing densities. The FCC phase has slightly higher free energy than the liquid at the lowest densities. The low-$\alpha$ branch of the BCC phase always has lower free energy than the liquid, although in the RSLT theory they are almost identical. In the RSLT, the simple cubic structure has much higher free energy than any other phase.

\begin{figure}[tbp]
\begin{center}
\resizebox{12cm}{!}{
{\includegraphics{fig6}}}
\end{center}
\caption{The free energy minima for the liquid phase and the various crystal structures
as functions of the average packing fraction calculated
using (a) the RSLT theory, (b) the Tarazona theory and (c) the White Bear theory. The lowest
curves are for the FCC structure, the intermediate curves are the low-$\alpha$ (full) and
high-$\alpha$ (dashed) BCC branches and the light curve that spans the width of the figures
are for the liquid phase. In (a), the upper curve is for the simple cubic structure.}
\label{fig6}
\end{figure}

\section{Conclusions}
\label{sec4}

In summary, the present calculations confirm the fact that the three FMT's discussed all give reasonable results for the FCC phase at all densities up to FCC close-packing with the only exception being the appearance of a spurious low-$\alpha$ minimum at high densities in the RSLT theory. On the other hand, when applied to the BCC phase, these results indicate several unphysical properties of the Fundamental Measure Theory functionals which must be carefully considered when using them, e.g., as the basis for thermodynamic perturbation theory. All of the theories appear to give reasonable results at moderate densities, say up to $\overline{\eta} \sim 0.56$.  However, the simplest model, the RSLT, gives multiple solid minima for all packing fractions in the BCC phase above about $0.56$. The Tensor theories do not give multiple minima for the FCC phase but they do for the BCC phase. As the density increases, the stable minima in the Tarazona theory switches from the unphysical branch to the physically well-behaved branch whereas the White Bear theory always picks out the unphysical branch. Only the RSLT and Tarazona  theories stablize the simple cubic phase, at least for $\alpha \sigma^{2} < 20,000$, but  the Lindemann parameter is, in this case, well-behaved and there are no spurious minima. Unfortunately for the goal of choosing a single theory to describe both the liquid and all solid phases, the RSLT and WB theories give poor results for the BCC phase Lindemann parameter while the Tarazona theory gives the worst quantitative description of FCC-liquid coexistence.

It is interesting to speculate on the resolution of the common deficiencies in these theories. It seems reasonable to attribute the unphysical behavior at high densities, which is manifested by the fact that the unphysical low-$\alpha$ minimum is favored, to a poor description of the liquid phase: the free energy of the liquid must clearly diverge at high densities and in fact it has long been debated whether the hard-sphere liquid-phase equation of state possesses a singularity at high densities\cite{McCoy_2006_1} - at, e.g., random close packing\cite{Ma_1985} or FCC close packing\cite{Sanchez_1_1994}.
If the free energy of the liquid did indeed diverge at random close packing, then the curves shown in Figs. \ref{fig1} and \ref{fig3} would be deformed with the low-$\alpha$ free energies ``pulled up''. This could conceivable eliminate altogether the low-$\alpha$ minima and thus give a reasonable picture of the BCC phase. However, as appealing as this scenario is, it cannot be implemented naively using eq.(\ref{f3}) to give the equivalent of the ``White Bear'' improvement to the Tarazona theory. This is because eq.(\ref{f3}) requires that the equation of state be evaluated at the local packing fraction $\overline{\eta}(\overrightarrow{r})$ and for the solid phase there are always points at which $\eta(\overrightarrow{r})$ is very nearly equal to one unless $\alpha$ is very small. (In particular,  it is easy to see that $\lim_{\alpha \rightarrow \infty}\eta(\overrightarrow{0}) = 1$.) Thus, simply inserting an equation of state with a divergence for $\overline{\eta} < 1$ is not an option since this will lead to divergent free energies in the solid phase, even for the already well-described FCC phase.

One approximation not tested here is the use of the Gaussian profile for the densities. Tarazona has performed calculations using more general parameterizations of the density and he reports that the generalization beyond the Gaussian is of negligble imortance for the equation of state\cite{tarazona_2000_1}. Groh and Mulder have also supplied evidence for the FCC phase that the Gaussian approximation is quite accurate, especially near the peaks of the density distribution\cite{Groh_Mulder}. This suggests that the effects reported here are not due to the Gaussian approximation but definitive proof will require further calculations using non-Gaussian profiles.

In summary, the Fundamental Measure Theories work well for the FCC phase at all densities while for the BCC phase, the FMTs are best applied to the solid phase for densities near liquid-solid coexistence. While they can in principle be used to model the BCC phase at higher densities, and while they do show some desirable features at high density like the vanishing of the Lindemann parameter at close packing, they also suffer from the same problem that plagued earlier DFT models: namely, the unphysical behavior of the Lindemann parameter for the BCC phase at intermediate densities. On the other hand, the SC phase is well modeled, at least using the RSLT theory, perhaps because it only exists at relatively low densities.

\begin{acknowledgments}
I am grateful to Marc Baus and Xeyeu Song for several stimulating discussions on this topic.
This work was supported in part by the European Space Agency/PRODEX
under contract number C90238.
\end{acknowledgments}

\bigskip

\bibliographystyle{apsrev}

redpajama_set_name: RedPajamaArXiv


The Board of Education for the Conejo Valley Unified School District announced at its regularly scheduled Tuesday, June 5 meeting that it has appointed Jeanne Valentine as the new Assistant Superintendent of Human Resources.
Ms. Valentine has a 26-year long tenure with the District, and most recently served as the District's Director of Elementary Education for the past four years. Prior to her role at the District office, Ms. Valentine served as the principal of Weathersfield Elementary, the principal of Cypress Elementary, an administrative intern at Lang Ranch Elementary, a teacher at Lang Ranch Elementary, and a teacher at Banyan Elementary.
Ms. Valentine has a Masters of Arts Degree in Educational Administration from California State University, Northridge, and a Bachelor of Arts Degree in Psychology from San Diego State University. She also completed the Association of California School Administrators (ACSA) Personnel/HR Academy in 2017.
This formal appointment comes after Robert Iezza, current Deputy Superintendent of Human Resources announced his retirement this June, after 30 years of dedicated service to the Conejo Valley Unified School District.


text:
\section{Introduction}

A Fatou-Bieberbach domain is a proper domain of $\mathbb{C}^k$ that is biholomorphic to $\mathbb{C}^k$. These domains have been extensively studied, but many questions about them are still open ~\cite{rr}.

One of these open question is the following: (\cite{rr}, p. 79)
\begin{question}
Is there a biholomorphic map from $\mathbb{C}^2$ into the set $\{zw \neq 0\}$ i.e. into the complement of the union of two intersecting complex lines?
\end{question}

One classical way of constructing Fatou-Bieberbach domains (i.e. biholomorphic maps from $\mathbb{C}^k$ into $\mathbb{C}^k$) is to find basins of attractions of automorphisms of $\mathbb{C}^k$ with a fixed point, as follows:

For $F \in {\mathrm{Aut}}(\mathbb{C}^k), F(p) = p, F'(p) = A$ we define
$$
\Omega_{F,p} = \{z \in \mathbb{C}^k \mid \lim_{n\to\infty}F^{n}(z) = p \}.
$$
We say $F$ has an \textit{attracting} fixed point when $A$ is a matrix with eigenvalues of modulus less than~$1$. In this case $\Omega_{F,p}$ is biholomorphic to $\mathbb{C}^k$ \cite{rr}.

In the \textit{semiattracting} case (eigenvalues of modulus smaller or equal than $1$), and for automorphisms \textit{tangent to the identity} (i.e., $A = Id$) then $\Omega_{F,p}$ can also be biholomorphically equivalent to $\mathbb{C}^l$ (where $l \leq k$). See \cite{ue} for semi-attracting case and \cite{hak1},\cite{hak2},\cite{we} for automorphisms tangent to the identity.

Although not all Fatou-Bieberbach domains are basins of attraction of an automorphism of $\mathbb{C}^k$ \cite{w5} these are the natural source of examples (and counter-examples) for various conjectures.

One natural approach, in order to answer Question 1 positively by using these  results, is the following (for the definitions see \cite{hak1} or Section 3 below):

\medskip

\textsl{
Find an automorphism $F$ of $\mathbb{C}^2$ such that:
\begin{itemize}
\item $F$ is tangent to the identity i.e. $F(0) = 0$ and $DF(0) = Id$.
\item $F$ fixes the coordinate axes i.e. $F(z,0) = (z',0)$ and $F(0,w) = (0,w')$.
\end{itemize}
Then:
\begin{enumerate}
\item There exists an attracting fixed point for $F$.
\\or
\item There exists $v$ non-degenerate characteristic direction of $F$ at the origin such that $\textrm{Re} A(v) > 0$, where $A(v)$ is the number associated to the direction $v$.
\end{enumerate}
Then, in the case of (a) we would have a basin of attraction associated to this fixed point \cite{rr}, or in the case of (b) we would a basin associated to $v$ (as in Hakim's notation) by \cite[Theorem 5.1]{hak1}. In either case, this basin $\Omega$ would be a Fatou Bieberbach domain  and $\Omega \subset \{zw \neq 0\}$.
}

\medskip

Our main result is that, assuming a well known conjecture about automorphisms of $\mathbb{C}^* \times \mathbb{C}^*$, this approach is not possible.

More precisely, we assume the following:

\begin{conjecture}
If $F \in {\mathrm{Aut}}(\mathbb{C}^* \times \mathbb{C}^*)$, then $F$ preserves the form:
\begin{eqnarray*}
\frac{dz \wedge dw}{zw}
\end{eqnarray*}
\end{conjecture}

Assuming this, we prove:

\begin{proposition}
If Conjecture 2 is valid, and $F$ is an automorphism of $\mathbb{C}^2$ tangent to the identity that fixes the coordinate axes, then (a) and (b) are both false.
\end{proposition}

Note that this does not answer Question 1 in general:  in principle, we could have basins associated to degenerate characteristic directions.

\bigskip

\noindent
{\bf Acknowledgments.} I would like to thanks Mattias Jonsson for helpful comments on a draft of this note and Berit Stens\o nes for useful conversations. Thanks also to the referee for useful remarks. Part of this work was done during a visit to the Institut Mittag-Leffler. Thanks to the Institut for its hospitality.

\section{Automorphisms of $\mathbb{C}^*\times\mathbb{C}^*$}

If $F$ is an automorphism of $\mathbb{C}^2$ that fixes the coordinate axes, then we have that $F|_{\mathbb{C}^*\times\mathbb{C}^*}$ is an automorphism of $\mathbb{C}^*\times\mathbb{C}^*$.

The automorphism group of $\mathbb{C}^* \times \mathbb{C}^*$ has been studied before \cite{ni},\cite{var1},\cite{an1}, but remains mysterious.

\begin{proposition}
Assume Conjecture 2 holds and let $F: \mathbb{C}^2 \to \mathbb{C}^2$ be an automorphism of $\mathbb{C}^2$ such that:
\enumerate
\item $F(0,0) = (0,0)$, $F'(0,0) = Id$
\item $F$ fixes the coordinate axes $\{zw=0\}$.

Then we can write $F$ as follows:
\begin{equation}\large{
F(z,w) = (z e^{wg(z,w)}, we^{zh(z,w)}) }
\end{equation}
where as power series we have:
\begin{eqnarray*}
g(z,w) = \sum_{\alpha+\beta \geq k} c_{\alpha,\beta}z^{\alpha}w^{\beta}\\
h(z,w) = \sum_{\alpha+\beta \geq k} d_{\alpha,\beta}z^{\alpha}w^{\beta}
\end{eqnarray*}
and
\begin{equation}
a_{\alpha-1,\beta} = -\frac{\alpha}{\beta}  b_{\alpha,\beta-1}
\end{equation}
for $k \leq \alpha + \beta \leq 2k$.
\end{proposition}

\begin{proof}
An easy computation shows that $F(z,0) = (z,0)$ since $F|_{(z,0)}$ is an automorphism of $\mathbb{C}$ and $F$ is tangent to the identity. The same argument implies $F(0,w) = (0,w)$.
If we write $F(z,w) = (f_1(z,w), f_2(z,w))$ then we have:
$f_1(z,0) = z$ and $f_2(z,0) = 0$. In the same way $f_1(0,w) = 0$ and $f_2(0,w) = w$. So we get $f_1(z,w) = z + zwr(z,w)$ and $f_2(z,w) = w + zws(z,w)$.
So, we can write:
$$
g(z,w) := \frac{\log(1 + wr(z,w))}{w}
$$
and
$$
h(z,w) := \frac{\log(1 + zs(z,w))}{z}
$$
where $g$ and $h$ will be well defined function in $\mathbb{C}^2$ and therefore we have (1).
Using Conjecture 3, we will have the following relationship between $g$, $h$ and their derivatives:
\begin{eqnarray}
g_z + h_w - gh - z gh_z - w hg_w - zw g_wh_z + zw g_z h_w= 0
\end{eqnarray}
Writing $g$ and $h$ as power series and looking at the terms of degree less than $2d$ in the left term we will have (2).
\end{proof}

\section{Basins of attraction}

We recall now Hakim's notation and results for automorphisms of $\mathbb{C}^2$ tangent to the identity.

Let $F: \mathbb{C}^2 \to \mathbb{C}^2$ be an automorphism of $\mathbb{C}^2$ tangent to the identity. We can write $F$ as power series around the origin as follows:
$$F(z,w) = (z + P_{k}(z,w) + P_{k+1}(z,w)+..., w + Q_{k}(z,w) + Q_{k+1}(z,w)+ ...)$$
where $P_{l}$ and $Q_l$ are homogeneous polynomials of degree $l$ (or identically 0).
The \textit{order} of $F$ is the smallest $k\geq 2$ such that $(P_{k}(z,w),Q_{k}(z,w))$ does not vanish identically.

\begin{defin}
A \textit{characteristic direction} is a direction $v \neq 0$ in $\mathbb{C}^2$ such that
$$(P_k(v), Q_k(v)) = \lambda v$$
for some $\lambda \in \mathbb{C}$, where $k$ is the order of $F$.
When $\lambda \neq 0$  then we call $v$ a \textit{non-degenerate} characteristic direction. Similarly if $\lambda =0$ then $v$ is a \textit{degenerate} characteristic direction
\end{defin}

For $v \in \mathbb{C}^2$ a non-degenerate characteristic direction we can assume without loss of generality $v = (1,u_0)$ with $P_{k}(1,u_0) \neq 0$.

If we define:
$$
r(u) := Q_k(1,u) - uP_k(1,u)
$$
then the non-degenerate characteristic directions of $F$ are the zeroes of the polynomial function~$r$.

To each non-degenerate characteristic direction we associate the number:
\begin{eqnarray*}
A(v) := \frac{r'(u_0)}{P_{k}(1,u_0)}
\end{eqnarray*}

Then Hakim proves \cite[Thm. 5.1 + Remark 5.3]{hak1}:

\begin{theorem}
Let $F$ be an automorphism of $\mathbb{C}^2$ tangent to the identity. Let $v$ be a non-degenerate characteristic direction. Assume that $\textrm{Re} A(v) > 0$. Then there exists an invariant attracting domain $D$ in which every point is attracted to the origin along a trajectory tangent to $v$. Then the open set:
$$
\mathcal{D} = \cup_{n=0}^{\infty}F^{-n}(D)
$$
is attracted to $0$ and $\mathcal{D}$ is biholomorphic to $\mathbb{C}^2$.
\end{theorem}

With this result in mind we prove Proposition 2.

\begin{proof} [Proof of Prop. 2]
We will first prove that there are no attracting fixed points.
If we assume Conjecture 2, an easy computation shows that the Jacobian of $F$ will be:
\begin{eqnarray}
JF(z,w) = e^{wg(z,w) + zf(z,w)}
\end{eqnarray}
The fixed points for $F$ are:
\begin{itemize}
\item $(0,0)$ where $DF(0,0) = Id$, therefore the origin is not attracting.
\item $(z_0,0)$ and $(0,w_0)$. For these points an easy computation shows that they are semi-attracting (or semi-repelling) fixed points (i.e. the eigenvalues are $1$ and $\lambda$); therefore not attracting either.
\item $(z_0,w_0)$ not necessarily on the axes, where $e^{w_0g(z_0,w_0)} = e^{z_0h(z_0,w_0)} = 1$. Using equation (4) we have: $JF(z_0,w_0) = 1$, therefore $(z_0,w_0)$ will not be an attracting fixed point either (in case $DF(z_0,w_0) = Id$ see Remark 7).
\\
\end{itemize}

Now we prove that (b) is not possible.
We have:

\begin{equation}\large{
F (z,w) = (z e^{wg(z,w)}, we^{zh(z,w)}) }
\end{equation}\\

We assume that the lowest degree in $g$ (and therefore in $h$) is $k$.
Then we have:
$$
g(z,w) = \sum_{\alpha + \beta =k} c_{\alpha,\beta}z^{\alpha}w^{\beta} + h.o.t.
$$
and
$$
h(z,w) = \sum_{\alpha + \beta =k} d_{\alpha,\beta}z^{\alpha}w^{\beta} + h.o.t.
$$
where
$$
d_{\alpha-1,\beta} = -\frac{\alpha}{\beta}  c_{\alpha,\beta-1}
$$

Therefore the order of $F$ is $k+2$:
\begin{eqnarray*}
F(z,w) &=& (z e^{wg(z,w)}, we^{zh(z,w)}) \\
&=& \left(z (1 + wg(z,w) + O(w^2g^2)),  w (1 + zh(z,w) + O(z^2h^2))\right)\\
&=& \left( z + zw\sum_{\alpha + \beta =k} c_{\alpha,\beta}z^{\alpha}w^{\beta} + O(|(z,w)|^{k+3}), w + zw\sum_{\alpha + \beta =k} c_{\alpha,\beta}z^{\alpha}w^{\beta} + O(|(z,w)|^{k+3}) \right)\\
\end{eqnarray*}

We will call the lowest degree homogeneous terms $P_{k+2}$ and $Q_{k+2}$ as in Hakim's notation:
\begin{eqnarray}
P_{k+2} (z,w) &=&  \sum_{\alpha + \beta =k} c_{\alpha,\beta}z^{\alpha+1}w^{\beta+1} \\
Q_{k+2} (z,w) &=&  \sum_{\alpha + \beta =k} d_{\alpha,\beta}z^{\alpha+1}w^{\beta+1}
\end{eqnarray}

Now we want to compute characteristic directions and the numbers associated to the non-degenerate ones. (Note that we have: $(P_{k+2}(1,0), Q_{k+2}(1,0)) = (0,0)$ and $(P_{k+2}(0,1), Q_{k+2}(0,1)) = (0,0)$; therefore $(1,0)$ and $(0,1)$ are degenerate characteristic directions).

From now on we assume that $v = (1, \theta)$ is a non-degenerate characteristic direction, with $\theta \neq 0$. So  we want to solve:
\begin{eqnarray*}
r(u) &=& Q_{k+2}(1,u) - u P_{k+2}(1,u) = 0
\end{eqnarray*}
Putting this back in (5) and (6) we get:
\begin{eqnarray*}
r(u) &=& u \left( \sum_{\beta =0}^k d_{k -\beta,\beta}u^{\beta} - \sum_{\beta= 0}^k c_{k-\beta,\beta}u^{\beta +1}\right)\\
&=& u\left( d_{k,0} + \sum_{\beta=1}^k(d_{k-\beta,\beta}-c_{k-\beta+1,\beta-1}) u^\beta - c_{0,k}\lambda^{k+1}\right)
\end{eqnarray*}

Therefore the characteristic directions will be $(1,\theta)$ where $s(\theta) = 0$ for $r(u) = u s(u)$ i.e.
\begin{eqnarray*}
s(u) = d_{k,0} + \sum_{\beta=1}^k(d_{k-\beta,\beta}-c_{k-\beta+1,\beta-1}) u^\beta - c_{0,k}\lambda^{k+1}
\end{eqnarray*}
and so we have:
\begin{equation}
s(\theta) = d_{k,0} + \sum_{\beta=1}^k(d_{k-\beta,\beta}-c_{k-\beta+1,\beta-1}) \theta^\beta - c_{0,k}\theta^{k+1} = 0
\end{equation}

Now we can compute the numbers associated to each direction with the following formula:
\begin{eqnarray}
A(\theta)&=& \frac{r'(\theta)}{P_{k+2}(1,\theta)}
\end{eqnarray}

Since we know:
\begin{equation*}
r(u) = d_{k,0}u + \sum_{\beta=1}^k(d_{k-\beta,\beta}-c_{k-\beta+1,\beta-1}) u^{\beta+1} - c_{0,k}u^{k+2}
\end{equation*}
then we can easily get:
\begin{equation*}
r ' (u) = d_{k,0} + \sum_{\beta=1}^k(\beta+1) (d_{k-\beta,\beta}-c_{k-\beta+1,\beta-1})u^{\beta} - (k+2)c_{0,k}u^{k+1}
\end{equation*}

Putting this back in (9) together with (6), we have:
\begin{eqnarray}
A(\theta) &=& \frac{d_{k,0} + \sum_{\beta=1}^k(\beta + 1) (d_{k-\beta,\beta}-c_{k-\beta+1,\beta-1}) \theta^{\beta} - (k+2)c_{0,k}\theta^{k+1}}{ \sum^k_{\beta = 0} c_{k-\beta,\beta}\theta^{\beta+1}}
\end{eqnarray}

(When $k=0$ the sum from $\beta = 1$ to $k$ is the empty sum.)

By (8) we have:
\begin{equation}
 d_{k,0} = - \sum_{\beta=1}^k(d_{k-\beta,\beta}-c_{k-\beta+1,\beta-1}) \theta^\beta + c_{0,k}\theta^{k+1}
\end{equation}
and putting this in (10) we can have some cancelations:
\begin{eqnarray*}
A(\theta) &=& \frac{- \sum_{\beta=1}^k(d_{k-\beta,\beta}-c_{k-\beta+1,\beta-1}) \theta^\beta + c_{0,k}\theta^{k+1} + \sum_{\beta=1}^k(\beta + 1) (d_{k-\beta,\beta}-c_{k-\beta+1,\beta-1})
\theta^{\beta} - (k+2)c_{0,k}\theta^{k+1}}{ \sum_{\beta = 0}^k c_{k-\beta,\beta}\theta^{\beta+1}}
\end{eqnarray*}

So after simplifying and canceling $\theta$ (since is not equal to $0$), we have:
\begin{eqnarray}
A(\theta)&=& \frac{\sum_{\beta=1}^k\beta (d_{k-\beta,\beta}-c_{k-\beta+1,\beta-1}) \theta^{\beta} - (k+1)c_{0,k}\theta^{k+1}}{ \sum_{\beta = 0}^k c_{k-\beta,\beta}\theta^{\beta+1}}\nonumber\\
&=& \frac{\theta \left( \sum_{\beta=1}^k \beta (d_{k-\beta,\beta}-c_{k-\beta+1,\beta-1}) \theta^{\beta-1} - (k+1)c_{0,k}\theta^k\right)}{ \theta\left(\sum_{\beta = 0}^k c_{k-\beta,\beta}\theta^\beta\right)}\nonumber\\
&=& \frac{\sum_{\beta=1}^k\beta (d_{k-\beta,\beta}-c_{k-\beta+1,\beta-1}) \theta^{\beta-1} - (k+1)c_{0,k}\theta^k}{\sum_{\beta = 0}^k c_{k-\beta,\beta}\theta^\beta}
\end{eqnarray}

If the conjecture holds true, then we have:
\begin{eqnarray*}
d_{\alpha-1, \beta} = -\frac{\alpha}{\beta}c_{\alpha, \beta-1}
\end{eqnarray*}
for $k \leq \alpha + \beta - 1 \leq 2k$. So, using $\alpha = k + 1 - \beta$ we have:
\begin{eqnarray*}
d_{k - \beta, \beta} = -\frac{k-\beta +1}{\beta}c_{k- \beta+1, \beta-1}
\end{eqnarray*}
(For $k = 0$ the condition is empty, but the result still holds.)
So, back in our equation (12):
\begin{eqnarray*}
A(\theta)&=& \frac{\sum_{\beta=1}^k \beta (-\frac{k-\beta +1}{\beta}c_{k- \beta+1, \beta-1}-c_{k-\beta+1,\beta-1}) \theta^{\beta-1} - (k+1)c_{0,k}\theta^k}{\sum^k_{\beta = 0} c_{k-\beta,\beta}\theta^\beta}\\
&=& \frac{-(k+1)\sum_{\beta = 0}^k c_{k-\beta,\beta}\theta^\beta}{\sum_{\beta = 0}^k c_{k-\beta,\beta}\theta^\beta}\\
&=& -(k+1)
\end{eqnarray*}

This finishes the proof of Proposition 2.
\end{proof}\

\begin{remark}
If $(z_0,w_0)$ is a fixed point different from the origin, where $DF(z_0,w_0) = Id$, then we can use the same technique to show that the result above still holds i.e. non-degenerate characteristic directions at $(z_0,w_0)$ will have a negative number associated to them.
\end{remark}


ASHBURN, Va. (AP) – The Washington Redskins have signed cornerback Quinton Dunbar to a multiyear deal. The team announced the contract Monday after finishing its season at 7-9. Dunbar was set to be a.
Travel guides. Starting at $32.49. Ready to go? Get to the heart of Skye with one of Lonely Planet's in-depth, award-winning guidebooks.
Statistics of Gull Lake From noon today Gull will be dropping petrol prices by 3 cents per litre at most outlets. If international prices remain low we hope that we are able to sustain this drop and that will be a big help f.Stoughton statistics Government of Brownlee Brownlee graduated from Minnesota State University Moorhead with a degree in accounting and became a certified public accountant in 1985. He was an audit manager for the Local Government Division at t.Stoughton, Wisconsin detailed profile.. Races in Stoughton detailed stats: ancestries, foreign born residents, place of birth. Mar. 2016 cost of living index in.Statistics of Belle Plaine 2017 belle plaine tiger statistics · 2016 Belle Plaine Tiger Statistics · 2015 belle plaine Tiger Statistics · Team 2013 State. 2014 Belle Plaine Tigers Statistics.
STOCKTON – On Sept. 15, the Lincoln high football team beat Golden Valley 41-14. There was plenty of scoring that Friday night, but the one that sticks out to Quinton Gaines was the first. "It was for.
Quinton Wall » Epic Travel Photography » Learning the Leica Q – What I've learned so far Aperture. The lens on this camera, 28mm fixed, is gorgeous. I'm comfortable shooting with a fixed lens as the 40mm.
Welcome to Mikelle Fetsch and Quinton Thiele's Wedding Website! View photos, directions, registry details and more at The Knot.
See distance to other cities from Birmingham – England – United Kingdom measured in kilometers (km), miles and nautical miles and their local time. distances are measured using a direct path, as the crow flies and the compass direction is shown as well.
9. Birmingham – Bearwood – Halesowen – Lye – Stourbridge via Broad Street, Five Ways, Hagley Road, Quinton and Colley Gate. Service Information.
· Manor Way Halesowen to Quinton – March 1981 Not great quality… but interesting. There is a factory next to Sandvik which is.


text:
ASUCR
Highlander Write-Off
R'Perspectives
Campus Cope
Under the Kilt
Building Culture
Concerts and Live Performances
Events and Profile
Highlander Hot Takes
Riverside, US
UCR moves to vaccinate front-line healthcare employees
Davina Garcia - January 18, 2021
As COVID-19 conditions worsen in Riverside, UCR announces decision to continue spring quarter instruction remotely
The Barn closes temporarily after a member of staff tests positive for COVID-19
Gov. Newsom updates California budget, prioritizing public education
UCR's Student Life prepares for remote Highlander Orientation
UCR provides assistance for undocumented and international students despite CARES Act restrictions
Laura Anaya-Morga - May 18, 2020
Last month, U.S. Secretary of Education Betsy DeVos issued guidelines detailing how colleges should distribute the $6.2 billion they were given through the Coronavirus Aid, Relief, and Economic Security (CARES) Act meant to ease...
UCR students respond to the campus closure and unchanging tuition and fees for spring...
Laura Anaya-Morga - March 30, 2020
UCR's campus closure due to COVID-19 means that all instruction will be delivered remotely for the entirety of spring quarter. This has left many students with questions regarding what will happen to...
The Highlander is committed to the pursuit of truth, the free exchange of information and ideas and maintaining a fair and independent student voice. The Highlander exists to serve its readership, comprised of the community at and around University of California, Riverside and the Inland Empire as a whole. In our pursuit of the truth, we will provide accurate information relevant to the experiences and interests of our readers. The Highlander seeks to foster an environment where student journalists learn the necessary skills to become probing, fair and critical in their writing and thought. We strive to go beyond basic reporting through vigorous investigation, analysis of the facts and creative thinking.


text:
Q: Java XStream - Serialize/deserialize multiple types of arrays at root level I have been looking for my scenario on SO and numerous other forums for two days now and tried several ideas in my code but nothing seems to produce my required xml, and deserialization has even more issues.
My app's user will have to save the project configuration to an xml file. Now the project has 3 different, distinct types of objects at root level, and not one. And there are multiple (yet fixed) number of objects of each root. Each object of each root further has other kinds of objects inside them (this part works perfectly well). So hierarchy is like this:
public static AA a[] = new AA[Global.MAX1];
public static BB b[] = new BB[Global.MAX2];
public static CC c[] = new CC[Global.MAX3];

Now as my app proceeds, user creates and populates all the objects of these arrays which are populated in a JTree. When he wishes to save or at app exit time, whole project's configuration (created objects) are to be saved to .xml file to be retrieved later as they're right now, for example to populate the same JTree.
@What I tried:
All the examples that I saw have just one top level element. Since I didn't want to create 3 different config files for each project, so I faked a top level element called Domain like this:
@XStreamAlias("Area")
public final class Domain {
    public static Domain domain = null;

    @XStreamAsAttribute
    public String name = "Area";

    @XStreamImplicit(itemFieldName = "aas")
    public static AA a[] = new AA[Global.MAX1];

    @XStreamImplicit(itemFieldName = "bbs")
    public static BB b[] = new BB[Global.MAX2];

    @XStreamImplicit(itemFieldName = "ccs")
    public static CC c[] = new CC[Global.MAX3];

    private Domain () { } // private so that it cannot be initialized by using {new domain();}

    public static Domain getInstance() { // singleton
        if (domain == null)
            domain = new Domain();
        return domain;
    }
}

But now at project save time, if I use following code to serialize:
String xml = "<?xml version=\"1.0\" encoding=\"UTF-8\"?>";
xml += lnSep + xStream.toXML(Domain.getInstance()); // or define a domain object and use
xml += lnSep + xStream.toXML(Domain.a);             // that to pass data to toXML()
xml += lnSep + lnSep + xStream.toXML(Domain.b);     // with same final effect
xml += lnSep + lnSep + xStream.toXML(Domain.c);

Then I get following xml (trimmed for brevity):
<?xml version="1.0" encoding="UTF-8"?>
<Area id="1" name="Area"/>
<aa-array id="1">
    <aa id="2">
    </aa>
</aa-array>

<bb-array id="1">
    <bb id="2">
    </bb>
</bb-array>

<cc-array id="1">
    <cc id="2">
    </cc>
</cc-array>

Look how it ends the domain object's xml on same line instead of closing after the cc-array (as required by me). Whereas if I use just the following code:
Domain domain = Domain.getInstance();
String xml2 = "<?xml version=\"1.0\" encoding=\"UTF-8\"?>";
xml2 += lnSep + xStream.toXML(domain);

It (annoyingly) produces just the following in XML file:
<?xml version="1.0" encoding="UTF-8"?>
<Area id="1" name="Area"/>

It seems that the arrays defined inside Domain class are not at all being treated as Domain's contained objects.
Thanks for your patience. Any ideas? Do I have to create marshal and unmarshall methods specifically? If yes... too cumbersome.. kindly guide for such a scenario... an object having one string and 3 array objects.
And Oh, btw, deserialization problems start after this serialization problem is done away with; as my aim is to get back the Domain object to fully populate my JTree.
A Side Question: Why itemFieldName = "aas" has no effect??

A: My bad... I made my arrays static so that they could be accessed from the whole app. And it is nowhere mentioned in XStream's documentation/on-site tutorials that static members are not serialized/deserialized.
On just an impulse, i made my arrays non-static. Many changes spurred up in the whole app wherever I was accessing these arrays. I had to use Domain.getInstance().ArrayName to access the array that was formerly being accessed as Domain.ArrayName. After I had removed all the errors... wow... same xstream code that i had written more than 2 days ago started working without any change. Now my whole project is being serialized to XML using
String xml = "<?xml version=\"1.0\" encoding=\"UTF-8\"?>";
xml += lnSep + xStream.toXML(domain);

and being de-serialized using
Domain domain = (Domain) xStream.fromXML(xml);

XStream is gr8! I wrote this answer to benefit any future users.

redpajama_set_name: RedPajamaStackExchange


Looking after my daily water intake is something that I am continuously working on. I never realised how much you really need to drink in order to feel good and be healthy. This was highlighted to me by my partner, who also happens to be a PT (handy!).
Most of us go through life not drinking the recommended amount of water and think that we feel fine, so why is drinking at least 2 litres of water a day so important?
Well, drinking water helps you both sleep better and it also increases your energy levels. This is because your body needs water to function, if you are not giving it the water it needs, your body will start to "slump" and then you will feel exhausted.
It's my habit to drink at least 2 glasses of water as soon as I wake up in the morning. Then, one glass shortly before and after taking a shower or bath.
Thanks for this advice going to try it.


Quick861dw hot air gun can repair iphone ipad cpu baseband. Quick 861DW hot air rework station is an orginal 1000W lead-free desoldering station. Quick 861DW is the right machine for moiblephone or other electrioncs maintence.
3. It is suitable for the situation needing different range airflow, softer or heavier.


I don't understand, because, in my wp-config.php, user and password are OK for the database (and the main site is working).
I dont understand where is the problem. Can you help me ? Thank you.
This is exactly the description of my problem (i transform a classic website to MU website and i have nginx) but i don't have nginx user and the solution described here doesn't work for me.
how are you adding sub domains?
When logged into the site under the main domain that you said works normally - do you go to network admin --> sites --> add new site?
or are you adding them through a Cpanel?
check if your wild card subdomain has been set up in cpanel *.yourwebsite.com.
My configuration is good. I think it's possibly a bug with wordpress ? I remark that when i go to a subdomains/wp-admin, it told me the database need to be repaired. I deactivate all plugins and repair tables, but it change nothing.
and if so the next thing would be to check the httpd.conf file or vhosts.conf file to see if the wildcard actually shows up in the alias's for the specified domain.
also is this on a shared IP by any chance?
@Korben00, I believe Joe's mention of the htaccess would be the best thing to check from here. Could you let us know how that goes?
Some great advice here, nice one guys!
I'm pretty sure the problem is that you have WP installed using WWW.
Basically, as mutli-site depends on wildcard sub-domains it has real issues when you install it on a domain using WWW.
I.e. instead of having WordPress installed at http://www.korben.info it should be installed at http://korben.info.
WordPress is going to be looking for http://subsite.www.korben.info instead of http://subsite.korben.info until you can remove WWW.
We've encountered this loads of times and the short answer is the only fix is to remove WWW.
if Phil is correct about this being a WWW problem which I have no doubt that he is ..... In my opinion the easiest way to fix this without reinstalling from the beginning is possibly to go to myPHPadmin or equivalent for your blog and look for the options table ..... open the table and change the setting item #1 "siteurl" to NOT contain the www by simply removing the "www." from the domain name and also item #37 removing the "www." from there also.
i pass my website without www. The main site is working but i have the same problem with subdomains.
In your sites list in the network admin area of your system - what do the URLs of the sites look like?
It probably won't do anything, but can you try a htaccess without the supercache rules in? Just so we can rule out any issues with those?
is there any chance that you added the domain to your web server and then deleted it and added it again at anytime?
well sir I was referring to the main domain i.e. korben.info if it had been added and deleted and then re-added to your hosting server?
On occasion I have seen where the removal of a domain from a server does not completely remove the domain, whereas it leaves the domain in the httpd.conf or vhosts.conf depending on the configuration of the server.
Then when you add the new domain it puts the domain info once again below the original which could very well cause this type of anomaly.
After making any changes be sure to restart apache so that the changes will take effect.
And the wildcard in my DNS zone is well configured yes.
no sir that is not what I meant.
What i am trying to figure out is two things - if there is a duplicate in the httpd.conf file or vhosts.conf file for korben.info that is causing your anomaly and also if there is an entry that says *.korben.info listed under server aliases for korben.info in the entry for the domain that is there.
besides, using http://web-sniffer.net/ I find out that test.korben.info is giving a 500 server error!
are you sure you have configured nginx properly for wordpress multisite?
I know nothing about nginx much less about setting it up for Multisite.
However sir, you made either a typo above in your post in Korben00 test URL or you tested the wrong url, I have no clue which.
some people shared their nginx rewrite rules, googe helps here... as far as I know, nginx doesn't support .htaccess files... they need to be rewritten so this whole thread is quite pointless if I am right.
@Korben: under what user does your nginx installation run if it is not nginxuser as you said in your second post here? Do you have the proper rewrite files in place?
My nginx is just a reverse proxy that use Apache processes, so i don't have to adapt my htaccess and stuff like that.
And when i create a real subdomain, it answers (why a 500 error, i can't explain) http://test.korben.info that says error with database.
It answer, but it tell me the database need to be repaired. I already repaired it and it doesn't change the problem.
you are right. the redirection works if I visit: somethingmadeup.korben.info I am allowed to sign up for that blog.
I realize you are trying to server static stuff with nginx and proxy everything else back to apache, right?
hm, just saw your other reply about the DB.... can you check what the nginx log files are saying? if you have any?
I seriously know nothing about nginx - but I do know that the sub-domain feature works via wordpress using redirection - but not redirection to a folder.
oh and regarding your DB, did you check it? maybe with mysqlcheck --all-databases followed by a mysqlrepair see here for syntax http://linuxcommand.org/man_pages/mysqlcheck1.html ?
and is it only happening with this test site? what if you create another one? same problem?
@Joe: here is where you can find explanations: http://wiki.nginx.org/VirtualHostExample thsi link shows a default catch all vhost so I think his directive is correct.
But your information, knowledge, and help was much more accurate concerning this issue then mine was from my apache references.
Hmmm... too much configuration files to edit... I will put the mess in my configuration.
The wp_users table is already optimized.
The wp_usermeta table is already optimized.
The wp_blogs table is okay.
The wp_signups table is okay.
The wp_site table is okay.
The wp_sitemeta table is okay.
The wp_registration_log table is okay.
The wp_blog_versions table is okay.
Have you tried manually creating those tables that don't exist and running the repair again?
@Phil : no, i'll try.
But i have always the SQL error bug.
I also try to duplicate my database to another one and switch it but same result.
I delete my wp-config.php and my .htaccess and i reconfigure from scratch and it works !!!