#asympotic

[positivelyprime]

Karatsuba multiplication

Everyone learnt how to multiply numbers in school, but did you know there are faster¹ ways (for computers)?

One good example of this is Karatsuba multiplication. Imagine you have two 'big' numbers you want to multiply, as an example I will take 1984 and 2023. How you would normally multiply these is to do 4*3, 4*20, 4*000, 4*2000, add them all up, and repeat for the other digits of 1964. In total, these are 16 'easy' (i.e. 1 digit times 1 digit) multiplications. However, Karatsuba jutsu multiplication takes another approach.

The first main idea is to split our numbers up into two halves. Then we first multiply the upper and lower halves. Multiplying upper halves gives 19*20 = 380, multiplying lower halves gives 64*23 = 1472.

Now we do something special, we add the two halves of our numbers and multiply those: (19+64)*(20+23) = 83*43 = 3569. From this number we substract the numbers obtained before: 3569 - 380 - 1472 = 1717.

Now I state that we only need to put the numbers in the right place (which we do by adding zeroes) and we are done!

380 0000 + 1717 00 + 1472 = 3,973,172 (I invite you to check the answer)

Neat trick right? It isnt only a trick though, the art comes from repeating this process. You see, we currently did multiplications of two 2-digit numbers, 3 times. But we can again split those numbers up using Karatsuba jutsu to only multiply 1-digit numbers (or 2-digit numbers starting with 1, which are close enough to be instant²). Using this, we can find our answer using only 9 easy multiplications instead of 16 which is a vast improvement! The working out of this is under the read more. I hope you find this tidbit of mathematics as cool as I do!

For one 2-digit multiplication, we need 3 multiplications: e.g.

84*43: 3 multiplications 8*4, 4*3 and 12*7

As we have three 2-digit multiplications, we need 9 in total.

¹ When I say faster I mean asympotically faster. Basically this method is better for larger numbers, and already starts to be better for numbers with 5 digits! There are of course many more faster methods but those are too difficult to cover in a tumblr post.

² The reason I can say this is that doing 12*4 is just like doing 2*4 and adding 4 at the right place. In real computer software a table may be used for 'small enough' multiplications, so then 12*4 is as fast as 2*4 anyways.

[learnerworld]

Time Complexity part_2 | What is it?

In the last post, we’ve talked about why is it important to understand time complexity of any program.

But wait! What exactly is time complexity? We’ve not talked about that yet.

Time complexity is basically the rate of growth of time taken by a program when the number of input increases. Please Note that time complexity is NOT the time taken by a machine to execute the task, it really depends on which machine is being used, single or multiple processors, available memory etc.

When we talk about time complexity, we basically talk about it’s dependency on the number of inputs.

 Let’s understand it with an intuitive example:

Suppose you’re playing a game with a kid. You’ve put a surprise gift for him in a box but there are 8 similar looking boxes and the kid has to find the gift.

He has tried a box and found it empty, he tried the next and alas! That’s empty too.

How many boxes will he need to open, if he randomly picks a box to find the gift? What do you think?

If he’s lucky, he can find it in the first attempt but in the worst case, he might need to open all 8 boxes to find the gift.

So, time taken by him to get a gift would be proportional to the number of boxes, isn’t it? [More boxes, more time in worst case to find the box with a gift]

Next day, again you played the same game. But this time the boy behaved smartly.

Instead of choosing from the 8 boxes, he divided them into two equal parts and asked you for a hint, if the gift is in one of the boxes on the left or on the right. And based on the answer, he will search. So, he has to find the gift form only four boxes now. He again did the same trick and divided the boxes into two parts and asked you, if the gift is in the left side boxes or right-side boxes. So now, he needs to choose from two only. Again, he repeated the process again & finally found the box with the gift.

So, by doing this kind of search (we can call it binary search where we divide the search space by two in every iteration), the boy found the gift in maximum 3 iteration.

Let me introduce the jargon here.

For n inputs (here n=8), linear search (first approach of the kid) takes time complexity order of n [O(n)] as in the worst case, it takes ‘n’ iteration to search for the desired result.

While for n inputs (here n=8), binary search (second approach of the kid), takes time complexity order of log2n [O(log n)] as in the worst case, it takes ‘log2n’ [Note that: log28 = 3] iteration to search for the desired result.

Don’t worry about the math here!

But what did we learn here is that time complexity talks about rate by which a program execution time increases based on input parameters [we are not considering system’s performance here]

But how do we actually compute time complexity? We will discuss that soon.

 Time for a quick quiz:

What would be the time complexity for the following programs?

1.    def say_hi():

        print(“hi, how are you?”)

     say_hi()

 2.    for i in range(1, n):

        print(“hi, how are you?”)

 Answer to previous quiz: 1_ it iterates ‘n’ times and second loop iterates √n (which is less than ‘n’) times

 Stay Tuned!

[ssuibian]

ssuibian’s mdzs fic rec

It looked like there was a few interested in some mdzs fic recs so here I am. I’ll divide this into mdzs verse and modern au so scroll to what you like. Some of these will be the infamous ones, but they’re known for a reason so on the list they go. 

Before we begin some info about my preference’s: I read a little bit of  everything, almost exclusively completed works and mostly wangxian-centric.

Code: personal fave - ♛ / explicit, contains smut - ♞ / fluff - ✿

All of these are on ao3, so check them out on there for more info.

Mdzs verse (set in the same era as the novel/cql)

Tame by rikke ♛✿

Tags: teenage wangxian, canon divergence, humor

Length: 11.7k words, oneshot 

Summary: When Wei Wuxian wakes up in Lotus Pier after the Xuanwu Cave it’s with a souvenir in the form of a Xuanwu egg. Wangxian end up as young parents.

A civil combpaign by ariaste ✿

Ariaste has a lot of good fics and is a wonderful writer so check out their other works too!

Tags: Jin Ling x Lan Sizhui, background wangxian, humor, teenage drama 

Length: 19.6k words, oneshot + a sidefic from wwx’s pov

Summary: Jin Ling tries to arrange his own marriage and thinks Lan Sizhui could be a suitable match. One problem: he’s a teenager, the matter is handled accordingly.

The lotus defense society by mondengel ✿ 

Another good writer with a lot of content!

Tags: humor, background wangxian, background yanxuan 

Length: 3.4k words, oneshot

Summary: Jin Zixuan and Lan Wangji start a club to help them ensure the Jiang siblings stay single until they can confess. 

Phantom leashes by mondengel ♛

Tags: vampire au, horror elements, set in canon

Length: 3.1k words, 2/2 chapters 

Summary: Lan Qiren can’t help but be cautious of his youngest nephew who inherited his mothers bloodthirstiness. 

An effect (without a cause) by astrobandit

Tags: post-canon, amnesia 

Length: 15.4k words, 10/10 chapters 

Summary: Lan Wangji knows that something is wrong. Maybe the black and red clad man saying his husband is missing has something to do with it.

Unexpected by nanodayolo ✿

Tags: princess and the pauper au, wangxian, mo xuanyu x xue yang, humor

Length: 68.2k words, 22/22 chapters

Summary: The Jin clan is on the borderline of bankruptcy and arranges a marriage between their third son Mo Xuanyu and the second master of Lan, Lan Wangji. Sabotage is in the works and Mo Xuanyu is kidnapped, luckily they have look-alike Wei Wuxian to step in as a stand in. No one will notice, right?

Baby of mine by pupeez4eva ♛✿

Tags: time travel, humor, a-yuan

Length: 3.3k words, oneshot

Summary: A-yuan has time traveled to his parents teenage days in Gusu. Cloud Recesses is in chaos by what their visitor from the future implies. 

Another writer with many good works. My personal faves also include The haunting of Wei Wuxian (ghost’s follow wwx around post resurrection and are witnesses to wangxian) 

and the Wangxian’s time-travelling shenanigans series (they time travel to Gusu school days in the middle of the Guanyin confession. doesn’t stop wwx)

Where the winds of change blow by merakily

Tags: post-canon, humor and a dash of angst, hurt/comfort, juniors, family 

Length: 17.4k words, oneshot

Summary: Lan Wangji and Jiang Cheng is always a second away from stabbing the other. Who would’ve know a discussion conference and sect leader Yao’s loud mouth was what it took to make them get along.

The twin jades’ mother by drjlecter

Tags: post-canon, mild angst, twin jades, canon compliant, TW! mentions of suicide

Length: 11.8k words, oneshot

Summary: Wei Wuxian does some experimenting and somehow manages to use inquiry to call upon a very dear spirit.

The father trap by jinko ✿

Tags: post-canon, juniors plotting, humor

Length: 3.6k words, oneshot

Summary: The juniors are frustrated that wangxian isn’t together. They start plotting to make it happen, overlooking the fact that they might be engaged.

Asympotic by chinxe

Tags: canon divergence, pining, angst w/ happy ending

Length: 26.6k words, 3/3 chapters

Summary: Lan’s have a history of having tragic love stories. Lan Wangji follows tradition by falling in love with a ghost.

Some days. by jollytortoise ✿

Tags: emotional hurt/comfort, post-canon

Length: 8.3k words, 4/4 chapters

Summary: Wei Wuxian is sick, and the entire Cloud Recesses is thrown into disarray. It’s a bad day and all he wants is his husband.

Conversations: this mending fences thing by alcove_words ✿

Tags: post-canon, moving on, family 

Length: 4k words, oneshot

Summary: Wei Wuxian and Lan Sizhui have to stop at Lotus Pier when wwx is injured on a night hunt. Jiang Cheng tries to pick up the pieces of the relationship he once had with his brother.

Lan Sizhui sees dead people by darkbrokenreaper ✿

Tags: background wangxian, canon divergence, humor

Length: series, 7 stand alone works, ongoing

Summary: The time a-yuan spent in the burial mounds has left its mark by giving him the ability to see ghosts. As a result he shocks the people around him when he talks about things he shouldn’t know.

Recompense for the martyr (wicked nights howl) by anemonii ♛

Tags: post-canon, family, emotional hurt/comfort, memories, brotherhood, forgiveness, canon typical violence

Length: series, 3/3 parts

Summary: 

Part 1: Jin Ling is cursed by a sword and witnesses the owner’s memories. As a result he’s taken back to the Lotus Pier of his uncles’ childhood, their teenage days in Gusu and their fallout when the Yiling Patriarch appeared.

Part 2: Wei Wuxian decides it’s time he and Jiang Cheng talked. 

Part 3: It’s time to lay Mo Xuanyu’s body to rest.

The Yiling Wei clan by scifigeek14 ♛✿

Tags: canon divergence, family of choice, fluff, everyone lives/nobody dies, getting together

Length: series, 6/6 works

Summary: Lan Wangji decided to stay at the end of the visit to Yiling. It changes everything, for the better.

Modern au 

Unstrictly ballroom by ariaste ♛♞ 

Tags: competitive ballroom dancing au, background songxiao 

Length: 47.5k words, 4/4 chapters 

Summary: 13 years ago Wei Wuxian was involved in a scandal and thrown out of the competitive ballroom dancing scene, and Lan Wangji never stopped looking. Faced with the heteronormativity within the field, Lan Wangji and his dance partner decide to make a change. It’s something so fitting Wei Wuxian, and Lan Wangji really wishes he was there now.

Lightning in a bottle by bigbabyjeno ♞

Tags: photographer lwj, wedding crasher wwx

Length: 10k words, oneshot

Summary: Lan Wangji is the photographer for a wedding and catches Wei Wuxian sneaking in, claiming to be the bride’s brother. They disguise him in as Lan Wangji’s assistant so he can see his sister get married. The night ends with some smutty times.

By mistake or design by allurie ♛♞

Tags: fashion designers au, slow-burn, sexual tension

Length: 51.7k words, oneshot

Summary: Lan Wangji is the head designer of the haute couture division of the fashion house Nuage, while Wei Wuxian is the head designer for ready-to-wear. Circumstances have them end up working together, and the result is an explosive collaboration.  

The best of you by sysrae ♞

Tags: university au, mutual pining, emotional hurt/comfort, mental health

Length: 41.9k words, 17/17 chapters 

Summary: Wei Wuxian rents Lan Wangji’s spare room after being thrown out by the Jiang household. Living together makes Lan Wangji learn to know Wei Wuxian on a much deeper level. 

So take my hand (take my whole life too) by cicer ♛✿♞

Tags: university, kid-fic, accidental baby acquisition, mutual pining, humor 

Length: 92.3k words, 13/13 chapters

Summary: Wei Wuxian was waiting for his package. What he finds on the doorstep instead is a baby in a car-seat with no one else in sight. Wei Wuxian juggles taking care of a baby and being in love with his co-parent/ roommate/ best friend Lan Wangji at the same time. 

Let’s you and me go for a ride by rinsled05 ♛✿

Tags: racing au, a-yuan, brotherly love, fluff, hurt/comfort

Length: 21.4k words, 5/5 chapters

Summary: Competitive motorcyclist Wei Wuxian meets his very handsome sponsor Lan Wangji, who despite not liking racing still comes back to watch. He wonders why. 

Love wakes me by dea_liberty ♛✿♞

Tags: childhood sweethearts, famous lwj, coffee shop owner wwx 

Length: 42.8k words, 4/4 chapters

Summary: A teenage bet was all it took for Wei Wuxian’s life to fall apart. As an adult he’s working in Yiling cafe, and he’s doing fine in his new life. That’s until Lan Wangji walk in the doors and make his heart ache again for everything in the past.

B-side by suspicious_popsicle ✿♞

Tags: band au, metalhead wwx, music teacher lwj, juniors

Length: 47.3k words, 15/15 chapters

Summary: Wei Wuxian disappeared right after graduation and Lan Wangji never saw him again. That’s until Lan Jingyi’s favorite band is performing nearby and Lan Wangji sees his first love again.   

My dreams are playing softer now by fleurdeliser ♛✿♞

Tags: modern w/ cultivation, drunk lwj, humor, fluff

Length: 8.2k words, oneshot

Summary: Lan Wangji is drunk and his older brother decides he doesn’t want to deal with that and makes Wei Wuxian take responsibility. What follows is a cupcake date. 

Congratulations, get rich by attila ♛✿♞

Tags: party, sibling shenanigans, big dumbass energy, humor

Length: 9,2k words, oneshot

Summary: Wei Wuxian really wants to kiss Lan Wangji at the chinese new years party. If mixing traditions and making his siblings go along with his scheme for it to happen, so be it.

The simplest way forward by harriet_vane ♛✿♞

Tags: kid-fic, accidental baby acquisition, green card marriage (except not), pining

Length: 71k words, 8/8 chapters

Summary: Wei Wuxian wakes up with an a-yuan in his care and he has no idea how. Lan Wangji helps out and Wei Wuxian tries not to fall in love with his husband.

Operation old men by chiharu ♛✿

Tags: juniors, matchmaking, family, doctor wwx, scheming

Length: 37.6k words, 6/6 chapters

Summary: It turns out Lan Sizhui’s father Lan Wangji and Jin Ling’s uncle Wei Wuxian know each other after a chaotic parent/teacher conference. The juniors decide to do some investigating that soon turns to matchmaking.

Your heartbeat, across the grass by fakeplasticlily ♛♞

Tags: football player lwj, photographer wwx, a-yuan, mutual pining, sexual tension 

Length: 44k words, oneshot

Summary: After graduation Wei Wuxian has done his best to avoid everything to do with Lan Wangji. Not easy when Lan Wangji has become a world known fotball player and a-yuan’s favorite. 

Other faves by this writer: Love on 35mm  (lwj is a film student and anonymous famous vlogger who’s in love with his best friends younger brother) 

and When we’re full of stories to be told (wwx went undercover for an article and cut contact with everyone. years later he ends up meeting childhood best friend lwj who has become a famous author.)

Wei Wuxian is the actual bogeyman (no really) by chatonnerie ♛✿

Tags: rise of the guardians crossover, humor, juniors, mystery

Length: 57.3k words, 6/6 chapters

Summary: Literal bogeyman Wei Wuxian was exiled by the guardians 1300 years ago. A mystery à la mdzs canon welcomes him back to the world for some buddy-cop holiday fun.

Our reflections as seen (when the water stills) by chatonnerie ♛✿♞ 

Tags: tokyo ghoul crossover, university, gore, humor, mystery

Length: 121k words, 10/10 chapters

Summary: Lan Wangji attends university and works for the ccg. His new course brings the annoyance that is Wei Wuxian into his life, who eventually turns out to be his new best friend. Lan Wangji is very concerned for his friends safety when he finds out Wei Wuxian’s brother is a ghoul. But that’s just the tip of the ice berg. 

[This one is hands down one of the best fic’s i’ve read ever. Incorporates the mdzs and tokyo ghoul verse so well.]

And they were roommates... by harriet_vane ✿♞

Tags: fem wangxian, college/university, roommates

Length: 41.2k words, 4/4 chapters

Summary: Wei Wuxian is without housing for the semester and fellow student Lan Wangji has a room available. Based on infamous reddit post "I (21f) have a crush on my roommate (20f). I can't figure out if she actually likes me back or not or is just being friendly. She cooks for me and knows all my favorite foods, and brings me lunch. (...)”

Jiang Cheng’s terrible summer by hellohln ♛✿

Tags: camp rock crossover, getting together, humor 

Length: 3k words, oneshot

Summary: Jiang Cheng tells his therapist about the disaster that was summer camp. Mostly because he had to witness his brother getting together with fellow camper Lan Wangji.

Let the streetlights guide you home by tellthemstories ♛✿♞

Tags: street racing au, pining, sexual tension

Length: 22.6k words, 5/6 chapters (steady updates so far so it should be done soon)

Summary: Wei Wuxian was the best street racer of his time and he was known for danger and recklessness. A fateful night with his sister beside him turns out to be his death. 10 years later Mo Xuanyu enters the scene. 

Window shopping by thunderwear ✿♞

Tags: quarantine fic, single dad wwx, a-yuan, “long distance”, phone sex

Length: 18k words, oneshot

Summary: Lan Wangji looks out of his apartment window to see a sign from the neighbor across the street asking what his bunny’s name is. The interaction turns into more than a one time occurrence and introduces him to the attractive single dad and his adorable son.   

A perfect match by elliemoran ♛✿

Tags: juniors, college/university, humor, gossip

Length:  2k words, oneshot

Summary: Professor Wei Wuxian overhears the students discussing in the library. It’s less about academics and more about how good he looks with professor Lan. But which Lan? 

Make it count by wearing_tearing ✿♞

Tags: actor wwx, lawyer lwj, pining, fluff, light angst, background songxiao, background yanxuan, TW: attempted rape, stalking

Length: 46.9k words, 12/12 chapters

Summary: Actor Wei Wuxian makes his way back into the acting world and prepares for his new role by working at a lawyers office. It brings him back to an old acquaintance. 

Grandmaster of demonic party games by trickster_angel ♛✿

Tags: college, paranormal, occult, light horror, mutual pining, humor

Length: 50k, 27/27 chapters 

Summary: Wei Wuxian likes to try out paranormal games but is always interrupted by Lan Wangji. He keeps wondering how the other knows what he’s up to. 

Scienter by synonemous ♛♞

Tags: serial killers au, dark, explicit, smut, flirting, crime, murder husbands, murder siblings

Length: series with multiple verses within the au, most can be read as stand alone’s

Summary: Serial killer Wei Wuxian aka the yiling patriarch has been flirting over text with the infamous fellow serial killer hanguang-jun. He has no idea who he is, but that doesn’t stop him.

[athwartjoe]

I have more than lost all will to live.

At this point I have surpassed the asympote and now spend everyday in the negative values of willingness to live. Really starting to get annoyed at the fact that I am still here.

[dotheygeteaten]

Steins;Gate 0 - Episode 15:  Recognition of the Asymptotic Line: Recognize Asympote

Do They Get Eaten? - No.

[pmvstump]

@pstumph RIGHT,,VERTICAL ASYMPOTES MY ASS YOU EXTRA PIECE OF CRAP

@pstumph 

[arxt1]

Numerical Relativity with Arbitrary Precision Arithmetic: Applications to Gravitational Collapse. (arXiv:1803.00858v3 [physics.comp-ph] UPDATED)

Numerical Relativity is a mature field with many applications in Astrophysics, Cosmology and even in Fundamental Physics. As such, we are entering a stage in which new sophisticated methods adapted to open problems are being developed. In this paper, we advocate the use of Pseudo-Spectral Collocation (PSC) methods in combination with high-order precision arithmetic for Numerical Relativity problems with high accuracy and performance requirements. The PSC method provides exponential convergence (for smooth problems, as is the case in many problems in Numerical Relativity) and we can use different bit precision without the need of changing the structure of the numerical algorithms. Moreover, the PSC method provides high-compression storage of the information. We introduce a series of techniques for combining these tools and show their potential in two problems in relativistic gravitational collapse: (i) The classical Choptuik collapse, estimating with arbitrary precision the location of the apparent horizon. (ii) Collapse in asympotically anti-de Sitter spacetimes, showing that the total energy is preserved by the numerical evolution to a very high degree of precision.

from gr-qc updates on arXiv.org https://ift.tt/2Ff0BqS

[mathematicianadda]

BCC at Birmingham, days 1-3

This week I am in Birmingham for the British Combinatorial Conference.

The organisation of the conference is outstanding. For one small example, yesterday, fifteen minutes before the Business Meeting was due to start, the Chairman noticed that we didn’t have the minutes of the previous Business Meeting to approve. The Secretary had the file on a laptop, and before the meeting started we had fifty printed copies to distribute.

After the excitement about ADE last week, these diagrams reappeared twice in the first couple of days, Hendrik Van Maldeghem (who talked about geometrical and combinatorial constructions of buildings) showed us all the crystallographic Coxeter–Dynkin diagrams. In a completely different context, Alexander Gavrilyuk mentioned the fact that connected simple graphs with spectral radius at most 2 are the ADE diagrams and the extended ADE diagrams. He attributed this to Smith (1969) and Lemmens and Seidel (1973). I think it would be not unjust to say that this result was part of the classification of the complex simple Lie algebras by Cartan and Killing in the last decade of the nineteenth century. That aside, Alexander was extending this to directed graphs, using a Hermitian adjacency matrix with entries 1 if there are arcs both ways between two vertices, while single arcs from v to w have i (the complex fourth root of 1) in position (v,w) and −i in position (w,v). This had been done by Guo and Mohar, but some small corrections were necessary; he used results of Greaves and McKee to achieve these. (As a footnote to this, it seems to be that to use τ, a complex 6th root of unity, in place of i would be more natural, since the sum of τ and its complex conjugate is 1 rather than 0.)

In fact, for the graph case, much more is known: the graphs whose greatest eigenvalue is at most 2 are the ADE diagrams and their extensions, but the graphs whose least eigenvalue is at least −2 can also be described.

The conference featured mini-symposia, and I organised one on “Designs and Finite Geometries”, which in my opinion has had some beautiful talks so far, from Ian Wanless on plexes in Latin squares, Rosemary Bailey on designs related to the Sylvester graphs (and the wrong turnings on the way to finding them), Peter Keevash on his and others’ results on existence of designs (including the fact that estimates for the number of Steiner systems, asymptotic in the logarithm, are now available, and hinting that he had constructions of large sets of Steiner systems for large admissible orders), and Moura Paterson on authentication schemes.

One of the most exciting talks was by Igor Pak. He has formulae, and good asymptotic estimates, for the numbers of standard Young tableaux for various skew Young diagrams. This was a mix of all kinds of things, including counting linear extensions of posets, rhombus tilings, plane partitions, counting disjoint paths, Vershik’s limiting tableau shapes, and a remarkable formula of Coxeter, which (if I copied it correctly) says

Σ(φn/n2) cos(2πn/5)  =  π2/100

(the sum over all positive integers n.)

Coxeter’s discovery of this formula was based on the existence of the 600-cell (a regular polytope in 4 dimensions) and some spherical geometry. As far as I can tell, the formula was not actually used in the talk, but the philosophy of it led to some of the things that came later.

Two things about the talk were a pity. First, there was no paper in the Proceedings. (In the history of the BCC, it has happened a few times that a speaker provided no talk; indeed I was the editor of the first “published-in-advance” volume, at Royal Holloway in 1975, where I failed to get papers from either Conway or Kasteleyn.) So I am unable to check these details. Second, Igor started in a bit of a rush, and some things were not clearly explained. For example, I think some nodding acquaintance with Plancherel measure is needed to make sense of the Vershik asympotic shape of a random Young diagram, and I didn’t find that in the talk. But it was so full of amazing stuff that it is perhaps churlish to complain.

Apart from these I will be very selective in my reporting. One contributed talk I really enjoyed was by Natasha Dobrinen, on the Ramsey theory of Henson’s homogeneous Kn-free graphs, which included a description of them in terms of trees. It went part rather fast (the talks were only 20 minutes), but I wonder whether this leads to a probabilistic approach to Henson’s graph. I have reported before how I laboured over this, and how Anatoly Vershik explained to me his construction with Petrov in a leisurely afternoon in Penderel’s Oak in London – a construction which is clearly related to the topic of graphons, the subject of Dan Král’s talk.

Then there was a sequence of three nice talks on quite different topics, but all related to permutations (in the combinatorial rather than the group-theoretic sense). Simon Blackburn proved a nice asymptotic result about random permutations for the uniform measure. At the end, Robert Johnson asked whether there were similar results for other measures. This was because Robert’s talk, which was next, was able to prove some of the results for wider classes of measures, though not for the Boltzmann measure, which he gave as an open problem. Then Fred Galvin talked. One of his results was that, far from being monotonic, the sequence of coefficients (excluding the constant term) in the independent set polynomial of a graph with independence number m can be any permutation of {1,…m}. This suggested to me another interesting measure on permutations. Choose n much larger than m, and choose a random graph on n vertices with independence number m; this induces a probability measure on the permutations. Does this measure tend to a limit as n→∞? If so, this could claim to be a “natural” measure on permutations. Fred thought this was an interesting question.

Any ideas?

We had a reception in the remarkable Barber Institute of Fine Arts. Guided tours of the gallery were offered. We went upstairs, and the first picture we saw was René Magritte’s famous picture “The flavour of tears”. Tuesday was the concert, and apart from having to move to a different room because the piano hadn’t been unlocked, we had a remarkable evening’s entertainment; there are several outstanding pianists at the conference. Today is the excursion, to the Museum of Black Country Living; but I have work to do …

from Peter Cameron's Blog https://ift.tt/2Kt7JnJ from Blogger https://ift.tt/317149m

[zethopiaa-blog]

Asympote. Bound to meet but will never be together.

[donttouchmyasymptote]

my brain is bad when I'm scrolling and reading and therefore messes up words but in my head, I read asymptote as asimpote. Idk what you'll do with this knowledge, but just thought it was relevant XD

once in a maths exam, i spelt asymptote as asympote

so if u ever feel bad for not spelling maths terms, i have once misspelt my own url

[arxt1]

Thermodynamics sheds light on black hole dynamics. (arXiv:1712.02672v1 [gr-qc])

We propose to unify two a priori distinct aspects of black hole physics : their thermodynamics, and their effective dynamics when they are "skeletonized" as point particles (a useful procedure when tackling, for example, their motion in a coalescing binary system). For that purpose, the Einstein- Maxwell-dilaton (EMD) theory, which contains simple examples of asympotically flat, hairy black hole solutions, will serve as a laboratory. We will find that, when reducing a black hole to a point particle endowed with its specific, scalar-field-sensitive, effective mass, one in fact describes a black hole satisfying the first law of thermodynamics, such that its global charges, and hence its entropy, remain constant. This shows that the integration constant entering the scalar-field dependent mass is its entropy.

from gr-qc updates on arXiv.org http://ift.tt/2k7FEoI

[arxt1]

Unique continuation from infinity in asympotically Anti-de Sitter spacetimes II: Non-static boundaries. (arXiv:1608.07521v2 [gr-qc] UPDATED)

We generalize our unique continuation results recently established for a class of linear and nonlinear wave equations $\Box_g \phi + \sigma \phi = \mathcal{G} ( \phi, \partial \phi )$ on asymptotically anti-de Sitter (aAdS) spacetimes to aAdS spacetimes admitting non-static boundary metrics. The new Carleman estimates established in this setting constitute an essential ingredient in proving unique continuation results for the full nonlinear Einstein equations, which will be addressed in forthcoming papers. Key to the proof is a new geometrically adapted construction of foliations of pseudoconvex hypersurfaces near the conformal boundary.

from gr-qc updates on arXiv.org http://ift.tt/2bt6LHs

[arxt1]

Null hypersurface quantization, electromagnetic duality and asympotic symmetries of Maxwell theory. (arXiv:1708.05606v1 [hep-th])

In this paper we consider introducing careful regularization in the quantization of Maxwell theory in the asymptotic null infinity. This allows systematic discussions of the commutators in various boundary conditions, and application of Dirac brackets accordingly in a controlled manner. This method is most useful when we consider asymptotic charges that are not localized at the boundary $u\to \pm \infty$ like large gauge transformations. We show that our method reproduces the operator algebra in known cases, and it can be applied to other space-time symmetry charges such as the BMS transformations. We also obtain the asymptotic form of the U(1) charge following from the electromagnetic duality in an explicitly EM symmetric Schwarz-Sen type action. Using our regularization method, we demonstrate that the charge generates the expected transformation of a helicity operator. Our method promises applications in more generic theories.

from gr-qc updates on arXiv.org http://ift.tt/2vVGvy0