Wednesday 29 May 2013

S5.157 - Chopping at the Gap

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Cosine: Root, what are you doing?

Root: I thought... wait, what's this session about?

Cosine: Chopping at the Gap. Between prime numbers.

Root: Oh, chopping! I heard something else.


Maud: We're already rolling.

Cosine: Gah! Okay. Big news about...


Maud: We couldn't get a feed to Sine either. Sorry.

Cosine: ...about twin primes. Sigh. I miss my twin.

Secant: I miss mine too.

ArcCos: At least yours haven't gone insane...

Sexi: Well, if no twins are available, I'll join Cosine.

Cosine: Really? Under what authority??

Sexi: The sexy primes. Primes separated by six, like 5 and 11, or 7 and 13.

Cosine: I feel like you're making that term up.

Sexi: Nope. There's also sexy foursomes.

Maud: Okaaay. Uh, back up to twin primes for a second, ladies. There's a song people can watch! It should get everyone up to speed.

Cosine: Indeed. Now then, if everyone is on board with twin - and sexy - primes, let's build up towards the most recent breakthrough.

Sexi: Maud's video referenced Goldston, who worked with Cem Yildirim and Janos Pintz in 2005, on what has become known as GPY. Interestingly, Pintz actually came on board last, after an error was found in the initial GY proof.

Cosine: But we can see how a math proof called GYP isn't the best idea...

Sexi: GPY's efforts regarding the smallest possible prime gap didn't close the door to infinity. Thus we fast forward to April 17, 2013, when Yitang Zhang submits a paper to Annals of Mathematics.

Cosine: The editors saw there was something to it, and fast tracked it through. REALLY fast tracked it. By May 13th, Zhang was scheduled to give a seminar at Harvard.

Sexi: In the following link, you can see how the news was getting out even on May 12th. Visit the Not Even Wrong blog, in the post "Number Theory News".

Cosine: Check out the comments there too; the speaker was listed as "Tom Zhang". Now, for a summary of the actual presentation, check out the n-Category Cafe here.

Sexi: If links don't thrill you though, in brief, Zhang showed that there are infinitely many primes that differ by N. Where N is a number smaller than 70 million. 

Secant: Let's add some context there.

Sexi: POINT ONE. We already knew there was NO upper bound. In other words N can be a number as large as you want. Mr. Honner provides a very simple proof of this fact, in his Infinite Prime Gaps post from May 15.

Cosine: In other words, you want a prime, then over 1,000,000,000,000 composite numbers, then the next prime... sure, we can do that.

Sexi: POINT TWO. We already know that the gaps get bigger as you count more and more numbers. In fact, the now hundred year old "Prime Number Theorem" states that among the first P numbers, about P/(ln P) are prime.

Cosine: Meaning among the first 100 numbers, 100/(ln 100) or about 22 are prime...

Expona: Actual value 25 primes!

Cosine: Among the first 1000 numbers, 1000/(ln 1000) or about 145 are prime...

Expona: Though the real value is 168 primes!

Cosine: Look, the formula is merely asymptotically approaching the function pi(x), so hush up.

Expona: You said log base e, I'm just filling in for Logan!

Sexi: POINT THREE. Given the gaps get bigger, and there is no upper bound on them - how can it be that you suddenly have two primes nearly side by side? Mind blown!

Cosine: Assuming you define "nearly" as 70 million.

Secant: That said, according to the latest Numberphile video, Gaps between Primes, people are already working on knocking the gap number even lower.

Maud: Now, there's a human element here too.

Cosine: Do tell?

Maud: Many news reports point out how Zhang was a relative unknown in the mathematics community. Moreover, he made the breakthrough by applying existing methods, those of GPY. So in a way, there's hope for everyone.

Sexi: Agreed. The idea even came to Zhang while taking a break from years of work. So sometimes the best thing you can do is take a break!

Maud: By the way, my information is from Simons Science News from May 19th: Unheralded Mathematician Bridges the Prime Gap. Many nice quotes there, like "There are a lot of chances in your career, but the important thing is to keep thinking."

Cosine: There's also an interesting writeup connecting primes to randomness in Slate from May 22nd! The Beauty of Bounded Gaps.

Sexi: And, to round things out, the prime gap news was in the New York Times. So now everyone knows about it. With that, I think we're done here!


Root: So, does anyone feel like going to a mall?


Sunday 26 May 2013

S5.156 - Sigh Fie

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"Now it's time for us to go, or so the timer tells me so," Con concluded. He pointed his remote out at the empty air. Sine caught a glimpse of a digital readout cycling down to zero, just before a couple of beams shot out, and a brilliant blue circle of energy appeared in the middle of the air.

Mink brought her umbrella up to rest the end on her shoulder.  "Mmm, I do so love this part, don't I?" she remarked to her husband.

"Wait, what's going on?" Cosecant asked, fingering her phaser again.

Mink turned and smiled. "What if you found a portal to another dimension? What if you could slide into a thousand different mathematical areas? Where it's the same year, and you're the same person, but everything else... is different?"

"Translation: To move, without rotating or flipping," Con interjected. "So thanks for the chat, but now we must be tripping."

"Try to stay out of the rain, won't you?" Mink added. Con reached out to grab Mink by the waist, and then the two functions leapt into the swirling portal.

Sine grabbed for her communicator. "Logan, did you hear that?  Take some readings at our location.  Pronto!"

"Already doing it," came the redhead's voice, from back on the gazebo's bridge.

Within a minute, the portal had disappeared.  As Cosecant walked around it's former location, the first few spatters of rain began to fall from the sky. Which helped to revive Hevvi. She sat up quickly, then held her hand to her head. "Agggh, what the heck?! Did Con hit me??"

"Um, yeah," Sine admitted.  She reached out a hand to help the step function up, but Hevvi smacked it aside, then fumbled for her glasses, which had fallen off.

"That jerk, hitting a woman!  You avenged me, right?!"  There was a slightly uncomfortable pause.  "Huh.  Did one of you at least check that I was okay?"  Sine and Cosecant exchanged a glance.

"We saw you were still breathing," Cosecant noted. "It's not like we thought he'd killed you."

Hevvi balled her hands into fists. "Oh, he did far worse than kill me. He hurt me. And he's still hurting me, in leaving me stuck here with you... marooned in the middle of nowhere... barely revived..." The function threw her head back to glare at the sky, face twisted in anger. "COOOOOOOONNNNNNN!"

Her scream echoed out into the sudden downpour.


Wednesday 22 May 2013

S5.155 - Plot Points

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Sine decided not to follow up by asking why Mink kept talking in questions. "Do the two of you live around here?" she asked instead.

"Oh no," Con responded. "This is just a quiet place where we come to think. Well, among doing other things, nudge nudge, wink wink."

"Then, if you know the area," Cosecant said quickly, waving her hand to prevent Con's gaze from slipping back to his wife, "Perhaps you can give us directions. Or draw us a map!"

"A map? Wait, Csc, do you not trust Signum to guide us correctly?" Sine asked of the security officer.

The reciprocal trig shrugged. "I'd rather not let her out of her quarters unless I have to now," Cosecant admitted. She couldn't help but think back to the 'I told you so' look that Radik had given her, after they'd found the problem with the gelpack. "Though even from the start, Signum said she'd be guiding us away from structure, rather than towards Probability. She had no exact heading."

"If that's her case, I'd say this space was valid navigation," Con remarked. "Me and Mink, we're on the brink of lacking all notation."  He rose from the devil's lap. She also stood up to fold her umbrella back together.

"Also, didn't you notice we're only defined from zero to one?" Mink added. "Is that not much like probability?"

Cosecant frowned. "Which is... not really giving us a heading."

"Oh, is THAT what you wanted?" Mink mused.

"Well, what's a heading when you can look into higher dimensions," Con stated. "For they can help you with your travel, if you've got the right inventions." He then reached into his pants pocket and pulled out a remote control, wiggling it back and forth.

Sine blinked. "We need one of those then?"

"What you need, trig ladies, is to check your logistics. For the probability you seek - it's alongside statistics."

"Still not a heading," Csc sighed.  "Please, can't you just plot us some coordinates? On a Cartesian plane, for instance?"

"A Cartesian plane?" Mink laughed. "Don't you know he's the box function?  Makes sense that he'd allude to a box plot, right?"


Sunday 19 May 2013

S5.154 - Uncommon Denominators

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Back in the gamma recording studio, Slope interjects with an important dialogue about DIVIDING FRACTIONS.

"Alternatively, 1/3 divided by 1/2... you're giving to a "whole", and
 doing it in "halves". 1/3 + 1/3 = 2/3, which is the answer.

Slope: Click to read, come back to comment!


Wednesday 15 May 2013

S5.153 - I Now Pronounce You

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"I was also able to deduce that she'd try her power to seduce," Con remarked, rubbing his knuckles on his sweater as he looked at the unconscious Heaviside on the ground.

"The real question is, will the two of you be more reasonable?" Mink asked. As she spoke, she unsnapped the handle on her umbrella, revealing that it folded out into two sections. She then planted the pointed end into the ground and sat back on the divided handle, folding her arms.

The two trig functions exchanged a glance. "Who ARE you?" Cosecant repeated as she looked back.

"Ahh, they're hearing deficient, I suppose?" Mink mused, looking to Con.

"Perhaps they've heard of your other fame, believing Slippery Devil's Staircase to be your name."

Mink looked back at Csc and Sin. "Oh, are you confusing me for my cousin, the Devil's Staircase?"

Sine shook her head. "No... we're not from around here, you see. We're on a quest to find probability and get ourselves back to mathematics."

"Get back? That's whack. I suggest you never left," Con opined. He moved to sit in Mink's lap. She circled her arms about his waist and pulled him close as he turned to her. "They must be from the curriculum of maths, blind to seeing the other paths."

"Oh, but doesn't that make you wonder?" Mink sighed. "Would they have let us into that curriculum, if only our notation could be vocalized without resorting to phonetic punctuation?"

Con smirked. "Well, I wouldn't want in. It's more fun with you, in sin."

Mink giggled. "Silly boy, it's not sin since we're married, right?" The devil girl pulled Con's head down and gave him a kiss. A long kiss. To the point where they seemed to be statues again.
? + [] = ...

Sine eventually cleared her throat, trying to think of a new topic. "Uh... so... why is it that Con rhymes whenever he talks?"

Mink finally released the blue function, her cheeks a deep red. After a second to catch her breath, she looked back to the primary trig. "Don't you know he's the box function?  Makes sense that a box knows rapping, right?"


Sunday 12 May 2013

S5.152 - Questionable Appearance

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Cosecant stared suspiciously at the two statues. The away team had found them within a few minutes of leaving the gazebo, after climbing out of the dry quarry where they had landed.

The one on the left looked a bit like a devil. Dressed in red, she had two horns sprouting out of her short orange hair. There also seemed to be a question mark motif on her outfit, given there were some on the collar of her blouse, another on her vest, and she carried an umbrella with a handle in the form of a question mark.

The one on the right looked more angelic. He was dressed in purples and blues, and had a halo hovering above his similarly blue hair - though it was square and box-like, rather than circular. The box theme continued on his collar and his sweater, and he carried a book which was of a similar colour to Logan's gazebo.

"They're very lifelike," Sine remarked.

The female devil abruptly turned to look at them. "If you think we're wax-works, you ought to pay - you know? Wax-works weren't made to be looked at for nothing, know how?"

"Contrariwise," the male chimed in. "If you think we're alive, you ought to speak, or else it implies you've got a real rude streak. That's logic, most cosmic."

By the time the two of them had finished speaking, Sine had stumbled back a few steps, Cosecant had whipped out her phaser, and Hevvi had pulled off her glasses to clean them. "Who are you?" Csc asked. "Identify yourselves!"

The devilish and angelic functions exchanged a glance, then looked back. "Would you believe our names are unpronounceable?" the orange haired one asked.

"Call her Mink, it's short for Minkowski," the blue haired one suggested. "I'm Con, for Conway, if you will agree." He then delivered a swift jab to the side of Hevvi's head, knocking the step function out cold.

Cosecant quickly stepped back out of range, phaser pointed at him. "Whoa! What was THAT about?!"

Mink rolled her eyes.  "Don't you know he's the box function?  Makes sense that he's able to box, right?"


Wednesday 8 May 2013

S5.151 - Hot Stuff

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The blue gazebo hit the ground of the quarry and bounced once, spinning around in the air twice before finally coming to rest, sheared slightly to the left. There was a momentary pause, then the trap door in the middle banged open, allowing the mathematical relations in Logan's base to emerge from the subscript.

Cosecant came out first, glancing around before giving the all clear. "Thank goodness, cooler air," Sine gasped, pulling at the collar of her shirt as she emerged next. She looked back down. "ParaB, you coming?"

"Mmm. I think I'd be of more use down here," the bunny conic called back. "While Radik is repairing any burned out components, I can construct a new gelpack. With some help from QT, that is. There's biological components."

The primary trig function nodded. "Okay, approved. Just keep an eye on the environmental temperature readings. I wouldn't want you two girls getting overheated."

"Don't worry, I'll pay close attention to QT's hotness." There was a pause. "That came out wrong."

"Did it?" came Radik's barely audible mumble.

Sine's forehead creased slightly, but before she could follow up, Hevvi was climbing out through the hatch. "Alright, finally a land that's not a steppe!" she said in delight. "Let's explore."

"Who said you were coming?" Cosecant challenged.

"The fact that I have the power to turn on functions, thereby changing your opinion with a wink of my eye?" Heaviside countered. She placed her hands on her hips. "A cynic might even say that the only reason you haven't confined me to quarters after the gelpack incident, like you did with Flora and Signum, is so that you can keep an eye on me."

"'Evvi could be useful," Sine admitted to her security officer. "We'll keep an open comm frequency with Logan on the bridge."

"Hmph. Very well," the reciprocal trig yielded. "But have someone sound the alarm down there if they no longer hear 'Evvi breathing."


Sunday 5 May 2013

S5.150 - Crash Course

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"150! No bones about it, I didn't think we'd survive this long!"

"Enough with the dramatic time updates Radik, we'll be landing momentarily," Logan sighed. He turned away from the radical function and leaned against the edge of his Captain's chair. "Sine, status?"


"Honestly?" the brunette executive officer began, peering at her iPad. "Our engineer has it right. Without that gelpack, your base is continuing to overheat. Lasting these 2.5 hours was luck, I estimate complete shutdown within 5 minutes." She threw a little glare over her shoulder.

The slightly heavyset woman in glasses met the gaze, folding her arms in response. "If the component was so vital to the ship," Hevvi shot back, "it shouldn't have been so easily accessible!"

"Hello? We'd removed the gelpack's covering panel for use in constructing our end of the Cone of Science!" ParaB called out. The conic immediately turned back to the gazebo rematerialization calculations, brushing some of her orange hair off her shoulder.

From the back of the bridge, Cosecant narrowed her eyes. "And you know 'Evvi, I still don't buy that Flora damaged the pack because she thought it was toothpaste! You step functions don't seem that stupid."

"You're the one being an idiot," Hevvi told the security officer. "We have no reason to cause sabotage, not when we'd be affected too!"

Off by the ladder, QT wrung her hands. "Oh, oh, can't we just get along here?" she pleaded. "I mean, look at me, a Polynomial, getting along with ParaB, a Conic! Doesn't that imply the rest of us can learn to accept the step functions too??"

"It's hardly the same... uh, hold up. What exactly do you mean by 'getting along'?" Radik III asked, raising an eyebrow. The quartic function's cheeks became a little darker in response.

"Uh oh, we're coming in too fast!" Sine interjected.  "Everyone hold onto something, I think we're about to crash!"


Wednesday 1 May 2013

S5.149 - Hyper Sensitive

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When Maud exited the bar, he found Para sitting down on the ground with the Hyperbola towering over her. "Watch where you're going!" the green haired scientist was saying, waggling her finger. For her part, the bunny girl simply looked up, her eyes misting slightly.

"Hyper!" Maud called out, drawing her attention. "Not the best time for Conic-Polynomial antagonism. Para's... had too much to drink."

"This excuses her plowing into me?!" Hyper said in irritation. She threw her arms out to her sides, her lab coat billowing around her. "Geez Maud, don't make it sound like she's the only one having a bad day!"

Maud lifted an eyebrow, curious despite himself. "What, are there problems in constructing your Cone of Science?"

"No! Well, yes," the eccentric relation amended. "But right now, the issue is that jerk ArcSin! He's wearing his Dr. Nisano glasses again, and has started completely redesigning my lab. The nerve!"

"Ah. I suppose it would be better to find him his own space."

"To heck with his own space, I'm worried about him stealing my archetype!" Hyper shot back. "Have you seen his twin around? I'm hoping ArcCos can rein Nis in."

"Er, no, I haven't..."

"Then of what use are you?!" Letting out a breath of exasperation, Hyper marched off down the road. It was only after she left that Maud realized that Para had vanished at some point too.