text:
## The Summer 2017 Mathematics A To Z: Height Function (elliptic curves)

I am one letter closer to the end of Gaurish’s main block of requests. They’re all good ones, mind you. This gets me back into elliptic curves and Diophantine equations. I might be writing about the wrong thing.

# Height Function.

My love’s father has a habit of asking us to rate our hobbies. This turned into a new running joke over a family vacation this summer. It’s a simple joke: I shuffled the comparables. “Which is better, Bon Jovi or a roller coaster?” It’s still a good question.

But as genial yet nasty as the spoof is, my love’s father asks natural questions. We always want to compare things. When we form a mathematical construct we look for ways to measure it. There’s typically something. We’ll put one together. We call this a height function.

We start with an elliptic curve. The coordinates of the points on this curve satisfy some equation. Well, there are many equations they satisfy. We pick one representation for convenience. The convenient thing is to have an easy-to-calculate height. We’ll write the equation for the curve as

$y^2 = x^3 + Ax + B$

Here both ‘A’ and ‘B’ are some integers. This form might be unique, depending on whether a slightly fussy condition on prime numbers hold. (Specifically, if ‘p’ is a prime number and ‘p4‘ divides into ‘A’, then ‘p6‘ must not divide into ‘B’. Yes, I know you realized that right away. But I write to a general audience, some of whom are learning how to see these things.) Then the height of this curve is whichever is the larger number, four times the cube of the absolute value of ‘A’, or 27 times the square of ‘B’. I ask you to just run with it. I don’t know the implications of the height function well enough to say why, oh, 25 times the square of ‘B’ wouldn’t do as well. The usual reason for something like that is that some obvious manipulation makes the 27 appear right away, or disappear right away.

This idea of height feeds in to a measure called rank. “Rank” is a term the young mathematician encounters first while learning matrices. It’s the number of rows in a matrix that aren’t equal to some sum or multiple of other rows. That is, it’s how many different things there are among a set. You can see why we might find that interesting. So many topics have something called “rank” and it measures how many different things there are in a set of things. In elliptic curves, the rank is a measure of how complicated the curve is. We can imagine the rational points on the elliptic curve as things generated by some small set of starter points. The starter points have to be of infinite order. Starter points that don’t, don’t count for the rank. Please don’t worry about what “infinite order” means here. I only mention this infinite-order business because if I don’t then something I have to say about two paragraphs from here will sound daft. So, the rank is how many of these starter points you need to generate the elliptic curve. (WARNING: Call them “generating points” or “generators” during your thesis defense.)

There’s no known way of guessing what the rank is if you just know ‘A’ and ‘B’. There are algorithms that can calculate the rank given a particular ‘A’ and ‘B’. But it’s not something like the quadratic formula where you can just do a quick calculation and know what you’re looking for. We don’t even know if the algorithms we have will work for every elliptic curve.

We think that there’s no limit to the height of elliptic curves. We don’t know this. We know there exist curves with ranks as high as 28. They seem to be rare [*]. I don’t know if that’s proven. But we do know there are elliptic curves with rank zero. A lot of them, in fact. (See what I meant two paragraphs back?) These are the elliptic curves that have only finitely many rational points on them.

And there’s a lot of those. There’s a well-respected that the average rank, of all the elliptic curves there are, is ½. It might be. What we have been able to prove is that the average rank is less than or equal to 1.17. Also that it should be larger than zero. So we’re maybe closing in on the ½ conjecture? At least we know something. I admit this essay I’ve started wondering what we do know of elliptic curves.

What do the height, and through it the rank, get us? I worry I’m repeating myself. By themselves they give us families of elliptic curves. Shapes that are similar in a particular and not-always-obvious way. And they feed into the Birch and Swinnerton-Dyer conjecture, which is the hipster’s Riemann Hypothesis. That is, it’s this big, unanswered, important problem that would, if answered, tell us things about a lot of questions that I’m not sure can be concisely explained. At least not why they’re interesting. We know some special cases, at least. Wikipedia tells me nothing’s proved for curves with rank greater than 1. Humanity’s ignorance on this point makes me feel slightly better pondering what I don’t know about elliptic curves.

(There are some other things within the field of elliptic curves called height functions. There’s particularly a height of individual points. I was unsure which height Gaurish found interesting so chose one. The other starts by measuring something different; it views, for example, $\frac{1}{2}$ as having a lower height than does $\frac{51}{101}$, even though the numbers are quite close in value. It develops along similar lines, trying to find classes of curves with similar behavior. And it gets into different unsolved conjectures. We have our ideas about how to think of fields.).

[*] Wikipedia seems to suggest we only know of one, provided by Professor Noam Elkies in 2006, and let me quote it in full. I apologize that it isn’t in the format I suggested at top was standard. Elkies way outranks me academically so we have to do things his way:

$y^2 + xy + y = x^3 - x^2 - 20,067,762,415,575,526,585,033,208,209,338,542,750,930,230,312,178,956,502 x + 34,481,611,795,030,556,467,032,985,690,390,720,374,855,944,359,319,180,361,266,008,296,291,939,448,732,243,429$

I can’t figure how to get WordPress to present that larger. I sympathize. I’m tired just looking at an equation like that. This page lists records of known elliptic curve ranks. I don’t know if the lack of any records more recent than 2006 reflects the page not having been updated or nobody having found a rank-29 curve. I fully accept the field might be more difficult than even doing maintenance on a web page’s content is.

• #### gaurish 6:45 pm on Thursday, 17 August, 2017 Permalink | Reply

Yet another beautiful post. You may like this lecture about the BSD conjecture: https://youtu.be/2gbQWIzb6Dg

Like

• #### Joseph Nebus 3:05 am on Saturday, 19 August, 2017 Permalink | Reply

Thank you! And also thank you for the link. I never think to look for videos that would explain topics. In many ways I still think of the Internet as being in about 1998, when video was nothing more than a theoretical possibility that might someday finish loading before it glitches out.

Like

## Did This German Retiree Solve A Decades-Old Conjecture?

And then this came across my desktop (my iPad’s too old to work with the Twitter client anymore):

The underlying news is that one Thomas Royen, a Frankfurt (Germany)-area retiree, seems to have proven the Gaussian Correlation Inequality. It wasn’t a conjecture that sounded familiar to me, but the sidebar (on the Quanta Magazine article to which I’ve linked there) explains it and reminds me that I had heard about it somewhere or other. It’s about random variables. That is, things that can take on one of a set of different values. If you think of them as the measurements of something that’s basically consistent but never homogenous you’re doing well.

Suppose you have two random variables, two things that can be measured. There’s a probability the first variable is in a particular range, greater than some minimum and less than some maximum. There’s a probability the second variable is in some other particular range. What’s the probability that both variables are simultaneously in these particular ranges? This is easy to answer for some specific cases. For example if the two variables have nothing to do with each other then everybody who’s taken a probability class knows. The probability of both variables being in their ranges is the probability the first is in its range times the probability the second is in its range. The challenge is telling whether it’s always true, whether the variables are related to each other or not. Or telling when it’s true if it isn’t always.

The article (and pop reporting on this) is largely about how the proof has gone unnoticed. There’s some interesting social dynamics going on there. Royen published in an obscure-for-the-field journal, one he was an editor for; this makes it look dodgy, at least. And the conjecture’s drawn “proofs” that were just wrong; this discourages people from looking for obscurely-published proofs.

Some of the articles I’ve seen on this make Royen out to be an amateur. And I suppose there is a bias against amateurs in professional mathematics. There is in every field. It’s true that mathematics doesn’t require professional training the way that, say, putting out oil rig fires does. Anyone capable of thinking through an argument rigorously is capable of doing important original work. But there are a lot of tricks to thinking an argument through that are important, and I’d bet on the person with training.

In any case, Royen isn’t a newcomer to the field who just heard of an interesting puzzle. He’d been a statistician, first for a pharmaceutical company and then for a technical university. He may not have a position or tie to a mathematics department or a research organization but he’s someone who would know roughly what to do.

So did he do it? I don’t know; I’m not versed enough in the field to say. It’s interesting to see if he has.

• #### mathtuition88 4:29 am on Thursday, 13 April, 2017 Permalink | Reply

He seems to have a PhD earned in 1975. (http://www.genealogy.ams.org/id.php?id=134663).

Like

• #### Joseph Nebus 5:32 am on Friday, 14 April, 2017 Permalink | Reply

Ah, thank you! I appreciate the reassurance that he wasn’t wholly an amateur or someone whose expertise came from on-the-job training.

Liked by 1 person

## The End 2016 Mathematics A To Z: Jordan Curve

I realize I used this thing in one of my Theorem Thursday posts but never quite said what it was. Let me fix that.

## Jordan Curve

Get a rubber band. Well, maybe you can’t just now, even if you wanted to after I gave orders like that. Imagine a rubber band. I apologize to anyone so offended by my imperious tone that they’re refusing. It’s the convention for pop mathematics or science.

Anyway, take your rubber band. Drop it on a table. Fiddle with it so it hasn’t got any loops in it and it doesn’t twist over any. I want the whole of one edge of the band touching the table. You can imagine the table too. That is a Jordan Curve, at least as long as the rubber band hasn’t broken.

This may not look much like a circle. It might be close, but I bet it’s got some wriggles in its curves. Maybe it even curves so much the thing looks more like a kidney bean than a circle. Maybe it pinches so much that it looks like a figure eight, a couple of loops connected by a tiny bridge on the interior. Doesn’t matter. You can bring out the circle. Put your finger inside the rubber band’s loops and spiral your finger around. Do this gently and the rubber band won’t jump off the table. It’ll round out to as perfect a circle as the limitations of matter allow.

And for that matter, if we wanted, we could take a rubber band laid down as a perfect circle. Then nudge it here and push it there and wrinkle it up into as complicated a figure as you like. Either way is as possible.

A Jordan Curve is a closed curve, a curve that loops around back to itself. And it’s simple. That is, it doesn’t cross over itself at any point. However weird and loopy this figure is, as long as it doesn’t cross over itself, it’s got in a sense the same shape as a circle. We can imagine a function that matches every point on a true circle to a point on the Jordan Curve. A set of points in order on the original circle will match to points in the same order on the Jordan Curve. There’s nothing missing and there’s no jumps or ambiguous points. And no point on the Jordan Curve matches to two or more on the original circle. (This is why we don’t let the curve to cross over itself.)

When I wrote about the Jordan Curve Theorem it was about how to tell how a curve divides a plane into two pieces, an inside and an outside. You can have some pretty complicated-looking figures. I have an example on the Jordan Curve Theorem essay, but you can make your own by doodling. And we can look at it as a circle, as a rubber band, twisted all around.

This all dips into topology, the study of how shapes connect when we don’t care about distance. But there are simple wondrous things to find about them. For example. Draw a Jordan Curve, please. Any that you like. Now draw a triangle. Again, any that you like.

There is some trio of points in your Jordan Curve which connect to a triangle the same shape as the one you drew. It may be bigger than your triangle, or smaller. But it’ll look similar. The angles inside will all be the same as the ones you started with. This should help make doodling during a dull meeting even more exciting.

There may be four points on your Jordan Curve that make a square. I don’t know. Nobody knows for sure. There certainly are if your curve is convex, that is, if no line between any two points on the curve goes outside the curve. And it’s true even for curves that aren’t complex if they are smooth enough. But generally? For an arbitrary curve? We don’t know. It might be true. It might be impossible to find a square in some Jordan Curve. It might be the Jordan Curve you drew. Good luck looking.

• #### gaurish 3:52 am on Thursday, 24 November, 2016 Permalink | Reply

Jordan curve theorem is again in news: http://wp.me/p3qzP-2tV

Like

• #### Joseph Nebus 11:11 pm on Friday, 25 November, 2016 Permalink | Reply

Ooh, thank you, that’s interesting stuff. And on that conjecture about squares, too, which is so neat.

Liked by 1 person

## What’s The Shortest Proof I’ve Done?

I didn’t figure to have a bookend for last week’s “What’s The Longest Proof I’ve Done? question. I don’t keep track of these things, after all. And the length of a proof must be a fluid concept. If I show something is a direct consequence of a previous theorem, is the proof’s length the two lines of new material? Or is it all the proof of the previous theorem plus two new lines?

I would think the shortest proof I’d done was showing that the logarithm of 1 is zero. This would be starting from the definition of the natural logarithm of a number x as the definite integral of 1/t on the interval from 1 to x. But that requires a bunch of analysis to support the proof. And the Intermediate Value Theorem. Does that stuff count? Why or why not?

But this happened to cross my desk: The Shortest-Known Paper Published in a Serious Math Journal: Two Succinct Sentences, an essay by Dan Colman. It reprints a paper by L J Lander and T R Parkin which appeared in the Bulletin of the American Mathematical Society in 1966.

It’s about Euler’s Sums of Powers Conjecture. This is a spinoff of Fermat’s Last Theorem. Leonhard Euler observed that you need at least two whole numbers so that their squares add up to a square. And you need three cubes of whole numbers to add up to the cube of a whole number. Euler speculated you needed four whole numbers so that their fourth powers add up to a fourth power, five whole numbers so that their fifth powers add up to a fifth power, and so on.

And it’s not so. Lander and Parkin found that this conjecture is false. They did it the new old-fashioned way: they set a computer to test cases. And they found four whole numbers whose fifth powers add up to a fifth power. So the quite short paper answers a long-standing question, and would be hard to beat for accessibility.

There is another famous short proof sometimes credited as the most wordless mathematical presentation. Frank Nelson Cole gave it on the 31st of October, 1903. It was about the Mersenne number 267-1, or in human notation, 147,573,952,589,676,412,927. It was already known the number wasn’t prime. (People wondered because numbers of the form 2n-1 often lead us to perfect numbers. And those are interesting.) But nobody knew which factors it was. Cole gave his talk by going up to the board, working out 267-1, and then moving to the other side of the board. There he wrote out 193,707,721 × 761,838,257,287, and showed what that was. Then, per legend, he sat down without ever saying a word, and took in the standing ovation.

I don’t want to cast aspersions on a great story like that. But mathematics is full of great stories that aren’t quite so. And I notice that one of Cole’s doctoral students was Eric Temple Bell. Bell gave us a great many tales of mathematics history that are grand and great stories that just weren’t so. So I want it noted that I don’t know where we get this story from, or how it may have changed in the retellings. But Cole’s proof is correct, at least according to Octave.

So not every proof is too long to fit in the universe. But then I notice that Mathworld’s page regarding the Euler Sum of Powers Conjecture doesn’t cite the 1966 paper. It cites instead Lander and Parkin’s “A Counterexample to Euler’s Sum of Powers Conjecture” from Mathematics of Computation volume 21, number 97, of 1967. There the paper has grown to three pages, although it’s only a couple paragraphs of one page and three lines of citation on the third. It’s not so easy to read either, but it does explain how they set about searching for counterexamples. But it may give you some better idea of how numerical mathematicians find things.

## Counting Things

I’ve been working on my little thread of posts about sports mathematics. But I’ve also had a rather busy week and I just didn’t have time to finish the next bit of pondering I had regarding baseball scores. Among other things I had the local pinball league’s post-season Split-Flipper Tournament to play in last night. I played lousy, too.

So I hope I may bring your attention to some interesting posts from Baking And Math. Yenergy started, last week, with a post about the Gauss Circle Problem. Carl Friedrich Gauss you may know as the mathematical genius who proved the Fundamental Theorem of Whatever Subfield Of Mathematics You’re Talking About. Circles are those same old things. The problem is quite old, and easy to understand, and not answered yet. Start with a grid of regularly spaced dots. Draw a circle centered on one of the dots. How many dots are inside the circle?

Obviously you can count. What we would like is a formula, though: if this is the radius then that function of the radius is the number of points. We don’t have that, remarkably. Yenergy describes some of that, and some ways to estimate the number of points. This is for the circle and for some other shapes.

Yesterday, Yenergy continued the discussion and got into partitions. Partitions sound boring; they’re about identifying ways to split something up into components. Yet they turn up everywhere. I’m most used to them in statistical mechanics, the study of physics problems where there’s too many things moving to keep track of them all. But it isn’t surprising they turn up in this sort of point-counting problem.

As a bonus Yenergy links to an article examining a famous story about Gauss. This is specifically the famous story about him, as a child, doing a quite long arithmetic problem at a glance. It’s a story that’s passed into legend and I had not known how much of it was legend.

• #### sheldonk2014 2:57 am on Monday, 6 June, 2016 Permalink | Reply

Thank you Joseph for visiting
Love the pinball league
Sheldon

Like

• #### Joseph Nebus 2:56 am on Saturday, 11 June, 2016 Permalink | Reply

Thank you for stopping in. Happy to be of service.

Like

## A Leap Day 2016 Mathematics A To Z: Conjecture

For today’s entry in the Leap Day 2016 Mathematics A To Z I have an actual request from from Elke Stangl. I’d had another ‘c’ request, for ‘continued fractions’. I’ve decided to address that by putting ‘Fractions, continued’ on the roster. If you have other requests, for letters not already committed, please let me know. I’ve got some letters I can use yet.

## Conjecture.

An old joke says a mathematician’s job is to turn coffee into theorems. I prefer tea, which may be why I’m not employed as a mathematician. A theorem is a logical argument that starts from something known to be true. Or we might start from something assumed to be true, if we think the setup interesting and plausible. And it uses laws of logical inference to draw a conclusion that’s also true and, hopefully, interesting. If it isn’t interesting, maybe it’s useful. If it isn’t either, maybe at least the argument is clever.

How does a mathematician know what theorems to try proving? We could assemble any combination of premises as the setup to a possible theorem. And we could imagine all sorts of possible conclusions. Most of them will be syntactically gibberish, the equivalent of our friends the monkeys banging away on keyboards. Of those that aren’t, most will be untrue, or at least impossible to argue. Of the rest, potential theorems that could be argued, many will be too long or too unfocused to follow. Only a tiny few potential combinations of premises and conclusions could form theorems of any value. How does a mathematician get a good idea where to spend her time?

She gets it from experience. In learning what theorems, what arguments, have been true in the past she develops a feeling for things that would plausibly be true. In playing with mathematical constructs she notices patterns that seem to be true. As she gains expertise she gets a sense for things that feel right. And she gets a feel for what would be a reasonable set of premises to bundle together. And what kinds of conclusions probably follow from an argument that people can follow.

This potential theorem, this thing that feels like it should be true, a conjecture.

Properly, we don’t know whether a conjecture is true or false. The most we can say is that we don’t have evidence that it’s false. New information might show that we’re wrong and we would have to give up the conjecture. Finding new examples that it’s true might reinforce our idea that it’s true, but that doesn’t prove it’s true.

For example, we have the Goldbach Conjecture. According to it every even number greater than two can be written as the sum of exactly two prime numbers. The evidence for it is very good: every even number we’ve tied has worked out, up through at least 4,000,000,000,000,000,000. But it isn’t proven. It’s possible that it’s impossible from the standard rules of arithmetic.

That’s a famous conjecture. It’s frustrated mathematicians for centuries. It’s easy to understand and nobody’s found a proof. Famous conjectures, the ones that get names, tend to do that. They looked nice and simple and had hidden depths.

Most conjectures aren’t so storied. They instead appear as notes at the end of a section in a journal article or a book chapter. Or they’re put on slides meant to refresh the audience’s interest where it’s needed. They are needed at the fifteen-minute park of a presentation, just after four slides full of dense equations. They are also needed at the 35-minute mark, in the middle of a field of plots with too many symbols and not enough labels. And one’s needed just before the summary of the talk, so that the audience can try to remember what the presentation was about and why they thought they could understand it. If the deadline were not so tight, if the conference were a month or so later, perhaps the mathematician would find a proof for these conjectures.

Perhaps. As above, some conjectures turn out to be hard. Fermat’s Last Theorem stood for four centuries as a conjecture. Its first proof turned out to be nothing like anything Fermat could have had in mind. Mathematics popularizers lost an easy hook when that was proven. We used to be able to start an essay on Fermat’s Last Theorem by huffing about how it was properly a conjecture but the wrong term stuck to it because English is a perverse language. Now we have to start by saying how it used to be a conjecture instead.

But few are like that. Most conjectures are ideas that feel like they ought to be true. They appear because a curious mind will look for new ideas that resemble old ones, or will notice patterns that seem to resemble old patterns.

And sometimes conjectures turn out to be false. Something can look like it ought to be true, or maybe would be true, and yet be false. Often we can prove something isn’t true by finding an example, just as you might expect. But that doesn’t mean it’s easy. Here’s a false conjecture, one that was put forth by Goldbach. All odd numbers are either prime, or can be written as the sum of a prime and twice a square number. (He considered 1 to be a prime number.) It’s not true, but it took over a century to show that. If you want to find a counterexample go ahead and have fun trying.

Still, if a mathematician turns coffee into theorems, it is through the step of finding conjectures, promising little paths in the forest of what is not yet known.

• #### elkement (Elke Stangl) 9:38 pm on Friday, 4 March, 2016 Permalink | Reply

Thanks :-) So you say that experts’ intuition that might look like magic to laymen is actually pattern recognition, correct? (I think I have read about this in pop-sci psychology books) And if an unproven theorem passes the pattern recognition filter it is promoted to conjecture.

Like

• #### Joseph Nebus 7:27 am on Wednesday, 9 March, 2016 Permalink | Reply

I think that there is a large aspect of it that’s pattern recognition, yes. But some of that may be that we look for things that resemble what’s already worked. So, like, if we already have a theorem about how a sequence of real-valued functions converges to a new real-valued function, then it’s natural to think about variants. Can we say something about sequences of complex-valued functions? If the original theorem demanded functions that were continuous and had infinitely many derivatives, can we loosen that to a function that’s continuous and has only finitely many derivatives? Can we lose the requirement that there be derivatives and still say something?

I realized at one point while taking real analysis in grad school that many of the theorems we were moving into looked a lot like what we already had with one or two variations, and could sometimes write out the next theorem almost by rote. There is certainly a kind of pattern recognition at work here, though sometimes it can feel like playing with the variations on a theme.

Liked by 1 person

• #### elkement (Elke Stangl) 7:37 am on Wednesday, 9 March, 2016 Permalink | Reply

Yes, I agree – I meant pattern recognition in exactly this way, in a very broad way … searching for a similar pattern in your own experiences, among things you have encountered and that worked. I was thinking in general terms and comparing to other skills and expertise, like what makes you successful in any kind of tech troubleshooting. It seems that you have an intuitive feeling about what may work but actually you draw on related scenarios or aspects of scenarios we had solved.

Like

• #### Pen & Shutter 1:09 pm on Saturday, 5 March, 2016 Permalink | Reply

I understood all that! I definitely deserve a prize … I am no mathematician … And I enjoyed every word! I love your use of English.

Like

• #### Joseph Nebus 7:42 am on Wednesday, 9 March, 2016 Permalink | Reply

You’re most kind, thank you. I’m glad you enjoyed.

Liked by 1 person

• #### davekingsbury 3:25 pm on Saturday, 5 March, 2016 Permalink | Reply

If you’ve nothing for Q, what about Quadratic Equations … though I start twitching whenever I think about them!

Like

• #### Joseph Nebus 7:43 am on Wednesday, 9 March, 2016 Permalink | Reply

I’m sorry to say Q already got claimed, by ‘quaternion’. But P got ‘polynomial’, which should be close enough to quadratic equations that there’s at least some help there.

Liked by 1 person

## The Short, Unhappy Life Of A Doomed Conjecture

So last month amongst the talk about the radius of a circle inscribed in a Pythagorean right triangle I mentioned that I had, briefly, floated a conjecture that might have spun off it. It didn’t, though I promised to describe the chain of thought I had while exploring it, on the grounds that the process of coming up with mathematical ideas doesn’t get described much, and certainly doesn’t get described for the sorts of fiddling little things that make up a trifle like this.

A triangle (meant to be a right triangle) with an inscribed circle of radius r. The triangle is divided into three smaller triangles meeting at the center of the inscribed circle.

The point from which I started was a question about the radius of a circle inscribed in the right triangle with legs of length 5, 12, and 13. This turns out to have a radius of 2, which is interesting because it’s a whole number. It turns out to be simple to show that for a Pythagorean right triangle, that is, a right triangle whose legs are a Pythagorean triple — like (3, 4, 5), or (5, 12, 13), any where the square of the biggest number is the same as what you get adding together the squares of the two smaller numbers — the inscribed circle has a radius that’s a whole number. For example, the circle you could inscribe in a triangle of sides 3, 4, and 5 would have radius 1. The circle inscribed in a triangle of sides 8, 15, and 17 would have radius 3; so does the circle inscribed in a triangle of sides 7, 24, and 25.

Since I now knew that (and in multiple ways: HowardAt58 had his own geometric solution, and you can also do this algebraically) I started to wonder about the converse. If a Pythagorean right triangle’s inscribed circle has a whole number for a radius, can does knowing a circle has a whole number for a radius tell us anything about the triangle it’s inscribed in? This is an easy way to build new conjectures: given that “if A is true, then B must be true”, can it also be that “if B is true, then A must be true”? Only rarely will that be so — it’s neat when it is — but we might be able to patch something up, like, “if B, C, and D are all simultaneously true, then A must be true”, or perhaps, “if B is true, then at least E must be true”, where E resembles A but maybe doesn’t make such a strong claim. Thus are tiny little advances in mathematics created.

• #### ivasallay 6:20 pm on Thursday, 11 December, 2014 Permalink | Reply

I’ve come up with a dead conjecture or two before, too. They are still fun to explore.

Like

• #### Joseph Nebus 6:52 am on Friday, 12 December, 2014 Permalink | Reply

They can be. I’m just disappointed I got to nowhere much; I was having so much fun with the triangle.

Like

## No Conjecture For 19,000

I failed to notice when it happened but my little blog here reached its 19,000th page view sometime on Saturday the 22nd. I’m sorry to have missed it; I like keeping track of these little milestones and I expect to be pretty self-confident if and when I hit 20,000. I know these aren’t enormous numbers, as even mathematics blogging goes, but I do feel like I’m getting a larger audience, that’s sometimes also more engaged, and that’s gratifying.

Since I don’t want to just seem to be bragging about a really minor accomplishment I hoped to include a conjecture that would be a nice little puzzle, following up on the right-triangle stuff done so recently here. But in writing out the problem exactly, I realized that the conjecture I had in mind was (a) false, and (b) really obviously false. Which is a shame, but so many attempts at figuring out something turn out that way. I’m deciding whether to swallow my pride and lay out the line of thought that ended up in disappointment, on the grounds that you never get to see a mathematician go through the stages of discovery only to come up with a flop; that’s not the sort of thing that gets into papers, much less textbooks, unless there’s a good juicy scandal behind it. We’ll see.

Wowie! Congrats! 🏆✨

Like

• #### Joseph Nebus 3:28 am on Friday, 28 November, 2014 Permalink | Reply

Aw, thank you.

Liked by 1 person

• #### howardat58 10:34 pm on Tuesday, 25 November, 2014 Permalink | Reply

Come on, then, lets have it !

Like

• #### Joseph Nebus 3:28 am on Friday, 28 November, 2014 Permalink | Reply

All right, all right, I’ll do it, but people have to promise not to laugh at me.

Like

• #### vagabondurges 2:05 am on Wednesday, 26 November, 2014 Permalink | Reply

I agree, I say post it. (And remember, you have readers who haven’t thought about math much since senior year of high school…ahem…)

Like

• #### Joseph Nebus 3:29 am on Friday, 28 November, 2014 Permalink | Reply

I am going to write it up, mostly as a way of explaining the course of thinking from initial impulse to exploration to sad realization there wasn’t so much there.

Like

## Radius of the inscribed circle of a right angled triangle

For that “About An Inscribed Circle” problem I posted the other day: HowardAt58 worked out one way of proving what the radius of the circle that just fits within the 5-12-13 right triangle has to be, and in a pretty neat geometric fashion. Worth the read. I recommend following his steps along by hand, writing each out, but that reflects that I’m much more likely to follow mathematical reasoning if I write it out, even if I don’t do something past what the original author does. HowardAt58 also includes a little conjecture, inspired by playing around with a couple of Pythagorean triangles (playing around with a couple of examples is a great way to find conjectures), which I at least believe to be true.

Interestingly, his proof isn’t the same geometric proof that I’d realized we could do, so, I’m thinking to include that as another follow-up around here when I can make a couple diagrams that explain it.

Like

c
Compose new post
j
Next post/Next comment
k
Previous post/Previous comment
r


text:
Structural Variation Detection On A Short Reads Simulation
2
1
Entering edit mode
10.5 years ago
Pascal ★ 1.5k

Hi,

I am trying to do the following exercise (still playing with indels):

1. simulate short reads (and indels) out of a human genome chromosome,
2. align the short reads simulated in (1),
3. call indels using a SV detection tool,
4. compare results in (3) with the indels generated in (1).

For this workflow, I'm using wgsim in (1), bowtie2 in (2) and dindel in (3). I've written at the end of this question the complete commands list.

So far I haven't been able to detect a single indel and I am wondering what I am doing wrong. Is it because I haven't simulated enough reads (1M reads)? Or something else? I am wondering if I misunderstood some concept or it is simply a matter of adjusting a parameter.

I tried to make the simulator to extend the indels (with -X parameter) but it didn't help.

Thanks in advance for any hint!

samtools faidx human_g1k_v37.fasta 20 > human_g1k_v37_chr20.fasta
bowtie-build human_g1k_v37_chr20.fasta homo_chr20
samtools view -bS homo_chr20.sam > homo_chr20.bam
./dindel_x86-64  --ref human_g1k_v37_chr20.fasta --outputFile 1 --bamFile homo_chr20.bam --analysis getCIGARindels

=> no variants detected here (in 2.variants file) so I don't go ahead :-(

structural indel bowtie • 3.2k views
0
Entering edit mode

Pascal, as Wolf noted bowtie(1) cannot be used here. to verify this, check the CIGAR strings in your SAM output if you can find a single "I" or "D" there.

4
Entering edit mode
10.5 years ago
Wolf ▴ 130

I may be missing something very basic here, but are you sure you are actually using bowtie2 and not bowtie1? the executables are bowtie2 and bowtie2-align and i think bowtie2 takes it's index with the -x switch.

0
Entering edit mode

Thanks Wolf ! You're definetly right: I'm actually using bowtie1 here :-O it's because I clicked the lastest version link and I got the lastest release from bowtie1 where bowtie2 is beta still... I think it won't help to fix my problem (I'm correctly using bowtie1) but you're right pointing it out!

0
Entering edit mode

but bowtie1 does not do gapped alignment, so there won't be any reads spanning your indels

0
Entering edit mode

Very good point Wolf! No gapped alignments, no indels, actually your comment contains the correct answer to the question. Pascal, try to replace bowtie with bwa, mosaik, bfast.

0
Entering edit mode

No, I think I will first use bowtie2. This was my initial objective as I read that "Bowtie 2 supports gapped alignment with affine gap penalties. Number of gaps and gap lengths are not restricted, except by way of the configurable scoring scheme. Bowtie 1 finds just ungapped alignments." Can you confirm this is what we are looking for or should I definitely use BWA? In any case I'm confused because I intended to use at first bowtie2 not 1 :-( Wolf, your're the greatest, thank you !!

0
Entering edit mode

Good morning. So good news, with a higher coverage and using bowtie2 (the real 2 :-) ), it seems that indels are being detected well now with dindel. At least I have thousands of candidates in my variants file and I am waiting for dindel to filter those. Thanks to both of you for your help.

For my records, I wrote a simple memo there (it may help other people):

http://biotechies.blogspot.com/2011/11/genome-structural-variation-on.html

3
Entering edit mode
10.5 years ago

I don't know what is the minimal coverage required by dindel, but with 1M reads 70bp paired, you have approx 2.2X coverage on chromosome 20. It might be that it is not enough to find indels, as it is very unlikely that the same indel is samples by two independent reads.

you should have approx 6000 indels, but half of them would be heterozygous...

0
Entering edit mode

Thanks Stefano, I will come back to you tomorrow after a new try.

format: plain


text:
# compactness / sequentially compact

I'm looking for two examples:

1. A space which is compact but not sequentially compact
2. A space which is sequentially compact but not compact

Explanations why the spaces are compact / not compact and sequentially compact / not sequentially compact would be appreciated. A reference would also be appreciated. So the conclusion would be, that there's no equivalence in general. Of course they are equivalent in a metric space.

math

-
en.wikipedia.org/wiki/Sequentially_compact_space –  user38268 Jun 1 '12 at 12:34
...and the canonical reference for answering such questions quickly and reliably is Steen and Seebach, Counterexamples in Topology. –  t.b. Jun 1 '12 at 12:57

The following examples are from $\pi$-Base, a searchable database of Steen and Seebach's Counterexamples in Topology.

• Stone-Cech Compactification of the Integers
• Uncountable Cartesian Product of Unit Interval ($I^I$)
• An Altered Long Line
• $[0, \omega_1)$ ($\omega_1$ is the first uncountable ordinal)
• The Long Line
• Tychonoff Corkscrew
-
@ Austin Mohr: My first attempt was Counterexamples in Topology. Without knowing that the Cartesian product of unit interval and $[0,\omega_1)$ are examples, it's hard to find something in there. So thanks for your answer. –  math Jun 3 '12 at 8:20
There are tables at the back of the book, where you can find such examples with patience. And isn't there a Venn diagram in the section on compactness at the beginning? Of course, the automated Spacebook is easier. –  GEdgar Sep 16 '12 at 0:50
@Austin Mohr I am interested how to prove on uncountable Cartesian product of unit interval is not sequentially compact –  math Mar 17 '13 at 22:12
@AustinMohr: This link is not work. Could you please check for me. Thank you so much! –  user52523 Nov 2 '13 at 12:06

Example 1 with proof: Stone-Čech Compactification of the Integers $\beta \omega$

Proof: It is compact obviously. We will prove that $\beta\omega$ is not sequentially compact. Note that every infinite set in $\beta\omega$ has $2^\mathfrak c$ cluster points, hence the only convergent sequences in $\beta\omega$ are those which are eventually constant; therefore if $X$ is a subspace of $\beta\omega$ and $X$ is sequentially compact, then $X$ is finite. So $\beta\omega$ cannot be sequentially compact.

-
Didn't you post this answer somewhere else today? –  Asaf Karagila May 11 '13 at 23:49


text:
### Sidebar

theorems:stone-von_neumann

# Stone-von Neumann Theorem

## Intuitive

The Stone-von Neumann theorem roughly says that for any operators $A$ and $B$ satisfying the canonical commutation relation, we can get away with using the standard representation $A \rightarrow u,\ B \rightarrow -i \hbar \frac{d}{du}$ without loss of generality. (More precisely, it says that any representation of the *exponentiated* canonical commutation relation on a sufficiently smooth Hilbert space is unitarily equivalent to the standard representation, so any other representation basically just describes the same physics in a different coordinate system.)https://physics.stackexchange.com/a/264587/37286

In typical physics quantum mechanics textbooks, one often sees calculations made just using the Heisenberg commutation relations, without picking a specific representation of the operators that satisfy these relations. This turns out to be justified by the remarkable fact that, for the Heisenberg group, once one picks the constant with which Z acts, all irreducible representations are unitarily equivalent. In a sense, the representation theory of the Heisenberg group is very simple: there’s only one irreducible representation. This is very different from the theory for even the simplest compact Lie groups (U(1) and SU(2)) which have an infinity of inequivalent irreducibles labeled by weight or by spin. Representations of a Heisenberg group will appear in different guises (we’ve seen two, will see another in the discussion of the harmonic oscillator, and there are yet others that appear in the theory of theta-functions), but they are all unitarily equivalent, a statement known as the Stone-von Neumann theorem. […] In the case of an infinite number of degrees of freedom, which is the case of interest in quantum field theory, the Stone-von Neumann theorem no longer holds and one has an infinity of inequivalent irreducible representations, leading to quite different phenomena. […]It is also important to note that the Stone-von Neumann theorem is formulated for Heisenberg group representations, not for Heisenberg Lie algebra representations. For infinite dimensional representations in cases like this, there are representations of the Lie algebra that are “non-integrable”: they aren’t the derivatives of Lie group representations. For such non-integrable representations of the Heisenberg Lie algebra (i.e., operators satisfying the Heisenberg commutation relations) there are counter-examples to the analog of the Stone von-Neumann theorem. It is only for integrable representations that the theorem holds and one has a unique sort of irreducible representation.https://www.math.columbia.edu/~woit/QM/qmbook.pdf

## Concrete

In the algebraic approach, the fundamental structure of a quantum theory is given by an abstract algebra of the canonical commutation relations, which can be given various different representations in terms of subalgebras of the bounded operators on a Hilbert space. Two such representations, each consisting of a Hilbert space $H$ and the set of bounded operators $\hat O$ defined on it, are unitarily equivalent if there is a (one-to-one, linear, norm preserving) map $U:G\to H'$ such that $(\forall i)(U^{-1}\hat{O}'_iU=\hat{O}_i)$. The Stone–von Neumann theorem guarantees that in the finite-dimensional case all irreducible representations of the abstract algebra are unitarily equivalent, but in the infinite-dimensional case there are unitarily inequivalent representations of the algebra.http://publish.uwo.ca/~csmeenk2/files/HiggsMechanism.pdf

Theorem (Stone-von Neumann). Any irreducible representation π of the group H3 on a Hilbert space, satisfying $$π 0 (Z) = -i1$$ is unitarily equivalent to the Schrödinger representation (ΓS, L2 (R)).https://www.math.columbia.edu/~woit/QM/qmbook.pdf

## Abstract

The motto in this section is: the higher the level of abstraction, the better.

## Why is it interesting?

The Stone-von Neumann theorem is important, for example, to understand why symmetry breaking is possible in quantum field theory.


text:
PDF e-Pub

## Section: New Results

### Porting Chameleon on top of OpenMP

Chameleon is a dense linear algebra software relying on sequential task-based algorithms where sub-tasks of the overall algorithms are submitted to a Runtime system. Algorithms were implemented on top of several task-based runtime systems: QUARK, PaRSEC, and StarPU (for which there is also an optional heterogeneous implementation). In the context of PRACE-5IP, we introduced OpenMP as an alternative backend for these linear algebra kernels.


text:
## CMU/Ford study assesses optimal mix of conventional, hybrid, plug-in hybrid and electric vehicles for minimizing GHG and cost

##### 31 October 2012
 Breakdown of (a) equivalent annualized life cycle cost and (b) life cycle GHG emissions for four independently cost-optimized vehicle designs. Traut et al. Click to enlarge.

In a new study, a team from Carnegie Mellon University (CMU) and Ford Research and Advanced Engineering set out to determine the optimal mix for the fleet of mid-size personal vehicles in the US—while maintaining current driving patterns—with the goals of minimizing greenhouse gas emissions (GHG) or cost. They also addressed the question of GHG or cost reduction with and without a workplace charging infrastructure.

Their study, they suggested in a paper analyzing the best possible outcomes and published in the journal Energy Policy, is a step towards understanding what should be incentivized by policy makers.

They developed an optimization problem to minimize life cycle cost or GHG emissions over the personal vehicle fleet by jointly determining (1) the optimal design of each CV (conventional vehicle), HEV (hybrid electric vehicle), PHEV (plug-in hybrid electric vehicle), and BEV (battery-electric vehicle); (2) the optimal allocation of each vehicle design in the fleet based on annual vehicle miles traveled (VMT); and (3) the optimal allocation of workplace charging infrastructure to PEVs in the fleet. Within the fleet, they considered only vehicles of similar size and acceleration performance to the Toyota Prius.

They also incorporated vehicle design constraints to ensure comparable acceleration performance and vehicle allocation constraints to ensure BEVs are assigned only if they have sufficient range to accommodate the vehicle’s driving distance on most days (base case 95% of days).

Prior studies compare and select among a small set of fixed vehicle configurations based on selected commercially available vehicles or a small set of simulated vehicle alternatives. However, interactions among engine sizing, motor sizing, and battery sizing can be important in comparing vehicle characteristics, and optimal battery sizing represents a compromise among drivers with different travel patterns. We follow Shiau et al. (2010) [earlier post] and pose a mixed-integer nonlinear programming (MINLP) formulation to determine the best configuration of vehicles in the design space in order to compare the best design of each conventional vehicle (CV), HEV, PHEV, and BEV model under acceleration performance constraints that ensure vehicles are comparable. We further incorporate charging infrastructure decisions that determine which of the PEVs should be only charged at home vs. charged both at home and at the workplace, given charging infrastructure costs and production emissions, and we use driving pattern data to model required BEV ranges and PHEV electricity and gasoline usage.

—Traut et al.

 Optimal vehicle allocations for minimizing life cycle GHG emissions (top) and cost (bottom) in selected scenarios. From Traut et al. Click to enlarge.

Among their findings were:

• In agreement with prior studies, without sufficient grid decarbonization, plug-in vehicles do not offer substantial GHG emissions reductions compared to HEVs. GHG reductions improve with low-carbon electricity. Thus, grid decarbonization is needed to make plug-in vehicles a relevant means of reducing GHG emissions beyond grid-independent HEVs.

• Compared to CVs, HEVs offer cost and emissions reductions in almost all scenarios and are an optimal or near-optimal solution for minimizing cost across many scenarios.

• Under the current US electricity generation mix, workplace charging availability provides no GHG emissions benefit in the optimized solution, but workplace charging does provide additional benefits of optimistically up to 21% in combination with low-carbon electricity.

• Workplace charging availability changes the GHG-minimized vehicle allocation slightly, allocating smaller capacity PHEVs and BEVs.

• Gas prices above $3.25/gal (plus 5.2% per year) cause PHEVs to appear in the minimum cost solution, but for plug-in vehicles to dominate over HEVs, either gas prices of$7/gal (plus 5.2% per year) or gas prices of $4.5/gal (plus 5.2% per year) in combination with low vehicle and battery costs (DOE 2030 program goal levels, including battery costs under$200/kWh) are needed. High carbon prices (over $100/tCO2e) do little to drive plug-in vehicles to appear in the cost-minimizing solution. • BEVs are restricted by range requirements from being a significant part of the minimum cost or GHG solutions. Even when range requirements are dramatically reduced, requiring BEV range adequate for only the average trip rather than the 95th percentile trip, a fleet of entirely BEVs is much more expensive and GHG-intensive than the other vehicle types, and BEVs are not GHG-minimizers for the full fleet even when charged with zero-emissions electricity. BEVs enter the GHG-optimal fleet only for short-range vehicles and only in cases with grid decarbonization. This formulation represents a best-case scenario for minimizing cost or GHG emissions with these vehicle technologies; market outcomes would likely deviate, and we do not attempt to predict firm or consumer behavior. —Traut et al. This research was supported by National Science Foundation grants from the Foundation’s Material Use, Science, Engineering and Society program, the CAREER program, and the Graduate Research Fellowship program. Support was also provided by Ford Motor Company, Toyota Motors of America, and the Steinbrenner Graduate Fellowship. Resources • Traut, E., et al. (2012) Optimal design and allocation of electrified vehicles and dedicated charging infrastructure for minimum life cycle greenhouse gas emissions and cost. Energy Policy doi: 10.1016/j.enpol.2012.08.061 ### Comments Harvey isn't going to like this study, especially the portion on BEV,s. I don't understand the graph to the right of the article but it appears to be a well thought out study. You are right..I don't like it because it does not account for major near term changes in batteries performance (3X to 10X) and much lower price (-70% to -80%) and possible increase in liquid fuel price (up to$10/gal)

I paid $59.95 for a spare 750 mAh Li-On camera battery a few years ago and$4.95 last week for a similar but much more powerful 1200 mAh Li-On unit for my new digicam. That's about a 92% reduction in price for 60% more.

I guess that study is OK for a quick current day window look but rather meaningless for mid and longer term outlooks. It would be much better with an extended time line?

I forgot to add that our electricity is from clean 100% Hydro/Wind at a uniformed 24/7 very low $0.06/kWh domestic rate. Public charging stations could get the same clean electricity at a uniformed 24/7 Industrial rate of$0.03/kWh and even less during off peak periods. However, our gasoline price is already between $5.15 to 5.75/gal. It normally goes up about$0.45/gal on weekends.

That would change their (current day) tables considerably.

To fill-up a large ICEV car/Pick-up can cost over $100, specially on weekends. Another useful info. Our sole nationalized Hydro, even with those very low uniformed rates, makes$3.5B net profit every year. About $1.5B/year is re-invested in new hydro/wind plants and about$2.0B/year is given to the local government (the people @ $950+ per family) as dividends. The Hydro dividends would be enough to give FREE e-energy for two BEVs per family? That would change the end results of the above study? Nothing really new here - BEVs on average grid electricity are not significantly different than HEVs or PHEVs from a cost or GHG perspective. But: 1. GHG emissions of current power plants is not sustainable and need to be drastically reduced regardless. 2. The "long tailpipe" of plug-ins greatly reduces pollution concentrations near population centers. 3. Oil is not getting any cheaper and it's not physically possible to get anywhere close to self-sufficient in the USA despite what drill-baby-drill proponents suggest. 4. BEVs are a joy to drive without the vibration, noise of an ICE. Efforts should be made to estimate the benefits of these items as well. It's the way the US generates power, not the cars which are the problem. France with 75% plus of its electricity from nuclear can run its BEVs just fine, with massive carbon savings. @HarveyD When will we get those batteries? 2113? Vaporware is vaporware and wackie tobacco is enlightening Any new battery technology will be a least a decade if not longer to get into a car. Too many lawyers. No big car company is going put anything into all of our stupid hands until they have tested it out every possible stupid thing we do like, what happens if i drive my electric car in flooded new york, stupid or not the lawyers will sue Plus it is not in the car companies benefit to pursue a BEV when they get more profit from HEV and HEV SUV, these are there new SUV's and reality is reality and that means lithium will come down until it becomes an in demand and it wll go back up in price as all limited minerals do Peter XXX....decent EV batteries should start to hit the market place around 2020/2022? Meanwhile, HEVs may be the best interim solution. Toyota's HEVs are probably the best. Nano-technologies will reduce the amount of lithium and/or similar material required. Much smaller, lower cost, more efficient FCs may become a worthwhile alternative? Peter...improved batteries could hit the market place before 2020? I forgot to add that our electricity is from clean 100% Hydro/Wind Except for Gentilly, you mean. The Hydro dividends would be enough to give FREE e-energy for two BEVs per family? Harvey seems to think that the whole world is like eastern Canada. I've tried breaking the bad news to him, but he goes blithely on. This article highlights yet another study that show that plain hybrid vehicles emit less CO2/mile than partial or full electric vehicles charged on grid mix average CO2/kWh. The key takeaways from the study are: BEVs enter the GHG-optimal fleet only for short-range vehicles and only in cases with grid decarbonization. Even when range requirements are dramatically reduced, requiring BEV range adequate for only the average trip rather than the 95th percentile trip, a fleet of entirely BEVs is much more expensive and GHG-intensive than the other vehicle types, and BEVs are not GHG-minimizers for the full fleet even when charged with zero-emissions electricity. But HarveyD keeps soldiering on about his electric car dreams, even though electric cars are actually WORSE for all of us than designing and using best-of-breed hybrid vehicles. And NO, this is not a matter of improving battery technology. Such misconceptions are really unfortunate, because politicians are very susceptible to magical thinking, and are prone to instituting laws that look good to the uninformed, but actually make things worse for all of us. Should I say it again? Ok, I will: The best solution to minimize GHG emissions AND fossile fuel consumption is diesel-hybrid cars. They can easily get 70mpg (and higher as technology and design improve) for a Prius-sized vehicle and are wastly superior to an electric car fleet running on any kind of realistic grid mix. I originally drank the koolaid I dreamed of solar panels on my roof charging my BEV Nobody is going to get rich from that so it isnt going to happen I have come to believe that a 200 mile range EV under$30,000 could change things a bit.

Why would anyone bet that EV batteries will not improve?

We can argue over whether it will take 2, 5 or 10 years before a much better/more affordable battery comes to market, but never?

Why would anyone bet that the "worldwide" grid will not get cleaner?

We can argue over whether nuclear will play a larger or smaller role or over how fast wind, solar, geothermal, hydro, and tidal will get installed, but a continued dirty grid?

We can even argue whether there's a role for hydrogen in our transportation system.

But to assume that today's reality is tomorrow's reality? I just don't see how that makes sense.

Right now PHEVs might make the most sense due to mature ICE technology and the current state of battery technology, but batteries are almost certain to improve and drop in price. As SJC says, a 200 mile EV for under $30 that was capable of at least 90% recharge in <20 minutes will shake things up. Anyone remember electric typewriters than had built in memory so that you could type a draft, make corrections, and then print out a finished copy? Those were the pre-desktop computer "PHEVs" of word processing. Bob Wallace, Your arguments for battery electric vehicles (BEV) and PHEV are all strawmen, wrong, or BOTH. It would be great if in 10 years we have a bunch of new nuclear plants online that can charge a fleet of BECs, but that does NOT mean we should purchase BEVs *today*. In fact, we are better off waiting and let plain hybrid and especially diesel hybrid vehicles provide the lowest resource usage and CO2/mile of all the current alternatives. Believing in the electron fairy or the hydrogen fairy does not make for lower CO2/mile. There is noisurplus of clean electrons available, and there is no cvleah hydrogen available AT ALL. It is just a waste of dear resources to buy PHEV or BEV today, and they pollute more than HEV and D-HEV (my new term for Diesel-Hybrid Electric Vehicle. (repost because I mistyped several words) Bob Wallace, Your arguments for battery electric vehicles (BEV) and PHEV are all strawmen, wrong, or BOTH. It would be great if in 10 years we have a bunch of new nuclear plants online that can charge a fleet of BEVs, but that does NOT mean we should purchase BEVs *today*. In fact, we are better off waiting and let plain hybrid and especially diesel hybrid vehicles provide the lowest resource usage and CO2/mile of all the current alternatives. Believing in the electron fairy or the hydrogen fairy does not make for lower CO2/mile. There is no surplus of clean electrons available, and there is no clean hydrogen available AT ALL. It is just a waste of dear resources to buy PHEV or BEV today, and they pollute more than HEV and D-HEV (my new term for Diesel-Hybrid Electric Vehicle. Topical joke for those who follow Paul Krugman in the NY Times Sometimes I feel like I am Paul Krugman, having to slay the electron zombies and the hydrogen zombies that just refuse to die, no matter how many time you refute them. You can say it twice but it still doesn't make you correct. Right now our grid is roughly 1/3rd coal, 1/3rd natural gas and 1/3rd "close to zero CO2". If a EV charged with 100% coal-electricity is only slightly more CO2 producing than a ICEV then right now, today, EVs are less CO2 producing than ICEVs. A 1/3rd, 1/3rd, 1/3rd grid is roughly half as CO2 producing as a 100% coal grid. If EV range works for you, why spend the extra money for an ICE? A Leaf EV is a lot cheaper than a Volt or Plug-in Prius. You're spending$4k to $11k for the ability to drive more than the EV range when you may have no need. And here's the sweet thing. More EVs (and PHEVs) charging at night boosts the off-peak electricity market and makes wind more profitable. That additional profit will bring more investment for wind farms. More wind farms mean more cheap wind power during peak hours and lower electricity bills for all. Someone else will have to argue the hydrogen issue with you. I can't see how hydrogen is going to play any significant role in our transportation future. Neither do I think nuclear energy has a future worth contemplating. Even existing nuclear reactors are struggling. Just some more precise numbers, first half of 2012: Coal 36% Natural gas 31% Almost zero CO2 32% Nuclear 19% Hydro 7.6% Non-hydro renewables 5.4% It would be great if in 10 years we have a bunch of new nuclear plants online that can charge a fleet of BEVs, but that does NOT mean we should purchase BEVs *today*. There are other reasons to buy PHEV/BEVs today: 1. Energy security: petroleum imports can be interrupted by boycotts or military action, but not domestic fuels. 2. The "long tailpipe": removing emissions from populated areas. 3. Pushing battery technology and building manufacturing capacity so it's there when we need it (preferably domestic, not another industry taken over by Asia). 4. Get integration with the Smart Grid going. Wind power is growing a lot faster than plug-in vehicle demand, and Smart Grid chargers can manage the variability of wind power. This isn't something where it makes any sense to sit on our hands; if the PHEV-20 is already at the sweet spot, that's the thing to build today. Bob Wallace, It would be great if you stopped misrepresenting the facts. Have you read the CMU study? Did you study the graphs? Did you understand the conclusions? Here is your first bug whopper of a lie: You say: If a EV charged with 100% coal-electricity is only slightly more CO2 producing than a ICEV then right now, today, EVs are less CO2 producing than ICEVs. An EV charged with 100% coal-electricity is nowhere near the the same CO2/mile as a same-size HEV. It is more than 2x worse. You say: A 1/3rd, 1/3rd, 1/3rd grid is roughly half as CO2 producing as a 100% coal grid. Yeah, sure, but you made a >2X mistake in the above, so now we are back to >1x when you divide by 2. Hence the EV on grid mix is still WORSE in terms of CO2/mile than the HEV. The numbers do not lie. Read the study again. These are real scientists from a real university. They are not fooling around. Also read again my comparison of a Nissan Leaf on grid mix versus a Prius on gasoline, The Prius has lower CO2/mile, and the advantage of the HEV will keep increasing faster than anyone can improve the CO2/kWh for the grid mix. If you look at graph "b" you will see that the tCO2e/vehicle/year for fuel/electricity is lower for the EV than the CV. I do not see where the study describes what they are using as "grid". The article is behind a pay wall. They attribute the extra CO2 for EVs to battery manufacturing. I'd like to see how they develop that number. Bob Wallace says: If you look at graph "b" you will see that the tCO2e/vehicle/year for fuel/electricity is lower for the EV than the CV. You need to get your vision checked, or something. Graph 1(b) shows BEV at 7.8 and CV at 6.5, so with 6.5 < 7.8 even a crummy CV (conventional vehicle) is better than EV when it comes to CO2 emissions !! But in any case, nobody in their right mind would compare CV (conventional vehicle) with an EV. The correct comparison is HEV versus BEV, and the numbers are 3.2 < 7.8 !! In other words, a plain hybrid emits less than half the CO2 of a BEV, when including all sources of emission (manufacture, driving, etc). Is this not plain for anyone to see? I cannot believe anyone can refer to the graph and then claim the exact opposite of what the graph shows. The mind boggles. Do you have any reply to the certainty that the battery industry (both traction and other) will be wholly owned by China if the rest of the world doesn't start building them and maintaining its own base of experience in the mean time? Do you place zero importance on economic security? The Chinese battery company BYD went into the auto business. This is not an accident. Try reading what I wrote and not what you imagine I wrote. "tCO2e/vehicle/year for fuel/electricity is lower" Fuel/electricity. Not overall. Furthermore, I'm free to compare CVs with BEVs if I wish. There's something very strange about their data. If you look at "a" they claim close to$10,000 per year costs for an EV. Over a 12 year span that's close to \$120,000. That's a screwy claim.

Over a 12 year period the cost of owning and operating a Leaf is somewhat higher than a Prius if one does not take advantage of the federal subsidy. About the amount of the subsidy.

They are claiming that battery manufacturing releases
~1.5 tons x 12 years CO2 but manufacturing a ICE releases ~0.1 tons x 12 years. Can anyone make sense of that? Are EV batteries slow cured over smoldering coal fires?

The comments to this entry are closed.


text:
Browse Questions

# Find the absolute maximum and absolute minimum values of $f$ on the given interval: $\;f(x)=\sqrt{9-x^{2}} ,[-1 , 2]$

Note: This is part 4th of a 7 part question, split as 7 separate questions here.

Toolbox:
• To find the absolute maximum and minimum values of a continuous function f on a closed interval $[a,b]$
• (i) Find the values of the critical numbers of f in $(a,b)$
• (ii) Find the value of $f(a)$ and $f(b)$
• (iii) The largest of the values from (i) and (ii) is the absolute maximun value, the smallest of these values is the absolute minimum value.
$f(x)=\sqrt {9-x^2}$ is continous in $[-1,2]$
Step 1:
$f'(x)= \large\frac{-2x}{2 \sqrt {9-x^2}}=\large\frac{-x}{\sqrt {9-x^2}}$ is defined in $[-1,2]$


text:
# Mathematics Questions and Answers – Three Dimensional Geometry – Angle between Two Planes – 1

«
»

This set of Mathematics Multiple Choice Questions & Answers (MCQs) focuses on “Three Dimensional Geometry – Angle between Two Planes – 1”.

1. Which of the following is the correct formula for the angle between two planes?
a) cosθ=$$\left |\frac{\vec{n_1}.\vec{n_2}}{|\vec{n_1}||\vec{n_2}|}\right |$$
b) sinθ=$$\left |\frac{\vec{n_1}.\vec{n_2}}{|\vec{n_1}||\vec{n_2}|}\right |$$
c) cosθ=$$\left |\frac{\vec{n_1}+\vec{n_2}}{|\vec{n_1}||\vec{n_2}|}\right |$$
d) sinθ=$$\left |\frac{\vec{n_1}+\vec{n_2}}{(|\vec{n_1}|+|\vec{n_2}|}\right |$$

Explanation: If two planes of the form $$\vec{r}.\vec{n_1}=d_1$$ and $$\vec{r}.\vec{n_2}=d_2$$ where $$\vec{n_1} \,and \,\vec{n_2}$$ are the normals to the plane, then the angle between them is given by
cosθ=$$\left |\frac{\vec{n_1}.\vec{n_2}}{|\vec{n_1}||\vec{n_2}|}\right |$$

2. Which of the following is the correct formula for the angle between two planes $$A_1 x+B_1 y+C_1 z+D_1$$=0 and $$A_2 x+B_2 y+C_2 z+D_2$$=0?
a) cosθ=$$\frac{A_1 B_1 C_1}{A_2 B_2 C_2}$$
b) cosθ=$$\left |\frac{A_1 A_2+B_1 B_2+C_1 C_2}{\sqrt{A_1^2+B_1^2+C_1^2}\sqrt{A_2^2+B_2^2+C_2^2}}\right |$$
c) sinθ=$$\left |\frac{A_1 A_2-B_1 B_2-C_1 C_2}{\sqrt{A_1^2+B_1^2+C_1^2}\sqrt{A_2^2+B_2^2+C_2^2}}\right |$$
d) cosθ=$$A_1 A_2+B_1 B_2+C_1 C_2$$

Explanation: If the planes are in the Cartesian form i.e. $$A_1 x+B_1 y+C_1 z+D_1$$=0 and $$A_2 x+B_2 y+C_2 z+D_2$$=0, where $$A_1,B_1,C_1 \,and \,A_2,B_2,C_2$$ are the direction ratios of the planes, then the angle between them is given by
cosθ=$$\left |\frac{A_1 A_2+B_1 B_2+C_1 C_2}{\sqrt{A_1^2+B_1^2+C_1^2} \sqrt{A_2^2+B_2^2+C_2^2}}\right |$$

3. Find the angle between two planes $$\vec{r}.(2\hat{i}-\hat{j}+\hat{k})=3$$ and $$\vec{r}.(3\hat{i}+2\hat{j}-3\hat{k})$$=5.
a) $$cos^{-1}\frac{1}{\sqrt{22}}$$
b) $$cos^{-1}\frac{1}{\sqrt{6}}$$
c) $$cos^{-1}\frac{1}{\sqrt{132}}$$
d) $$cos^{-1}\frac{1}{\sqrt{13}}$$

Explanation: Given that, the normal to the planes are $$\vec{n_1}=2\hat{i}-\hat{j}+\hat{k}$$ and $$\vec{n_2}=3\hat{i}+2\hat{j}-3\hat{k}$$
cosθ=$$\left |\frac{\vec{n_1}.\vec{n_2}}{|\vec{n_1}||\vec{n_2}|}\right |$$
$$|\vec{n_1}|=\sqrt{2^2+(-1)^2+1^2}=\sqrt{6}$$
$$|\vec{n_2}|=\sqrt{3^2+2^2+(-3)^2}=\sqrt{22}$$
$$\vec{n_1}.\vec{n_2}$$=2(3)-1(2)+1(-3)=6-2-3=1
cosθ=$$\frac{1}{\sqrt{22}.\sqrt{6}}=\frac{1}{\sqrt{132}}$$
∴θ=$$cos^{-1}\frac{1}{\sqrt{132}}$$.

4. If the planes $$A_1 x+B_1 y+C_1 z+D_1$$=0 and $$A_2 x+B_2 y+C_2 z+D_2$$=0 are at right angles to each other, then which of the following is true?
a) $$\frac{A_1+B_1+C_1}{A_2+B_2+C_2}$$=0
b) $$A_1+A_2+B_1 +B_2+C_1+C_2$$=0
c) $$A_1+B_1+C_1=A_2 B_2 C_2$$
d) $$A_1 A_2+B_1 B_2+C_1 C_2$$=0

Explanation: We know that the angle between two planes is given by
cosθ=$$\left |\frac{A_1 A_2+B_1 B_2+C_1 C_2}{\sqrt{A_1^2+B_1^2+C_1^2} \sqrt{A_2^2+B_2^2+C_2^2}}\right |$$
Given that, θ=90°
∴cos90°=$$\left |\frac{A_1 A_2+B_1 B_2+C_1 C_2}{\sqrt{A_1^2+B_1^2+C_1^2}\sqrt{A_2^2+B_2^2+C_2^2}}\right |$$
0=$$\left |\frac{A_1 A_2+B_1 B_2+C_1 C_2}{\sqrt{A_1^2+B_1^2+C_1^2}\sqrt{A_2^2+B_2^2+C_2^2}}\right |$$
⇒$$A_1 A_2+B_1 B_2+C_1 C_2$$=0.

5. Find the angle between the planes 6x-3y+7z=8 and 2x+3y-2z=5?
a) $$cos^{-1}\frac{11}{\sqrt{98}}$$
b) $$cos^{-1}\frac{11}{\sqrt{1598}}$$
c) $$cos^{-1}\frac{13}{\sqrt{198}}$$
d) $$cos^{-1}\frac{11}{1598}$$

Explanation: We know that, the angle between two planes of the form $$A_1 x+B_1 y+C_1 z+D_1$$=0 and $$A_2 x+B_2 y+C_2 z+D_2$$=0 is given by
cosθ=$$\left |\frac{A_1 A_2+B_1 B_2+C_1 C_2}{\sqrt{A_1^2+B_1^2+C_1^2} \sqrt{A_2^2+B_2^2+C_2^2}}\right |$$
Given that, $$A_1=6,B_1=-3,C_1=7$$ and $$A_2=2,B_2=3,C_2=-2$$
cosθ=$$\left |\frac{6(2)-3(3)+7(-2)}{|\sqrt{6^2+(-3)^2+7^2} \sqrt{2^2+3^2+(-2)^2}|}\right |$$
cosθ=$$|\frac{-11}{\sqrt{94}.\sqrt{17}}|$$
θ=$$cos^{-1}\frac{11}{\sqrt{1598}}$$.
Participate in Mathematics - Class 12 Certification Contest of the Month Now!

6. If two vectors $$\vec{r}.\vec{n_1}=d_1$$ and $$\vec{r}.\vec{n_2}=d_2$$ are such that $$\vec{n_1}.\vec{n_2}$$=0, then which of the following is true?
a) The planes are perpendicular to each other
b) The planes are parallel to each other
c) Depends on the value of the vector
d) The planes are at an angle greater than 90°

Explanation: We know that if the scalar or dot product of vectors is 0, then they are at right angles to each other. Here the dot product of the normal vectors of the plane is 0, i.e. $$\vec{n_1}.\vec{n_2}$$=0. Hence, the planes will be perpendicular to each other.

7. Find the angle between the two planes 2x+2y+z=2 and x-y+z=1?
a) $$cos^{-1}\frac{1}{3}$$
b) $$cos^{-1}\sqrt{3}$$
c) $$cos^{-1}\frac{1}{3}$$
d) $$cos^{-1}\frac{1}{3\sqrt{3}}$$

Explanation: The angle between two planes of the form $$A_1 x+B_1 y+C_1 z+D_1$$=0 and $$A_2 x+B_2 y+C_2 z+D_2$$=0 is given by
cosθ=$$\left |\frac{A_1 A_2+B_1 B_2+C_1 C_2}{\sqrt{A_1^2+B_1^2+C_1^2} \sqrt{A_2^2+B_2^2+C_2^2}}\right |$$
According to the given question, $$A_1=2,B_1=2,C_1=1 \,and \,A_2=1,B_2=-1,C_2=1$$
cosθ=$$\left |\frac{2(1)+2(-1)+1(1)}{|\sqrt{2^2+2^2+1^2} \sqrt{1^2+(-1)^2+1^2}|}\right |$$
cosθ=$$\left |\frac{1}{\sqrt{9}.\sqrt{3}}\right |$$
θ=$$cos^{-1}\frac{1}{3\sqrt{3}}$$.

8. Which of the given set of planes are perpendicular to each other?
a) $$\vec{r}.(2\hat{i}+2\hat{j}+\hat{k})$$=5 and $$\vec{r}.(\hat{i}+2\hat{j}+2\hat{k})$$=5
b) $$\vec{r}.(\hat{i}-2\hat{j}+\hat{k})$$=7 and $$\vec{r}.(\hat{i}+\hat{j}+2\hat{k})$$=2
c) $$\vec{r}.(2\hat{i}-2\hat{j}+\hat{k})$$=4 and $$\vec{r}.(\hat{i}+2\hat{j}+2\hat{k})$$=5
d) $$\vec{r}.(3\hat{i}-2\hat{j}+\hat{k})$$=2 and $$\vec{r}.(\hat{i}+2\hat{j}+8\hat{k})$$=8

Explanation: Consider the set of planes $$\vec{r}.(2\hat{i}-2\hat{j}+\hat{k})=4 \,and \,\vec{r}.(\hat{i}+2\hat{j}+2\hat{k})$$=5
For the set of planes to be perpendicular $$\vec{n_1.}\vec{n_2}$$=0
In the above set of planes, $$\vec{n_1}=2\hat{i}-2\hat{j}+\hat{k}$$ and $$\vec{n_2}=\hat{i}+2\hat{j}+2\hat{k}$$
∴$$\vec{n_1}.\vec{n_2}$$=2(1)-2(2)+1(2)=0
Hence, they are perpendicular.

9. Find the angle between the planes $$\vec{r}.(4\hat{i}+\hat{j}-2\hat{k})$$=6 and $$\vec{r}.(5\hat{i}-6\hat{j}+\hat{k})$$=7?
a) $$cos^{-1}\frac{12}{\sqrt{1302}}$$
b) $$cos^{-1}\frac{1}{\sqrt{1392}}$$
c) $$cos^{-1}\frac{23}{\sqrt{102}}$$
d) $$cos^{-1}\frac{15}{\sqrt{134}}$$

Explanation: Given that, the normal to the planes are $$\vec{n_1}=4\hat{i}+\hat{j}-2\hat{k} \,and \,\vec{n_2}=5\hat{i}-6\hat{j}+\hat{k}$$
The angle between two planes of the form $$\vec{r}.\vec{n_1}=d_1 \,and \,\vec{r}.\vec{n_2}=d_2$$ is given by
cosθ=$$\left |\frac{\vec{n_1}.\vec{n_2}}{|\vec{n_1}||\vec{n_2}|}\right |$$
$$|\vec{n_1}|=\sqrt{4^2+1^2+(-2)^2}=\sqrt{21}$$
$$|\vec{n_2}|=\sqrt{5^2+(-6)^2+1^2}=\sqrt{62}$$
$$\vec{n_1}.\vec{n_2}$$=4(5)+1(-6)-2(1)=20-6-2=12
cosθ=$$\frac{12}{\sqrt{21}.\sqrt{62}}=\frac{12}{\sqrt{1302}}$$ ∴θ=$$cos^{-1}\frac{12}{\sqrt{1302}}$$.

10. Which of the following sets of planes are parallel to each other?
a) 2x+3y+4z=8 and 3x+9y+12z=7
b) 2x+3y+4z=2 and 4x+6y+8z=9
c) 3x+2y+4z=0 and 3x+4y+2z=0
d) 2x+4y+8z=9 and 4x+2y+7z=0

Explanation: If two planes are of the form $$A_1 x+B_1 y+C_1 z+D_1$$=0 and $$A_2 x+B_2 y+C_2 z+D_2$$=0, then they will be parallel if $$\frac{A_1}{A_2}=\frac{B_1}{B_2}=\frac{C_1}{C_2}$$
Consider the planes 2x+3y+4z=2 and 4x+6y+8z=9
$$\frac{A_1}{A_2}=\frac{2}{4}$$
$$\frac{B_1}{B_2}=\frac{3}{6}$$
$$\frac{C_1}{C_2}=\frac{4}{8}$$
$$\frac{2}{4}=\frac{3}{6}=\frac{4}{8}=\frac{1}{2}$$
Hence, the above set of planes are parallel.

Sanfoundry Global Education & Learning Series – Mathematics – Class 12.

To practice all areas of Mathematics, here is complete set of 1000+ Multiple Choice Questions and Answers.


text:
# A Young Quant’s Illustrated Primer: the SABR Model

## What is the SABR model?

The SABR model (pronounced as in light saber) is a popular stochastic volatility model. A somewhat more general version than the one we discuss here was introduced by P. Hagan and co-authors in 2002, with applications to interest rate-related derivatives. The model became popular because a (quite difficult) small-time analysis resulted in some easy-to-apply, approximate, “smile” formulas that could fit the authors’ target markets.

While equity applications are rare, the model makes mathematical sense for equities, where the stochastic process pair is \{S_t,\sigma_t\}. Here S_t \ge 0 is a stock price and \sigma_t > 0 is the associated stochastic volatility. Using that equity language, the SABR model, as an SDE (stochastic differential equation) system, reads

In equation (1), \beta is any real number, and \nu > 0 is the “volatility of volatility”. The latter can be set equal to one without loss of generality, so we’ll do so for the remainder of this post. Although any real \beta makes mathematical sense, fits to market data will almost always have -\infty < \beta \le 1, so we’ll stick to that range also. Also \rho \in (-1,1) is a correlation parameter. Finally, \tau_0 is the first time, if any, that the stock price hits 0.

## Boundary conditions

The stock price can hit 0 with a positive probability whenever \beta is strictly less than 1. What happens after that requires a boundary condition. Of course, for equities, to avoid an arbitrage opportunity, once a stock price hits zero, it must stay there (unless the company is recapitalized). So, an additional specification of the equity version of the model is that S_t = 0 for all t \ge \tau_0.

For the original fixed income applications, the underlying security is a (forward) interest rate and other boundary conditions are possible.

## Exact solutions

By a “solution” to equation (1), I mean a solution for the associated transition probability density. Once you have that, option values, for example, are found by an integration. Exact solutions are known for the following special cases:

• Normal SABR model: \beta = 0 with \rho = 0.
• Lognormal SABR model: \beta = 1 with any \rho.

## McKean’s formula

The Normal SABR model solution is found by adapting a famous formula for the transition density found by Henry P. McKean in 1970. For that case, it makes more sense to switch gears slightly in the notation and use X_t instead of S_t, now X_t the location of a “particle” which can hit 0 and keep on going to arbitrarily negative values. With that new notation, the Normal SABR model process is:

Now B_t and W_t are two independent Brownian motions. Equation (2) also describes a generalized type of Brownian motion, namely Brownian motion on a two-dimensional hyperbolic space.

Suppose we start the particle at X_0 = x_0 = 0 and \sigma_0 = 1. Then, the probability of finding the particle at subsequent times is given by Mckean’s formula for G(t,X,\sigma|x_0,\sigma_0). If we let t range from 0.01 to 1, we can plot McKean’s formula to findÂ the following evolution:

If you watch closely, you’ll see that the probability density starts with a spike near (X,\sigma) = (0,1) and then spreads out in the X direction. In the \sigma direction, mass begins to accumulate near \sigma \rightarrow 0. That’s due to lack of mean reversion in the volatility process and is an ‘unphysical’ aspect of the model.

Now that we’ve talked about it so much, you may be curious as to what this mysterious formula looks like. Here it is:

\displaystyle G(t,X,Y|x,y) = \frac{e^{-t/8}}{2 \, Y^2 \, (\pi t)^{3/2}} \int_d^{\infty} \frac{s \, e^{-s^2/(2 t)}}{\sqrt{\cosh s - \cosh d}} \, ds,

\displaystyle  where \quad d = \cosh^{-1}z, \quad using \quad z = 1 + \frac{(X-x)^2 + (Y-y)^2}{2 y Y}

To adapt the McKean solution for an equity model, you can use the well-known “method of images”. The method will ensure that the associated equity process S_t, which starts at S_0 > 0, is absorbed should it hit S=0.

(There’s much more to the model than we have touched upon, and we’ll certainly return to it in future blog posts. This post has been adapted from material in Ch. 8 (“A Closer Look at the SABR Model”) from “Option Valuation under Stochastic Volatility II”).

Animation code

# Started adding book codes from ‘Option Valuation under Stochastic Volatility II’

I have started adding book codes to this site. Almost all will be Mathematica notebooks (ending in .nb) , although there will be some C/C++. This may be a slow process since most files will require some clean-up before posting.

One file has been added so far: from Ch. 1: CIRJumpStart.nb, which was used to generate Fig. 1.6 in the book and below:

The figure shows a fit of the CIR bond process with a delayed jump-start (solid curve) to the US Treasury Zero Coupon yield curve (dashed curve), using Feb. 10, 2012 data. The delayed jump-start is/was a mechanism to model the end of the ZIRP (Zero Interest Rate Policy) regime. Details and related models are found in the book Chapter 1: “Slow Reflection, Jump-returns, and Short-term Interest Rates”.

The code file location is the Latest Book link (Just click on ‘Selected Codes’)

I will probably just work my way slowly through the book, but if you are a reader and particularly want to see certain book codes, please let me know by a comment or email. I will try to accommodate if possible.


text:
Predict a sequence given many sequences

I'm trying to find an algorithm that would fit this use case:

My data: a bunch of fixed-size integer arrays, e.g.

[0,2,3,4,5]
[1,2,3,1,5]
[4,1,2,4,5]
...

Input: an array of integers, and output: prediction for the rest of the array, e.g.

[0,1] -> [2,4,2]
[3] -> [1,3,2,5,5]
[3,2,4,1] -> [4]

• cool question! wish i had the answer... – Dave Kielpinski Apr 2 '20 at 1:51
• This is really too broad a question to answer without some knowledge of the domain (ie how the data was constructed). For example, is there significance to the position in the array - is the probability of 2 following an initial zero related to the possibility of 2 after a zero in second place? – Josh Friedlander Apr 2 '20 at 21:18
• If so, this sounds parallel to next word/ next n-gram models in NLP, except with a vocab of only 10 words. HMMs and LSTMs could work for this. – Josh Friedlander Apr 2 '20 at 21:20
• Are you using a rolling window as your input data? – Donald S Jun 14 '20 at 5:04
• If so, do you expect the same performance for each test example? – Donald S Jun 14 '20 at 5:05


text:
# A certain block weight 15 N in air. But it weight only 12 N when completely immersed in water…

Q: A certain block weight 15 N in air. But it weight only 12 N when completely immersed in water. When immersed completely in another liquid, it weight 13 N. Calculate the relative density of (i) the block and (ii) the liquid.

Sol:(i) $\large Relative\; density\; of\; body = \frac{W_{air}}{W_{air}-W_{water}}$

Where Wair = 15 N (weight of the body in air)

And Wwater = 12 N (weight of the body in water)

$\large RD_{block} = \frac{15}{15-12} = 5$

(ii) $\large RD_L = \frac{Loss \;of \;weight \;in \;liquid}{Loss \;of \;weight\; in \;water}$

$\large = \frac{15-13}{15-12} = \frac{2}{3}$


text:
2014
02-12

# find the nearest station

Since dandelion has left the hometown so long,she finds it’s difficult to find the station in the city.So she needs you ,a clear programmer, to help her.
Now you know the map of the city, which has showed every station in the city.You are asked to find the shortest distance between every grid and the stations.You should notice that the road in dandelion’s hometown is vertical or horizontal,so the distance of two girds is defined as |x1-x2|+|y1-y2|.

The input consists of several test cases. Each test case start with a line containing two number, n, m(1 <= n, m ≤ 182), the rows and the columns of city. Then n lines follow, each contain exact m characters, representing the type of block in it. (0 for empty place ,1 for station).The data will contains at least one station.

The input consists of several test cases. Each test case start with a line containing two number, n, m(1 <= n, m ≤ 182), the rows and the columns of city. Then n lines follow, each contain exact m characters, representing the type of block in it. (0 for empty place ,1 for station).The data will contains at least one station.

3 4
0001
0011
0110

3 2 1 0
2 1 0 0
1 0 0 1

/*

水题，BFS或者暴力都行。
竟然排第一额-、-II。暴力的方法就不说了，可以

2013-01-23
*/

#include"stdio.h"
#include"string.h"
#include"stdlib.h"
#include"queue"
#define N 333
using namespace std;

int n,m;
char map[N][N];
int dis[N][N],flag[N][N];
int dir[4][2]={1,0, -1,0, 0,1, 0,-1};

struct node
{
int x,y;
};
void solve()
{
queue<node>q;
node now,next;
memset(dis,0,sizeof(dis));
memset(flag,0,sizeof(flag));
int i,l;
for(i=0;i<n;i++)
{
scanf("%s",map[i]);
for(l=0;l<m;l++)
{
if(map[i][l]=='1')
{
now.x=i,now.y=l;
q.push(now);
flag[i][l]=1;
}
}
}
while(!q.empty())
{
now=q.front();
q.pop();
for(i=0;i<4;i++)
{
next.x=now.x+dir[i][0];
next.y=now.y+dir[i][1];
if(next.x<0 || next.x>=n || next.y<0 || next.y>=m)    continue;
if(flag[next.x][next.y])                            continue;
q.push(next);
flag[next.x][next.y]=1;
dis[next.x][next.y]=dis[now.x][now.y]+1;
}
}
}
int main()
{
int i,l;
while(scanf("%d%d",&n,&m)!=-1)
{
solve();
for(i=0;i<n;i++)
{
printf("%d",dis[i][0]);
for(l=1;l<m;l++)    printf(" %d",dis[i][l]);
printf("\n");
}
}
return 0;
}

1. 因为是要把从字符串s的start位到当前位在hash中重置，修改提交后能accept，但是不修改居然也能accept

2. 有一点问题。。后面动态规划的程序中
int dp[n+1][W+1];
会报错 提示表达式必须含有常量值。该怎么修改呢。。


text:
# Exergoeconomic valuation of a waste-based integrated combined cycle (WICC) for heat and power production

CASAS, Y.; GONZÁLEZ, P.; CONCHA, S.; ZAROR, C.; ARTEAGA, L.:
Energy, 114, 239-252 (2016).

DOI: 10.1016/j.energy.2016.07.165

### Abstract

Exergy and exergoeconomics assessment tools were applied to a municipal solid waste (MSW) gasification system integrated with a combined cycle (WICC). The techno-economic feasibility of the system was studied using a typical MSW produced in southern Chile and a combined experimental-modeling approach. Results suggested that gasifier should operate between 850 °C and 950 °C, in order to produce a syngas with the required energy content (LHV>3 MJ/kg) to feed power and steam cycles. Operation at lower temperatures and equivalence ratios, reduced the exergy efficiencies (<49%) due to the low carbon conversion, syngas composition and to the low heat recovery potential of exhausted gas. Exergy losses at gasifier amounted nearly 60% of the whole system due to the effect of chemical reactions and heat transfer in this stage. Above 850 °C, the total production cost (Ctot), exergy destruction cost (CD) and environmental cost (Cenv) were reduced, leading to a positive economic framework. The thermoeconomic cost of electricity (TCOE) using MSW as feedstock varied between 0.07 and 0.13 US$kWh-1, which is competitive with the electricity price in the Chilean market (0.12 US$ kWh-1).

If you have found a spelling error, please, notify us by selecting that text and pressing Ctrl+Enter.