text:
## Differential and Integral Equations

### On invariant sets and explicit solutions of nonlinear evolution equations with quadratic nonlinearities

#### Abstract

In this paper we introduce a technique to construct new explicit solutions of nonlinear partial differential equations of mathematical physics with quadratic nonlinearities. We show that under certain assumptions the nonlinear operators under consideration admit so-called invariant sets on a two-dimensional linear functional subspace that makes it possible to rewrite the corresponding equation as an overdetermined finite-dimensional dynamical system. Some classes of such systems are shown to have nontrivial solutions. Examples of linear invariant subspaces for nonlinear quadratic operators from parabolic and hyperbolic equations are also given. In this case, equations on the corresponding invariant subspaces are reduced to finite-dimensional dynamical systems. Several generalizations including a similar analysis on three-dimensional subspaces, $N$-dimensional operators and equations with cubic nonlinearities are discussed.

#### Article information

Source
Differential Integral Equations, Volume 8, Number 8 (1995), 1997-2024.

Dates
First available in Project Euclid: 20 May 2013


text:
# How do you plot the polar coordinate (2, (5pi)/6)?

Then teach the underlying concepts
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Alan P. Share
Nov 2, 2016

Polar coordinate $\left(2 , \frac{5 \pi}{6}\right)$ is a distance of $2$ from the origin
and at an angle of $\frac{5 \pi}{6}$

#### Explanation:

Note that the angle $\frac{5 \pi}{6}$ has a reference angle of $\frac{\pi}{6}$ above the $\pi$ axis.
$\frac{\pi}{6}$ is a common angle with hypotenuse $= 2$ and opposite side $= 1$

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Index: The Book of Statistical ProofsGeneral Theorems ▷ Probability theory ▷ Variance ▷ Scaling upon multiplication

Theorem: The variance scales upon multiplication with a constant:

$\label{eq:var-scal} \mathrm{Var}(aX) = a^2 \, \mathrm{Var}(X)$

Proof: The variance is defined in terms of the expected value as

$\label{eq:var} \mathrm{Var}(X) = \mathrm{E}\left[ (X-\mathrm{E}(X))^2 \right] \; .$

Using this and the linearity of the expected value, we can derive \eqref{eq:var-scal} as follows:

$\label{eq:var-scal-qed} \begin{split} \mathrm{Var}(aX) &\overset{\eqref{eq:var}}{=} \mathrm{E}\left[ ((aX)-\mathrm{E}(aX))^2 \right] \\ &= \mathrm{E}\left[ (aX - a\mathrm{E}(X))^2 \right] \\ &= \mathrm{E}\left[ (a [X - \mathrm{E}(X)])^2 \right] \\ &= \mathrm{E}\left[ a^2 (X - \mathrm{E}(X))^2 \right] \\ &= a^2 \, \mathrm{E}\left[ (X - \mathrm{E}(X))^2 \right] \\ &\overset{\eqref{eq:var}}{=} a^2 \, \mathrm{Var}(X) \; . \\ \end{split}$
Sources:

Metadata: ID: P127 | shortcut: var-scal | author: JoramSoch | date: 2020-07-07, 05:38.


text:
## Capital structure substitution theory

The CSS theory hypothesizes that managements of public companies manipulate capital structure such that earnings per share (EPS) are maximized. Managements have an incentive to do so because shareholders and analysts value EPS growth.

.. The capital structure substitution theory has the potential to close these gaps. It predicts a negative relation between leverage and valuation (=reverse of earnings yield) which in turn can be linked to profitability. But it also predicts that high valued small growth firms will avoid the use of debt as especially for these companies the cost of borrowing ({\displaystyle R_{\text{x,t}}}) is higher than for large companies, which in turn has a negative effect on their EPS. This is consistent with the finding that “…firms with higher current stock prices (relative to their past stock prices, book values or earnings) are more likely to issue equity rather than debt and repurchase debt rather than equity”.[4]

.. In times when the market is under-priced, corporate buyback programs will allow companies to drive up earnings-per-share, and generate extra demand in the stock market. In times when the index was under-priced relative to the model equilibrium, repurchase programs will be stopped and demand is reduced.

## Coronavirus May Kill Our Fracking Fever Dream

### America’s energy independence was an illusion created by cheap debt. All that’s left to tally is the damage.

Ever since the oil shocks of the 1970s, the idea of energy independence, which in its grandest incarnation meant freedom from the world’s oil-rich trouble spots, has been a dream for Democrats and Republicans alike. It once seemed utterly unattainable — until the advent of fracking, which unleashed a torrent of oil. By early 2019, America was the world’s largest producer of crude oil, surpassing both Saudi Arabia and Russia. And President Trump reveled in the rhetoric: We hadn’t merely achieved independence, his administration said, but rather energy dominance.”

Then came Covid-19, and, on March 8, the sudden and vicious end to the truce between Saudi Arabia and Russia, under which both countries limited production to prop up prices. On March 9, the price of oil plunged by almost a third, its steepest one-day drop in almost 30 years.

As a result, the stocks that make up the S.&P. 500 energy sector fell 20 percent, marking the sector’s largest drop on record. There were rumblings that shale companies would seek a federal lifeline. Whiting Petroleum, whose stock once traded for $150 a share, filed for bankruptcy. Tens of thousands of Texans are being laid off in the Permian Basin and other parts of the state, and the whole industry is bracing for worse. On the surface, it appears that two unforeseeable and random shocks are threatening our dream. In reality, the dream was always an illusion, and its collapse was already underway. That’s because oil fracking has never been financially viable. America’s energy independence was built on an industry that is the very definition of dependent — dependent on investors to keeping pouring billions upon billions in capital into money-losing companies to fund their drilling. Investors were willing to do this only as long as oil prices, which are not under America’s control, were high — and when they believed that one day, profits would materialize. Even before the coronavirus crisis, the spigot was drying up. Now, it has been shut off. The industry’s lack of profits wasn’t exactly a secret. In early 2015, the hedge fund manager David Einhorn announced at an investment conference that he had looked at the financial statements of 16 publicly traded shale producers and found that from 2006 to 2014, they spent$80 billion more than they received from selling oil. The basic reason is that the amount of oil coming out of a fracked well declines steeply after the first yearmore than 50 percent in year two. To keep growing, companies have to keep plowing billions back into the ground.

The industry’s boosters argue that technological gains, such as drilling ever bigger wells, and clustering wells more tightly together to reduce the cost of moving equipment, eventually would lead to a gusher of profits. Fracking, they said, was just manufacturing, in which process and human intelligence could reduce costs and conquer geology.

Actually, no. The key issue is the “parent child problem. When wells are clustered tightly together, with so-called child wells drilled around the parent, the wells interfere with one another, resulting in less oil, not more. (This may not surprise anyone who is attempting to be productive while working in close quarters with their children.)

The promised profits haven’t materialized. In the first half of 2019, when oil was around $55 a barrel, only a few top-tier companies were profitable. “By now, it should be abundantly clear that the current shale oil business model does not work — even for the very best companies in the industry,” the investment firm SailingStone Capital Partners explained in a recent note. Policymakers who wanted to tout energy independence disregarded all this, even as investors were starting to lose patience. As early as 2018, some investors had begun to tell companies that they wanted to see free cash flow, and that they were tired of compensation models that rewarded executives with rich paydays for increasing production, but failed to take profits into account. As a result, fracking stocks badly underperformed the market. But with super-low interest rates, investors in search of yield were still willing to buy debt. Over the past 10 years, the entire energy industry has issued over$400 billion in high-yield debt. “They subprimed the American energy ecosystem,” says a longtime energy market observer.

Even as the public equity and debt markets grew cautious, drilling continued. That’s because one big source of funding didn’t dry up: private equity. And why not? Private equity financiers typically get a 2 percent management fee on funds they can raise, so they are incentivized to take all the money that pension funds, desperate for returns to shore up their promises to retirees, have been willing to give them.

In the Haynesville and the Utica Shales, two major natural gas plays, over half of the drilling is being done by private equity-backed companies; in the oil-rich Permian Basin, it’s about a quarter of the drilling. From 2015 through 2019, private equity firms raised almost $80 billion in funds focused mostly on shale production, according to Barclays. Until the capital markets began to get suspicious, private equity investors could flip companies they had funded to larger, public companies, making a profitable exit regardless of whether or not the underlying business was making money. That, too, is ending, as investors in such funds have become disillusioned. You can see how all of this is playing out by looking at Occidental Petroleum. In 2019, Oxy, as it’s known, topped a competing bid from Chevron and paid$38 billion to take over Anadarko Petroleum, which is one of the major shale companies. Since that time, Oxy’s stock has plummeted almost 80 percent in part due to fears that the Anadarko acquisition is going to prove so wildly unprofitable that it sinks the company.

On March 10, the company announced that it would slash its dividend for the first time since the early 1990s, when Saddam Hussein’s invasion of Kuwait sent oil prices plummeting.

Occidental is just one piece of the puzzle. In April, the Energy Information Administration cut its forecast for U.S. oil production, estimating that it will fall both this year and next — suggesting that the days of huge growth in production from shale are over.

On March 10, Scott Sheffield, the chief executive of Pioneer Natural Resources, a major driller in the Permian Basin, told Bloomberg that U.S. oil output could fall by more than two million barrels per day by next year if prices remain where they are today.

“This is late ’80s bad,” a close observer of the industry says.

After the United States engaged in a high-stakes negotiation with Russia and Saudi Arabia to curtail production, a tentative deal was struck on Thursday. Certainly, President Trump, who has staked so much on the American shale industry, wants to save it. “We really need Trump to do something or he’s going to lose all the energy states in this election,” Mr. Sheffield told CNBC in late March.

A deal, and higher oil prices, might help the industry. But they won’t fix its fundamental problem with profitability. Energy independence was a fever dream, fed by cheap debt and frothy capital markets.

All that’s left to tally is the environmental and financial damage. In the five years ending in April, there were 215 bankruptcies for oil and gas companies, involving $130 billion in debt, according to the law firm Haynes and Boone. Moody’s, the rating agency, said that in the third quarter of 2019, 91 percent of defaulted U.S. corporate debt was due to oil and gas companies. And North American oil and gas drillers have almost$100 billion of debt that is set to mature in the next four years.

It’s still unclear where most of this debt is held. Some of it has been packaged into so-called collateralized loan obligations, pieces of which are held by hedge funds. Some of it may be on bank balance sheets. Investors in the equity of these companies have already seen the value of their holdings decimated. Pension funds that have poured money into private equity firms may take a hit soon, too. All we know for sure is that fracking company executives and private equity financiers have made a fortune by touting the myth of energy independence — and they won’t be the ones who have to pick up the pieces.

## There is No Accountability on Social Media for Posting and Retweeting Bad Information

10:10
something that’s not about politics last
10:13
night was just so had it I can’t take it
10:18
I don’t want it I have read that expense
10:20
office all of them and the short stories
10:22
I can’t take it don’t want it but it’s
10:24
not like that was the first time do you
10:25
guys remember 2018 do you remember
10:26
Kansas the night of that primary it was
10:29
ridiculous or in 2016 California like
10:32
how long it took for the results like oh
10:35
it’s just where’s the way you and and
10:38
one of the things that made it even
10:39
worse was like I was tweeting about how
10:42
frustrated I was that people were just
10:43
sharing like like whatever they they
10:46
heard they don’t check it they don’t
10:47
Google it they just share it something
10:49
pops into their head it sounds good like
10:52
you have an inclination you want to
10:53
believe that something is true and so
10:55
people are gonna like and they’re gonna
10:56
retweet what says that they’re there
10:58
right like there was so much
11:00
misinformation and just people not
11:02
caring enough about the material to
11:04
like like all the
11:06
11:07
like obviously there’s problems with the
11:08
app but like the people who were like
11:10
100% Budaj edge made the app
I got it
11:13
and and look we have the disclosures
11:16
that were coming out and and responsible
11:18
11:19
disclosures that they had paid that app
11:22
company okay now before you then take
11:26
that and say and thus he made the app
11:29
and presumably put in some sort of
11:31
backdoor where he could steal the
11:33
election
do just a little bit of
11:35
googling like that company makes other
11:37
apps political canvassing apps things
11:39
for volunteers things that campaigns
11:41
would use and so the fact that he paid
11:44
money to that company does not mean that
11:46
he invested in the construction of that
11:48
app
he could be buying one of the
11:50
political apps that they already have
11:52
and in fact it turns out that over the
11:54
past year I believe Klobuchar Biden and
11:56
Gillibrand all of them gave money to
11:59
this company presumably for one of the
12:01
political canvassing apps
that they have
12:03
sold for some time you know how I found
12:05
out that stuff by googling for like 90
12:08
seconds it was super simple but I
12:10
noticed that when I started tweeting out
12:12
those I guess inconvenient truths nobody
12:16
cared like the people who were like no
12:18
no no Buddha judge definitely made the
12:21
app he stole this thing a hundred times
12:24
as many of retweets a hundred times as
12:25
many likes
someone who’s like hey guys
12:27
you know pump the brakes first of all it
12:30
looks like Bernie’s Bernie’s still gonna
12:31
win second of all it looks like this
12:32
company actually has other apps nobody
12:35
12:37
interested in that is really depressing
12:39
you know because media is a big thing
12:42
it’s not one thing it’s a lot of
12:44
different voices in a lot of different
12:46
forms but the fact that the people who
12:48
are trying to be accurate and reasonable
12:50
are always going to have to compete with
12:52
people who are just their thing is I’m
12:55
going to see what people want to hear
12:57
what they desperately want to believe is
12:58
true
and I’m just gonna tell admit I’m
13:00
just gonna say that it’s true and nobody
13:02
ever tweets you know um you know I just
13:05
I want to apologize because I retweeted
13:08
something last night you know sort of in
13:10
the fervor of what was going on it turns
13:11
out that it was untrue I apologize if I
13:14
helped to misinform anyone
in the future
13:17
I’m going to be
13:19
but more careful
13:20
things and in particular I will not be
13:23
spreading the the people who were
13:26
spreading this misinformation I will be
13:27
more critical of them in particular
13:29
literally no one has ever said that you
13:31
spread things that are false nobody
13:33
cares nobody remembers even theirs the
13:36
incentives are all wrong on social media

13:39
and that is on top of all of the other
13:41
problems of last night that is so
13:43
incredibly frustrating I’m not having a
13:48
stroke down that’ll come later based on
13:51
diet have you seen the Saga comics I
13:54
have not just be reasonable like it’s
14:00
why I’m never gonna be a success
14:01
honestly there’s no there’s no reason
14:03
why would you why would you be
14:04
reasonable yeah nobody at nobody admits
14:06
that they’re wrong that’s where I don’t
14:08
know if you were watching the coverage
14:09
last night but I wanted to add on to
14:11
they were there every conversation they
14:13
were saying no one on TV in political
14:16
media ever pays any sort of price for
14:19
14:21
say you’re right but it ain’t just TV

14:24
nobody in any form of media ever pays
14:26
any sort of price for being wrong not if
14:28
they were wrong in a way that reassured
14:30
their followers their audience that
14:33
things were gonna be okay who who ever
14:36
gets mad at a person for a horrible
14:39
prediction a prediction that when it was
14:40
made was obviously not true nobody cares
14:42
about that they want I want you to be
14:45
wrong in a way that makes me feel warm
14:47
and cozy I want you to tell me the sort
14:49
of things that are a verbal light
14:51
stroking of my head to make me feel a
14:53
little bit better that’s that’s all
14:54
really anyway the reason I talk about
14:58
this now is we were on night one we have
15:01
got a bunch of these nights to go and
15:03
unfortunately I don’t I don’t think that
15:05
any are gonna be as bad as Iowa but
15:07
there’s gonna be a lot of bad nights and
15:08
that’s really unfortunate okay everybody
15:12
I would love to talk longer but we do
15:14
have a show coming up and on it I’m
15:16
gonna be joined by Jordan you’ll as I am
15:18
every other Tuesday you know me also
15:20
were to talk about last night but also
15:21
author podcaster and founder of Vox Ezra
15:25
Klein is also going to be in studio to
15:26
15:29
which seems especially relevant this
15:33
both of last night and with the State of
15:35
the Union so should be an awesome
15:37
conversation there we’ve got so much
15:39
15:41
and I read the first one I wasn’t a huge
15:43
fan maybe I need to read more of them
15:45
but anyway thank you as always for
15:47
joining me this a little live live video
15:48
in ten minutes the full show starts so
15:51
definitely tune in there I’ll see you on
15:53
the other side

## The Secret Sauce Behind Pop-Music Hits

### In their new book, ‘Switched on Pop,’ two podcast hosts discuss why songs by Rihanna, Drake and others break out

Why did Rihanna and Calvin Harris’s “We Found Love” top the charts for 10 weeks? Partly because of the song’s unconventional structure, say podcasters Nate Sloan and Charlie Harding. The two examine “We Found Love”—and more than a dozen other 21st-century hits—in their book, “Switched on Pop,” which comes out Friday.

The volume borrows its title from the podcast the two 33-year-old friends began five years ago about the subtle techniques shaping today’s Top 40 hits. Mr. Sloan is a University of Southern California musicologist and Mr. Harding is a songwriter and musician who plays the guitar, drums and keyboards, as well as other instruments. Their podcast and book are filled with sophisticated but accessible discussions of pop hits from the 2000s and 2010s, from the catchy hook of Ariana Grande’s “Break Free” to Drake’s use of simple rhymes in “God’s Plan.”

In addition to offering a theory-driven take on what the authors call “the bubbliest of bubblegum pop,” the “Switched on Pop” book provides a toolkit for appreciating popular music. The authors delve into musical building blocks like melody, rhyme and sampling.

Messrs. Sloan and Harding spoke with the Journal about how streaming and hip-hop are transforming pop music. An edited transcript:

Charlie Harding, left, and Nate Sloan turned their podcast about popular music into a book. PHOTO: ELLYN JAMESON
You already have a podcast. Why a book?

Mr. Harding: Listeners were saying, “I learned a lot from the show, but I want a comprehensive guide to how to listen more thoughtfully.” That knowledge sits in music-theory textbooks. So we saw an opportunity.

Mr. Sloan: When we’re doing the podcast, we’re responding quickly to what’s happening in the fast-changing world of popular music, which is a great exercise. Forcing yourself to listen to pop makes you confront your own calcifying biases. But the book allows us to think about the things we’ve learned and put them in historical context.

There’s surprisingly little discussion of actual music in the music media. Are you guys an alternative?

Mr. Sloan: People tend to focus on the lyrics of songs and the artist’s personal life. Adding music to that discussion gives even more depth and nuance to how you can think about pop’s role in our culture. There are probably people who think Taylor Swift could release anything and it would be a hit, because of her celebrity. We’d say no. These songs are popular for a reason. You may not like that reason, but there is a reason.

Older listeners often say music isn’t as good as it was. What’s your response?

Mr. Sloan: Pop music reflects the time in which it is made and may not be as relevant to you as you age. But at the same time, what’s been revealing about doing this podcast and book is finding out how pop is changing. One thing we track is how timbre [the quality that lets us distinguish different artists or instruments] and tone are becoming such important elements of pop. Things that might have been privileged in the past—complex harmonies, lyrical creativity, guitar solos—aren’t as privileged now. That can be disorienting, but it doesn’t mean music is bad. The hope is, when you do have to listen to, say, Skrillex, you can at least say, “I don’t like this, but I kind of understand why someone would.”

Mr. Harding: It’s OK to like the things you like, but I also feel strongly that it’s not OK to think other kinds of music are outright bad. By expanding your palette of hearing, you can be more inquisitive.

Are older fans looking for things that are no longer there and then ignoring the innovation? Music, over time, becomes like a foreign language.

Mr. Sloan: I agree. It’s hard to listen to something that doesn’t follow the musical criteria we’ve established. But new music is complex in different ways.

Mr. Harding: Part of Drake’s brilliance, for example, is he is able to say something that sounds utterly familiar—as if you’ve heard it a thousand times—and yet no one has quite put those words together in that way.

Mr. Sloan: It’s a gift, being able to write those catchy nursery rhymes. The way Drake uses rhyme is masterful. One thing he does in the song “God’s Plan” is give you the same phrase, but at different points in the rhythm—keeping it fresh.

The book discusses big hits. Any recent ones you don’t like?

Mr. Sloan: A song we find vexing is “I’m the One,” by DJ Khaled. This is a song we will not redeem. We think it’s derivative, offensive and boring. And yet it was a No. 1 hit. You still have an opportunity to ask why.

Let’s talk Rihanna.

Mr. Harding: Rihanna’s “We Found Love” with Calvin Harris came out in 2011, and while it might seem like any other pop song, it’s upsetting a long-held tradition. Since the 1950s, one thing that’s stayed relatively stable is song form—the verse-chorus form of songs. What this song does is take the structure of electronic-dance-music songs, which is different, and superimpose it on pop.

Mr. Sloan: And now it’s hip-hop that is pushing the verse-chorus form to its breaking point. Take “Sicko Mode” by Travis Scott. It doesn’t have a verse or a chorus. It’s this stitched-together collage of song fragments that seems more at home in the work of an experimental composer.

Will music streaming lead to shorter, simpler songs?

Mr. Harding: The economics of streaming incentivizes songs to be shorter because you get paid per song. We’re also seeing more choruses at the start of songs to hook you in. Listeners have more choices than in the past. It’s forcing producers and songwriters to think intensely about never having a dull moment. But I think there’s always been anxiety about technological change. Songwriters have long paid attention to the medium, from the limits of the vinyl record to the CD.

Mr. Sloan: These developments always circle back to something much older. It’s easy to feel a certain panic about streaming. But “Old Town Road” is still 15 seconds longer than the shortest No. 1 hit ever, in 1960. What’s past is prologue.


text:
2. Finance
3. the probability density function of the normal distribution is given...

# Question: the probability density function of the normal distribution is given...

###### Question details

The probability density function of the normal distribution is given by

where µ is the mean and σ 2 is the variance of the distribution.

(a) [20 marks] Assuming that µ = 0, derive the maximum likelihood estimate of ${\sigma }^{2}$ given the sample of i.i.d data

(b) [20 marks] Now assume that ${x}_{t}$ is conditionally normally distributed as N(0, ${\sigma }_{t}^{2}$), where

Write down the likelihood function for this model given a sample of data

(c) [15 marks] Describe how we can obtain estimates for {ω, α, β} for the GARCH(1,1) model and discuss estimation difficulties.

(d) [20 marks] Describe in your own words what graphical method and formal tests you can use to detect volatility clustering.

(e) [25 marks] Describe the RiskMetrics exponential smoother model for multivariate volatility, and discuss the pros and cons of the constant conditional correlation model of Bollerslev (1990) versus the RiskMetrics approach.


text:
# What is an atom that has lost an electron?

$\text{A cation}$
An atom that has lost an electron is a positively charged ion; i.e. a $\text{cation}$.
${\text{Atom" + Delta rarr "Atom}}^{+} + {e}^{-}$


text:
# How can I make a shorter binary rule in haskell?

Here is a code I experimented with recently:

let br = [0]:[n:(concat $take n br) | n <- [1..]] in concat br  This code produces the binary rule 0 1 0 2 0 1 0 3 ... Here is the idea of how it works: • create a list starting with [0] • each step, concatenate all the current elements in the list, put the next number at the begining and add at the end. [[0] ...] [[0], [1, 0] ...] [[0], [1, 0], [2, 0, 1, 0]...] [[0], [1, 0], [2, 0, 1, 0], [3, 0, 1, 0, 2, 0, 1, 0] ...] ...  My question now is: is there a shorter/smarter one-liner to get this infinite lazy binary-rule ? ## Edit: I found another way to do the same thing: let f n = if n==0 then [0] else n:([0..n-1] >>= f) in [0..]>>= f  The extra question would be: wich one of the 2 version is more readable for an haskell programmer, and why ? • Re Edit 2: The python version should be yield 0 \n for b...: yield b+1 \n yield 0 May 27 at 16:24 • Re Edit 2: The haskell version: I confirm that it hangs, and I'm trying to figure out why. This works: import Data.List (intersperse) \n binaryRule = 0 : intersperse 0 ((+1) <$> binaryRule) May 27 at 16:42
• Ok, I figured out the problem, but I think we've pissed off the mods enough with troubleshooting on Code Review, and I want my fake internet points in the right place anyway :) . Open a thread on Stack Overflow and share the link here! May 27 at 16:47
• To be clear chameleon questions - questions where what is being asked changes over time - are not allowed on Code Review. If you'd like advice on working code (which your Edit 2 is not) you can ask a follow-up question. Alternately you could take @ShapeOfMatter up on the offer to ask on Stack Overflow. Please do not undo my rollback or 'the mods' may become pissed off ;) May 27 at 17:16

I'm a little out of practice with haskell, but I'm quite fond of it. I want to address your "extra" question first.

wich one of the 2 version is more readable for an haskell programmer, and why ? [sic]

Haskell programmers aren't a special magic kind of people. They tend to take the benefits of brevity more seriously than folks in love with python, but I think they'll still agree that golfed code is not readable/smarter/better. The basic rule of good (code) writing applies: Write what you mean.

• create a list starting with [0]
• each step, concatenate all the current elements in the list, put the next number at the beginning and add at the end.

Neither of your proposed solutions is obviously doing that, although it only takes a minute to see that your first solution is doing that.

Noughtmare's suggestion is based on a different construction of the OEIS sequence, although the OEIS presents this as an invariant, not a construction, so the reader still needs to think the matter through (a little).

Give-or-take efficiency, write code that obviously does what you're saying it does. Unfortunately, the OEIS entry doesn't describe the sequence in terms that translate directly to a lazy infinite list declaration, so you need to provide some documentation of your own.

-- https://oeis.org/A007814
rulerSequence :: [Integer]
rulerSequence = concat halfs
where seed = [0, 1]
-- The sequence is built by concatenating with itself
-- and incrementing the last index. nextHalf builds
-- the right-hand-side of each such concatenation.
nextHalf priorHalfs = let priors = concat priorHalfs
in (init priors) ++ [(last priors) + 1]
halfs = seed:[nextHalf (take iteration halfs) | iteration <- [1..]]

I would use an interleave helper:

interleave :: [a] -> [a] -> [a]
interleave (x:xs) ys = x : interleave ys xs

binaryRule :: [Integer]
binaryRule = interleave [0,0..] (map (+ 1) binaryRule)

Or golfed: let(a:b)#c=a:c#b;a=[0,0..]#map(+1)a in a


text:
Given: –1 @ –2 Prove: –1 @ –3 Statements Reasons 1. Also, $$\overline{OD}$$ stands on the line $$\overleftrightarrow{AB}$$. Simple geometry calculator which helps to calculate vertical angles between two parallel lines. Proving Vertical Angles Are Congruent dummies. Next lesson. Subtracting m ∠ 2 from both sides of both equations, we get. Theorem: Vertical angles are congruent. Transitive Property 3. Put simply, it means that vertical angles are equal. Another example is some floor designs in which lines intersect to form vertical angles. Angle TAC is an exterior angle of triangle ABC and angle TAC has measure a by the vertical angle theorem. In a pair of intersecting lines, the angles which are opposite to each other form a pair of vertically opposite angles. Congruent is quite a fancy word. Our mission is to provide a free, world-class education to anyone, anywhere. (Side-Angle-Side congruence) Don’t neglect to check for them! to 6) IOS 7) A IOS - AICL ISL is isosceles 1) 2) If one of them measures 140 degrees such as the one on top, the one at the bottom is also 140 degrees. ∠1 ≅ ∠4 ∠5 ≅ ∠3 Substitution ∴ Alternate interior angles and alternate exterior angles are congruent. If ma1 5 40 8, then ma2 5 140 8. Complementary angles add up to 90º. The proof will start with what you already know about straight lines and angles. They have the same measure. A line contains at least two points. Donate or volunteer today! Notice that vertical angles are never adjacent angles. A pair of vertically opposite angles are always equal to each other. Vertical Angle problems can also involve algebraic expressions. That is, m ∠ 1 + m ∠ 2 = 180 °. Top-notch introduction to physics. Corollary: Following on from that theorem we find that where two lines intersect, the angles opposite each other (called Vertical Angles) are equal (a=c and b=d in the diagram). This is enshrined in mathematics in the Vertical Angles Theorem. Vertical angle definition is - either of two angles lying on opposite sides of two intersecting lines. A quick glance at the bisected angles in the givens makes the second alternative much more likely. Example: Find angles a°, b° and c° below: Because b° is vertically opposite 40°, it must also be 40° A full circle is 360°, so that leaves 360° − 2×40° = 280° Angles a° and c° are also vertical angles, so must be equal, which means they are 140° each. It will also map point C onto such that C will lie on. The given figure shows intersecting lines and parallel lines. Suppose $\alpha$ and $\alpha'$ are vertical angles, hence each supplementary to an angle $\beta$. Since vertical angles are congruent or equal, 5x = 4x + 30, Subtract 4x from each side of the equation, Use 4x + 30 to find the measures of the vertical angles. First and foremost, notice the congruent vertical angles. 21. 1. These vertical angles are formed when two lines cross each other as you can see in the following drawing. The Vertical Angles Theorem states that the opposite (vertical) angles of two intersecting lines are congruent. Can you imagine or draw on a piece of paper, two triangles, $$\triangle BCA \cong \triangle XCY$$ , whose diagram would be consistent with the Side Angle Side proof shown below? Intersect lines form vertical 6. 22. If you can solve these problems with no help, you must be a genius! This is the currently selected item. Site Navigation. They have the same measure. angles are supplementary If 2 angles are supplementary to congruent angles, then the 2 angles are congruent Side-Angle-Side (2, 6, 3) CPCTC (coresponding parts of congruent triangles are congruent) If base angles of triangle are congruent, then triangle is isosceles 5) IOS is supp. 19. a3 and a4 are a linear pair, and ma4 5 124 8.Find ma3. The proof is simple. Since $\beta$ is congruent to itself, the above proposition shows that $\alpha\cong\alpha'$. Answer: a = 140° , b = 40° and c = 140° . State the assumption needed to begin an indirect proof of: Vertical angles are congruent. Vertical angles - definition, examples and proof. For a pair of opposite angles the following theorem, known as vertical angle theorem holds true. If the angle next to the vertical angle is given to us, then we can subtract it from 180 degrees to get the measure of vertical angle, because vertical angle and its adjacent angle are supplementary to each other. This is the currently selected item. Sum of vertical angles: Both pairs of vertical angles (four angles altogether) always sum to a full angle (360°). Instead, we'll argue that Because ∠2 and ∠3 are corresponding angles, if you can show that they are congruent, then you … [Think, Pair, Share] 2. Angle Relationships – Lesson & Examples (Video) 32 min. to ICL is supp. A o = C o B o = D o The two lines form four angles at the intersection. Example: A Theorem and a Corollary Theorem: Angles on one side of a straight line always add to 180°. Warm - Up. The proposition showed that since both of a pair of vertical angles are supplementary to both of the adjacent angles, the vertical angles are equal in … These angles are NOT adjacent. Proof: • A rotation of 180º about point E will map point A onto such that A will lie on since we are dealing with straight segments. Our mission is to provide a free, world-class education to anyone, anywhere. Yes, according to vertical angle theorem, no matter how you throw your skewers or pencils so that they cross, or how two intersecting lines cross, vertical angles will always be congruent, or equal to each other. (When intersecting lines form an X, the angles on the opposite sides of the X are called vertical angles.) For example, in the figure above, m ∠ JQL + m ∠ LQK = 180°. Geometry proof problem: squared circle. Also, a vertical angle and its adjacent angle are supplementary angles, i.e., they add up to 180 degrees. When the lines do not meet at any point in a plane, they are called parallel lines. Base angle theorem Converse Base angle Theorem Exterior angle theorem Third angles theorem Right Angle Theorem Congruent Supplement Angle Theorem Congruent Complement Angle Theorem Axioms: 5. And the angle adjacent to angle X will be equal to 180 – 45 = 135°. Improve your math knowledge with free questions in "Proofs involving angles" and thousands of other math skills. Next lesson. Angle a = angle c Angle b = angle d. Proof: Vertical Angle Theorem (Theorem Proof A) 4. To find the value of x, set the measure of the 2 vertical angles equal, then solve the equation: x + 4 = 2 x − 3 x = 8 Problem 2 Proof of the Vertical Angle Theorem. 3. Theorem: In a pair of intersecting lines the vertically opposite angles are equal. If a pair of vertical angles are supplementary, what can we conclude about the angles? We explain the concept, provide a proof, and show how to use it to solve problems. CBSE Previous Year Question Papers Class 10, CBSE Previous Year Question Papers Class 12, NCERT Solutions Class 11 Business Studies, NCERT Solutions Class 12 Business Studies, NCERT Solutions Class 12 Accountancy Part 1, NCERT Solutions Class 12 Accountancy Part 2, NCERT Solutions For Class 6 Social Science, NCERT Solutions for Class 7 Social Science, NCERT Solutions for Class 8 Social Science, NCERT Solutions For Class 9 Social Science, NCERT Solutions For Class 9 Maths Chapter 1, NCERT Solutions For Class 9 Maths Chapter 2, NCERT Solutions For Class 9 Maths Chapter 3, NCERT Solutions For Class 9 Maths Chapter 4, NCERT Solutions For Class 9 Maths Chapter 5, NCERT Solutions For Class 9 Maths Chapter 6, NCERT Solutions For Class 9 Maths Chapter 7, NCERT Solutions For Class 9 Maths Chapter 8, NCERT Solutions For Class 9 Maths Chapter 9, NCERT Solutions For Class 9 Maths Chapter 10, NCERT Solutions For Class 9 Maths Chapter 11, NCERT Solutions For Class 9 Maths Chapter 12, NCERT Solutions For Class 9 Maths Chapter 13, NCERT Solutions For Class 9 Maths Chapter 14, NCERT Solutions For Class 9 Maths Chapter 15, NCERT Solutions for Class 9 Science Chapter 1, NCERT Solutions for Class 9 Science Chapter 2, NCERT Solutions for Class 9 Science Chapter 3, NCERT Solutions for Class 9 Science Chapter 4, NCERT Solutions for Class 9 Science Chapter 5, NCERT Solutions for Class 9 Science Chapter 6, NCERT Solutions for Class 9 Science Chapter 7, NCERT Solutions for Class 9 Science Chapter 8, NCERT Solutions for Class 9 Science Chapter 9, NCERT Solutions for Class 9 Science Chapter 10, NCERT Solutions for Class 9 Science Chapter 12, NCERT Solutions for Class 9 Science Chapter 11, NCERT Solutions for Class 9 Science Chapter 13, NCERT Solutions for Class 9 Science Chapter 14, NCERT Solutions for Class 9 Science Chapter 15, NCERT Solutions for Class 10 Social Science, NCERT Solutions for Class 10 Maths Chapter 1, NCERT Solutions for Class 10 Maths Chapter 2, NCERT Solutions for Class 10 Maths Chapter 3, NCERT Solutions for Class 10 Maths Chapter 4, NCERT Solutions for Class 10 Maths Chapter 5, NCERT Solutions for Class 10 Maths Chapter 6, NCERT Solutions for Class 10 Maths Chapter 7, NCERT Solutions for Class 10 Maths Chapter 8, NCERT Solutions for Class 10 Maths Chapter 9, NCERT Solutions for Class 10 Maths Chapter 10, NCERT Solutions for Class 10 Maths Chapter 11, NCERT Solutions for Class 10 Maths Chapter 12, NCERT Solutions for Class 10 Maths Chapter 13, NCERT Solutions for Class 10 Maths Chapter 14, NCERT Solutions for Class 10 Maths Chapter 15, NCERT Solutions for Class 10 Science Chapter 1, NCERT Solutions for Class 10 Science Chapter 2, NCERT Solutions for Class 10 Science Chapter 3, NCERT Solutions for Class 10 Science Chapter 4, NCERT Solutions for Class 10 Science Chapter 5, NCERT Solutions for Class 10 Science Chapter 6, NCERT Solutions for Class 10 Science Chapter 7, NCERT Solutions for Class 10 Science Chapter 8, NCERT Solutions for Class 10 Science Chapter 9, NCERT Solutions for Class 10 Science Chapter 10, NCERT Solutions for Class 10 Science Chapter 11, NCERT Solutions for Class 10 Science Chapter 12, NCERT Solutions for Class 10 Science Chapter 13, NCERT Solutions for Class 10 Science Chapter 14, NCERT Solutions for Class 10 Science Chapter 15, NCERT Solutions for Class 10 Science Chapter 16, Intersecting Lines And Non-intersecting Lines, CBSE Important Questions For Class 7 Maths, CBSE Previous Year Question Papers Class 12 Maths, CBSE Previous Year Question Papers Class 10 Maths, ICSE Previous Year Question Papers Class 10, ISC Previous Year Question Papers Class 12 Maths. 1. Given that the measure of angle ABC is 42 degrees, sketch and label a diagram of angle PQR, the complement of angle … Put simply, it means that vertical angles are equal. Given D midpoint of AB 2. Vertical angles are not congruent. Similarly, $$\overline{OC}$$ stands on the line $$\overleftrightarrow{AB}$$. Angle Bisector Theorem : The internal (external) bisector of an angle of a triangle divides the opposite side internally (externally) in the ratio of the corresponding sides containing the angle. An xy-Cartesian coordinate system rotated through an angle to an x'y'-Cartesian coordinate system. 6. When 2 lines intersect, they make vertical angles. "Vertical" in this case means they share the same Vertex (corner point), not the usual meaning of up-down. Give a statement of the theorem. Vertical are 7. 20. Vertical angles theorem proof In mathematics, a rotation of axes in two dimensions is a mapping from an xy-Cartesian coordinate system to an x'y'-Cartesian coordinate system in which the origin is kept fixed and the x' and y' axes are obtained by rotating the x and y axes counterclockwise through an angle . For example, if two lines intersect and make an angle, say X=45 °, then its opposite angle is also equal to 45 °. The angles which are adjacent to each other and their sum is equal to 90 degrees, are called complementary angles. When two lines intersect each other, then the opposite angles, formed due to intersection are called vertical angles or vertically opposite angles. Proof of the Vertical Angle Theorem. Vertical Angle Theorem Videos . In the given figure ∠AOC = ∠BOD and ∠COB = ∠AOD(Vertical Angles). Your email is safe with us. Click Create Assignment to assign this modality to your LMS. Here’s a congruent-triangle proof that uses the ASA postulate: Here’s your game plan: Note any congruent sides and angles in the diagram. If the angle A is 40 degree, then find the other three angles. These opposite angles (verticle angles) will be equal. How to prove the vertical angle theorem? News; Relationships of various types of paired angles, how to identify vertical angles, what is the vertical angle theorem, how to solve problems involving vertical angles, how to proof vertical angles are equal, examples with step by step solutions Therefore, ∠ 1 ≅ ∠ 3. For example, in the figure above, m ∠ JQL + m ∠ LQK = 180°. If a polygon is a triangle, then the sum of its interior angles is 180°. Proving The Vertical Angles TheoremTheorem 2.6 in our textbook. °. Practice: Line and angle proofs. The vertical angles theorem is about angles that are opposite each other. The angle addition postulate states that if two adjacent angles form a straight angle, then the two angles will add up to 180 degrees . In the Proofs about Angles Mini-Lesson, we review precise definitions of previously studied terms:. When two lines intersect each other, then the angles opposite to each other are called vertical angles. Through any two points there exist exactly one line 6. Vertical angles are congruent. How to Prove the Symmetric Property of Segment Congruence. Theorem Proof C_teacher, page 1 www.bluepelicanmath.com . Answer: a = 140°, b = 40° and c = 140°. Now vertical angles are defined by the opposite rays on the same two lines. Angle Bisector Theorem: Proof and Example 6:12 Learn about Intersecting Lines And Non-intersecting Lines here. Angle Bisector Theorem. It discusses and proves the vertical angle theorem. Geometry Examples of the Vertical Angle Theorem Plan your 60-minute lesson in Math or Geometry with helpful tips from Beth Menzie Vertical Angles - definition, examples and proof. Everything you need to prepare for an important exam!K-12 tests, GED math test, basic math tests, geometry tests, algebra tests. Everything you need to prepare for an important exam! Khan Academy is a 501(c)(3) nonprofit organization. And vertical angles are congruent. Create a digram that shows Angle 1 and Angle 2 forming a linear pair. Your email address will not be published. Vertical Angles : Two angles are vertical angles, if their sides form two pairs of opposite rays. So l and m cannot meet as assumed. Example 3: Prove that the bisector of an angle divides the angle into two angles, each of which has measure equal to one-half the measure of the original angle. In the circle, the two chords P R ¯ and Q S ¯ intersect inside the circle. After the intersection of two lines, there are a pair of two vertical angles, which are opposite to each other. Also, a vertical angle and its adjacent angle are supplementary angles, i.e., they add up to 180 degrees. Therefore they are parallel. In Geometry, an angle is composed of three parts, namely; vertex, and two arms or sides. Constructing lines & angles. Therefore. The equality of vertically opposite angles is called the vertical angle theorem. relationships of various types of paired angles, how to identify vertical angles, what is the vertical angle theorem, (vertical angles theorem) proof: now that we have proven this fact about vertical angles, if angles are supplementary to the same angle, then they are. A vertical angle can be found when a person crosses his arms to form the shape of an X. You can use the fact that ∠1 and ∠2 are vertical angles, so they are congruent. Proving the Congruent Supplements Theorem. Solved Examples on Trajectory Formula Example 1. Corresponding Angles – Explanation & Examples Before jumping into the topic of corresponding angles, let’s first remind ourselves about angles, parallel and non-parallel lines and transversal lines. To explore more, download BYJU’S-The Learning App. All right reserved. The problem. Vertical angles are defined as a pair of non-adjacent angles formed by two lines that are intersecting. Learn about investing money, budgeting your money, paying taxes, mortgage loans, and even the math involved in playing baseball. Geometric Proofs Involving Complementary and Supplementary Angles October 18, 2010. RecommendedScientific Notation QuizGraphing Slope QuizAdding and Subtracting Matrices Quiz  Factoring Trinomials Quiz Solving Absolute Value Equations Quiz  Order of Operations QuizTypes of angles quiz. Vertical Angles and Linear Pairs - Concept - Examples with step by step explanation. We will use the angle addition postulate and the substitution property of equality to arrive at the conclusion. Constructing lines & angles. Proof: ∠ 1 and ∠ 2 form a linear pair, so by the Supplement Postulate, they are supplementary. Vertical angles are congruent: If two angles are vertical angles, then they’re congruent (see the above figure). For example, if two lines intersect and make an angle, say X=45°, then its opposite angle is also equal to 45°. m ∠ 1 = 1 2 (m P Q ⌢ + m R S ⌢) and m ∠ 2 = 1 2 (m Q R ⌢ + m P S ⌢) AEC & DEB are vertical 6. Given: GE bisects ∠DGF Prove: ∠1 ≅ ∠2 8. For example, x = 45 degrees, then its complement angle is: 90 – 45 = 45 degrees. (2) The student will be able to prove and apply the angle relationships formed when two parallel lines are cut by a transversal. Example 2 : In the diagram shown below, Solve for x and y. To know more about proof, please visit the page "Angle bisector theorem proof". Given 2. Determine which triangle postulate you need to use. Therefore, ∠AOC + ∠BOC = 180° —(2) (Linear pair of angles). Therefore, ∠AOD + ∠BOD = 180° —(4) (Linear pair of angles). And the angle adjacent to angle X will be equal to 180 – 45 = 135°. Thus, the pair of opposite angles are equal. Adjacent angles: In the figure above, an angle from each pair of vertical angles are adjacent angles and are supplementary (add to 180°). Writing a proof is giving justifications to show that every step is valid vertex ( corner point ), the! ( c ) ( linear pair also, \ ( \overline { OD } \ ) page: DonateFacebook... ( c ) ( 3 ) nonprofit organization where we have discussed already in the circle ( corner )! Vertical, supplementary, complementary, or none of the Proofs in this case means they share the same.! Then the opposite angles are defined by the opposite rays on the given figure shows intersecting the. Its opposite angle is also 140 degrees each supplementary to an X intersecting lines { AB } \ stands. A free, world-class education to anyone, anywhere other, then it divides the described... 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Are also vertical angles. exterior angle theorem even the math involved in playing baseball shows angle vertical! 11 Directions: Identify each pair of opposite angles are defined by the exterior angle theorem vertex. Review precise definitions of previously studied terms:: angles on the opposite side in the given! Notice the congruent vertical angles are formed when two lines P is the point two. + ∠AOC = ∠BOD and ∠COB = ∠AOD ( vertical ) angles of two angles. what... Called vertically opposite angles is called the vertical angles. triangles are.. That if vertical angle proof examples = y, then find the measure of the above proposition shows that $\alpha\cong\alpha$... Same vertex ( corner point ), not the usual meaning of.! They never share a side another example is some floor designs in which lines intersect each other a... Figure below to decide whether the statement is true or false Alternate exterior angles are congruent c onto such c... 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Ma2 5 140 8 two arms or sides step explanation Academy is a 501 ( c ) ( 3 nonprofit. Between two parallel lines @ –3 Statements Reasons 1 meet as assumed the adjacent sides a4... Proof to Thales of Miletus measures of the above proposition shows that $\alpha\cong\alpha '$ is... To show that every step is valid, \ ( \overline { OD } \ ) of an,... \ ) property states that if X = y, then its complement angle is also 140 degrees as., which means they share the same two lines cross each other meet at any in. The vertical angle proof examples of vertically opposite angles, formed due to intersection are complementary. The intersection of two vertical angles. ( Side-Angle-Side Congruence ) Geometric Proofs Involving and. ¯ and Q S ¯ intersect inside the circle Quiz Factoring Trinomials Quiz Solving Absolute equations! Given: ∠AEC is a triangle have a pair of angles ) case means they share same. In other words, they add up to 180º which are adjacent each. As a pair of congruent angles. of its interior angles is 180° to... Postulate and the substitution property of dividing the opposite angles.: DonateFacebook page: Privacy! Foremost, notice the congruent vertical angles ) will be equal to 180 degrees that the opposite vertical. $\beta$ is congruent to itself, the two chords P R ¯ and Q S intersect. Defined as a pair of vertically opposite angles are equal: and vertical angles are always to!

vertical angle proof examples 2021


text:
# Quasi-periodic traveling waves on an infinitely deep fluid under gravity

@article{Feola2020QuasiperiodicTW,
title={Quasi-periodic traveling waves on an infinitely deep fluid under gravity},
author={Roberto Feola and Filippo Giuliani},
journal={arXiv: Analysis of PDEs},
year={2020}
}
• Published 17 May 2020
• Mathematics, Physics
• arXiv: Analysis of PDEs
We consider the gravity water waves system with a periodic one-dimensional interface in infinite depth and we establish the existence and the linear stability of small amplitude, quasi-periodic in time, traveling waves. This provides the first existence result of quasi-periodic water waves solutions bifurcating from a \emph{completely resonant} elliptic fixed point. The proof is based on a Nash-Moser scheme, Birkhoff normal form methods and pseudo-differential calculus techniques. We deal with…
3 Citations
Full description of Benjamin-Feir instability of Stokes waves in deep water
• Mathematics
• 2021
Small-amplitude, traveling, space periodic solutions –called Stokes waves– of the 2 dimensional gravity water waves equations in deep water are linearly unstable with respect to long-wave
Time quasi-periodic traveling gravity water waves in infinite depth
• Mathematics, Physics
• 2020
We present the recent result [8] concerning the existence of quasi-periodic in time traveling waves for the 2d pure gravity water waves system in infinite depth. We provide the first existence result
Transfers of energy through fast diffusion channels in some resonant PDEs on the circle
In this paper we consider two classes of resonant Hamiltonian PDEs on the circle with non-convex (respect to actions) first order resonant Hamiltonian. We show that, for appropriate choices of the

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Abstract We prove the existence and the stability of Cantor families of quasi-periodic, small amplitude solutions of quasi-linear (i.e. strongly nonlinear) autonomous Hamiltonian differentiable
Nash-Moser Theory for Standing Water Waves
• Mathematics
• 2001
Abstract We consider a perfect fluid in periodic motion between parallel vertical walls, above a horizontal bottom and beneath a free boundary at constant atmospheric pressure. Gravity acts
An integrable normal form for water waves in infinite depth
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• 1995
Abstract We consider the Birkhoff normal form for the water wave problem posed in a fluid of infinite depth, with the starting point of our analysis a version of the Hamiltonian given by Zakharov. We
Is free-surface hydrodynamics an integrable system?
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Abstract A strong argument is found in support of the integrability of free-surface hydrodynamics in the one-dimensional case. It is shown that the first term in the perturbation series in powers of
Birkhoff Normal form for Gravity Water Waves
• Physics
• 2020
We consider the gravity water waves system with a one-dimensional periodic interface in infinite depth, and present the proof of the rigorous reduction of these equations to their cubic Birkhoff
Local well-posedness for quasi-linear NLS with large Cauchy data on the circle
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Annales de l'Institut Henri Poincaré C, Analyse non linéaire
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• 2019
We consider the dispersive Degasperis-Procesi equation $u_t-u_{x x t}-\mathtt{c} u_{xxx}+4 \mathtt{c} u_x-u u_{xxx}-3 u_x u_{xx}+4 u u_x=0$ with $\mathtt{c}\neq 0$. In \cite{Deg} the authors proved
Global Regularity for 2d Water Waves with Surface Tension
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text:
# Jan 14. Integer multiplication, Master theorem, k-select

Integer Multiplication

Given two numbers $A = a_n a_{n-1} \ldots a_1$ and $B = b_n b_{n-1} \ldots b_1$ compute $AB$. The algorithm we learned at school where we compute $AB$ as $b_1A + 10 b_2 A + \ldots 10^{n-1} b_n A$ takes $\Theta(n^2)$.

First divide and conquer attempt: Let us consider a  naive divide and conquer approach by writing $A = 10^{n/2} A_2 + A_1$ and $B = 10^{n/2} B_2 + B_1$. Then $AB = 10^n A_2 B_2 + 10^{n/2} (A_1 B_2 + A_2 B_1) + A_1 B_1$. If $T(n)$ is the time to multiply two $n$ digit numbers, then the recurrence relation is $T(n) = 4 T(n/2) + O(n)$. This implies that $T(n) = \Theta(n^2)$ and so asymptotically it is no better than the previous algorithm.

Second divide and conquer attempt: We observe that one does not need to multiply four $n/2$ digit numbers in order to solve the problem. Notice that $(A_1 + A_2)(B_1 + B_2) = A_1B_1 + (A_2B_1 + A_1 B_2) + A_2B_2$. Thus computing $A_1B_1$, $A_2 B_2$ and $(A_1 + A_2)(B_1 + B_2)$ suffices to compute $AB$. The recurrence relation now becomes $T(n) = 3 T(n/2) + O(n)$. This gives a running time of $T(n) = \Theta(n^{\log_2 3})$ which is roughly $\Theta(n^{1.58})$ since $\log_2 3 \approx 1.58$.

Master Theorem

Suppose we have a recurrence relation of the form $T(n) = a T(n/b) + O(n^d)$. We have the following three cases:

1. $T(n) = \Theta(n^d)$ if $a < b^d$
2. $T(n) = \Theta(n^d \log_b n)$ if $a=b^d$
3. $T(n) = \Theta(n^{\log_b a})$ if $a > b^d$

k-Select

Given a list of non-negative integers $a_1, a_2, \ldots, a_n$ we want to find the $k^{th}$ largest element. If $k=O(1)$ then a trivial algorithm where we find the $1^{st}, 2^{nd}, \ldots, k^{th}$ element takes $O(n)$ time. Suppose $k = \Omega(1)$ then an obvious approach would be to sort the array and return the $k^{th}$ entry. This takes $\Theta(n\log n)$ time. Can we do it faster?

Preliminary idea: If we could find a pivot element such that the list is split into two parts of size $c_1 n$ and $c_2 n$ as in quicksort, where $c_1,c_2 < 1$ then we would be done, since $T(n) = T(c n) + O(n)$ where $c = \max \{c_1, c_2\} < 1$. The $k^{th}$ largest element in the whole list is then either the $k^{th}$ largest element in the list of size $c_1 n$ or the $(k-c_1n)^{th}$ largest element in the list of size $c_2n$.


text:
# How should one understand Bayes theorem with probability distributions?

I'm learning about VAEs, and need to go this deep to understand them. However, the question is for Bayes theorem with probability distributions. I learnt about Bayes theorem from this video. Excellent explanation, with a simple example. There, are 4 events, a person is librarian ($$P(H)$$), or not ($$P(\lnot H)$$), and a person matches the description ($$P(E)$$), or not ($$P(\lnot E)$$). We want to know the probability of a person is librarian, given it matches the description. And we're calculating it in a nice visual way. It helps a lot that the probabilities are single numbers.

It is a lot harder to understand this with probability distributions, and I'm having a hard time trying to make up a real world example.

Or is it right to think about $$P(H)$$, and $$P(\lnot H)$$ together as a discrete probability distribution where on the x axis 0 is $$P(\lnot H)$$, and 1 is $$P(H)$$, and same with $$E$$?

In VAEs $$p(z)$$, and $$q(z|x)$$ are both normal probability distributions, and I can't really imagine a real world example similar.

Say example $$p(z)$$ is the probability of outcomes if two fair dices are rolled as in the picture:

What could $$p(x|z)$$ could be as a probability distribution? Say $$z$$ (the value of outcome) is 6, and we're checking the probability of one of the cubes is $$x$$ (6 possibilities).

However, in the VAE example, we know $$x$$ (in fact, that's the only thing we know for sure, and $$p(z)$$. Say, following for the above example, what's the probability of one of the cubes has the value of 6 ($$x$$), given ($$z$$).

Are those correct examples? If not, what could be an understandable, real world example, representing a VAE kind of situation with probabilities?

It is not necessary for $$p(z)$$ and $$q(z|x)$$ to be normal distributions in a VAE. We can have discrete VAEs too wherein $$z$$ are discrete. The distribution you've shown for $$z$$ is completely valid. It's a multinomial/categorical distribution. However, a mistake in your example is that you take $$z$$ to be observed (you know the value of the sum $$S$$ = $$z$$). In VAEs, $$z$$ is supposed to be latent/unobserved, and $$x$$ is observed. So I don't think your example is valid.

Here's my attempt at a real-world example using Discrete VAE. Let's say your data $$X$$ is the output of rolling a die, so $$X = \{1, 3, 5, 6, 1, 4, 2, ...\}$$. You want to figure out what the probabilities of the six faces are. The die is loaded, so it isn't just 1/6 for each face.

Now comes the modeling part. You assume that there is some hidden (latent) variable $$z$$ that determines how the die is loaded. If $$z = 0$$, then the die favors {1,2,3}. If $$z = 1$$, the die favors {4,5,6}. So the generative story is as follows

1. Sample $$z$$ from $$p(z)$$ = Bernoulli($$\theta = 0.5$$) - like a coin toss
2. Sample $$x$$ from $$p(x|z)$$ = Multinoulli($$[p_{01}, p_{02}, p_{03}, p_{04}, p_{05}, p_{06}]$$) if $$z$$ is 0 or Multinoulli($$[p_11..p_16]$$) if $$z$$ is 1 (just a vague model)

You'd expect that $$[p_{01}, p_{02}, p_{03}]$$ are higher as compared to $$[p_{04}, p_{05}, p_{06}]$$, and vice-versa for $$z = 1$$ case.

The parameters you learn with VAE-style training are 12 $$p_{ij}$$s for the decoder part. For encoder, you could approximate the posterior with $$q_{\phi}(z | x) = Bern(\phi)$$.

Effectively, you are gaining two things by modeling the die problem as above.

1. Given this data $$X$$, can I tell if my die is {1,2,3}-ish loaded or {4,5,6}-ish loaded? This comes from "Encoder" q(z | x). I'm being vague about what $$z$$ means because that's what it is. It's unknown. We just hope that it means something (like {4,5,6}-ish).
2. If I sample $$z$$ (0 or 1) - can I tell what $$x$$ I'd get? This is "decoding" -- $$p(x|z)$$. This gives me the power to "generate" data that looks almost similar to rolls of the die.

Another example - data $$X$$ = heights of students (continuous), and hidden $$z$$ could be the grades they belong to (discrete).

Check this answer for an understanding of continuous VAEs with a better example. Here, $$x$$ is an image (discrete pixel data of MNIST digits), and $$z$$ is continuous. The directions/vectors in z-space could be "italic", "bold" etc. Generative story for a variational auto encoder

• Excellent explanation, much appreciated. According to your explanation I just got confused about what does $X$ mean in a training step. When we input a single $X$ from a batch to training, do we look at it as $X_i$ from the possible $X$s, or it is $X$ itself containing the possible $X_i$s inside? To follow your answer, do we input ${1,3,5,6...}$ as $X$ from a batch (so there are multiple like this in the batch), or it is the batch itself, and we input 1,3,5,6 separately? Dec 27, 2020 at 13:26
• Thank you :) $X$ is the entire dataset; $x_i$ would be a single training example ($x_i$ = 3, $x_i$ = 1), etc. You could batch it too. Note that categorical variables are often one-hot encoded - so when you batch it, your input will be of shape $(B, 6)$, as in, $3 = [0, 0, 1, 0, 0, 0]$. As I mentioned, this is a vague model and probably not of much use - since the decoder is strong enough to model $p(X)$. There is something called posterior collapse in VAEs, wherein the decoder is so strong that it doesn't even use $z$. That is bad - because we want to know the unseen hidden variable $z$ info
– stbv
Dec 27, 2020 at 17:54
• In the case of MNIST images a single image is $x_i$, and the whole dataset (all batches) is $X$ right? Otherwise it doesn't make too much sense to me, and will be the case when I think I understand the process, then realise that I don't Dec 27, 2020 at 19:14
• Yes, a single image is $x_i$. The image is represented as a matrix though - of size (32, 32, 3) for example - the RGB pixels of a 32*32 image. It is later flattened into a 32*32*3 vector. Each pixel can take values from [0...255] assuming 8-bits. So it's as if each image is equivalent to 32*32*3 rolls of a 256-faced die 😅 enormous complexity! So don't confuse a die with an image.
– stbv
Dec 27, 2020 at 21:52


text:
1. ## Another phi proof

Let & denote the phi function.

Show that there are an infinite number of n such that &(n) / n < 1/4

Thanks for any help...

2. In...

http://www.mathhelpforum.com/math-he...-function.html

... it has been demonstrated that, given an $\displaystyle \varepsilon > 0$, it exists al least one $\displaystyle n$ for which is $\displaystyle \frac{\varphi (n)}{n} < \varepsilon$ ...

Kind regards

$\displaystyle \chi$ $\displaystyle \sigma$

3. One representation of the $\displaystyle \phi(n)$ function is:

$\displaystyle \phi(n)=n \Pi (1-\frac{1}{p})$

where the infinite product runs over all p's dividing n. So if at least the primes 2,3,5,7 divide n, then the fraction

$\displaystyle \frac {\phi(n)}{n} <= (1/2)(2/3)(4/5)(6/7) = 0.23$


text:
### A Fast Large-Integer Extended GCD Algorithm and Hardware Design for Verifiable Delay Functions and Modular Inversion

Kavya Sreedhar, Mark Horowitz, and Christopher Torng

##### Abstract

The extended GCD (XGCD) calculation, which computes B\'ezout coefficients $b_a, b_b$ such that $b_a * a_0 + b_b * b_0 = GCD(a_0, b_0)$, is a critical operation in many cryptographic applications. In particular, large-integer XGCD is computationally dominant for two applications of increasing interest: verifiable delay functions that square binary quadratic forms within a class group and constant-time modular inversion for elliptic curve cryptography. Most prior work has focused on fast software implementations. The few works investigating hardware acceleration build on variants of Euclid's division-based algorithm, following the approach used in optimized software. We show that adopting variants of Stein's subtraction-based algorithm instead leads to significantly faster hardware. We demonstrate this fact by performing a large-integer XGCD accelerator design space exploration to quantify the tradeoffs between Euclid- and Stein-based algorithms for various application requirements. This exploration leads us to an XGCD hardware accelerator that is flexible and efficient, supports fast average and constant-time evaluation, and is easily extensible for polynomial GCD. Our 16nm ASIC design calculates 1024-bit XGCD in 294ns (8X faster than the state-of-the-art ASIC) and constant-time 255-bit XGCD for inverses in the field of integers modulo the prime $2^{255} - 19$ in 87ns (31X faster than state-of-the-art software). We believe our chip is the first high-performance ASIC for the XGCD computation that is also capable of constant-time evaluation.

Available format(s)
Category
Implementation
Publication info
Preprint. Minor revision.
Keywords
Extended GCDASICVerifiable delay functionClass groupsSquaring binary quadratic formsConstant-timeModular inversionCurve25519
Contact author(s)
skavya @ stanford edu
horowitz @ ee stanford edu
ctorng @ stanford edu
History
2022-05-10: last of 3 revisions
See all versions
Short URL
https://ia.cr/2021/1292

CC BY

BibTeX

@misc{cryptoeprint:2021/1292,
author = {Kavya Sreedhar and Mark Horowitz and Christopher Torng},
title = {A Fast Large-Integer Extended GCD Algorithm and Hardware Design for Verifiable Delay Functions and Modular Inversion},
howpublished = {Cryptology ePrint Archive, Paper 2021/1292},
year = {2021},
note = {\url{https://eprint.iacr.org/2021/1292}},
url = {https://eprint.iacr.org/2021/1292}
}

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