Youtube daily Feb 5 2019

Face Off (1997) Cast - Then and Now 2019

Press the Pelican on the update

Oh, my baby

So, I'm crazy once you've been that bad for me me pasta got your feet both fly

Get to flap with me. I

Can take you down fall, right

Take you over

Crosswalk instead lucky

Oh

I'm crazy

In

This room, so I'm crazy, but you've been that bad

We musta got Sufi

You get to fly with me. I

Can take you down for

Take you around

Fuck it

Oh

My baby

Cuz they've written it you so I'm crazy, but you been that bad for me

Meatballs done got your feet

you get to

I

Can take you down for a ride

Take you around

Crosswalking read lucky

For more infomation >> Face Off (1997) Cast - Then and Now 2019 - Duration: 4:09.

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Sanders And Schumer Team Up To Stop Corporate Stock Buybacks At Greedy Companies - Duration: 4:47.

In a very unlikely Senate teamup, a establishment Chuck Schumer has teamed up with Progressive

Bernie Sanders to draft legislation that would effectively put an end to corporate stock

buybacks if those corporations are not paying their employees at least a $15 minimum wage.

Now, I am not surprised in the least that this is legislation that Bernie Sanders would

help draft.

I am, however, really surprised that Chuck Schumer would be involved in this because

when you think about corporate accountability and progressive policies, the named Chuck

Schumer is not the first thing that springs into your head, but gotta give credit where

it's due and give Schumer credit for signing onto this because honestly, this is the kind

of legislation that we have needed in this country for a very long time.

Here's how bad the issue is.

Right after the Republicans passed their tax cut package right at the end of 2017, incorporations

in 2018 started getting those tax cuts.

They did not reinvest in their own companies.

They did not go out there and make new factories.

They did not expand their their businesses.

They did not hire more people and they did not give people raises, at least not in the

ways that Republicans promised us was going to happen.

Instead, they spent $1,000,000,000,000 in 2018 buying back their own stock and the only

thing that does is enrich the shareholders of those corporations.

You know the already wealthy people calling the shots.

They got even more money because they bought back their own stock.

Therefore driving up the price, even higher employee has got nothing.

Most of them did not get raises.

Most of them did not get bonuses.

You know, in fact, we've actually had a lot of the corporations who engaged in stock buybacks

laid off thousands upon thousands of workers.

So not only did some workers get nothing, other workers literally lost everything.

The companies made a lot more money and sanders and Schumer, and I'm sure we're going to have

a lot more democrats signing onto this.

They've come together and say, we're stopping this, and our legislation says, if you're

not paying your employees a living wage, if you don't give them, raises your not allowed

to buy back your own stocks.

We're putting limits on you and these are the kinds of regulations that are sorely needed

in the United States because it doesn't say that they can't buy back their stocks.

It just puts prerequisites on that company.

You cannot enrich yourself if you have not taken care of your workers.

You cannot put more money in your pocket if you haven't put any more in your workers'

pockets.

That's how we create a strong working class here in the United States.

That's how companies use to show their appreciation for the hard work that their employees did.

They got rewarded.

I know this was back in the day long before most us were born, but that's actually how

it happened.

When the company did better, the workers did better in a lot of the reason that happened

is because we had strong labor unions.

Then Reagan came along, started the era of deregulation and union busting, and now we

are where we are today.

Concentrated Wealth, a squeezed middle and working class here in the United States who

keep getting less, and yet those corporations keep squeezing them and every drop they can

get out of them goes right back into the pockets of the wealthy elite.

This bill would help curb that to an extent.

I mean, after all, these are the companies who, again, they spend a trillion dollars

on stock buybacks last year, spent 140 times more money buying back their own stock than

they did on their employees.

So yes, it's time to put an into this and even if it takes a weird alliance between

Bernie Sanders and Chuck Schumer, who's pretty much always been a corporatist, if that's

what it takes to get it done, then so be it.

Maybe Chuck Schumer's trying to say, you know what?

Maybe I've been wrong for the last few years of my career and maybe maybe I need to start

doing something to help American workers.

Whatever his motivation is, congrats to him.

And as always, congrats to Bernie Sanders for being the true leader

of this legislation.

For more infomation >> Sanders And Schumer Team Up To Stop Corporate Stock Buybacks At Greedy Companies - Duration: 4:47.

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Mel, Limão e Babosa 🍋 Se Você Beber 1 Colher Todo Dia, Vai me Agradecer Pelo Resto da Sua Vida - Duration: 2:50.

For more infomation >> Mel, Limão e Babosa 🍋 Se Você Beber 1 Colher Todo Dia, Vai me Agradecer Pelo Resto da Sua Vida - Duration: 2:50.

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Dayanara Torres reveló que fue diagnosticada con cáncer | Suelta La Sopa | Entretenimiento - Duration: 1:47.

For more infomation >> Dayanara Torres reveló que fue diagnosticada con cáncer | Suelta La Sopa | Entretenimiento - Duration: 1:47.

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Solar Energy Incentives in Irvine | Why You Should Invest in Solar Today! - Duration: 1:41.

Semper Solaris

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For more infomation >> Solar Energy Incentives in Irvine | Why You Should Invest in Solar Today! - Duration: 1:41.

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LEITE DE AMENDOIM CASEIRO - Duration: 6:39.

hi guys The peanut is originated from here, our South America and is very present

here in our country both in sweet and salty recipes Peanut is considered

villain because he has high caloric taxes and also high lipid content, but the truth

is that if properly prepared and consumed in moderation the peanut helps control

cholesterol and triglycerides and helps regular the metabolism The peanut is also

rich in proteins that maintains the muscular system healthy and goes far beyond.

It is rich in vitamin E and vitamin B complex and also revesterol which is a substance that protects

the cardiovascular system.

But beware due to the anti nutrients present in peanuts can cause allergic reactions

for some people and I'm one of them.But some care is needed in the preparation of

of peanuts.

Of course, this does not include people who has very serious reactions. So for these

people the consumption of peanuts is prohibited. stay there and follow the recipe of

peanut milk and indeed is the one that has preference here at home.

Mooo, cococo, oinc, auuu, miauu.To do O

The peanut milk we are using peanuts with bark, plus or minus 1 liter of water

filtered, a funnel and a small piece of voal so we can strain after. Then the first

thing that we will do is wash the peanut.Here is already washed in filtered tap water.

We're going to put on the glass, put 800 ml of water and cover with the voal.

important to leave the seeds and grains breathe and you will leave to soak for

12 hours.

Overnight.

After leaving the peanut soak for overnight You are going to rinse and go

to let it dry in a pan for take out the excess water and then you

will spread on a tray and take to the oven in 180 degrees or less to dry.

Now for those who want a more accurate taste We're going to roast peanuts in a frying pan.

anti adherent. and they always stir until stay golden and try to

feel the taste and see if it is good. They are listening these stalks?

That's when you start toasting the peanuts, So be careful, they will stir, if not they can

burn.

Then the roasted peanuts, ready.

We tried it, saw it is crunchy, So that's good. Our peanuts are ready.

So now just take the peanuts. roasted to the blender by the water

and beat well. With content already beaten, people Now it's going to scratch. I'm using a glass, a

funnel and a small piece of voal to pass only the liquid. As you go strain

the milk is getting this rest here.

Then you just press with yours hands

This rest here you reserve because you can use it in other recipes.

ready our peanut milk.

And whoever you want can sweeten, you can put a pinch of salt and can store in the refrigerator

for a maximum of three days. Dou you like of revenue?

so be sure to sign up for the our channel here at this link and leave your like

and your comment.Thank you !!! Moooo

For more infomation >> LEITE DE AMENDOIM CASEIRO - Duration: 6:39.

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Sulla Bellezza - MakeUp e bellezza - La bocca dell'abisso e il desiderio di ordine - Video 2 di 4 - Duration: 5:32.

For more infomation >> Sulla Bellezza - MakeUp e bellezza - La bocca dell'abisso e il desiderio di ordine - Video 2 di 4 - Duration: 5:32.

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Your Relationship Questions Answered! - Duration: 10:45.

Well, my only real relationships have all been long-distance relationships,

so I guess you can consider me some sort of expert in this field.

(WARPING ELECTRONIC TUNE)

Hello and welcome back to the channel.

Today, I want to answer your questions about relationships,

more specifically, long-distance ones.

So, I went to the Community tab on my channel,

which is something I've been using a lot lately,

and I requested some questions and some of you came up with some questions.

Take a shot for every time I just said "questions".

But we're responsible and drinking water while I'm filming.

"What are some little ways you can make your partner feel loved?

"Not like huge shows of affection.

"Just small things you can do every day."

So, I like to send my boyfriend little gifs on Facebook Messenger.

My most popular ones seem to be the Sailor Venus kiss thing.

You know, that one, I can't remember what attack it was,

but, you know, it's just really cute.

Most recently, I did "Will Chu be my Valentine?"

It was a little Pikachu holding a little heart, you know, for Valentine's Day,

and I thought it was absolutely adorable.

And most recently, I've left a Facebook post on his wall

that said "Hey, you're pretty nice-looking" or whatever.

Not only that but, you know, just sending some snaps.

You know, we send little "I love you" snaps,

and, if he's working, you know, just check in every once in a while.

Just ask 'em how their day is going, you know?

"I'm kind of in a long-distance relationship.

"How can you show them you love them since you can't give them hugs or go on dates?

"I've been trying to write Open When letters or letters in general to show my affection or care.

"I guess, even though we can text, I would say it's more sentimental.

"That's why I'm putting in this effort."

I actually find that really cute

because it's something that I've been considering myself,

and I even went to Barnes & Noble and I actually saw what I'm assuming

you're using is the actual, like, product that says "Open when" or whatever.

I mean, yeah, those are really cute.

I kind of already answered this in the other question.

You know, just leaving nice little messages.

Texts are great.

My boyfriend and I, we tend to send more snaps than we do text messages.

I think it's because, with text messages,

you can't see their face, but, with snaps, you can, so it's just...

For us, we just kind of like it better that way,

although sometimes you want to keep some messages,

which you can't really do on Snapchat, unless you screenshot, which I do now,

because it's not as creepy now because we're together.

But, yeah, I think the letters is really, really nice.

I've been considering that myself.

I'm glad you brought that up,

because, when I was getting ready to leave for my trip during Halloween week,

he actually bought me a little notebook from The Container Store

because I was looking at it and I'm a sucker for notebooks and it was my favourite color,

and he just grabbed it and went off with it.

And I was like, "You don't have to do that."

But, then, just recently, I was thinking, "What if I took this book and I wrote letters in it?"

Because I think that would be something cute to read.

And, you know, I did make him a memory book for Christmas

but I think this is a little bit different than that, but I think that's a really cute idea.

Otherwise, you know, just keep sending, you know, little...maybe cute gifs,

maybe cute graphics that you've seen.

Just send really nice, reassuring messages.

Just something simple like an "I love you" is just fine.

"With online dating sites, has being deaf made it hard for you to have a partner?"

Well, I'm in a relationship right now so we don't have to worry about that,

but I actually did do a video not too long ago about...

It's called "Will I be loved despite my disability?"

Something like that, so I would say just go watch that video 'cause I don't...

Just in fear of repeating myself too much,

but, to try to shorten it as much as possible,

in the beginning, when I was first trying online dating,

yeah, I thought it was a little bit frightening and that it really wouldn't work out,

but, I guess, at the end of the day, in some ways, it did,

although it wasn't really online dating websites.

It just so happened that I met my last boyfriend and this boyfriend over the internet,

over YouTube and Twitch.

"Why do people not like that I never want to get in a relationship?"

I think, because in this day and age,

we've been conditioned to think that, if you're not graduating high school

and then going to college and then getting a job

and then going to get married and then getting kids,

that we're failures in life or whatever or we're not doing what it is expected of us,

which is annoying and I completely understand your frustrations,

but, you know, if somebody just tells you, "Why aren't you gonna get in a relationship?"

Or whatever, whether it's somebody

that wants to be in a relationship with you or is just wondering, like,

I don't know, your mom, because, you know, moms want to nag about grandchildren, right?

Just "I don't want to, end of story, done."

Some people are probably going to try to nag forever about it

but all you can do is just ignore them.

"How do you cope with a long-distance relationship?

"I imagine it wouldn't feel as strong as if you were together often."

I would say that our relationship is pretty strong.

At times, it can feel a little bit lonely,

to which my response really is try to set up, like, online dates.

So, like, I used to use Rabbit a lot

and you could watch Netflix or Crunchyroll or anything else,

or you could just sit there and talk.

It was essentially the same thing as Facetime or Skype

but you got to do a little bit more with it because you could have movie dates, right?

Communication is a must.

There are some days when, you know, it's the actually saying goodbye part that is hard

and then, when you get back into your, like, 'separate lives', it sucks, it feels lonely

but it's not as horrible, it's not as heart-wrenching

as when you're saying goodbye.

I mean, I feel like it might be difficult to explain this

because I don't want to say that being apart isn't sad, because it is, but, you know, just...

If you're in a long-distance relationship, you probably get what I'm trying to say here,

but, yeah, when you're just feeling really lonely,

you get a Facetime date, a Rabbit date,

just some sort of online date, it'll help a lot

because, if you're not seeing each other,

even if it's, like, in text messages or Snapchat, whatever...

There have been times when I've text my boyfriend

getting really upset about the fact

that, when there were times that we really couldn't talk as much

because he would be really, really busy or I would be really, really busy, you know,

and it just really, really sucked.

Communicate your feelings and get it out - it'll help a lot.

Another thing that I've been doing is,

because my boyfriend and I really started getting together because of Twitch,

he visits my streams often and we play video games together on there - that's always fun.

"How do you balance your own life and spending time with your partner?"

Finding your own hobbies, I think, for one.

You don't really have to spend every single day with your partner.

Now, for me, because I am in a long-distance relationship,

when we're together, I want to be together all the time.

I mean, there are some times, yes, I may want an hour to myself,

but, really, for the most part, I just...I want to hold on to him

and I'm like, "No, you're not going anywhere," which doesn't always work

because my boyfriend has a 'mainstream job' so I can't always keep him to myself,

but, yeah, get your own hobbies and you can set your own schedule that way.

You can say a Friday night, you all can hang out together

or this many nights you hang out together,

and then this many nights you do your own thing.

I think it's really just all about communication,

finding out both of your schedules and just coming up with a plan together.

And this one isn't so much a question as it is a statement

but it's something that I agree with and I kinda want to talk about.

"I'll say a relationship is all about trust

"and you have to be honest with one another and I'll say, for me, that I'm still learning."

And I agree with that.

There are definitely some things I've learned in the past year.

I've learned to be a lot more open about communication and honest about my feelings

because one of the bad traits about myself is that I am terrible at expressing my feelings,

so what ends up happening is I let things build up,

which only ends up getting worse, so I end up getting resentment for my partner

and also resentment for myself because I'm letting it happen and grow and it gets toxic.

So, I've been finally learning to open up about things

and, yeah, sometimes, it may be a conversation that you don't want to have

and it may kind of go badly for a little bit,

but what matters is you get your feelings out there and you be honest.

Relationships aren't puppies and kittens and rainbows and glitter and sparkles 24/7,

as great as that would be, so you just...

You gotta kinda work through it, you know?

But not just being honest

about if you have some sort of problem that's going on with a relationship,

but also just, like, if I'm feeling, like, really...

If I'm having a really bad depression episode,

I used to not ever want to talk about it because of potential backlash,

which has occurred in the past,

but now I'm, you know, being much more open, or attempting to be much more open.

Doesn't always work, but, if I'm asking for help,

even if the help cannot be given because it might be out of someone's control,

it's been a life-changer.

But, yeah, in any relationship, honesty and communication is very, very important.

If you don't have that, it's kind of basically a recipe for disaster.

Alright, so, those are all the questions and comments that I received

in the Community tab.

Thank you for your questions.

I hope I helped you out with something.

Admittedly, I don't know much about relationships where you live in the same spot

and you see each other often, but, hopefully, I helped out with the long-distance ones.

If you have any other questions, I guess, you know, feel free to leave them down below.

I'll try my best to get to you

or maybe somebody else might see a comment and reply.

Everybody's welcome to, you know, help each other out with stuff.

As always, I very much appreciate you watching,

and, if you want to consider translating this video into other languages,

I'll have a translation link down below, and thank you if you do.

Again, thank you very much for watching and I will see you later. Bye.

(DYNAMIC POP TUNE)

For more infomation >> Your Relationship Questions Answered! - Duration: 10:45.

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Dynamical Systems And Chaos: Universality in Maps Part 1 - Duration: 14:21.

In this lecture we'll examine

the period-doubling route to chaos in the logistic equation in considerable detail.

So here's the bifurcation for the logistic equation

R goes from zero to four

Period one goes to period two to four and so on

and then all this interesting behavior in here.

Here's a closer view of that. Again, you've seen these before.

This is from three to four. Period two to four to eight,

then regions of chaos

periodic windows

more chaos

and so on.

And what I want to do now is focus on just this particular portion of the bifurcation diagram,

where the periods double

and then eventually give us chaos.

So I'm going to zoom in on that even more

and we'll look at this bifurcation diagram.

So now this goes just from 3 to 3.6.

It's period one over here.

At 3.0 it becomes period two.

Then period four, then period eight, and so on,

and by the time we're over here

in these dark regions,

the function is chaotic

and it has sensitive dependence on initial conditions.

So if one looked for the butterfly effect over here,

one would see that yes, at 3.59 or whatever that is,

there is indeed chaos, aperiodicity, and the butterfly effect.

Over here there is not a butterfly effect.

The easiest way actually to test for that

is with a Lyapunov exponent.

Over here the Lyapunov exponent is positive.

And over here it's negative,

meaning that there's not sensitive dependence.

In any event,

there's a transition from periodic nonchaotic behavior

to chaotic behavior,

somewhere right around here.

So we'll look at what happens as we approach

that chaos-order transition point.

So I want to start by looking at the length of this.

So how long--in the sense of,

what range of r values gives me period two behavior,

what range of r values give me period four,

what range of r values give me period eight?

So I want to look at what these r values are

where these transitions occur.

And

I'll call the r value for this first transition r1

and that's 3.0.

So this transition

--I'll put a little arrow there--

occurs right when r is 3.0.

Then there's a transition that occurs here.

This is transition from period two to period four.

And it turns out that this occurs at

r is about 3.44948.

So I figured that out

by zooming in on the bifurcation diagram

looking more and more closely at this point

and trying as accurate a value for

where this line splits in two

as possible.

This was the intermediate,

one of the beginner-intermediate problems

from last unit's homework.

OK, there's also a transition then over here.

And this is where the behavior goes

from period four to period eight.

And I'll call that r3--

It's the third bifurcation.

And that turns out to be

at around 3.544089.

So that's where this transition occurs.

To the left of this transition we have period four.

A little to the right of this number

we have period eight.

And then lastly,

there's a transition from period eight to sixteen.

And that's really hard to see

on this diagram.

We'll have to zoom in a lot.

But if we did,

we would find that

that transition occurs where r is about 3.564407.

So we have four bifurcations,

from period one to two,

two to four,

four to eight,

and eight to sixteen.

And they occur approximately at these r-values.

And I'll call the first r-value r1

and then r2, r3, and r4.

We're interested in this length.

What range of r-values give us

period two behavior?

And I'll call that length

capital delta1.

So let me draw that here.

So I'll do it in the middle.

So this triangle is the capital Greek delta

and its just r2 minus r1.

This length minus that length.

So this is the range of r-values

for which we have period two behavior.

Then I'll...

Can do a similar thing for

period four

and then also

the period eight region.

So this line segment here,

this length right where my finger is,

is delta two.

and this small line segment here,

notice the regions are getting smaller,

is delta three.

And delta two will be r3 minus r2,

delta three is r4 minus r3.

Let me write that.

This is r4 minus r3,

and delta two is r3 minus r2.

OK,

so we've got these deltas.

This length, that length,

and that length.

And we observe,

as we've seen the bifurcation diagram,

these lengths are getting smaller.

So what we'll do next is

we'll calculate the ratio of this length

to that length.

And we'll call that lowercase delta.

So,

delta one is just

over delta two.

So little delta

--that's the lowercase Greek letter delta--

is just this length divided by that length.

And in terms of the R's,

let's see what'll that be?

That will be delta one,

which is r2 minus r1

over r3 minus r2.

So we have numbers for these

and we can plug these numbers in

and we can get a numerical value for delta.

So

let me do that.

So this will be

So I've just plugged in

the numbers I have for r2, r1, and r3.

And there they are.

And now, its subtraction

--I always like doing subtraction

on the calculator--

well let's see this one

I don't need to do.

3.44 blah blah blah

minus 3.0

is just .44 blah blah blah.

So this on top is 0.44

and blah blah blah is 948

And then this subtraction on the bottom

I better do on a calculator.

Let's see here

So I've got 3.544089

minus 3.44948

and that gives me 0.094609.

And the last thing to do

is to carry out that division.

I'll do that quickly.

Let's see

0.44948 divided by0.094609

and I get 4.751.

Okay, so

that's a bunch of calculator work

not tremendously exciting I don't think

but let's just interpret this number 4.751.

What that says is

is that this length is 4.751 times longer

than this length.

That's all it means.

We can do a similar thing now,

comparing this length to that length.

And if I do that.

Delta two is just

capital delta two over delta three.

It's this length

divided by that smaller length.

And I can define that in terms of R.

Carry out all this stuff,

there's no point in redoing all this subtraction.

If I do that,

I get the number 4.6564.

So I've skipped a lot of

intermediate steps here that you can just

do on a calculator reading off these numbers.

And we get this number.

And what this means

is that this length

is 4.65--about 4.66--times larger

than this length.

And we could keep doing this,

we could look at delta three,

delta three over delta four,

delta four, delta four, delta five,

and so on.

So we can ask ourselves

what happens

--we can try to do an experiment--

what happens,

as I go deeper and deeper into this transition,

the periods double more and more,

what happens to this ratio?

So here's a simplified view of what we just did

here's a schematic of the bifurcation diagram

period two to four to eight to sixteen

delta 1 is this length

delta 2 is that length

and so on.

And we're looking at the ratios

How much larger is this than that?

You call that delta 1,

and that turned out to be about 4.751

there's an approximate here because

this was an experimental result

and there's likely some numerical error.

It's very hard to pin down the exact value

of this transition

But this is pretty good.

And then delta 2...

that's how much longer is this length

than that length.

And that was about 4.6564.

And we can keep going

and define delta n as follows:

So it's just the ratio of

one periodic region to that

Let me say that again:

It's the ratio of these two deltas,

this is the length of one periodic region

this is the length of the doubled periodic region.

And what one finds

that as n gets large,

this approaches the number 4.669201.

So as we go deeper and deeper

into this period-doubling,

the lengths of these pitchforks

or branches

approaches a constant ratio

so that each branch--

this branch is about 4.669 times

as large as that one.

This is 4.669 times as large as that one,

and so on.

This doesn't become exact until n becomes large,

but its a pretty good approximation

even for small n,

even for the first set of period doublings.

So this says that as we get

closer and closer to this transition point--

the transition from periodic to chaotic behavior--

there's a regularity to that,

that we see new periods appearing

at this constant ratio.

So now we have a way of characterizing

the period-doubling route to chaos

in the logistic equation and

we can see if

the same thing holds true for

other bifurcation diagrams.

There are two quizzes following this lecture,

where you'll carry out some of this analysis,

but for the cubic equation

instead of the logistic equation.

In order to do that,

you'll need a program that makes

bifurcation diagrams for the cubic equation,

and there's a link to just such a program

in the section titled,

"Bifurcation programs" or

"Links to bifurcation programs"

on the navigation bar on the complexity explorer site.

So find that program and open it up

and give the next two quizzes a try

and see how these deltas work for another equation.

Subtitles by the Amara.org community

For more infomation >> Dynamical Systems And Chaos: Universality in Maps Part 1 - Duration: 14:21.

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Dynamical Systems And Chaos: Universality in Maps Part 2 - Duration: 11:11.

So we've been looking at period-doubling

and here again in this picture

we have period-doubling

from two to four, four to eight, and so on.

Capital delta is the range of parameter values

for which we have period 2.

Delta two is the next range,

delta three is the next range,

and we define these ratios

this divided by this,

this divided by that,

to be these little deltas.

And for the logistic equation,

we found that delta one is about 4.75,

and delta two is about 4.65/4.66.

In the quiz that you just did,

you did the same thing for the cubic equation.

And you found 4.419 and 4.618.

So the numbers for delta one and delta two

for these equations aren't similar--

sorry, ARE similar, but not identical.

However, if we kept going on and calculate delta three

and delta four and delta five and so on

these numbers would get closer and closer to each other.

So let me write that,

we're interested in delta n--

which is capital delta n over capital delta n plus one.

And this is going to go to the number 4.669201.

So for both of these equations,

as we let n get larger and larger,

we get closer and closer to this transition point,

these ratios approach 4.669201.

So in the large n limit, this number is just known as delta.

And we would say that delta is universal.

so delta is universal,

and what that means is

that it has the same value for

a very large family or class of functions.

Let me state this a little bit more carefully,

since it's a crucial result.

So again, the phenomenon of universality:

We have delta n approaching a number delta,

and we say that delta is universal.

And the result is

that this number delta is the same,

4.669201 for all iterated functions

that map an interval to itself

and have a single quadratic maximum.

So let me say a little bit about these conditions.

So if the function maps an interval to itself,

that just excludes the possibility that orbits go off to infinity,

either positive or negative infinity.

So the logistic equation is an example.

It mapped the unit interval to itself,

meaning numbers between 0 and 1

remain between 0 and 1.

Let me draw some pictures to illustrate this:

What does it mean

for a function to have a single quadratic maximum?

So single quadratic maximum...

So our favorite example, the logistic equation,

that's just an upside down parabola.

It has a single maximum right here

and it's quadratic because well,

it's a parabola, its a quadratic function.

Here's another example though.

This is let's see...

the cubic function looks something like this.

It's not symmetric

and the function is not exactly a parabola.

However, if we look closely near the maximum,

it would become more and more parabola-like.

So it's locally parabolic.

And in calculus terms, this just means

that the second derivative doesn't disappear.

So most functions that have a peak like this,

will behave this way.

Let me give some counterexamples.

This function does not have a single quadratic maximum

because it does not have a single maximum--

it has two: a maximum here and then there.

So this function does not meet the criteria

in that statement I gave on the previous piece of paper.

A few more counterexamples:

A function like this,

this has a single maximum, a nice peak,

but it's not a quadratic maximum.

If you zoom in on this point,

it doesn't start to look like a parabola,

it keeps looking like a point.

The idea is that this is infinitely-sharp.

Here if you zoom in on this,

you can make it look as close to a parabola as you wish.

So this does not have a single quadratic maximum

because, well, it has a maximum,

but it's not quadratic--it's pointy!

Here's one more example.

It has a maximum, but it's flat,

so it doesn't really have a single maximum value.

And if you zoom in on this line,

again, it won't look like a parabola,

it will look like a line.

So this is also not a function

with a single quadratic maximum.

So any function with a single quadratic maximum

it could be a parabola, a cubic function,

sine functions, various exponentials,

there are many many examples of such functions.

And this is a pretty generic criteria

in that if you just draw a function by hand

or cook something up that has a single maximum,

odds are it'll be smooth

and this criteria will hold.

So this isn't a very restrictive criteria.

And there's a vast number of functions that meet this criteria.

So let me state this once more

This property delta,

which is a feature of how the

sideways used those shapes on the bifurcation diagram,

how they're related to each other,

how much smaller each one gets

as we get closer to this transition,

that this geometric quantity delta is universal.

It has the same value, 4.669201,

and it goes on and on for all iterated functions f(x)

that map an interval to itself and

have a single quadratic maximum.

So again, this is not a very restrictive criteria.

So any function you come up with

that meets this very mild criteria,

you can make a bifurcation diagram

find the bifurcation points like we did,

calculate the deltas,

go deeper and deeper

and let n get larger and larger,

and this number will appear.

So this number is a property of all of those functions.

So I want to emphasize

just how amazing this result is.

We started with the logistic equation,

just about as simple a nonlinear equation

as one could imagine,

and we saw that we had this bifurcation diagram

with an amazing amount of complexity,

but also with some regularity--

we saw those pitchforks,

those sideways Us,

repeating again and again and again,

and we noticed that there's

sort of a geometric similarity to them,

that the next pitchfork in the sequence

is smaller than the previous one,

but maybe smaller by the same factor.

So that led us to investigate that idea quantitatively.

And we defined these lowercase deltas

as that ratio of one pitchfork length

to the next pitchfork length.

And we found that that ratio approaches a constant number,

this number 4.669.

And that's an interesting result for the logistic equation.

It's a practical mathematical oddities tells us about

the geometry of this particular bifurcation diagram.

Then we looked at bifurcation diagrams

for other equations--quite different equations:

sines and cubes as well as parabolas--

and we saw the same general features of the bifurcation diagram,

but then if we calculate these deltas,

and we kept calculating them

to larger and larger periods,

we would find again the same number appearing--4.669.

So that's where things started to get really weird--

that we're seeing the same number appearing

in very different equations.

So that tells us,

at some level the equations,

or the details of the equations don't matter.

There's some broader or overarching

feature of these systems thats

independent of the particular equation

that we use.

So that's something deep and

something pretty surprising I think.

Let me mention just a tiny bit about

the history of this idea and this realization.

The results I presented date from 1978.

They're usually attributed to Mitchell Feigenbaum,

and American physicist who discovered

this property and then did some

analytical work to try to understand it further.

But it was also discovered independently by

Charles Tresser and Pierre Coule,

around the same time,

also published in 1978.

So this is a relatively new result.

Another thing I want to mention is that

I presented these results experimentally--

the result of doing some numerical work on a computer,

but there's a lot of very elegant

and very powerful analytic work

that calculates these numbers--4.669201--

and explains why this number appears again and again.

The mathematical framework for

carrying out that analysis is know as

a renormalization group,

or just renormalization,

and I'll say a little bit about that

in a subsequent unit.

I can't explain it in detail.

It's just too much math for the level of this course.

But I can maybe give some sort of an argument for that.

But before I do that, I want to mention the next thing,

which is,

that this result is not just mathematics,

So it's an amazing mathematical result,

these one-dimensional functions

have these beautiful bifurcation diagrams

with the same branching,

or same sort of fork-ratio in them.

But this is physics as well.

Period-doubling occurs in real physical systems,

and one can go out and measure the rates

at which those period-doublings occur.

And one finds again, this number 4.669.

So I'll describe that result

in the next lecture.

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