Waching daily Jan 29 2019

Here's an overview of Euler's method

for solving differential equations of this form.

The derivative of some unknown function X is just a function of X,

and we assume that we know the starting value of this function X,

- so this could be temperature, this could be position, could be anything.

So, we start - we need to choose a step size,

- that's a choice that we get to make, and we'll start at time t equal 0,

where we know the X value.

So, then using the current value of X,

the equation - the differential equation - this function,

tells us the rate of change.

So, we know how fast X is changing.

We then use this rate of change

to determine the next value for X,

This is the current value for X,

this is how much X changes, in the time interval delta t.

This is a little bit of make-believe

- we're pretending that the rate of change is constant

for this time interval delta t, and we can use that

to figure out the next value of X,

then, we increased t by delta t

and we go back to this step, and we figure out the derivative again.

The derivative tells us how to go forward a little bit in time

to figure out the next X, then we update time, figure out the derivative again,

move forward to figure out X.

So 2 and 3 - these are the key steps here,

- wer'e constantly shuffling back and forth,

the function - the differential equation - tells us the derivative,

- here we use the derivative to figure out the value of X,

and then we go back, X gives us the derivative from the differential equation,

- we use a derivative to figure out X, and so on.

So, one repeats through these processes

until you have enough of a solution.

So, in practice, one would choose smaller and smaller delta t

until the solution curve stops changing.

So, if you chose a delta t of 2, and then 1, and then .01, and .001,

doing this on a computer, or a spreadsheet,

and eventually one would see that your table of values, if you plotted them,

your X's would stop changing, and that would be an indication

that you're delta t was small enough.

So, this is Euler's method in a nutshell.

It gets at the heart of what the differential equation means,

- the rate of change is given by the quantity X,

we use that rate of change to figure out X,

so, again we're thinking of this as a dynamical system,

- it's a rule that specifies how X changes,

the rule is written in terms of the derivative,

- the rate of change of X, rather than X directly,

but, not a problem, Euler's method, or things like it,

let us convert from rates of change in to the function itself.

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For more infomation >> Dynamical Systems And Chaos: Computational Solutions Part 4 - Duration: 2:56.

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Dynamical Systems And Chaos: Computational Solutions Part 2 - Duration: 4:23.

So, let's continue with this example.

We just found the T(2) was 11, or approximately 11

because we had to do some make-believe to get this,

but now let's see if we can figure out T(4).

I can figure out how fast the temperature is changing

at time 2, assuming that the temperature is 11.

What's the rate of change? Well I just ask the equation

- that's what the differential equation does - it's a rule that tells me how fast

the temperature is changing, if we know the temperature.

So let's do that.

So we use the equation - we ask the equation:

When the temperature is 11, what's the rate of change? what's the derivative?

So, when time is 2, we plug in 11, so capital T is 11, 20 -11 is 9,

times .2 is 1.8

So now we know that when the temperature is 11,

it is warming up at 1.8 degrees per minute.

So now suppose we want to know T(4), 4 minutes in,

again, we have the same problem

- this rate isn't constant - it's changing all the time,

as soon as a temperature changes we get a new rate,

but as before, we'll ignore the problem

and pretend that it's constant.

So, again the problem is: the rate is not constant

- our solution is to ignore the problem

- not always a good way to go about things

but for Euler's method, it turns out to work okay

- we'll ignore the problem - pretend it is constant

and then we can figure out the temperature at time 4, 4 minutes in,

in these 2 minutes, that we're pretending:

how much temperature increase do we have,

well at 1.8 degrees per minute for 2 minutes, that's 3.6,

3.6 +11, where we started, gives us 14.6

So now, I know the temperature at T equals 4 minutes.

We can keep doing this,

continue along with this process, and we'll get

a series of temperature values for a series of times.

So, we continue this process,

and we can put our results in a table.

So these first 3 entries we've already figured out

- the initial temperature is 5, then at time 2 it was 11,

at 4, it was 14.6, and at 6,

if when one follow this process along, one would get 16.76,

and we could keep on going.

So, let's make a graph - let's make a plot of these numbers

and see what it looks like, and compare it to the exact solution.

So, for this equation, it turns out one can use calculus to figure out

an exact solution for this differential equation,

and that shown as this solid line here.

Towards the end of this sub unit, I'll talk a little bit about

how one would get this solid line.

The Euler solution - that's what we're doing here

- are these squares - so we start at

the initial condition, and then here at 11,

a little bit less than 15, almost 17, and so on.

So we can see that the Euler solution

- the squares connected by the dotted line

is not that close to the exact solution.

It's not that bad, but it's not a perfect match

and we wouldn't expect a perfect match

because we had to do some pretending in order to get this.

So, as is often the case, ignoring the problem

- remember the problem was that:

the derivative - the rate of change wasn't constant.

Ignoring the problem actually wasn't a great solution

because we have these errors here.

For this example, I'd chose a step size of 2, a delta t of 2.

I said: let's figure out the temperature, capital T, every 2 minutes,

but it's this step size that got us into trouble

because I had to pretend that a constantly changing rate

was actually constant over this time of 2 minutes,

and that's clearly not true,

so, a way we could do better with this Euler method is to use a smaller delta t.

For more infomation >> Dynamical Systems And Chaos: Computational Solutions Part 2 - Duration: 4:23.

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Real Life Paranormal Experiences | Paranormal Story Time - Duration: 11:54.

real life paranormal experiences, a paranormal story time series

so today's video is real life paranormal experiences paranormal story time super super cool because for the first time ever I'm

actually gonna be sharing with you a paranormal story or a few paranormal

stories actually that came from one of you guys so Leslie sent me an email

sharing some of her absolute craziest paranormal experiences and I was blown

away reading the stories I just thought these are so cool I've got to share them

with you guys so I hope that reading them I do them justice let's see thank

you so so much again for sending them in to me I'm so

grateful that you've taken the time to write them out so get comfortable here

we go so here are my experiences my first experience was when I was 12

living with my mom and aunt as well as the two dogs that we had and cats back

then we all lived north played Nebraska back then I wasn't into the paranormal

although I've always believed in demons and ghosts since I grew up in a

Christian home I normally always close my closet door before I went to bed but

one night I forgot to close it I realized I forgot to close it when I saw

that both of my closet sliding doors were open when I looked to see that I

saw what today I would say was what looked like the Grim Reaper black cloak

with a hood over its head hands that were skeleton and it was holding a stick

like you would when you're hiking and need a stick to help you hike up a hill

it was scary to me having never seen a demon in person before so I rolled over

so I couldn't see it and prayed for it to go away when I looked over again when

I got the sense that it was okay to look it was gone the strange part is that one

it was around the time that my Nana died so I saw that thing before she died of

pneumonia I never told my mom about it she was highly religious and I didn't

think it was necessary for her to know because I was hoping I'd never have to

see that thing again and I never did see it ever again the other

thing is that my lab crossed German wire-haired dog perdy slopes to beside

me in bed every night and she never reacted to what I saw my second

experience didn't happen until I moved in with my sister and brother-in-law by

law my mother was seen as an unfit parent to raise me so when I aged out at

18 I was asked where I wanted to live I chose my sister and brother-in-law at

the time they lived in Scottsbluff Nebraska they had two dogs one was a

German Shepherd Kraus Rottweiler named storm whom was my brother-in-law's and

the other was destiny whom was my sister's dog assume he has seizures when

her anxiety level is up too high she's a rare breed a flat coated black retriever

after quite a while of living in Scottsbluff Nebraska with them in my

high school years my sister began feeling very ill in the

house it was every single time she was in that house the doctors in Nebraska

didn't know what was wrong with her the dogs every single night whenever my

sister and brother-in-law kenneled them in the basement and their cages they

wouldn't stop barking they really hated that basement they were always barking

to be let out one time when I went down there when I was sent to go quiet them

they were growling and looking at the middle of the basement then one night

when I got myself a coffee mug of milk or something since I have stared into

space seizures and have to take something with my prescription I went

into the living room and saw three ghost children standing in front of the couch

I just looked at them it was probably somewhere between eight and ten at night

since that was my bedtime when I was in high school there were three different

ages I'm not a good judge of Ages but I can just assume maybe seven ten and

thirteen or something like that all three of them were boys and they weren't

transparent I didn't get any sense that they were evil or bad it just seemed

that they were sad that was the only time I saw them they probably wanted

help but they never spoke to me after he moved out of that house and went to go

live in Torrington Wyoming my sister did some digging she convinced the landlord

of the Scottsbluff house to give her information about that house and that

information is spooky too even this the landlord said that they had to

replace the windows and get rid of a pentagram on the basement floor the

windows had been all spray-painted black we all knew that something was wrong

about that house but we don't know what all happened there I could only assume

that whatever was bad in the house or evil is why when we moved my sister was

diagnosed with fibromyalgia and Wyoming said that that's what she has shortly

after we move out of that house I could only assume to that the ghost

children were sad because they probably were sacrifices or trapped there or both

and the way storm and destiny were growling in their kennels and me feeling

that something bad was there that night in the basement makes me believe that

destiny and storm probably saw some kind of demon while after we moved into the

Torrington Wyoming house and my sister was still trying to sort things out I

was the first to know we had a basement ghost my sister had boxes and what we

called the big room where she was still trying to sort out what to keep and what

not to what bloomed in her office there was a

door on the back of the kitchen that opened up to a couple steps landing and

stairs after the stairs there was a laundry room to your left and straight

ahead was the big room then when you went around the corner to your right he

would see a hallway in some rooms first one was my sister's office second one

was the guest bathroom and third one was my bedroom the landing I mentioned was

where there was the back door in the backyard although it wasn't fenced in

the dogs had to go on leashes instead the basement was all carpet apart from

the laundry room one day my sister was doing I don't remember what upstairs she

asked me to go fetch something amongst the boxes in the big room I didn't know

what box that was in so I had to look after not finding it I got up from the

floor and that's when I saw what looked to be maybe a floating skull it was just

like oh we have a basement ghost now and went upstairs to tell her that I

couldn't find but she wanted me to get for her a while after that incident I

would be in the basement and feel a hand on my shoulder or the

feeling of a ghost behind me it always listened to me though when I play it

would ask it to go away and stop touching me then my brother-in-law told

my sister and I his experience back then he worked on the EOP railroad so he was

usually up at all hours wee hours in the morning he was sitting in the living

room watching TV as usual one night and we out were in the morning when he kept

hearing the sound of someone running up and down the basement stairs when it

stopped at the landing my brother-in-law pretty much told the ghost well come on

in a moment later he watched the door open slowly that gave it an invitation

to come upstairs that's when Strom and Daisy would start

seeing the ghost storm would stare into space at nothing while destiny would

bark at nothing one evening my brother-in-law and sister were watching

TV on the couch in the living room the dogs were both sleeping by them when

destiny bolted right out of dead sleep and stood on her paws looking at midair

and barking frantically that Ghost spooked her when my sister told it to go

downstairs though destiny stopped barking one afternoon my sister and I

were talking and doing dishes when destiny started barking again at midair

and my sister told it to go back downstairs and again she stopped barking

my sister actually gave that Ghost a name George I don't know why she did

that a lot later my sister and brother-in-law nearly got divorced my

brother-in-law even bought a house in the country in Lingle Wyoming and took

strong with him destiny stayed with us while we stayed in the house we were

separated for a while but after they decided to get back together again we

moved into the house my brother-in-law had bought in the house in Torrington

that was where destiny would have a lot of seizures and we didn't know why and

it started happening a lot around the time that my sister and brother-in-law

were having marriage problems and almost got divorced after moving into the house

and Lingle I immediately didn't like a certain area in my room I felt like a

demon was in the corner by the left side of my closet during the day and evening

and it felt like when it was late the demon would stand in front of my closet

when I went to bed I would sleep with my Bible and pray and cover my head I

didn't want to see anything I always felt that when I went to bed it would

watch me sleep and sometimes it felt like it was standby where I was sleeping

but I just pray and recite a scripture and close my eyes so I wouldn't see

anything and I'd pray for protection against it also I should note that I

don't know if it was just because we lived in the country or what but we were

always getting mice in the house also in that house my sisters from George died

there was a lot of things that happen in the house I was happy when we all moved

back into the Torrington house after I graduated from high school we all moved

out here to Arizona even though I didn't want to go my sister is a city girl and

my brother-in-law's a country boy so they compromised and chose Maricopa as

our home nothing happened for a while but probably about I'd say a year after

we moved in I started hearing noises up in the attic

that we literally never used it's the kind of attic that you have to pull the

string to open it I'm never one to investigate if I was in a horror film

I'd be the quiet one that doesn't go up to the attic when everyone else wants to

I know better than that it's not that I'm afraid of any ghosts but I don't

want to see anything scary anyways after maybe a year of living there I adhere

things in the Attic it would sound like someone was up in the attic walking on

all fours sometimes it was quick and sometimes it

was slow then shortly after my brother-in-law's dad died of cancer we'd

start noticing things like finding certain pictures facing down on the

shelf instead of the way they're supposed to be or some of them facing

towards the wall instead of outward so you can see the picture

my sister uses an IKEA shelf for the pictures and other things so it doesn't

have a back into it you can see the wall that the shelf is against when you look

at the shelf if that makes sense about four or five years ago I moved out of

Maricopa into Glendale about ten blocks away from Phoenix and my sister says

that the pictures still move around I live in a crappy complex infested with

roaches since I'm low-income and Phoenix is

expensive as well as Glendale I have a ghost now too it was here when I moved

in I noticed it when it started making noise

I had hear the sound of someone in cowboy boots walking from one end of my

bedroom to the other end or an unidentified banging noise and when my

guitar would be in the corner and when you'd walk past one of the strings would

play I had even here knocking from behind my fridge in my fridge isn't on

any wall to where it could have been one of my neighbors and once in a while I'll

hear what sounds like someone squeezing an empty water bottle even though I

don't keep empty water bottles my ghost is usually quiet now though so I think

my ghost has accepted that noises don't scare me

also there was one afternoon about a year or two after I moved in that I had

to hunt down my apartment key finally I found it neatly placed under my bed I

also saw that a picture of a friend of mine that I went to north plate high

school with was moved around a little now my ghost normally just closes my

door to a crack I have a weird apartment where there's a door in the middle of it

if you close the door you can't see my bedroom or tiny hallway or where my

bathroom is I call my ghost an introvert because it closes my door and whenever

I'm not in the back part of my apartment I assume that it goes to the back part

of my apartment because I have to turn the heat on a while before I go to bed

because the back of my apartment will be chilly or even freezing no matter what

season it is so that's the end of my experiences of the paranormal if I have

any more I'll update you in the future about them if you have your own

paranormal experience that you would like to share with everyone please send

it to the email address on the screen

For more infomation >> Real Life Paranormal Experiences | Paranormal Story Time - Duration: 11:54.

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Dynamical Systems And Chaos: Differential Equations Summary Part 2 - Duration: 8:19.

Let's summarize and review the key results and ideas

from unit 2, on differential equations.

A differential equation is a type of dynamical system,

and we've looked at differential equation of this form,

again, the derivative of some variable,

is just a function of that variable.

Remember, a dynamical system is a system that changes in time,

according to a well specified rule,

and a differential equation is such a dynamical system.

The differential equation specifies the derivative of X,

as a function of X - that's the rule,

and the derivative, which if you haven't had calculus,

might be a new term,

- is just the instantaneous rate of change of X.

It's how fast X is changing, at a particular moment, or instant in time.

There are three main classes, or types of solution methods for differential equations:

first, is qualitative, or geometric techniques.

- there, we might sketch the right hand side of the differential equation,

and from that we can figure out fixed points, and their stability;

we can draw a phase line, as I've done here,

and we can sketch the general shape of solution,

the directions in which the solutions are going.

We can't however get an exact form for X of t,

we don't necessarily know how fast the solutions go,

but this method is good for giving an overall feel

for the long-term behaviour of solutions,

- where do orbits go? how many fixed points are there?

and what are there stabilities?

Another approach to solving a differential equation, is: computational.

So, I presented Euler's method,

and one could use Euler's method, or a fancier version - one of these Runge-Kutta methods,

to figure out to X of t, and one does so, step-by-step.

We start with an X value - we use that X value to figure out the derivative,

the derivative tells us the X value a little while later,

and we can go back here, plug X in, the new X,

and figure out the new derivative,

- a derivative we can use to figure out the X value a little later,

and so on, and so we're constantly swapping back and forth

between these two sides of the equation.

I think Euler's method gets at the heart of what a differential equation is:

a rule specifying how a quantity changes.

Computational techniques like this are reliable

and they work for all well posed equations.

It does require the use of some computer software, or spreadsheet.

These results are often called numerical results

because the end result is a table of numbers, not a formula,

but it's very easy to plot that table of numbers,

and get a feel for what the solutions are doing.

The last type of solution method is

one that I actually have only hinted at so far,

and that's a type of method called analytic.

So, here, the task is to find a formula

for the solution X of t, using calculus.

So calculus - there's a well-developed machinery

for finding derivatives, doing derivatives backwards, and so on.

So, depending on your point of view this can be

a lot of fun, or not so much fun at all.

I myself had mixed feelings about it.

My first differential equations class in college

felt like all the worst parts of calculus 2 coming back to haunt me.

Later on, in grad school, learning some other techniques

for solving differential equations: power series, and Laplace transforms,

- I enjoyed a lot, but anyway, it's a very different approach from the first two.

It uses all the machinery of calculus.

The bad news is, is that most nonlinear equations,

and that's what we'll be studying in this course by an large,

cannot be solved analytically.

In some cases, one can even prove that there isn't an analytical solution,

so, geometric, or numerical and computational solutions are necessary.

Moreover, even for equations that can be solved analytically,

doing so does not always lead to intuition, or understanding.

You might do a bunch of calculus tricks, and get a weird-looking formula,

and that's certainly a valuable thing,

but it might not give you a feel, or intuition,

for what the equations are doing.

In my view, many certainly not all, but many differential equations text books

place too much of an emphasis on analytic techniques.

This course will focus on qualitative and computational solutions,

- I think they're much better suited for dynamical systems and chaos,

particularly for an introductory course like this.

OK, let me review some of the key terminology from this unit,

and it's actually almost the same as the terminology from the previous one.

So, differential equations, just like iterated functions,

have fixed points, and a fixed point is a point that doesn't change.

In differential equations one often calls a fixed point an equilibrium point,

but it means the same thing.

A fixed point X is fixed, if it's derivative is 0.

If your derivative is 0, you aren't changing,

and if you aren't changing, you're a fixed point.

Fixed points have stability as well - they can be stable, or unstable.

A fixed point is stable if near by points

move closer to the fixed point when iterated,

or, perhaps I should say, if you have an initial condition

near the fixed point, and you solve the differential equation

it moves closer to that - to the fixed point.

A stable fixed point is also called an attracting fixed point,

or an attractor, or in differential equations, it is sometimes called a sink,

because you can imagine a lot of solutions all head into this one point,

so it looks like water going down a drain.

A fixed point is unstable if near by points move further away from it,

and an unstable fixed point is called a repelling fixed point, or a repellor,

- it's also called a source: you can imagine a lot of solution lines

or solution curves emanating from this repelling fixed point.

So, source and sink - I don't think I'll use those terms much,

- but they're pretty standard, so you might encounter them elsewhere.

Just as we did for iterated functions,

we can draw a phase line for differential equations,

and it lets us see, all at once, the long-term behaviour for all initial conditions.

In a phase line we lose time information, so for example here,

I know that solutions move towards 9, but I don't know how fast.

The phase line here is for a differential equation that has an attractor at 9.

- things are getting closer to 9, and a repellor at 1,

- things are getting pushed away from 1.

So, differential equations are a type of dynamical system,

and in dynamical systems, one of the goals of study,

is to classify, and characterize, the sorts of behaviours that we see.

So, for differential equations, what have we seen?

- well we've seen: Fixed points (stable and unstable).

- Orbits can approach a fixed point.

- Orbits can tend towards infinity,

or tend towards negative infinity - they can move off the ends of the phase line,

and that's pretty much it,

and additionally, an orbit cannot increase and then decrease,

- it's rate of change is only a function of X - it's value,

so that means that cycles or oscillations are not possible.

In the next several units we'll see that

differential equations are capable of doing some more interesting things

but we'll have to go to higher dimensions

in order to see this more interesting and exciting behaviour.

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For more infomation >> Dynamical Systems And Chaos: Differential Equations Summary Part 2 - Duration: 8:19.

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Dynamical Systems And Chaos: Computational Solutions Part 5 - Duration: 5:49.

I'll conclude this sub unit on computational methods

for solving differential equations, with some more general remarks.

So, we've been discussing Euler's method,

- Euler's method is known as a numerical

or computational method for solving the differential equation.

It's computational because it involves doing computations

using, almost always, a computer,

and it's called numerical - that's its may be more traditional term,

because the result of Euler's method is not a formula for a function

but it's a list of numbers, so we call it a numerical method.

Euler's method is quite simple conceptually,

and it gets at the heart of what difference equations are:

- a rule for how something changes,

and that rule is written in terms of the derivative,

- the rate of change, not the function itself.

So, I think if you understand Euler's method,

and may be code it up yourself,

then you really have a good understanding

of what differential equations mean,

so, I definitely recommend if you have some programming skills,

- the language doesn't matter at all,

and you can use a spreadsheet,

to try coding up Euler's method yourself.

I'll list this in the homework for this unit.

So, Euler's method is very nice conceptually,

however it's not very efficient computationally,

so, it's not used much in practice.

Let me say a little bit about how one can improve on Euler's method.

So, to improve on Euler's method,

there are two things one might like to do:

first, there's a family of techniques,

a group of techniques, known as Runge-Kutta methods,

and here's the idea behind them:

I'm not going to go into this in details,

but It's worth mentioning.

So, in Euler's method,

we pretend that the rate of change is constant, over an interval delta t,

and then we have to choose what rate of change

we're going to use, and we just choose the rate of change

at the left part of that interval.

We just take the starting rate of change in that interval

and pretend that it's constant for all of delta t.

Runge-Kutta methods say,

well, rather than using the rate of change at the start of the interval,

what if we use the rate of change at the start of the interval,

and at the end of the interval, and average those two.

That would probably be a more fairer representation

of what's happening in that interval,

or better still, maybe we could sample the rate of change

at three different points along that interval,

and there are different schemes for doing that sampling

and different ways of averaging the different derivatives

that one estimates, but that's the general idea.

Rather than just use one derivative,

sample a couple derivatives, and average those.

So, that's one way to improve on Euler's method.

It's not immediately obvious, but this turns out to be

more efficient, in the sense that, with less computational effort

you can get an equally accurate answer.

The other thing that one typically does

is something called adaptive step size,

and this is: we have the program automatically adjust delta t

on the fly - as it's doing, as it's trying to find a solution.

So, delta t needs to be small

when the derivative is changing rapidly.

We get into trouble with these methods

when we pretend the derivative is

constant over delta t, but it actually changes a lot.

So, if we have a situation where, sometimes,

as time goes on, the derivative is changing rapidly,

and other times, it's not changing rapidly,

then we don't need to use the same delta t.

If the change in derivative is slow,

we can use a large delta t,

when it's fast we need to use a small delta t,

and so these adaptive step-size methods

figure that out on the fly.

I sort of think about it - adaptive step size - as this way,

and this is a rough analogy but it may be gives the right idea,

- imagine you're walking across a landscape,

and you're blindfolded - you can't see.

If the landscape is flat, you can take very big steps,

and you're not going to miss anything,

but if the landscape is very bumpy,

then you'll need to take small steps

to make sure you don't trip or miss something.

So, if you're blindfolded, you might adjust your step size

depending on what you sense the terrain to be.

OK, in any event, the standard way to improve on Euler's method

is to do these two things:

some type of Runge-Kutta method,

and some type of adaptive step size.

Almost all numerical programming environments, that I'm familiar with:

MATLAB, Octave, Maple, Mathematica, Python

- have some sort of built-in Runge-Kutta, adaptive step size solver.

So, perhaps in the form, some of you who have experience

with these different things,

and maybe have solved differential equations before

can post some examples

for how to use these different built-in functions.

OK, lastly, may be just take a look ahead.

So, in the next units, I'll be frequently presenting

solutions to differential equations,

and almost always, the solutions that I present

and show you and discuss will be numerical solutions.

In order to do this course you don't need to

solve differential equations on your own,

you don't need to code up your own algorithms,

you don't need to use other peoples' algorithms,

however, I think it's important that you have

a sense of where these numerical solutions come from.

So, if I show you a solution to a differential equation

you have some idea where that solution came from,

- it's not magic, it's just from using something like

Euler's method - a very simple, but repetitive way

of solving differential equations.

Subtitles by the Amara.org community

For more infomation >> Dynamical Systems And Chaos: Computational Solutions Part 5 - Duration: 5:49.

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"White Hat Hackers" PUSH Progressive Causes - Duration: 11:13.

For more infomation >> "White Hat Hackers" PUSH Progressive Causes - Duration: 11:13.

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Dynamical Systems And Chaos: Qualitative Solutions Part 1B - Duration: 5:09.

Here's a version of Newton's Law of Cooling that could apply to an object that is in a room with the temperature of 20.

Now, in this equation capital T is temperature, and lower case t is time.

And, we 're interested in how capital T changes over time. So, given that the initial temperature of the beverage is 5 degrees,

I would like to know what is the temperature at all other times.

So, given that T(0) is five. That means the temperature at time little t=0, is 5.

That's the initial temperature. We would like to know what is T(t). How does the temperature vary; how does it change as a function of time?

So, if we can find this, we'll say we've solved the equation or found a solution.

Finding this would be like finding the orbit or itinerary of an iterated function.

So, in this subunit, I'll describe some qualitative methods for figuring out the general behavior of these types of differential equations.

And the key is going to be to graph the right hand side of this equation.

So, let me do that and we'll see what it looks like and what it tells us.

So, what I've done is I've drawn a plot/a graph of the right hand side of this equation.

So this purple line is this function.

If we don't see how to make this graph right away, don't worry. You can graph it on a computer, but for this course I'll provide you graphs like this.

Okay! So, let's think about what this tells us. On the horizontal axis is temperature in degrees Celsius.

So that's T as we vary t.

And then we interpret this as the derivative, the rate of change of the temperature. So this would have units of degree Celcius per minute.

How fast is the temperature warming up or cooling off.

So, if we have water that's right around 0, then it would be warming up at 4 degrees per minute.

So, we use this graph to read not the direct value

of the temperature. But if we know the temperature, we can

figure out how fast the temperature is changing.

If we are at 10 degrees C, then this says, and sorry the scale is off a little bit,

but, if we are at 10 degrees C, then we are warming up at 2 degrees per minute.

If we are at 20 degrees C then we are not warming up at all.

The rate of change of the temperature is 0, because the purple line

goes through 0.

Or, if you plug 20 here, you get 20 minus 20; that's 0, so the rate of change

, i.e. the left hand side of this equation, is 0.

If we had something at 30, maybe a warm cup of tea,

then its derivative, its rate of change according to this function is -2.

So, it's warming up at -2 degrees per minute.

Or you would say it's cooling off at 2 degrees per minute.

Its temperature is decreasing at that instant at 2 degrees per minute.

So, from this type of graph we can go immediately to a phase line for the solutions for this differential equation.

So, let me draw that.

There is one fixed point or equilibrium value, and that's at 20.

20 is fixed because the rate of change when the temperature is 20, is 0.

If you put a glass of water that's at room temperature in a room of 20 degrees, it will stay at 20 degrees.

If we are below 20, cooler than 20, we know that the water, the object will warm up. And we know that from everyday experience.

We can also see that from this graph. The graph, which we interpret as the derivative, is positive,

that means that capital T, the temperature is increasing so we move this way. And it increases until we get to 20.

And if capital T is over here, so at 30 or 40 degrees, we know that the temperature will decrease, will cool off to room temperature.

And we know that from everyday experience, but we also can see that from the function.

The purple line is negative here, that means that dT/dt, the rate of change of temperature, is negative.

The temperature is getting smaller. If I plug in 30 here, I'll get a negative number,meaning the temperature was decreasing.

So, immediately from this graph, we can get a lot of information.

We see that there is a fixed point, or an equilibrium at 20.

And we see that it's stable or attracting.

Subtitles by the Amara.org community

For more infomation >> Dynamical Systems And Chaos: Qualitative Solutions Part 1B - Duration: 5:09.

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Dynamical Systems And Chaos: Qualitative Solutions Part 1A - Duration: 2:21.

In this sub unit, I'll present a

particular example of a differential

equation that we'll solve and analyze a

number of different ways.

The differential equation that I'll use is known as: Newton's Law of Cooling.

It describes how objects warm up or cool off.

You're probably familiar with it already even if you haven't studied physics.

For example, suppose you're home, after a

long hard day of recording videos about

derivatives and you open a cold beer....

So, I just took this beer out of my refrigerator, where, let's say, it's 5 degrees.

The room temperature is probably around 20 degrees.

So, as we probably have all experienced, this cold beverage is going to warm up,

and it will warm up very quickly at first.

The rate of warming up is largest when the difference

between the beer temperature and the room temperature is the largest.

So it will warm up very quickly at first, and then less quickly, and less quickly,

until it reaches room temperature at 20 degrees.

So, I'm aware that this beer is warming up, so I think I'm going to focus on this, and will finish this lesson tomorrow morning...

The differential equation I mentioned last night is known as Newton's Law of Cooling,

and here's what this equation says in words: the rate of cooling or warming,

either way, is some function of the temperature T, the temperature of the object that is warming or cooling.

And, in particular, that function is proportional to the difference between the object's temperature and the room temperature.

So, if the object is a lot hotter, or a lot colder, than the room temperature, its temperature will change very quickly.

If the difference between the object's temperature and the room temperature

is very little, it won't be warming up or cooling off very quickly.

So, this is Newton's Law of Cooling in words. Let's take a look at the equation.

For more infomation >> Dynamical Systems And Chaos: Qualitative Solutions Part 1A - Duration: 2:21.

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Dynamical Systems And Chaos: Newton, Laplace, and Determinism Part 1 - Duration: 7:02.

I'll end this unit by talking a little bit about some ideas

from the history of philosophy and science,

focusing on Newton and Laplace.

This isn't a definitive or comprehensive history by any means.

I just want to highlight a few ideas

that will help set the stage for the next unit,

when we encounter chaos and the butterfly effect.

My main goal is not so much to get a historical account,

but to look at some historical ideas

to help maybe elucidate, or bring to the forefront,

some ideas about science, and how science goes about explaining the world,

and some assumptions about the world, that I think are still with us today.

And these are some ideas that the discovery of chaos

and various related phenomena

cause us, I don't think to throw out completely,

but maybe to re-examine or reconsider a little bit.

In any event, our story begins with Isaac Newton.

In 1687, Newton published the Principia Mathematica,

in many ways the crowning achievement of the Scientific Revolution.

So, there's a lot that Newton did in this,

but one of the main things he did was, he laid out a theory of motion.

He explained why things move.

And this is most clearly embodied

in what's now known as Newton's Second Law of Motion.

You've likely seen this before

if you've taken a physics or physical science class.

So the equation says –

On the left-hand side we have: F⃗_{net}.

F is the forces, and net means total.

So we're looking at the total forces that are acting on an object.

The arrow on top of F means that force is a vector quantity.

When considering the total forces,

we need to take into account not just how strong a push or pull it is,

but the direction of that push or pull.

And vectors are a mathematical way for doing that.

The right-hand side of the equation is: ma⃗.

a is acceleration.

It's the rate of change, or the rate of change of the position,

or the rate of change of velocity –

how fast something's slowing down or speeding up.

Like forces, acceleration is a vector quantity.

In order to describe motion

I need to say not only the magnitude of acceleration but also the direction.

And m is an object's mass.

So what Newton's second law says is

that if I know the total forces acting on an object,

and I know its object's mass,

I can figure out its acceleration,

and from its acceleration I can figure out how it will move.

The basic idea is that forces determine motion

in the same sense that the functions,

iterative functions and differential equations,

that we've been studying so far,

are deterministic.

So Newton's second law is a rule;

it tells us how things move if you know the forces that act on them.

So the second law is – I think of it as a theory of motion,

it says why things move.

Another key part of Newton's Principia

was that he put forth the idea that the laws of physics are universal.

And in many ways I think this is the most revolutionary part of Newton's work.

So he said that the laws of physics

that describe how an object falls in Cambridge, England,

those laws are at the same in Brussels or Beijing or Bogota, anywhere.

Then not only anywhere on the Earth, anywhere in the Universe.

So the same law that describes how an apple falls in England

describes how the Moon orbits the Earth or how the Earth orbits the Sun.

So, as I said before, Newton's Principia is in many ways

a crowning achievement of the Scientific Revolution.

And to summarize in very brief strokes,

the idea of the Scientific Revolution

is that knowledge can be generated through careful observation

and repeated experiment,

building on the work of others,

and that that knowledge is expressed

in a coherent and consistent logical framework

that's often mathematical in nature.

So Newton in the Scientific Revolution

gives way to a period of intellectual history called the Enlightenment,

that I think of as spanning most of the1700's.

The Enlightenment is associated

with expanding individual rights and democracy,

continued scientific advances,

a belief in the power of logic and reason over authority and doctrine.

So during the Enlightenment

a scientific method and scientific worldview starts to solidify.

And I sort of think of this as a Newtonian framework.

So, some ideas about the world and about science

that aren't directly contained in Newton's Laws

but sort of flow naturally from them.

So, in the Newtonian framework,

the world is mathematical, mechanistic and material.

It's mechanistic because things happen for a reason,

objects move because forces act on them.

So motion can be explained,

things happen not just because, but for particular reasons.

It's mechanistic in that the world is seen

as being made up of stuff, material objects.

Even forces that act apparently at a distance, like gravity,

were viewed as being transmitted by a tangible physical medium.

So it's a world of things, of stuff, in which things happen for a reason.

And lastly the world is seen as being mathematical.

The laws of nature, and maybe nature itself,

is seen as being mathematical.

And Galileo has a quote that summarizes this idea really nicely.

So Galileo says: "Philosophy is written in this grand book of the Universe,

which stands continually open to our gaze.

But the book cannot be understood

unless one first learns to comprehend the language

and to read the alphabet in which it is composed.

It is written in the language of mathematics.

So although that quotation is hundreds of years old,

I could imagine a physics professor saying that today.

So in the Newtonian framework

the world is mechanistic, material and mathematical.

The picture that emerges

is of the Universe as a giant clock, or a giant machine.

Gears and levers pushing on each other,

making things happen, making other things happen,

and then making other things happen,

but always following the laws of motion.

The Newtonian world is one of cause and effect.

There are laws and a certain orderliness out there.

And even if we don't understand all of those laws,

the Newtonian world holds out the promise

that the world is fundamentally an understandable and logical place.

For more infomation >> Dynamical Systems And Chaos: Newton, Laplace, and Determinism Part 1 - Duration: 7:02.

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Arrow | Black Lightning: Best Of Week 13 | The CW - Duration: 0:41.

For more infomation >> Arrow | Black Lightning: Best Of Week 13 | The CW - Duration: 0:41.

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Dynamical Systems and Chaos: The Idea of the Derivative Part 1B - Duration: 4:18.

Let me generalize this idea of average speed.

So, average speed from time 1 (t₁) to time 2 (t₂),

some time interval along your journey,

is just the distance you traveled during that time interval,

from time 1 (t₁) to time 2 (t₂), divided by how long that time took,

this final time minus the initial time (t₂ - t₁).

It's still a distance over a time.

And this distance is how far you traveled during this time.

Okay, so the problem with this is we still haven't answered the question:

How fast were you going exactly 10 minutes into the ride?

So the question is: how fast were you going exactly 10 minutes into your ride...

not "What's your average speed from 9 to 11?"

but, "At exactly t=10, what is your speed?"

Well, we know how to calculate the speed using this formula.

But this formula requires two times and we're really only given one.

So this is over some time interval from t₁ to t₂,

but I want to know the time at exactly 10.

So, we're in a little bit of a bind, and calculus

and this idea of a derivative is going to get us out of the bind.

So here's what we do: We take a time interval

maybe from t=10 to t=11

and then we calculate the average speed for that.

So we might say that the speed at exactly t=10

is the average speed from t=10 to t=11

But you might object and say, "This can't

be right because the speed might be changing.

Maybe you're speeding up or slowing down a lot

as you go from minute 10 to minute 11.

So this really isn't your speed at exactly t=10, this could be changing

and this is not an accurate way of looking at it."

So that's a reasonable objection, actually.

And here's one way we could address it.

You could say, okay, yeah, you're right

the speed might be changing

from 10 to 11, so what if I consider a smaller interval?

I'll go from 10 to 10.1

That's just a tenth of a minute.

Surely the speed isn't changing much.

So the average speed is a good approximation

to the speed exactly at t=10.

And you might object again:

"Well, how do you know? Maybe you're speeding up or slowing down

a lot and maybe you stopped for just an instant in

here so that this average speed is not

a good representation, is not a good approximation to this exact speed."

Okay, that's again a reasonable objection.

So all right, fine, I'll try this.

Maybe I'll just calculate the average speed

from t=10 to t=10.01

And again you might object

"Well, yes, this is a really small time interval

but still, your speed could be changing in

this time interval, so it's not a good approximation or

representation of the speed at this particular

instant." So again I could meet this objection

and we could continue arguing back and forth

And so on and so on and so on

considering a smaller and smaller and smaller

time interval here. And so the way out of this bind

(it seems we could play this game forever)

is to agree that if the right hand side is getting closer and closer

to something, then we say that's the speed exactly at t=10.

For more infomation >> Dynamical Systems and Chaos: The Idea of the Derivative Part 1B - Duration: 4:18.

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Cover Template for KDP Print Using GIMP (2019) - Duration: 13:48.

- Do you wanna know how to use the KDP Print cover template?

Well, in this video, I'm gonna show you how

to use the cover template for KDP Print using GIMP.

Stayed tuned.

(distorted guitar strums)

Welcome to Self-Publishing with Dale,

where you'll learn how to publish books that sell

and build an unstoppable brand.

Today, we're gonna discuss how to use the cover template

for KDP Print books.

But first, there's a few items that you're gonna need,

and it's going to be your ebook cover,

GIMP, also known as GNU Image Manipulation Program,

your book description, and this an optional thing,

get your biography ready for the book.

Let's get right to it.

Alright, I've already downloaded GIMP,

I'm just gonna go ahead and open it up with double-clicking.

Alright, now that we've got GIMP open,

I wanna go ahead, and we're just gonna do just a very simple

Google search, and it's gonna be KDP Print cover template,

and we're gonna go right on to kdp.amazon.com.

Okay, so I know right away that my trim size

is six by nine.

I'm gonna select an option.

We're gonna choose six by nine.

I have 232 pages in my book.

Now, based on paper color,

it's either white, cream, or color.

It's not a full-color book.

I actually selected white when I printed this out.

This is gonna vary based on the project

that you're going with.

We're just gonna go and hit Download Cover Template.

Let's go ahead, we're gonna unzip this file,

gonna Extract All, just gonna keep it right

in the Downloads folder, let's double-click and open that.

I just need the PNG file, so here's what we're gonna do,

is we're gonna minimize all the other windows,

then we're gonna just drag-and-drop

that over on to the surface.

Keep same as before, we're just gonna drag

and drop it on to the surface.

Gonna keep.

Alright, so we're gonna take the Alignment tool.

This is perfectly fit for nine inches high.

I'm gonna select the Alignment tool,

we're gonna point it right at the cover.

Let's go over into the Tool Options,

and we're gonna move it all the way over to the right.

Now in the event that when you drag

and dropped it into place, it was out of position,

we're just gonna go ahead and center it with the cover,

and then we'll right-align it.

Now, here's a little tricky part.

We're just gonna grab the Move tool,

we're gonna left-click, just left-click once on the cover.

We don't wanna move it at all.

And with your arrow key, we're just gonna press over

to the left until we get

right to that dotted line at the spine.

Now the next thing we're gonna want to do,

is I wanna color pick.

We're gonna grab the Color Picker tool,

gonna come over to the cover again,

and I'm gonna grab one of these dark ones

'cause I wanna make sure we blend it really nicely.

So we've selected that,

it's already picked that black color there.

Now what I'm gonna do is I'm gonna keep this template here,

and I'm gonna duplicate that.

We're gonna select and Duplicate Layer.

I'm just gonna shut that layer, the bottom layer off.

Now for this top one, what we're gonna do is,

since we have the foreground color with that Color Pick tool

we're just gonna go ahead and press Control + Comma.

We're gonna shut this off just for right now,

'cause we're gonna need to see where the bleed lines are at.

Our next step, let's see about putting

all the information on the back cover.

So now that you've got your book description,

we're just gonna go ahead and,

I'm just gonna click it right into there.

I like Bebas Neue, it's probably one of my favorite ones.

It matches the fonts there,

so the very first thing is, let's get that headline,

that hook, Control + C.

Okay, we're gonna hit Control + V.

Now, don't worry about formatting it just now.

We're gonna come back to that in just a moment.

We're gonna create another box.

Now, let's drop that bio in there.

Now the bio is not mandatory,

and if you find that you're hard-pressed

for room on your back cover, you can probably go without it,

but I think it's a nice touch,

and especially if you're the type of person

that goes to networking meetings, you can hand that out,

and that bio being on the back cover,

it makes it easy for people to kinda go,

oh, okay, this guy's pretty important.

I'm gonna go ahead and, in order to change out the font,

what we can do is, the very first thing we wanna do,

is I wanna select our hook,

we're gonna come up to Tool Options.

I wanna center that,

and then we're gonna make that just a little bit bigger,

so you see at the very top of this, right there,

we're gonna go ahead and just kinda make it big.

In the event that we want the box

to kind of stick with the text itself,

we're gonna hit Dynamic.

Now, I won't use that for the larger portion

of the description.

Again, we're gonna go ahead and select the next box.

Select all, Control + A.

Right, I need something a little bit more subtle,

probably something a little closer to like, Sans Serif.

Now, in order to kinda check this out, what we'll do

is from time to time, we can probably turn on our background

but you'll notice that since it's a black background,

it's not gonna quite work out in our favor.

Here we go, let's do this.

I'm gonna grab the Shape tool.

Click and drag over the area within the bleed,

so that way I know where I wanna be.

We're gonna select the layer there,

we're gonna hit Transparent, hit OK, awesome.

Now, let's just go ahead and fill it with white.

Nothing spectacular here,

this is just to kinda help us out.

We're gonna hit Control + Period.

Now, you'll notice some of the text is down here,

and let's just, we're gonna just drag this open

just a little bit more.

I wanna make sure that it is lined up with that bleed,

there we go.

Awesome, 'cause we don't write on top of the bleed,

because that means that it's in danger

of going either on the spine or being cut,

so we don't wanna bother with that.

I keep a little bit of wiggle room.

Alright, same thing, let's go ahead, we're gonna select all,

and let's just keep it right in theme with everything else.

Sans Serif.

Okay, right away you can notice that

we've got a lot of, we've got probably too much things

going on here, so what we're gonna do,

is we're gonna take this,

and I wanna kick down the size just a bit.

We're gonna hit 35, Enter.

Not bad.

Not bad, I'm very happy with this.

But in the same instance, this kinda looks

a little bit out of place, so here's what I'm gonna do,

is I'm gonna just italicize this.

Yeah, there we go.

And since this is all italicized, that's the title,

we're gonna go ahead and bold that title.

Okay, let's just drag this box,

kinda like the one we did above.

Spread it out a little bit.

So it looks like we have just a little bit of space,

and from a graphic design standpoint,

I'm sure people are probably flipping out,

like, oh my gosh, that's way, way too big,

way too small compared to the other ones,

so all we're gonna do is we're just gonna try

to stretch this out, oops, undo.

Now let's go ahead and select our closing line here.

I'm gonna do the same thing like I did with the hook.

I'm gonna make it Dynamic.

And we're gonna go ahead

and let's make it about the same size, 106.

And we're gonna highlight the whole thing.

106, Enter, okay, that's a little bit too big for my liking.

So we're probably just gonna just squeeze it right on down.

Alright, so now all we gotta do,

is we're gonna just scoot this on up.

I wanna get the bio kinda formatted,

so let's kill this layer here,

I wanna see where that bar code's at.

Again, I'm gonna go ahead and select all this,

and we're gonna go Sans Serif

so we can match the other font.

Look at that, that works almost perfectly.

And let's go ahead, and let's do it the same size

as the other one, should be 48, okay, awesome.

And I'm gonna put a little bit of emphasis

on the author name.

Gonna go a little bigger.

I wanna take up a good amount of real estate

right about there.

We're gonna scoot it over, there we go.

If we just turn this on, we'll see,

like, the layout looks alright,

I'm gonna adjust a few things,

so we're just gonna get the Move tool.

Okay, now that we have everything laid out,

the real challenge is going to be making it look good.

We're gonna get the Color Picker tool.

Let's go back to that front cover,

and I'm just gonna pick aqua blue color, there we go.

Now we're gonna switch this,

I'm gonna go ahead and we're just gonna grab

just a little bit of that lighter aqua blue.

We're gonna grab headline,

and let's make it that aqua blue color,

so we're gonna just pick that again here.

Boom, okay.

Awesome, looks good.

Let's go ahead, and we're gonna do

the rest of this in white for right now.

It's all gonna disappear real briefly,

but then we'll have it pop back up.

And let's just, for the fun of it,

we're gonna grab this line, we're gonna right-click on it,

Alpha to Selection, let's grab the Gradient tool.

And now, where we drag it,

it's gonna start with the foreground color,

and then it'll go into the background color,

so watch this, we're gonna just click and drag this on up.

Okay, Control + Shift + A.

Ah, that's much better, much, much, much better.

I'm gonna select just this area, let's make it white.

And then I'm gonna select my name,

and let's make it that dominant color there.

Okay, so if we shut this off and we turn this on,

I'm gonna select that, we're gonna hit

the Minus key to move out.

See how clean that looks?

It's not too bad.

I feel like we can probably frame out before the content

and then possibly after the content.

So let's do this, we're gonna grab the Pencil tool,

we're gonna go into Tool Options,

why don't we choose the line instead?

So we can keep it nice and uniform.

I'm gonna zoom in, let's hit a transparency.

Okay, so this is gonna be our line layer.

Okay, here we go, we're just gonna kinda hit that, click,

Shift, great.

Let's hit the Fuzzy Wand tool,

we're gonna select that line.

I'm gonna hit Control + C to copy, Control + V to paste.

And now we're probably gonna have

to zoom in just a little bit.

Okay, let's zoom out.

Let's do another paste.

Just gonna move this into place.

Looks pretty good, and why don't we do just one more,

just for the fun of it.

Anchor, good.

I'm gonna turn off that black background

just to make sure that I'm good with the barcode yet.

So here's what I'm gonna do is let's select the bio layer,

I'm gonna alpha this selection

so I can see where my text is at,

and I'm just gonna move this down in position here,

just a little bit.

I'm making sure that I'm not covering up the bar code,

'cause that's a big no-no.

Control + Shift + A will deselect it.

Let's turn that background layer on.

The very next, and very last thing we need to do

is get a spine.

Now bare in mind that in order to get a spine,

you have to have 100 pages or more.

That roughly works out to,

and I believe Julie Broad of Book Launchers put it this way,

a 30,000 word book is perfect to actually have a spine.

Let's just shut off that layer.

It looks like our hook is probably perfect

for fitting on the spine, so here's what we're gonna do.

I'm gonna select that hook.

Keep in mind, you can always just do a new text box,

but I'm gonna make this easy on myself,

because then I can just Duplicate,

and let's bring it forward a few layers,

so here's we're gonna kinda come up here.

So let's select all of this here,

and I'm gonna start to type in,

we're going to go up into Layer, Transform,

and I wanna flip this clockwise,

so that way it goes right on the spine.

Okay, next thing, let's take the Alignment tool.

We're gonna select that,

And let's just go ahead and center it, see that?

Alright, so now let's turn it back on,

let's see how this looks.

Alright, last but not least,

let's see about exporting this

so that way we can upload it over onto KDP Print.

What we're gonna do is flatten out

all of your visible layers,

so here's what we're gonna do,

we're gonna right-click the very top,

Merge Visible Layers, we're gonna Merge.

Now, let's hit Control + Shift and the letter E,

and we'll put our title, and we need to save this as a PDF.

Select the file and location that you wish to put it on.

Here we go, we're gonna hit Export.

Let's see how this looks.

Not bad, I'm very happy with this.

But what exactly can you expect

when it comes to the KDP Print quality of books?

Good news, if you check out this video

that is up in the top-right corner here,

it's gonna show you an unboxing of the KDP Print book.

I'll see you in the next video.

We send it in that bubble wrap packaging,

which leads me to what we can expect

from shipping and handling.

I am not too happy when it comes to sending.

For more infomation >> Cover Template for KDP Print Using GIMP (2019) - Duration: 13:48.

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Funny And Lucky Moments - Hearthstone - Ep. 417 - Duration: 10:36.

For more infomation >> Funny And Lucky Moments - Hearthstone - Ep. 417 - Duration: 10:36.

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YOU RAP/SING ► YOU LOSE !😷 (JANUARY 2K19) - Duration: 10:45.

No

best in the group chat my niggaz wanna admit that 31 hoes in my face time and

I'm no no that rockin and rollin like Jack Black schooling you niggas like

Jack black how I said you got like Jack Black

I put that on my mama nigga no cat

me and Drake came for it when I got I disappeared before I tried turning on

the I really here now they always say congratulations to the kid and this is

not a funny but I'm part of this shit you serve it up when I got mine

following me he's beat there we go I love then I start with

we got everything suddenly butts into anything and we got fishin for taking us

seriously

but this body never let the beat you codes

put up with your block that I bleed like if I would I have these lucid dreams

where I can't move ain't a gain of you in my bed

you were my F I walked down the aisle and I doubt the priests cuz I'm so fly

might marry me will you accept this ring free hell yeah let's do this thing

I don't wanna

I

I think it'll be too much I'm all in my bag gets hot as a kid I think I saw

powder I might take a sip I might hit the puffs but I'm not gonna trip I ain't

poppin no pill but you do as you wish I wrote some things I love him to death

I got a few mil but not all of them rich what good is the bread of my niggas is

broke forget us just keep breathing

heartbreak and low love it hey-ya love our $5.00 up do you make it bitch made

the best cardio

no master P hey ten bad bitches na after me Hey

one bad bitter like a masterpiece looking for a dunk like an ethylene Oh

be true what you call my

lately I think I want you to be happier I want you to be happier even though I

might not

how much money you - Greta how much money you - shut up how much money you

got a lot how much money you got a lot how many problems you got a lot how many

people in doubted the last left you had to write a lot

how many predator

sorry I needed I would trust tell me

like million dollar deals with my pet bitch some Sun I like those but let's

see yeah the ones that look like that I like going to a bit cooler

I put bucks all in my watch I like Texas from my exes when they wanna take intent

still going bad on your anyway I can feel like a tear s in my Mary Lynn

Jersey back the back is getting scary man if you fuckin with my

up in the sky you can't call me I

seven are not get jumping on an artist it jump is run another date

tapping on another date stepping on another date jumping on another date

jumping on another date

For more infomation >> YOU RAP/SING ► YOU LOSE !😷 (JANUARY 2K19) - Duration: 10:45.

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Yêu Lắm Thái Bình ƠI song ca Nguyễn Văn Diệu & Thanh Huyền [NVD thuychinh] - Duration: 4:13.

For more infomation >> Yêu Lắm Thái Bình ƠI song ca Nguyễn Văn Diệu & Thanh Huyền [NVD thuychinh] - Duration: 4:13.

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[ENG Sub] 스파이더맨 파프롬홈 예고편 리액션 SPIDER-MAN: FAR FROM HOME - Official Teaser Trailer- REACTION - Duration: 2:46.

Hello, everyone.

Enter happy Chubby Land.

Everyone, it's finally here!

spiderman far from home trailer~~!!

Wow!!! That sounds like fun!!!

As soon as this trailer was released, I heard it recorded 130 million views in 24 hours.

As soon as this trailer was released, I heard it recorded 130 million views in 24 hours.

Among the trailers of Sony Corporation, it's now the highest number of views ever!

Come on! Let's take a look.

Ah.. May!! May aunt !! May knows he's Spiderman! (In Homecoming, May was get a secret)

kkk Ah.... banana....kkk !! Bananas were a problem kkk

The crew thought he was doing cosplay as a hobby

Two weeks in Europe....

He must have gone out to play

Does he fall in love with her?

He's dating that woman

Nick Fury...!!

wow....

Oh!! Actions!!

Spiderman movies are very interesting in action scene!!

Ah...... that sounds like a lot of fun!! OHHH

Uh... That sounds fun

It's water..water..?! water monster! water monster!

Who is he?

He look strong

OOHHHH!! That sounds fun!!!

Ah... I can't wait to see this movie !!

Oh kkk~ he's Spiderman, and no one knows kkkk~

Ah~ That sounds really fun!

Ah......how can I wait until July?