We can repeat this process at every point, and if we then shift our perspective to two dimensions, look, what we've got is the gravitational field of the star. Mathematically , we say that the gravitational field G is equal to the negative gradient of V.- So Lagrange had found a way to switch the problem back and forth between one of vectors and one of scalars.
Matrices are arrays of numbers which encapsulate these transformations, telling us how to reflect, rotate, stretch, or squish across each dimension. Mathematically , the order we multiply them matters.Since we multiply the rows of the first matrix by the columns of the second,
Now imagine taking all possible values of alpha simultaneously, and then averaging out all of those vectors. Mathematically , we can write this as the integral from 0, to 1 of E of the I 2 pi alpha, D alpha.So what happens? Well, this vector cancels with this one on the opposite side, and this one with that one.
And this phase is what determines the amplitude of the wave at that point. Mathematically , we can write the amplitude our stopwatch as e to the I phi, where phi is the phase.As the particle wave follows a path, its phase increases.
So in conclusion, just to say, we're at this point in the course of human and social evolution where the demands of survival converge Mathematically , that just defines having past the inflection point on the curve.
foreign land to writing about mathematics and I see myself as the kind of foreign correspondent in the world of mathematics you know I'm kind of mathematically enumerate mathematically I have a sense of it understand it and I try and explain it to people who mightnot like it might not understand it so that was the idea of my first book where I went around the world interviewing
And not all of them were. Mathematically , it's impossible. And I could tell you the license plate numbers of the ones who were the worst.The bottom line is, people need constant, immediate feedback in a positive way to begin to realize how they exist in this framework.
Eventually, you can figure out what the shape of the asteroid may be. Mathematically difficult-- it takes, again, a lot of telescope time.And so as we focused our energy into detection, because that's the first problem we need to solve, it's expensive
both ways, you know -- together or separately. And so, when I heard about the mathematician in the film, Tom Hall, who was making, as you saw, some modular pieces that were so mathematically -based, but also so beautiful to look at, I thought that that would be a nice, short film. So, I decided to visit him, and I had hopedto make a 7- or 8-minute movie about the mathematics of origami.
And statistically, it's extremely unlikely. But mathematically , it's possible that this can happen and we can't rule that out. And this is a situation that shows up a lot in mathematics.
I don't have these special configurations of water. But mathematically they have some similar features.
Yeah, that's exactly right. And mathematically speaking, checkers is simpler than chess, of course. But its top players make up for that difference by looking deeper into the game.
If the dominoes are playing the role of the professor, the classes are going to be the individual squares of the board. And mathematically , this is a little bit of a pain to write down. You have to define a set, Pij that's the set of all these quintuples, m, n, o, i prime, j prime, that cover a particular square of the board.
So the question is, what is it that is interfering? So mathematically , there is an explanation. So mathematically , each photon or each particle of light is represented by what physicists call the "wave function."
So mathematically , there is an explanation. So mathematically , each photon or each particle of light is represented by what physicists call the "wave function." It's a mathematical abstraction.
And this function evolves according to something called the "Schrodinger equation." So there is an equation that tells you how this wave function changes with time. And mathematically , the way you model this system is you let this wave function propagate towards the two slits. It hits the two slits.
So now they're giving a reason but it's adding no more information. So mathematically speaking, or in terms of information content, it's just like the first way, where you just say, can I get in front of you without giving a reason? So you should get 60% compliance.
And everything is so simple, just because I'm not up to drawing these beautiful funnel shapes for Schwarzschild, but otherwise, it is actually a mathematically accurate depiction of the Schwarzschild metric. Now, let's have one more thought experiment where Franz and I start at the same elevation
So mathematically , that is the same knot.
The mathematically -minded will see it immediately -- the bigger T becomes in the denominator, the smaller the environmental impacts become. But what must happen in society for such a
it got the mass and the magnetic moment of the electron. The theory is mathematically sound.
It has to be mathematically consistent.
Why? Well, it has the tendency to lead to a lot of draws. So this is not mathematically proven, but many suspect that perfectly played chess is a guaranteed draw, much like perfectly played tic-tac-toe or checkers is a guaranteed draw.
I understand it mathematically why it's this, why it's that, but it just doesn't touch or move me the same kind of way.
It was modeled mathematically and then in computer software and is now, of course, amazingly successful in the world of machine learning because now we
You can show mathematically this will happen very, very quickly.
and transmitted with 50% probability when it hits a beamsplitter. So these two waves mathematically are in phase, and they constructively interfere.
It is mathematically impossible to have confidence in yourself if you can't even carry out the commitments you
These scenarios are mathematically similar, both exchanging one life for five.
and extending that mathematically entirely through the network, is an idea that was not invented by the founders of Google.
And that's just mathematically wrong.
He's mathematically gifted. He's arithmetically gifted.
That's pretty simple mathematically .
It may be mathematically accurate, but it is useless for the purposes of prediction.
This is well established mathematically . And there are many theories as to why this growth and success has not occurred. One would be immigrants taking your jobs. Anti-Muslim feelings,
So it just mathematically doesn't work I think.
It's beautiful mathematically .
And it's mathematically not possible to really get both the false positives and the false negatives, the likelihood of both, down to 0.
And he was mathematically challenged.
It was kind of mathematically proved and then, decades later, it was discovered.
And he was very mathematically -oriented, and he knew how to count pretty early on.
There may be a mathematically valid center.
That's in a mathematically describable, but physically inaccessible dimension.
even though that's mathematically impossible and uh we'll get through this okay it's not a matter of look to your left
that theory is only mathematically consistent if there are precisely six
and and mathematically provable and
So, the order matters. How do we express this mathematically ? We need some way to represent transformations, like reflections as a single coefficient, which means we need matrices.
Some of you might be super duper mathematically sophisticated.
So let's make it mathematically consistent.
We can sort of characterize this mathematically .