So it's still array-oriented, but it's not doing things a scalar at a time. Not everything is a scalar function. And then we have, again, slightly different terminology in APL.
A lot of the terminology was invented back in the late '50s and '60s. And all scalar functions apply in the same way. So anything that takes two arguments, you can do a reduction, and it's going to just do the work.
So there's a power operator-- which you see is a very similar symbol to the mathematical exponentiation, it just has a dieresis above it-- They are all scalar functions.
What Lagrange had developed was the gravitational potential V. And what's important to note is that V is a scalar , it has a magnitude but no direction. So the genius in Lagrange's idea is this.
Now, not all of the functions are these scalar functions. So all the mathematical and logical functions are scalar . But the iota, if you give it two arguments, is a look-up.
So again, for a matrix on the right, and a single number on the left, we can take the base 2 logarithm of all of these numbers. Now, not all of the functions are these scalar functions. So all the mathematical and logical functions are scalar .
- So Lagrange had found a way to switch the problem back and forth between one of vectors and one of scalars . And while adding up vectors is hard, adding scalars is a piece of cake. To find the combined potential landscape of any number of bodies, you just add up their individual potentials.
It might look something a little bit more like jazz as an analogy. So it's not like a higher dimensional version of our 12-tone scale that contains scalar relationships in the geometry.
Yes, we have lost a bit of real estate here. OK, so itemwise map is implicit for all of what we call scalar functions, which is about half-- at least half of the functions. So if you have an array, it just applies the function to every element of the array without you having to say, map, or for each, or anything like that.
But it's looking each of the items up in the left. So it's still array-oriented, but it's not doing things a scalar at a time. Not everything is a scalar function.
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 . And while adding up vectors is hard, adding scalars is a piece of cake.
He really developed tensors. Those of you who aren't with a technical background, scalars are numbers and stuff, and vectors are a number with a direction like an arrow, and tensors just generalize that a little bit more.
And just by juxtaposing things, you create lists. So here we have a bunch of numbers, three scalars , and a vector. And you can see the structure.
You can do it if you're good, and people who are good at mechanics can do it. But with the Lagrangian approach, you could just write down the energy, which is a scalar not a vector, plug it into the Euler-Lagrange Equation, and you get the right equation to motion and you don't have to be a good physicist.
many elementary particles, just as you're saying, their mass through their interaction with it. The Higgs boson is the particle associated with ripples or excitations of this field. In modern particle physics, every type of particle corresponds to a field that exists everywhere. The Higgs field is one such scalar field, meaning at each point in space, it has a single numerical value rather than a direction. The Higgs field differs from most other fields because even in empty space,
And other times, he would say that no, he was just saying that as an analogy. I think that we have to, like you were saying, recognize that life is this multi-scalar phenomenon. So for example, the cells that compose our bodies, they're not organisms in and of themselves.
In India, only a couple of places-- at least, I know only one place where this work is going on, but it needs to be done, I think, They can have different temperatures even in the direction-- in one direction, plasma and the temperature is scalar .
Now, here's this image that I've shown before and as I promised I would reference. existed but has these interesting features because-- sorry, that was the next question-- because we can have this level of scalar precision
It might look something a little bit more like jazz as an analogy. So Coltrane tired to-- I believe he read everything he could get his hands on about Einstein-- he was trying to find, OK, you know these different scalar
Now, if there is not the same number on both sides, you get what we call a length error. But if there's exactly one number on the one side and an array on the other, we do something called scalar extension, and the single number is applied to all of the items.
And they could just barely do it with natural, with the natural things that are in the environment. The, actually the importance of the exponential curve is not the curve so much as the fact that if a certain scale or, in many cases if a certain scalar gets larger than a certain