@SineoftheTime More than anything else if this procedure is correct (without checking calculations etc.), and then the choice of the values of the free variables, for example I chose $x=1$, for the 1st system however I was wondering if
@Thorgott Not me, but there's a user in the hbar whose studies things from there and he usually comes up with the most abstract, weirdest things I've ever heard of and it's absolutely hilarious hahaha
@AlessandroCodenotti I am wondering which LCH (or Tychonoff) spaces $Y$ have the property that if $Y\to Z$ is a perfect map onto LCH (or Tychonoff) space $Z$, then $Z$ is zero-dimensional. For example, if $Y$ is countable, then $Z$ must also be countable, so zero-dimensional.
@SineoftheTime Oh ok, but if for example the variable x doesn't appear in the system, can I assign any value to it, but without creating a row of all zeros?
this is the kind of thing you should ask an instructor, if you are able. if you are following a standard recipe in this realm, an equality like M = PDP^{-1} will follow from the way you are identifying P and D, without you performing the computation.
the question for the instructor is whether independently verifying that is a part of what you are supposed to do
if someone just popped up out of nowhere and told me to compute that, i would absolutely compute it. in context, it seems like slightly unnecessary work because all of the previous stuff is set up so that that equality will hold.
if you subdivide the square into 9 squares (like a tic tac toe kind of thing) the blue region feels like it's covering at least one of them. that's my argument why it isn't 1/10
putting visual intution to words is always a challenge
or that it has enough area to cover at least one of them, if it doesn't literally cover at least one of them
So, by recognizing the 2:1 division and the symmetry of the triangles within the square, you can deduce that the shaded area is $\frac{1}{3}$​ of the total area?
Divide the square into three triangles by looking at the lines AE and BD, which intersect at point F. These lines split the square into three triangles: the shaded triangle ABF, the triangle BCF, and the triangle DEF.
Because CE is twice as long as DE, triangle DEF is half the area of triangle BCF. This means that triangle BCF is twice the size of triangle DEF.
Together, these three triangles cover the entire square. Since triangle DEF is one-third of the area of the square, triangle BCF must also be one-third. This leaves the shaded triangle ABF as the remaining one-third of the square.
I often see memes and jokes about $\sqrt{-1}$ not being "real". I've seen part of the movie The Well 2 (Netflix), the fat dude is a mathematician that was troubled about $\sqrt{-1}$ not being real and stuff... I can't take this anymore
Why people care so much about $\sqrt{-1}$ when ALL numbers are "imaginary"? (just abstract concepts without any concrete/material reality? Counting is also abstract, etc nothing real about it)
They did a scene with the fat guy writing $\sqrt{-1}$ on the wall inside the prision, and then crossing it with an "x" mark, like "not real" lol so cringe
@BenSteffan Yeah, but like, all number sets are created so solve some problem. Of course $\mathbb{C}$ is more sofisticated, but they are all abstract, from the very beginning ($\mathbb{N}$)
No! That's the point! There is no platonic world where math concepts lie. Waiting to be discovered. They're just "in our heads", like, we create them, but they don't really exist
Of course, I joke a bit when I say that I am a three-ist, but I think that there is a nugget of truth there. And I think that it is helpful, when teaching, to help students to see this.
When I see those pictures, about golden ratio everywhere, I also get so annoyed, because it's so "forced"(? I don't know how to say this in English)... I mean, it's aproximated, not exact so... It's simply not there
What is "real" (as far as I am concerned) is encapsulated by the distinction between what is subitizable, and what requires further construction or cognition at a higher level.
I've given a definition of what I consider to be "real", and explained the consequences of that definition. But you seem to be attempting to impose your idea about what is real on me.
In order to be continuous, given an $\epsilon>0$, you need to find a $\delta>0$ such that around the point $x$, something happens. For uniform continuity, this $\delta>0$ can be "uniformly" chosen, i.e. it fits for every point, not just around $x$.
The idea of uniform continuity is that it gives a uniform delta. Usually, to show that a function is continuous, you show that it is continuous at each point in its domain. To do that, for each point in the domain, you show that, given some epsilon, you can find a delta such that blah blah blah.
But the value of delta corresponding to a particular epsilon may depend on the point being studied.
A function is uniformly continuous if, given some epsilon, it is possible to find a uniform delta which makes all the good stuff happen, independent of any particular point in the domain.
I'm not disputing the definitional differences b/t continuity & uniform continuity (anymore), I just mean that for a nonlinear/nonconstant function don't we have $\delta$ dependent on $\varepsilon,x$
If we have two incomplete measures $\mu$ and $\nu$, associated to two measure spaces. Is it always the case that $$\overline{\mu\times\nu}=\overline{\overline{\mu}\times\overline{\nu}}?$$Here overline denotes the complete measure.
The difference between $\mu\times \nu$ and $\overline{\mu\times\nu}$ is, fundamentally, that $\overline{\mu\times\nu}$ is defined on a larger sigma-algebra.
Otherwise they agree.
Ditto $\mu$ and $\overline{\mu}$; $\nu$ and $\overline{\nu}$; etc.
So, first question: what sigma-algebras are these measures defined on?
(which is what I meant by the above)
If the sigma-algebras are different, then you know that the measures are different.
@XanderHenderson Ok, so let us define $(X,\mathcal M,\mu)$ and $(Y,\mathcal N,\nu)$. So for $\overline{\mu\times\nu}$, the domain is $\overline{\mathcal M\otimes\mathcal N}$ and for $\overline{\overline{\mu}\times\overline{\nu}}$ it is (I think) $\overline{\overline{\mathcal M}\otimes\overline{\mathcal N}}$.
Is there a general guiding principle for taking a geometric object and letting the object be a point in some space?
for example say my geometric object is a cube with different colored faces and each lattice point in a 10X10 grid is a specific configuration of the cube at this point.
there is nothing stopping anybody from defining whatever you want, the hard part is proving useful results about whatever 'spaces' that result from that
the challenges you run into are very subject and purpose specific
it isn't inherent in math that interesting properties of one thing be a model for 'nearby' or 'similar' things of the 'same type,' although a lot of stuff maybe could be regarded as trying to make that happen with limited success
I was thinking along the lines of the objects on the grid form discrete groups, so I can use geometric group theory to study the grid as a Cayley graph, where each point represents a group element or cell. Transformations of geometric shapes in the grid then correspond to group actions, which could potentially lead to a moduli-like space with group-theoretic structure
