Mathematics

Zahir

12:10 AM

hello if we have f(x,y)

then f'y (x,0)

what does it mean

Ashley Davies

12:38 AM

Would it be acceptable to finish a question asking me to solve a system of equations (which I do using a matrix) as a matrix/vector form result? i.e. $\begin{bmatrix} x \\ y \\ z \end{bmatrix} = \frac{1}{5} \begin{bmatrix} -3 \\ 5b - 8 \\ 5b - 1 \end{bmatrix}$

Karim Mansour

12:52 AM

Hilbert's paradox of the Grand Hotel

Hilbert's paradox of the Grand Hotel is a veridical paradox (a valid argument with a seemingly absurd conclusion, as opposed to a falsidical paradox, which is a seemingly valid demonstration of an actual contradiction) about infinite sets meant to illustrate certain counterintuitive properties of infinite sets. The idea was introduced by David Hilbert in a lecture he gave in 1924 and was popularized through George Gamow's 1947 book One Two Three... Infinity. == The paradox == Consider a hypothetical hotel with a countably infinite number of rooms, all of which are occupied. One might be tempted...

cool

Soham Chowdhury

1:11 AM

@Paul, are you sure?

In Aluffi there were no functors there.

And I'm pretty sure a slice category is a special case of a comma category (or v.v.?)

Exo Primes

1:29 AM

Any method to prove the following?A square was divided into 1000 triangles with equal areas. Then the same square
was divided into 1000 rectangles with equal areas. Prove that there exist 1000 points inside the square such that every triangle from the first division and every rectangle from the second division.
contains some of these points inside.

please advice

Soham Chowdhury

What do you mean by "first" and "second division"?

Exo Primes

first division of triangles and second division for rectangles

first 1000 triangles from a square, and then 1000 rectangles from a square

Soham Chowdhury

1:46 AM

The "probabilistic method" might be useful here.

bayanbox.ir/view/3802662114912891085/probability-20problems.pdf

Morning, @Paul.

Paul Plummer

Slice is a special case of comma, yours is a special case of comma, and aluffi's is a special case of comma

Hey

Soham Chowdhury

Hmm.

So the one with functors I showed you, what's that called?

$\require{AMScd}
\begin{CD}
F(X) @>{F(f)}>> F(Y)\\
@VsVV @VtVV\\
A @= A
\end{CD}$

Paul Plummer

Pretty sure slice category, or you would just specify the type of comma category

Not everything needs a fancy name

Actually, maybe just comma, at least looking at the wiki def

Like I said, not everything needs a special name

Soham Chowdhury

Oh, yes!

This is a kind of comma category.

I should've checked wiki before :/

Paul Plummer

Categories for the Working Mathematician just calls this, $F$-over $A$

or $(F \downarrow A)$

Soham Chowdhury

1:58 AM

Hmm, thanks.

Paul Plummer

(and they point out that it is a special comma category)

Exo Primes

2:14 AM

I am trying to prove the following, but I wonder if this relates to an identity matrix.

would you please help... All entries of an n × n matrix are non-negative. It is known that the sum of numbers
at any row and at any column of this matrix is exactly 1. Prove that you can choose n positive
entries such that all of them are from different rows and from different columns.

Soham Chowdhury

2:30 AM

Uh, the diagonal?

I think you meant n positive entries summing to 1.