I already read this, and so wish to intuit 3 without relying on (only rearranging) the definition of Conditional Probability. I pursue only intuition; do not answer with formal proofs. Which region in my 2D Venn Diagram below matches $Pr(A|B)$? On the left, the green = $\Pr(A\cap B)$. The right is herefrom, but fails to […]

I was wondering whether someone could provide me with an easy definition of a skyscraper sheaf and, more importantly, with a visualization of it. Thanks in advance.

Let $V$ be a vector space over the field $\mathbb{K}$ and let $T(V)$ be it’s tensor algebra. We usually define the exterior algebra $\Lambda (V)$ by the following process: we consider the bilateral ideal $I$ generated by all elements of $T(V)$ of the form $u\otimes v+v\otimes u$ for all $u,v\in V$ and define the exterior […]

Take $S^3$ to be the three-sphere, that is, $S^3=\lbrace (x_1,x_2,x_3,x_4):x_1^2+x_2^2+x_3^2+x_4^4=1\rbrace$. Using the stereographic projection, $S^3=\mathbb{R}^3\cup \lbrace \infty \rbrace.$ Can someone explain how the complement of the solid torus (centered at the origin) $S^1\times D^2$, where $D^2$ is a 2-disk, is also a torus? I am reading Milnor’s paper “On Manifolds Homeomorphic to the 7-Sphere,” and […]

The question I am working on asks me to construct a homeomorphism $\phi : T^2/A \to X/B$ where $T^2$, $A$, $X$ and $B$ are given as follows: $T^2=S^1 \times S^1$ and $A \subset T^2$ is given by $A=S^1 \times\{1\}$. $X=S^1 \times [-1, 1]$ and $B = S^1 \times\{-1, 1\}$. But first of all, I do […]

As a follow-up question to What is the norm measuring in function spaces I just had an idea: How about visualizing function spaces as normal planes. What I have in mind is to have an orthogonal coordinate system where the two axes represent two orthogonal functions (e.g. sin and cos). Different functions, that can be […]

Complex numbers $\mathbb{C}$ are usually constructed as $\mathbb{R}^2$ together with a suitable multiplication. But this is not the only possible way, one can get to the complex numbers. One alternative would be via $\mathbb{R}[x]/{\left< x^2+1\right>}$, but I have never seen this one be used in practice. When visualizing complex number one usually uses the fact […]

Name $r$ = clockwise 90 deg. rotation and $f$ = flip across the square’s vertical axis = the brown $\color{brown}{f}$ in my picture underneath. Zev Chonoles’s $f$ is different. Carter fleshes out why $frf = r^{-1} $ intuitively: (1.) Can someone please unfold, like Carter, why $fr = r^{-1}f $? I see why for this […]

As regards $ f(x, y) = 4 + x^3 + y^3 – 3xy$, I computed that (0,0) is a saddle point, and (1,1) is a local minimum. So I’m not asking about this, and am asking only about sketching contours. $1.$ Here, I’m referring to the middle sketch entitled “Why not this?” in red. How […]

If we take the definition of a real projective space $\mathbb{R}\mathrm{P}^n$ as the space $S^n$ modulo the antipodal map ($x\sim -x$), it is possible to see that $\mathbb{R}\mathrm{P}^1$ is topologically equivalent to the circle. It is equivalent to the upper half of the circle where the two end points are glued together – i.e. another […]

Intereting Posts

$E_n =\lbrace X_n > X_m \ \forall m < n \rbrace $ are independent
Contour Integral: $\int^{\infty}_{0}(1+z^n)^{-1}dz$
Regular in codim one scheme and DVR
Solving $x+2y+5z=100$ in nonnegative integers
How to prove that $ \sum_{n \in \mathbb{N} } | \frac{\sin( n)}{n} | $ diverges?
Distance between a point and a m-dimensional space in n-dimensional space ($m<n$)
Finding the n-th lexicographic permutation of a string
What goes wrong when you try to reflect infinitely many formulas?
If $A$ is a complex matrix of size $n$ of finite order then is $A$ diagonalizable ?
Where can I learn about the lattice of partitions?
Want to understand the epsilon-delta definition of a limit, please help me!
Conjecture: the sequence of sums of all consecutive primes contains an infinite number of primes
$\sum_{n=0}^{\infty}3^{-n} (z-1)^{2n}$ converges when
Find the Jordan form of this Matrix
Why the primitive of a Riemann functions must be continuous?