All Questions

3k views

Dual graph: Simple example

If I have a graph consisting of 2 disjoint triangles, which are connected by an additional edge, then I have difficulties understanding how its dual graph looks like.
87 views

Non zero components in DFT

I wanted to do a simple example of DFT computation using the following python code (numpy + scipy). I am posting here because I am sure my problem is more related to my comprehension of the DFT ...
127 views

How to work with Connections

I am currently reading a book which deals with complex manifolds. Since I am fairly new to the topic I don't know exactly the meaning of the followinig: Suppose we have a holomorphic vector bundle ...
244 views

Relationship between different L-functions

What's the relationship between between Artin $L$-functions and Dirichlet or Hecke $L$-functions if $L/K$ is an abelian extension? I've been told that one can interpret the Artin $L$-functions as ...
104 views

Finding the norm of $x \in \mathbb{R}^2$ if the unit ball is defined in a specific way

I need to find the norm of $x \in \mathbb{R}^2$ if the unit ball is defined by this inequality: $B=(\begin{pmatrix} x_1\\ x_2 \end{pmatrix}: -a_1\leq x_1\leq a_1, -a_2\leq x_2\leq a_2 )$. What ...
365 views

Injection from the set of countable ordinals $\Omega$ into $\mathbb{R}$

I'm reading through this and I'd like to define an injective function from the set of countable ordinals $\Omega$ into $\mathbb{R}$ using transfinite induction (or maybe transfinite recursion?). ...
279 views

495 views

Complexification of Tangent Bundle

I am currently reading a book where the author says that the tangent and cotangent bundles $TM$ and $T^*M$ of a manifold $M$ are complexified. I am not familiar with Complex Manifolds so looked it ...
232 views

362 views

Proof: Matrix exponential maps from tangent space to Matrix (Lie) group

Let us assume we have a definition of the tangent space (e.g. as in Proof: Tangent space of the general linear group is the set of all squared matrices). Furthermore, we already verified that the ...
102 views

Why does every finite subgroup of $\mathrm{Aut}(F_n)$ acts on a graph of Euler characteristic $n-1$?

My question is the following: In a paper I read that: Any finite subgroup of $\mathrm{Aut}(F_n)$ can be realised as agroup of baspoint-preserving isometries of a graph of Euler characteristic $1-n$. ...
134 views

570 views

Fast $L^{1}$ Convergence implies almost uniform convergence

$\sum_{n \in \mathbb{N}} ||f_{n}-f||_{1} < \infty$ implies $f_{n}$ converges almost uniformly to $f$, how to show this? EDIT: Egorov's theorem is available. I have been able to show pointwise ...
178 views

Counting the number of Strings.

I encountered this question in a coding competition. The question: Given a string, calculate the number of permutations of that string such that no two identical letters lie adjacent to each ...
86 views

477 views

Derivation of the Riccati Differential Equation

I am attempting to derive the Riccati Equation for linear-quadratic control. The original equation is: $-\partial V/\partial t = \min_{u(t)} \{x^TQx + u^TRu + \partial V^T/\partial x(Ax + Bu) \}$ \$x ...
1k views

Do you need real analysis to understand complex analysis?

I'm debating whether I should take a course, in complex analysis (using Bak as a text). I've already taken Munkres level topology and "very light" real analysis (proving the basic theorems about ...
55 views

arrangements of a platoon

Ten men from a platoon are arranged in two rows. Each row has the men arranged by increasing height from left to right, and every man in the back row must be taller than the man in front of him. In ...
3k views

Mathematical Induction and “the product of odd numbers is odd”

I am extremely poor at proofs and logical manipulation so I am stuck on a lot of these questions especially induction. The question below I have been stuck at for a little over 1 hour and I can't ...
115 views

Calculating the percent of a person's life that a certain time takes up?

A while ago I read an article on Cracked.com about why you wouldn't want to be immortal. Among many other reasons was that time would speed up for you after so many years and it would make every ...
All I know is that it uses the fundamental theorem of calculus. $$\large\frac{d}{dx}\int_{x^2}^{\sin x} e^{xt^2}dt = e^{x\;\sin^2 x}\cos x - e^{x^5}2x+\int_{x^2}^{\sin x} t^2e^{xt^2}dt$$