Newest Questions
1,698,195 questions
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isometry group of an orbifold is the quotient of the isometry group of its cover
I have a 2-dimensional orbifold that is the quotient of $\mathbb{R}^2$ under a group of isometries $\Gamma$ generated by $180^\circ$ rotations. I would like to say that $\text{Isom}(\Gamma \backslash ...
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Let $f$ be defined by $f(x) = 1$ for $x = 1/n, n \in \mathbb{N}^*$ and $f(x) = 0$ otherwise. Show that $f$ is Riemann integrable on $[0,1]$. [duplicate]
I am having trouble solving this problem. The first thing to note is that I would like to solve this without using Lebesgue's theorem, since on my exam I will not be able to use that theorem.
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Proof of Euler identity using ODE
I am writing an article to prove Euler identity :$e^{i\pi}+1=0$
Here the main part:
Consider the function :$ \mathbb{R} \rightarrow \mathbb{C}, f(x)=e^{ix}$
Differentiating twice,we get : $f''(x)=-f(x)...
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Book Recommendation for Vector Bundles
I am interested in learning more about general vector bundle theory. More specifically, vector bundles of class $C^k$ for $k\in\mathbb{N}$ or $C^\infty$ or real-analytic whose fibers can be given the ...
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The Intuition behind the nim game and the XOR?
The game of nim is played with two players againts each other ,by removing 1 or many stones from only one pile in each turn from n piles each pile with $n_1,...,n_k$ and a player cannot skip a turn.
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Central Limit Theorem via Fixed Point Theorem and Entropy
I'd like to work out the details of a proof of the Central Limit Theorem that utilizes the Banach Fixed Point Theorem and possibly also entropy. The rough idea is:
The average $\bar{X} = \frac{1}{n} \...
- 1,987
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Any easier ways to prove an explicit form for a generated $\sigma$-algebra besides transfinite induction?
This question is a follow-up to an answer to a previous question, and motivated by my laziness in not wanting to learn about transfinite induction or how to write proofs using transfinite induction ...
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Decomposition of von Neumann algebra in semi-finite and purely infinite part
I am reading the proof of type decomposition of a von Neumann algebra $M$ in the book 'Lectures on von Neumann algebras' by Stratila and Zsido (Theorem 4.17).
The proof starts with the following claim:...
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Prove that the iterate $f^{n}$ is a constant function.
Let $A \subset \mathbb{R}$ be a finite set with $|A| = n$ and let $f : A \to A$ satisfy the strict contraction condition $|f(x) - f(y)| < |x - y|$ for all $x \neq y$ in $A$. Prove that $f$ is not ...
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Is my Euclidean-style proof valid? It is for a summation of infinitely many line segments equaling to a finite length without calculus, or limits. [closed]
I have attempted to prove that the sum of infinitely many quanities can still equal a finite quantity without using calculus, measure theory, or any other modern mathematical tool such as set theory.
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Showing that the trace of the difference of the square of the curvatures of two connections on a vector bundle is exact
Let $E$ be a rank 2 complex vector bundle over a 4-dimensional manifold $X$. (I believe that the argument below does not depend on the rank of $E$ and the dimension of $X$.)
I want to try to show, for ...
- 1,792
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What are applications of inprimitive Dirichlet characters and associated L-functions?
For simplicity, I'm gonna call L-functions associated with primitive Dirichlet characters 'primitive' and same with inprimitive.
GRH (Generalized Riemann Hypothesis) says that all non-trivial zeros of ...
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Matrices that are related to their transpose via diagonalizable matrices
I'm currently working with matrices having the following property:
Let $A \in M_n(\mathbb Z)$ be square matrix such that there exist diagonalizable matrices $S,T \in M_n(\mathbb C)$ with $A = S A^t T$,...
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Prove $\frac{x^{2^n}-1}{x^{2^m}-1}$ is not a perfect square when $n, m$ have different parity
I am trying to prove the following number theory problem:
Problem:
Let $n, m$ be positive integers with different parity (one is even, the other is odd) and $n > m$. Prove that there is no integer $...
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Number of integer solutions to $\sum_{i=1}^{10} x_i = 100$ with $2 \le x_i \le 21$
I am trying to solve a combinatorial problem involving finding the number of integer solutions to the following equation:
$$ x_1 + x_2 + \dots + x_{10} = 100 $$
Subject to the constraints:
$$ 2 \le ...