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How many vertices does $(\Bbb{T}^2)^{\#g}$ have?
Here is a description in a topology lecture note that I need help.
Since $(\Bbb{T}^2)^{\#g}$ can be represented by a $4g$-gon with edges identified according to the word $a_1b_1a_1^{-1}b_1^{-1}\cdots ...
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On the proof of Buckingham π theorem
This is going to be a slightly longer post, but it will be a part of my masters work, so its important to me to get all the details right.
I am working on the Buckingham $\pi$ theorem and there is a ...
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Applying Leibniz's Rule to Double Integrals with Variable Limits
Consider the following double integrals:
$$G_1(z_1, z_2) = \int^{z_1}_{0} \int_{0}^{z_2 + \frac{\alpha_1}{\alpha_2}(z_1 - x_1)} \varphi(x_1, x_2) \, dx_2 \, dx_1$$
$$G_2(z_1, z_2) = \int^{z_2}_{0} \...
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How to find the maximum length of chord cut by a right angle inside a circle
How to find the maximum length of chord AB in the figure below?
P is a fixed point inside circle centered at O (P is not O). PA and PB form a right angle. Imagine this right angle rotates inside the ...
- 5,408
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Conditions for a Fourier-windowed quadratic form to define a finite-range or exponentially decaying interaction?
Let $A: \mathbb T^4 \to \mathfrak{su}(2)$ be a smooth gauge potential on the 4-torus.We consider a quadratic functional $Q[A]$ defined via a decomposition using a wavelet-type frame $\{\psi_{j,k}\}$:$$...
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Changing the order of integration in an iterated integral with a single varible function
I am trying to consider a double integral:
$$
\int_t^\infty \int_s^\infty f(r) dr ds <+\infty
$$
where $f:\mathbb{R} \to \mathbb{R}$ is a smooth function, but NOT a non-negaitive function. And the ...
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Multinomial Distribution to Binomial Distribution and Joint Probability Function
I'm having trouble with a few multinomial questions. I know that for a multinomial distribution we have:
$$p_1+p_2+\dots+p_k=1$$
$$X_1+X_2+\dots+X_k=n$$
For $(X_1,\dots,X_{10}) ∼$ Mult$(n,p_1,··· ,...
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Restrictions of Knight's Tour on Circular Board
Does a closed knight’s tour exist on an n-vertex “circular” chessboard with wrap-around moves?
I’m interested in variants of the knight’s tour, but on a “circular board” rather than a rectangular one. ...
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If $a^2+b^2+c^2=1$ so $\sum\limits_{cyc}\frac{1}{(1-ab)^2}\leq\frac{27}{4}$
Let $a$, $b$ and $c$ be real numbers such that $a^2+b^2+c^2=1$. Prove that:
$$\frac{1}{(1-ab)^2}+\frac{1}{(1-ac)^2}+\frac{1}{(1-bc)^2}\leq\frac{27}{4}$$
This inequality is stronger than $\sum\...
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Example of the $\mathrm{dim}A > \mathrm{depth}A$
In commutative ring theory of Matsumura, Theorem 17.4
" If A is a Cohen-Macaulay ring with $\mathrm{dim}A=n$, the following is equivalent.
(1) A sequence $a_1,\cdots,a_n$ is a regular sequence.
(...
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Any 'locally butterfly' graph be constructed as line graph from a 3-regular, triangle-free graph: Can this be generalized to locally windmill graphs?
A simple graph is called locally butterfly, if the closed neighbourhoods of all vertices (set of all vertices at distance 1 and adjacencies between these and the original vertex) are isomorphic to the ...
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If $\varphi(f)$ is riemann integrable for each $\varphi \in E^*$, then is $f$ riemann integrable?
Let $I=[0,1]$, $E$ be a banach space and $f:I \rightarrow E$ be a map.
Suppose that for every continuous functional $\varphi\in E^*$, the map $\varphi(f):I\rightarrow \mathbb{R}$ is riemann integrable....
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Quintic residues and $p=x^2+125y^2$
Euler conjectured (and Gauss later proved) that:
If $p\equiv 1\pmod 3$, $2$ is a cubic residue mod $p$ iff $p=x^2+27y^2$ for some integer $x$ and $y$.
If $p\equiv 1\pmod 4$, $2$ is a quartic residue ...
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Complex logarithm base 1
Is a logarithm with base 1 defined in the field of complex numbers? I have not found any information about this. In real numbers, this is uncertain because $ \ln(1) = 0 $ and
$ \log_a(b)= \frac {\ln(...
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Solving $\sin(x) + \sin\left(2^{(n+12)/12}x\right) = \sin(x+w) + \sin\left(2^{(n+12)/12}(x+w)\right)$ for $w$
I want to solve the following equation here for $w$ (ie, $w = \dots$).
$$\sin(x) + \sin\left(2^{(n+12)/12}x\right) = \sin(x+w) + \sin\left(2^{(n+12)/12}(x+w)\right)$$
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