# All Questions

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### I need some application examples of the nonlinear spreading of sound waves. [on hold]

In which branches it's being used?
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### How do I see that function $e^{-4x}$ is not bounded?

How do I see that function $e^{-4x}$ is not bounded? How do I proove that?
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### Count the expected value from mimimum [on hold]

Random variable $S_{N}=X_{1}+\dots+X_{N}$ has a Poisson distribution(I assume that the author mean than $N$ has a Poisson distribution). wih $\lambda=5$. Random variable $X$ takes two values ...
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### Isomorphism of group algebras $k[G\times H]\cong k[G]\otimes k[H]$ and interpretation

could someone check it the following is correct? I want to show the isomorphism of $k$-modules ($k$ a Ring) as mentioned in the title. I would like to simplyfy the situation to two finite groups ...
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### Infimum of distance between point and (closed) set

I'm having a little trouble with the following exercise: Let $V \subset\mathbb{R}^p$ be a non-empty, closed set and $a \in \mathbb{R}^p$. For $x, y \in \mathbb{R}^p$ we note $d(x,y) = \|x-y\|$. ...
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### $p,q$ coprime polynomials - are $p^n,q^m$ coprime? [on hold]

Suppose that $p,q \in K[x]$ are coprime (there is no polynomial that divides both) and let $n,m \in\mathbb{N}$. Are $p^n$ and $q^n$ coprime and, if so, how to prove it?
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### Good introduction to cohomology of spaces?

I'm trying to read chapter $3$ of Hatcher but I find it a bit difficult to read. I really only made it through the first two chapters because I had in-class lectures to go along with the reading. Does ...
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### If $S \subset X$ does a subsequence of $S$ converge in $S$ or in $X$?

Let $X$ be a metric space and let $S\subset X$ be a compact space. By definition, $S$ is compact implies that for all sequences $(x_n)$ of $S$, there exists a subsequence $(x_{n_{\alpha}})$ that ...
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### A definition of the Legendre transform from Zorich

This is from exercise 8.5.5.2 from Mathematical Analysis I by Zorich. The Legendre transform of a (presumably differentiable) function $f:\mathbb R^n\to\mathbb R$ is "the transformation to the new ...
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### Gradient and Laplacian of a submanifold

let $M^n$ be a smooth an $n$-dimensional sub-manifold of a $\mathbb{R}^{m}$. Denote by $\nabla^{M}$ and $\nabla^{\mathbb{R}^{m}}$ be the gradient of $M$ and $\mathbb{R}^{m}$ respectively. Similarly ...
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### Image of base change of immersion

I'm revising for my exams now and struggling with the following exercise: Let $f: X \to S$ be an open or closed immersion and $g: S' \to S$ another morphism where $X,S,S'$ are schemes. Then the ...
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### Gaining Mathematical Maturity [on hold]

I was redirected here by a kind fellow from math.overflow. This is not a typical math question, so I apologize if that is discourteous. I am currently a sophomore in my undergraduate mathematics ...
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### Reference for $L$ functions

What is the best possible reference to study $L$ functions for a beginner in this field? I have studied up to the Chebotarev Density Theorem in Algebraic Number Theory. Depending on that, which ...
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### What is the boundary of $\mathbb{Q} \times \mathbb{Q}$ in $\mathbb{R} \times \mathbb{Q}$?

Consider the space $X = \mathbb{R} \times \mathbb{Q}$ with the standard Euclidean topology, and the subset $A = \mathbb{Q} \times \mathbb{Q} \subset X$. I am trying to determine the boundary of A. ...
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### Complexity of Munkres Algorithm

At the moment I'm trying to derive the complexity of the Munkres algorithm in the worst case. For this I'm working with this website and currently I have following complexities for the steps: ...
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### Condition probability distributions: Two people flipping fair coins

Suppose that two people are playing a game where they each flip a fair coin 100 times. The winner of this game is the person who has flipped the most heads. What is the expected number of heads ...
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### What happens when $\beta_1 + \beta_2=1$ and when $0<\beta_1 + \beta_2<1$?

I have the following example of the Scwarz-Christoffel integral formula: $$S(z)=\int_0^z w^{-\beta_1}(1-w)^{-\beta_2}dw$$ with $0<β_1 <1, 0<β_2 <1$, and $1<β_1 +β_2 <2$ and I know ...
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### The signature of an inner product space does not depend on its basis

In R.W.R. Darling's "Differential Forms and Connections" an inner product is defined for a vector space $V$ as a bilinear, symmetric and nondegenerate (but not necessarily positive-definite) map from ...
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### Hilbert curve: how can I find the image of an irrational point in $[0,1]$?

I have a question on the construction of the Hilbert curve: how can I find the image of an irrational point in $[0,1]$? Consider the Hilbert curve $$f_h:[0,1]\rightarrow [0,1]^2$$ Consider a ...
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### Are these integrations the same?

For the integration of: $$v^2 + 6k^2 = -Mv \frac{dv}{dx}$$ I rearranged to get: $$\int \frac{1}{-M} dx = \frac{1}{2} \int \frac{2v}{v^2 + 6k^2} dv$$ Is this the same as the following integral that is ...
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### Let $A\in M_{n \times n}(\mathbb{C})$, $P_A(x)=\prod_{j=1}^{k} (x - \lambda_j)^{n_j}$, and $g\in \mathbb{C}[X]$. Find a C.P. for $g(A)$.

Let $A\in M_{n \times n}(\mathbb{C})$. Its characteristic polynomial $P_A(x)=\prod_{j=1}^{k} (x - \lambda_j)^{n_j}$, and $g\in \mathbb{C}[X]$. Find a characteristic polynomial for $g(A)$. I believe ...
I need help with proving / disproving something: Look at the map $$\Phi: (C([0,1], \mathbb R), ||\cdot||_{\infty}) \to (\mathbb R, |\cdot|); \,\,\,\,\,\,\Phi(u) := \int_0^1 u²(t) dt$$ a) ...
### Evaluating $\int R(X)sin(x) dx$ with residue theorem.
The integral I am trying to evaluate is: $$I = \int_{-\infty}^\infty \frac{x}{1+x^2}\sin x\ dx = \int_{-\infty}^\infty f(x)\ dx$$ The standard approach to this is to realise $\sin x$ as the complex ...