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Sub $x=\tanh{u}$, $dx = \operatorname{sech^2}{u} \, du$. Then the integral is $$\int_{-\infty}^{\infty} du \, \frac{\operatorname{sech^2}{u}}{\pi^2+4 u^2}$$ Now, use Parseval. The Fourier transforms of the pieces of the integrand are $$\int_{-\infty}^{\infty} du \, \frac{e^{i u k}}{\pi^2+4 u^2} = \frac14 \frac{\pi}{\pi/2} e^{-\pi |k|/2}$$ ...

5

I believe that A. Ya. Helemskii, Lectures and Exercises on Functional Analysis, Translations of Mathematical Monographs , vol. 233 (2006). in conjunction with some standard book such as G. K. Pedersen, Analysis Now, Springer-Verlag, (1989). should suit you well. Note that Pedersen's book itself contains lots of basic information about Banach and ...

3

For the second question, the finite rings are precisely those for which every finitely generated module has finitely many submodules. For a ring $R$ and $r\in R$, let $M_r$ be the submodule of $R\oplus R$ generated by $(1,r)$. Then $(1,r)$ is the only element of $M_r$ whose first coordinate is $1$, and so $M_r\neq M_s$ for $r\neq s$, and so if $R$ is ...

3

$\mathcal F\setminus U$ is dense in $\mathcal F$ because every $f:\mathbb R\to\mathbb R$ can be approximated pointwise (and even uniformly) by functions with nowhere dense image: $$f_n(x) = \frac1n \lfloor nf(x) \rfloor$$ Therefore the only subset of $U$ that is open in $\mathcal F$ is $\varnothing$, which is obviously not dense. Just for completeness, ...

3

It is a consequence of the co-area formula. In particular, I studied it in the books by Giaquinta and Modica, volumes 4 and 5.

2

Let $Y\subset X$ be closed and $E$ a vector bundle on $X$. If $E$ restricted to $Y$ is trivial and $H^0(X,E)\to H^0(Y,E_{|Y})$ is onto, then $E$ is trivial in a neighbourhood of $Y$. In particular this is true if $X$ is affine. On the other hand, in general, this is not true. For example, take $C$ to be an elliptic curve and let $V$ be the non-split ...

2

The infinitude of the twin primes is an open problem, so currently proving anything about the asymptotics of this function is out of reach. However, the first Hardy-Littlewood conjecture would imply that your sum is asymptotic to $$4\Pi_2 \frac{x}{\log x}$$ where $$\Pi_2=\prod_{p\geq 3} \frac{p(p-2)}{(p-1)^2}$$ is the twin prime constant.

2

I'll give you half the answer, and maybe you can work out the other half for yourself. As I said in a comment a moment ago, there is no canonical isomorphism between $Y_1(5)$ or $Y(5)$ and an open subset of $\mathbf{P}^1$, so $C(5)$ and $C_1(5)$ will depend on your choice of uniformisation. So the real question here is "what are explicit modular functions ...

2

Let's prove it. Suppose $k$ is in the kernel of $A$ and $x_0$ is a specific solution, so that $Ax_0=b$. We have $$A(x_0+k)=Ax_0+Ak=Ax_0=b.$$ Conversely, suppose that $x_0$ and $y_0$ are two solutions. Then $$A(x_0-y_0)=0$$ implying that $x_0-y_0=:k$ is in the kernel. Therefore any two solutions differ by an element of the kernel.

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You should read about these topics: Vector spaces (I think you already know what this is) Metric spaces (A vector space + a norm) Hilbert spaces (A metric space with a finite or infinite number of dimensions) Fourier transform (The projection of a vector belonging to a Hilbert space in an orthogonal basis) Spherical coordinates (Really easy to understand, ...

2

For $n\geq k\in\mathbb{N}$, you can prove that $\frac{n!}{k!\cdot(n-k)!}$ is an integer from a combinatorial/counting argument. Establish that the formula gives the number of ways to choose a subset of $k$ from a set of $n$, and that automatically makes it an integer. Or you can prove that $\frac{n!}{k!\cdot(n-k)!}$ is an integer by showing that the ...

1

Have a look at $$\frac{dy}{dx}=\frac{y^k}{x}$$ for an example where you really can't swap orders of taking the limit. For one order the answer is $$y = \ln x$$ and for the other order it is a peculiar expression which depends on whether $x$ is less than or greater than $1$.

1

It seems like your best bet for both questions will be to consider finite rings and their finitely generated modules. These at least will be closed under products.

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Let the initial set of points be $\mathbf P_i, i = 0,\dots, n$. Let also $S$ be the B-spline of second degree with knot vector $$\mathbf T = \left\{u_{-1}, u_0 = 0, \dots, u_{n+1} = n+1, u_{n+2}\right\}, \quad u_0 \leq 0, \quad u_{n+2} \geq n+1$$ and $\mathbf P_i$ as its control polygon. $u_{-1}$ and $u_{n+2}$ are boundary knots and can be defined ...

1

Here is a sketch (and there are other methods too) for jorst. For any point $p\in\mathbb{P}^2$, one has an exact sequence, $0\to\mathcal{O}_{\mathbb{P}^2}(-2)\to\mathcal{O}_{\mathbb{P}^2}(-1)\oplus \mathcal{O}_{\mathbb{P}^2}(-1)\to \mathfrak{m}_p\to 0$, where $\mathfrak{m}_p$ is the ideal sheaf defining $p$. Take a quadric $Q$ not passing through $p$ and ...

1

Area of regular $n$-gon in terms of circumradius $b$ is $$A=\frac{1}{2} nb^2 \sin\left(\frac{2\pi}{n}\right),$$so to avoid the sine function expand in a taylor series: $$A=\frac{1}{2} nb^2\sum_{k=0}^\infty \frac{(-1)^k\left(\frac{2\pi}{n}\right)^{1+2k}}{(1+2k)!}.$$

1

There are many books for graph theory, and you should select the topics that suit the education goals of the course. For example, in computer science the Graph theory algorithms is highly considered as main topic in each graph theory course. Personally, I prefer the following two books: Introduction to Graph Theory Graph Theory (Graduate Texts in ...

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