# All Questions

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### Morphism of affine varieties.

I'm confused by the following thing I'm reading on the Wikipedia page Affine variety: Let $X = \operatorname{spec} A$, $Y = \operatorname{spec} B$ where $A$, $B$ are integral domains that are ...
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### Biyective graph homomorphism implies isomorphism.

Given this definition: If $G_1=(V_1,E_1),G_2=(V_2,E_2)$ are graphs, then $\varphi:V_1\rightarrow V_2$ is a homomorphism iff $\{v_1,v_2\}\in E_1\Rightarrow \{\varphi(v_1),\varphi(v_2)\}\in E_2$ I ...
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### How could I prove whether the infinite series $n!/n^n$ converges or diverges? [duplicate]

How would you go about proving the infinite series $\frac{n!}{n^n}$ converges or diverges?
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### Derivative of a function on a manifold

I want to show that: Given $f,g \in C^\infty(M)$ defined in a differential manifold of dimension $n$ and $a \in M$, we have $$(dfg)_a=f(a)(dg)_a+g(a)(df)_a,$$ using the following proposition: ...
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### how to calculate sum of a series? (me or Wangenmakers is wrong)

Wagenmakers in his critical article about p-values wrote that: $$\sum_{i=12}^{\infty} {{n-1} \choose {2}} \cdot \left(\frac{1}{2}\right)^n \approx .033$$ How could he do his calculations if the D'...
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### Product of functions in $H^1(B)$ where $B \subset \mathbb{R}^2$

I'm rather new to Sobolev spaces and finding myself rather deficient of intuition. So when given a problem like the below where I need to "prove or disprove", I'm finding myself stuck. Suppose $B$ is ...
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### Standardising Bivariate Normal

I don't understand why $\sigma_{Z_1}=\sigma_{Z_2}=1$ and $\mu_{Z_1}=\mu_{Z_2}=0$. I would understand if: $X_1$~$N(\mu_{x_1},\sigma_{x_1}^2)$ and $X_2$~$N(\mu_{x_2},\sigma_{x_2}^2)$ but this is not ...
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### Describe the image of following map $w= \frac{1}{z}$ with $|z-\frac {1}{2}| \leq \frac {1}{2}$

Describe the image of following map $w= \frac{1}{z}$ with $|z-\frac {1}{2}| \leq \frac {1}{2}$ I know that the inverse function map the inside of the circl to the outside of the circle and vice versa....
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### Eigenvalues for $y''+2y'=\lambda y$

I must find the eigenvalues and eigenfunction for $$y''+2y'=\lambda y$$ with initial conditions $y(0)=0$, $y'(1)=0$. I have found the non-trivial case, and made an attempt to solve for $\lambda$, but ...
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### Set with infinitely many limit points not contained in S

I'm trying to find a set S with infinitely many limit points but none of the limit points themselves can be contained in S.
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### Solve $\text{Minimize} \max\{|x-a_i|, i=1,..,n\}$

I have to solve $$\text{Minimize} \max\{|x-a_i|, i=1,..,n\}$$ For $a_1 \leq a_2 \leq ...\leq a_n$ My intuition says that this x is a point in the middle of the $a_i's$ but I am not sure that it is ...
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### integration of $\frac{x}{(x^2-1)^\frac{1}{2}}$

When I use the substitution $u=\cosh x$ to integrate $\frac{x}{(x^2-1)^\frac{1}{2}}$, I get $\frac{1}{2}(x+\sqrt{x^2-1}-\frac{1}{x+\sqrt{x^2-1}})$ but when I check online the answer is $\sqrt{x^2-1}$ ...
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### Odd number of students in odd number of classes

In a school there are an odd number of classes, and each class has an odd number of students. We want to choose a school council consisting of one student from each class. Prove that the following are ...
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### If $p$ is prime and congruent to $1$, then show $((\frac{p-1}{2})!)^2 \equiv -1 \pmod p$ [closed]

I got another one. Quadratic residues are completely new to me... Thanks!
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### If for any $\varepsilon$ exists $\delta$, does that mean that for every $\delta$ exists $\varepsilon$? [closed]
For any $\varepsilon \gt 0$ exists $\delta \gt 0$, does that mean that for any $\delta \gt 0$ exists $\varepsilon \gt 0$? If $\delta$ depends on $\varepsilon$ such as $\delta = \frac 1 \varepsilon$, ...