# Tagged Questions

Use this tag for questions asking about "problem books", "exercise books", and their solutions.

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### Aspherical homology class

I am completely stuck on the following algebraic topology exercise: Let $X$ and $Y$ be CW complexes and $\alpha \in H_p(X)$, $\beta \in H_q(Y)$, $p, q > 0$, homology classes such that the homology ...
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### How to prove that $x \rightarrow e^{1/x}$ is not a restriction of any real distribution to $\mathbb {R}_+$?

This is an excercise 2.2 from Hormander, vol. I: Does there exist a distribution $u$ on $\mathbb{R}$ with the restriction $x \rightarrow e^{1/x}$ to $\mathbb{R}_+$? The answer, provided in the book, ...
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### Probability solution verification

On a statistics trial exam I encountered the following tricky exercise: Assume that there are two types of car drivers in a country. Safe drivers constitute $70$% of the population and they have a ...
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### How to show $nP\{|X|>n\}\to 0$ as $n\to\infty$, but $X$ is not integrable.

How can I construct a random variable $X$ such that: $nP\{|X|>n\}\to 0$ as $n\to\infty$, but $X$ is not integrable.
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### Is there a solution manual for Royden fourth edition?

I bought the fourth edition of Royden Real Analysis, this book is awesome and is quite different of third edition that has less excersices. I have the solution manual for the third edition. Is there ...
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### Functional analysis problem involving maximal ideal space

For the past week, I've been trying to solve, as a practice homework exercise, Problem 6 of Chapter 11 in Rudin's Functional Analysis, but have not gotten very far it seems. The problem is as follows: ...
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### Prove the FHHF theorem using as much abstract non-sense as possible

This is my second attempt to solve exercise 1.6H from Vakil: Assume $F$ is a covariant right-exact functor and show that we get a map $HF^i \rightarrow H^iF$. Attempt to a solution: Apply $F$ to the ...
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### Exercise about the function field of an (irreducible) affine variety

Look at the following definition: I have problems to solve part iii) and iv) of the following exercise taken from Gathmann's notes: In particular for the part iii) given a class ...
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### Prove that if W is a subspace of a vector space V

Prove that if $W$ is a subspace of a vector space $V$ and $w_1, w_2, ..., w_n$ are in $W$, then $a_1w_1 + a_2w_2 + ... + a_nw_n \in W$ for any scalars $a_1, a_2, ..., a_n$. My solution is we have ...
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### Miranda book Proposition 3.10 pag. 40

Let $F$ and $G$ be two holomorphic maps between Riemann surfaces $X$ and $Y$. If $F=G$ on a subset $S$ of $X$ with a limit point in $X$, then $F=G$. Choose two charts $(U_\alpha,\phi_\alpha)$ and ...
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### A fact about independent exponentially distributed RVs

Let's say $Y_i$ are independent exponentially distributed with rates $c_i$ which can be assumed to be all in $(0, \infty)$ but not necessarily the same. Whenever $t\geq0, K\geq0$ let ...
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### Solvable Lie algebra with codimension 1 ideal

There is an exercise in Humphreys's An Introduction to Lie Algebras and Representation Theory: "Any nilpotent Lie algebra contains a codimension 1 ideal". The proof I am thinking of is the following. ...
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### Composition of a subharmonic function and a conformal mapping

this is q.4 of p.248 of Ahlfors book: Prove that a subharmonic function remains subharmonic after a composition with a conformal mapping. What I'v tried: Let $u:\Gamma \rightarrow \mathbb{R}$ and ...
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### Exercises on nets

well I'm learning nets with Munkres, but I'd like to do more exercises than those in this book. Any web site or reference would be welcome. Thanks in advance
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### Tricky calculus exercise (Now finding a tangent space)

Let $f(x,y,z)=z^3-g(x,y)z+h(x,y)$, with $g,h:\mathbb{R}^2\to \mathbb{R}$ of class $C^1$ and $g(x,y)>0\; \forall (x,y)\in\mathbb{R}^2$. Consider $S=\{(a,b,c)\in\mathbb{R}^3:f(a,b,c)=0\}$, ...
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### Vakil's exercise 5.5.#

I'd like to check that my solution to the following exercise of Vakil's FOAG is correct. I am bothered that we have to use (B) in it, and I want to make sure that I used it in the end for the correct ...
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### Finite p-primary group question

Problem is rotman's Intrdouction to theory of group, Exercise 6.8. Let G be a direct sum of b cyclic groups of order $p^{m}$. If $n< m$, then $p^{n}G/p^{n+1}G$is elementary and ...
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### How to solve this second order PDE $(\nabla\cdot k \nabla F+F=0 )$?

I am trying to follow an example of finding a fixed solution of PDE using method of manufactured solutions (p. 58). At some point there is an equation $$\nabla\cdot k \nabla F+F=0$$ where ...
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### Show that in a quasi-compact scheme every point has a closed point in its closure

Vakil 5.1 E Show that in a quasi-compact scheme every point has a closed point in its closure Solution: Let $X$ be a quasi-compact scheme so that it has a finite cover by open affines $U_i$. ...
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### Large deviations rate for binomial distributions

The problem is from Varadhan's Probability Theory, p.39, EXERCISE 3.7. Can you calculate the geometric ratio \rho(x)=\lim_{n\rightarrow \infty}\left(\sum_{r\geq nx} ...
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### Proving Fernbahnhof theorem (FHHF) using only concepts of abelian categories

Statement: let $F$ be a right exact functor. Describe a map $FH \rightarrow HF$. (from Vakil's notes 1.6H) attempt of a solution: Let $K$ be the kernel of $FA^i \rightarrow FA^{i+1}$. Since F is ...
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### Condition number for sample variance — summation question

I'm trying to understand the solution to Problem 1.7 from Accuracy and Stability of Numerical Algorithms by Higham: The sample variance is defined as: ...
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### Constant sheaves associated with locally closed subsets

I'm studying P. Schapira's notes Algebra and Topology, available online here, and I'm having trouble understanding sheaves associated with locally closed subsets, in particular constant sheaves. For ...
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### Analyticity of a function in $x$ and $y$, without employing the Cauchy-Riemann eqns

Exercise from Saff & Snider's Complex Analysis: How to determine the analyticity of this function, without using the Cauchy-Riemann equations? I tried to work from first principles (taking the ...