# All Questions

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### Algebra extension of modules question

Let $M$ be an extension of modules of $A$ by $B$ (modules), i.e. $A \leqslant M$ and $M/A \cong B$. Show that if $M$ is finitely generated then so is $B$. In the solution it just says: If $M = XR$ ...
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### Exercise 2.2.1 in Charles A. Weibel's book An Introduction To Homological Algebra

How to prove that a chain complex is a projective object in ${Ch}$ (chain complexes of $R$-modules) iff it is a split exact complex of projectives? A chain complex of projectives means a chain ...
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### Connected spaces: is there a mistake in the example below from James Munkres' Topology?

A subspace $Y$ of $X$ is seperable if it can be written as the union of nonempty, disjoint, open sets in $Y$, neither containing the limit point of the other. $Y$ should then not be connected, yet ...
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### Show that both these Prove $f$ is differentiable from the right at $0$

Let $f:[0,1) \rightarrow \mathbb{R}$ be continuous on [0,1) and differentiable on $(0,1)$. Suppose the limit $\lim_{x \to +0} f'(x)$ exists. Prove that $f$ is differentiable from the right at $0$. ...
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### Question about Big O Notation

I don't seem to understand big-O notation very well. If someone would explain it to me as well as explain how this problem would work Let f(n) = (3$^n$$^+$$^1$ - 3)/2. For each of the following ...
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### Why a function satifies certain conditions is a norm?

Let $E$ a linear normed space over $\mathbb{R}$ and $n:E\to \mathbb{R}_+$ an application such that: $n(x)=0\Leftrightarrow x=0$, $n(\alpha x)= |\alpha|n(x)$ and the set $$A=\{ x\in E:n(x)\leq 1\}$$ is ...
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I need some help understanding the steps to take to prove subsets. Question: For each of the following universal statements regarding any three finite sets $X, Y$, and $Z$, determine whether it is ...
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### Lebesgue Integrable functions and calculating the limit

$$\lim_{n\rightarrow \infty} \int_{\frac 1 n }^1 \frac { 1+nx }{ (1+x)^n } \, dx$$ How can I solve this problem using Bounded convergence theorem?
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### General Solution for Cosine (negative angles)

cos2(x+pi/3)=1/2 2(x+pi/3)=pi/3 x+pi/3=pi/6 x+2pi/6=pi/6 x=-pi/6 x=5pi/6 (is this step correct) ... ?? x = +/- pi/6 +kpi , k is a subset of Z x = +/- 5pi/6 +kpi , k is a subset of Z can someone ...
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### Linear Optimization Study Material

I've recently enrolled in a linear optimization course, and it's been a while since I've taken linear algebra. I do not yet have access to the book for the course or I would skim it to see what I need ...
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### Question about sums of limit superiors

Let $(s_n)$ and $(t_n)$ be sequences defined on $\mathbb{R}$. Prove that lim sup $s_n$ + lim sup $t_n \geq$ lim sup $(s_n+t_n)$ Proof (can someone please verify it?): Set $\alpha =$ lim ...
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### Proof for inequality with $a,b,c,d$ with $d =\max(a,b,c,d)$

Let $a,b,c,d$ positive real numbers with $d= \max(a,b,c,d)$. Proof that $$a(d-c)+b(d-a)+c(d-b)\leq d^2$$ I believe that the GM-AM inequality with $n=4$ variables might be helpful. ...
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### Cross-Validation

Does anyone understand the paragraph below? The paragraph comes from Cross-valiation explanation at wikipedia. "It can be shown under mild assumptions that the expected value of the MSE for the ...
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### Integral Bessel recurrence relation

I want to show that $\int x^vJ_{v-1}(x)dx = x^vJ_v(x) + C$. Now I know the recurrence relations of the Bessel equation/function and the one I need to use is $x^vJ_v(x) = x^vJ_{v-1}(x)$ I'm just ...
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### Sum of divergent and convergent sequence proof.

Suppose $(s_n)$ is a convergent sequence and $(t_n)$ diverges to $\infty$. Prove that lim $s_n+t_n = \infty$. Proof (can someone verify it?): Pick $N_1$ such that $\forall n > N_1$, ...
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### Is integration/differentiation an inverse relation?

A: $\frac{d}{dx}(\sin(x)) = \cos(x)$, $\int(\cos(x)) = \sin(x) + C$ B: $\sin(\arcsin(x)) = x$ Both A and B are inverse relations? A goes full circle with a FUNCTION, and B goes full circle with a ...
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### Probability question and how to approach it

Nine tiles are numbered 1, 2, 3, . . . , 9. Each of three players randomly selects and keeps three of the tiles, and sums those three values. Find the probability that all three players obtain an ...
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### Definition of Cyclic subgroup

The above is a theorem from my book. What I don't understand is the second sentence when it says $b$ generates $H$ with $n/d$ elements. I thought that since $b = a^s$ generates $H$, it would ...
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### Black Derman & Toy Model

The BDT model is given by $$d(\ln\,r)=\left(\theta(t)-\frac {d(\ln\sigma(t)}{dt}\ln r\right)\,dt+\sigma(t) \, dW.$$ How can one rewrite the BDT model as $dr=A\,dt+B\, dW$, using It$\hat o$?
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### Minimizing the sum of vectors

I have this problem: Given a set of unit vectors $\{ \vec{v_i} \}$, I want to determine another set, $W$, the element of which are in $\{ \vec{v_i} \}$(repeating allowed), so that the module of the ...
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### Borel sigma algebra in topology and R

This is a very basic question about which I am concerned. What is the difference between the Borel $\sigma$-algebra for a topological space and for $\mathbb R$? or they are same?
I'm thinking about the image under the adjoint representation $\mathrm{Ad}$ of the proper (identity connected component) Lorentz group $SO^+(1,3)$. Since this group has a trivial centre (it contains, ...
### Why does $\frac{4}{2} = \frac{2}{1}$?
I take for granted that $\frac{4}{2} = \frac{2}{1}$. Today, I thought about why it must be the case. My best answers amounted to $\frac{4}{2}=2$ and $\frac{2}{1}=2$; therefore ...