# All Questions

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### Finitely generated modules over principal ideal domain

Let A be principal ideal domain with field of fractions K. L is finite separable extension of K and B is integral closure of A in L. It is obvious that there exists constant d in A, such that dB is ...
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### Demonstration with fitch notation and quantifiers

I'm tryng to demonstrate with fitch notation this: {∀x (A(x) ↔ B(x)), ∀x (A(x))} |= ∀x (B(x)) Here what I tried: http://i.stack.imgur.com/7S5Zy.png Someone can explain me how i can obtain ∀x (B(x)? ...
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### Rational Funtion Integration

This looks to be a simple problem, but it has me stumped. I already have the answer, but a step-by-step solution would be appreciated. $$\int\frac{x+4}{x^2+2x+5}$$
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### What's the answer to this number and digit problem? (I'm confused at “exceeds three times the tens digit by 3”)

The units digit of a two digit number exceed three times the tens digit by 3. If the tens digit is subtracted from the units digit, the difference is 7. Find the number.
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### Find tangent to trigonometric function

I want to find the tangent to the curve: $x\sin{y} + y\sin{x} = \frac{\pi}{4}(1+\sqrt{2})$ through the point $(\frac{\pi}{2}, \frac{\pi}{4})$ Now I know I can fill certain information into this ...
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### Total derivative vs. partial derivative Legendre transformation

My question is about how to compute the total derivative for the function $f(x,y)$. In theory we have: $$df=\frac{\partial f}{\partial x}dx+\frac{\partial f}{\partial y}dy$$ So, if somebody asks ...
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### Stoke's formula for a sphere

I have a question, raised form kinetic theory (pure mathematical). Imagine, that $\Psi (\overrightarrow{r},\overrightarrow{p},t)$ - sufficiently smooth function, where $\overrightarrow{r}$ - radius ...
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### Confused About Definition of a Limit Proof

I'm working on $\epsilon-\delta$ limit proofs, and there's something about the proof I don't get. Currently doing a proof for $\lim_{x\to 3} (2x-1) = 5$. The first part of the definition says "if ...
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### Graph Theory for Dummies Book

Does anyone have a good book on Graph Theory that will introduce me to some of the basic concepts without being so filled with terminology that it's hard to read? I have taken an introductory course ...
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### Find isomorphism between $D_6$ symmetries of regular triangle, $S_3$, and $GL_2(F_2)$

problem: Find isomorphism between $D_6$ symmetries of regular triangle, $S_3$, and $GL_2(F_2)$. I already proved $D_6$ is isomorphic to $S_3$. And $S_3$ is isomorphic to $GL_2(F_2)$ Am I suppose ...
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### Probability of no 6 or no 5 when dice is rolled n times

Can anyone guide me in the general direction of the answer to the following: A die is rolled $n$ times $$A = \text{no 6s}$$ $$B = \text{no 5s}$$ $$P(A\cup B) = \;?$$ I am first finding $P(A)$ ...
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### Determine measurability of E(X|N) or even $\sigma(E(X|N))$?

Suppose $(\Omega, F, P)$ is a sample space, $X$ a random variable, and $N$ a sub sigma algebra of $F$. How can we determine $\sigma(E(X|N))$? How is $\sigma(E(X|N))$ related to $\sigma(X)$ and ...
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### Question: outer measure

To define the outer measure of an arbitrary subset $E\subset \mathbb{R}^n$, cover $E$ by a countable collection $S$ of intervals $I_k$, and let $$\sigma(S) = \sum_{I_k \in S} v(I_k)$$ where $v(I)$ is ...
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### Non computable numbers are normals?

We know that almost all real number are normal and almost all real number are non computable. This does not suffice to deduce that all non computable numbers are normals but , intuitively (??) this ...
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### $\pi_7(S^4)$ contains an element of infinite order.

Show that $\pi_7(S^4)$ contains an element of infinite order. Now, I know that I should probably use the Hopf bundle here somewhere. I also know that $\pi_3(S^7) = 0$. But I am stuck. Can anyone ...
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### How can I find the length to this geometry problem?

A person 6 feet tall is standing at the base of a lamp post that is 25 feet tall and then begins to walk away from the lamp post. When the person is 10 feet from the lamp post, what is the length of ...
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### Is the maximum of the eigenvalues of any symmetric positive?

Let $A$ be a symmetric matrix having dimension $n \times n \;, \; \; n\geq 2$. If one wants to pick the maximum of its eigenvalues, will the value be positive? Suppose A was an adjacency matrix, ...
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### Grothendiek and singular points

My question relates to the following thread I opened some weeks ago: A question regarding Grothendieck , topos and (adelic??) points Specifically, consider this paragraph: At 1:14:30 and after, ...
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### Proving $(1-x)\cdot (1-x^2)\cdots(1-x^{n-1})=n$ if $x^n=1$ and $x\neq 1$

If we have a equation $x^n=1$, then how can we prove $$(1-x)\cdot (1-x^2)\cdots (1-x^{n-1})=n$$ when $x$ is not $1$? I know that $x= e^{(2\pi + 2k\pi)/n}$ and we can get different value of $x$ when ...
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### $Y$ be the group such that it is define as $Y = < u, v | u^4 = v^3 = u= v=1, uv = v^2u^2, v^2 = v^{-1}>$ .

Let $Y$ be the group such that it is define as $Y = < u, v | u^4 = v^3 = u= v=1, uv = v^2u^2, v^2 = v^{-1}>$ . a) Show that $v$ commutes with $u^3$. [Show that $v^2u^3v = u^3$ by writing the ...
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### Rank of an $m$ by $n$ matrix?

Can anyone state, in plain English, how to find the rank of an $m$ by $n$ matrix? Is it necessary to perform Gaussian elimination first, or translate it into upper triangle form (or however it is ...
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### Examples of Lp spaces in Applied Math

I was wondering if there are examples of exotic Lp spaces being used in applied mathematics. I know that the "special" p's (1,2 , infinity ) are of use, for example in statistics, L1 is mean, L2 is ...
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### Proving an identity of the resolvent set

$\{T(t)\}_{t\ge 0}$ is a $C_{0}$-semigroup with infinitesimal operator $A$. I'm trying to prove that the set $\{ z|\text{ Re }z>\omega_{0}\}$ belongs to $\rho(A)$, and for $z$ in this set, the ...
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### Primal and dual feasible = optimal?

Is any primal feasible and dual feasible point of a convex function, a global min of that function? If yes, why? If no, do we need any more conditions?
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### Problem on vertical light elastic string

A mass of $4$ lbs suspended from a light elastic string of natural length $3$ feet extends it to a distance $2$ ft. One end of the string is fixed and a mass of $2$ lbs is attached to other. The ...
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### Supersolubility of $G/H$ and $H$ not deduced supersolubility of $G$.

I want show $S_{4}$ isn't supersoluble group. For this suppose $1 \leq B_{4} \leq A_{4} \leq S_{4}$ be a normal serie of $S_{4}$, that $B_{4}$ is Klein’s four-group. Since $B_{4}$ isn't ...
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### Every integer $n > 1$ can be written in one and only one way with a certain property

Every integer $n > 1$ can be written in one and only one way in the form $n = p_1p_2p_3 \ldots p_r$ where $p_i$ are positive primes s.t. $p_1 \le p_2 \le p_3 \le \ldots \le p_r$. $n$ is unique ...
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### Maximal compactifications without the Tychonoff theorem

I once saw a neat proof in American Mathematical Monthly of the Tychonoff theorem (The Tychonoff product topology of a family of compact spaces is compact) for the special case of the product of ...
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### Does every compact simply-connected subset of $\mathbb{R}^n$ have an efficient $r$-covering path for all $r>0$?

Let $A$ denote a subset of $\mathbb{R}^n$. Definition 0. Given a positive real number $r$, an $r$-covering path of $A$ is a non-negative real number $T$ together with a differentiable function ...
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In the book Dynamic Reported - The definition of the solenoidal sets is: Let $I_{0} \supset I_{1}\supset I_{2}\dots$ be periodic intervals with periods $m_{0}$, $m_{1}$,$\dots$. If $m_{i} \to \infty$ ...
We are given any arbitrary ellipse with focii $F1$ and $F2$ , $T$ is the unit tangent to the ellipse through a point $P$. Let $f$ be the sum of the distances of the of $F1$ and $F2$ from $P$ , we ...