# All Questions

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### Proving Existence of Field Extension

This seems so simple that it doesn't warrant a proof, but how would one prove this if asked to? If $K$ is a field and $f(x)\in K[x]$ is monic, then prove that there exists some field $L\supseteq K$ ...
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### Examples of manifolds foliated by $S^2$

I have come across the Frobenius theorem in my study of GR, which for the special case of $S^2$ roughly means, that every point of a manifold with spherical symmetry can be foliated by spheres. I know ...
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### Probability of selecting four letters from ENCYCLOPAEDIA

From a maths textbook: "Four letters are randomly selected from the word ENCYCLOPAEDIA. Find the probability that one letter E will occur in the selection of 4 letters." Clarification - as I ...
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### What are some physical, geometric, or otherwise useful interpretations of divergent sums?

Is there any practical application to discussing the 'sum' of sequences that are not convergent under Cesàro summation? Why would we want to assign a value to an otherwise divergent sequence and ...
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### Mirror images of knots and Kauffman and HOMFLY polynomial

Let $K$ is a knot and let $\bar{K}$ be the mirror image of $K$. I want to confirm this relationships. Let $f_K(t)$ be the Kauffman polynomial of $K$. To get the mirror image we swap every right ...
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### What is the relationship between the spectrum of a matrix and its image under a polynomial function?

Clearly if $\lambda$ is an eigenvalue of $A$ then $p(\lambda)$ is an eigenvalue of $p(A)$ where $p$ is a polynomial. And there are cases where $A$ may have eigenvalues other than these. Is there a ...
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### How to solve $(2x+2, 6x) = x+1$

I'm looking at an old discrete mathematics test preparing for my test tomorrow and one question says solve for $x$ and gives the above. I'm thinking that this is the $\gcd(2x+2, 6x)$ but have never ...
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### Proof of geometric congruence using linear algebra

We may assume some set $T$ of all triangles within the same plane. Let $R$ be defined on $T$ where $a\ R\ b$ if the triangles $a, b$ are congruent. We may assume congruence to be defined as follows ...
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### A group theory exercise in Brown's group cohomology book

My question is taken from exercise IV.3.4(b) in Brown's group cohomology book. Let $E$ be a finitely generated group, and suppose that $C$, the center of $E$, has finite index in $E$, say $[E:C]=n$. ...
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### Proving a limit exists

Supposing that a function $g(x)$ is differentiable on the interval $(0,1]$, and $g'(x)$ is bounded on $(0,1]$: show that the limit of $g(1/n)$, as $n$ goes to infinity, exists. My initial guess was ...
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### The theorem on formal functions

Recall the Theorem on Formal Functions [Hartshorne, III.11.1] Let $f:X \to Y$ be a projective morphism of noetherian schemes, let $\mathcal{F}$ be a coherent sheaf on $X$ and let $y\in Y$. Then the ...
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### Find a quantity that is constant along the solutions of the ODE

I have the following problem. Find, for any $r > 0$, a quantity that is conserved along the trajectories of the system $\ddot{x} = r - e^x$ How do I exactly solve this question? I had a ...
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### Sum of absolute values and the absolute value of the sum of these values?

I'm working on a proof and I need some help with this: I determined that for some situations ($x$ or $y$ are negative but not both): $|x| + |y| > x + y$ How can I conclude using that statement ...
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### Expected value of a number of events

If I have a set of random events say ${\{A_{1}, A_{2}, ... A_{n}\}}$ and a number ${N=\sum_{1}^{n} I(A_{i})}$ where ${I(A_{i})}$ is the indicator function (basically ${N}$ is the number of events that ...
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### Group theory: Let $H=\{0,\pm 3, \pm 6, \pm9,\ldots\}$ Find all the left cosets of $H$ in $\Bbb Z$.

I am having trouble understanding the following homework question, Let $H=\{0,\pm 3, \pm6, \pm9,\ldots\}$ Find all the left cosets of $H$ in $\Bbb Z$. I know the answer is $H$, $1+H$, and $2+H$ but ...
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### help with nand circuit

I tried to make a circuit from this expression but it's not working right. Here's the expression and circuit: Expression ...
### Determination of the prime ideals lying over $2$ in a quadratic order
Let $K$ be a quadratic number field. Let $R$ be an order of $K$, $D$ its discriminant. By this question, $1, \omega = \frac{(D + \sqrt D)}{2}$ is a basis of $R$ as a $\mathbb{Z}$-module. Let ...
### Why is the sequence $a_n = \left(1+\frac{1}{n}\right)^n$ Cauchy?
I was looking at the post: Cauchy Sequence that Does Not Converge And the top answer was this sequence: $a_n = \left(1+\frac{1}{n}\right)^n$. I understand that this sequence converges to $e$, ...