# All Questions

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### Show there is no measure on $\mathbb{N}$ such that $\mu(\{0,k,2k,\ldots\})=\frac{1}{k}$ for all $k\ge 1$

For $k\ge1$, let $A_k=\{0,k,2k,\ldots\}.$ Show that there is no measure $\mu$ on $\mathbb{N}$ satisfying $\mu(A_k)=\frac{1}{k}$ for all $k\ge1$. What I have done so far: I am trying to apply ...
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### Determinant of the sum of matrices: $\det (A + B^T) = \det(A^T + B)$

How can you show that $$\det(A + B^T) = \det(A^T + B)$$ for any $n\times n$ matrices $A$ and $B.$
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### How to prove positive definiteness?

$B_{(n+1)(n+1)}$ = $\begin{bmatrix} A & u \\ u^T & 1 \\ \end{bmatrix}$ is given, and $A$ is a positive definite matrix where its Cholesky factorization is given ...
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### Trying to price options using infinite series

If you are trying to price an option if the stock surges you can reap a very large return, but most of the time the return is $-p_1$ where $p_1$ is the amount you invested The problem i'm running ...
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### Non-trivial homomorphism between multiplicative group of rationals and integers

Let $\mathbb{Q}^{\times}$ be the multiplicative group of non-zero rationals. Is there a non-trivial homomorphism $\mathbb{Q}^{\times} \to \mathbb{Z}$? In the same spirit, is there a homomorphism ...
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### Limit points question

Let $X$ be any topological space and let $C \subseteq X$ be any subset. If $C$ is a closed subset of $X$ does it follow that the set of all limit points of $C$ is closed as well?
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### 20 options. 10 must be chosen. how many combinations exist

I have a list of twenty options and each option is different. If ten options must be chosen, how many combinations exist? Can someone show me how the math on this? Example, if i have 20 different ...
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### Is $Z$ also independent??

In a problem I am asked to find $\Bbb P(X=1|\frac{X+Y}{2}=2)$, and $X$ and $Y$ are independent random variables. In a previous part of the problem I defined $X+Y$ to be $Z$. So I simplified the ...
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### How many times does this expression give the value 0 as modulus?

$S=\{1,2,3\ldots,19\}$ $(5k + 5) \mod 20$ $\gcd(20,5) = 5$ $20$ and $5$ are divisible by $5$ and $1$. thus the expression gives the value $0$ $2$ times between $1$ and $19$?
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### What is the probability one card from each suit will be represented when 5 cards are dealt?

If each suit is represented then we will have two cards from one suit and one card each from the remaining suits. So I am counting the ways this can happen like so - ${52 \choose 1}$ ways of ...
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### Analytic function on convergent sequence

Let $f:U\to\mathbb{C}$ analytic function, where $U$ is a region. $x_n \to x_0 \in U$ is a real convergent sequence, it is known that $f(x_n)$ is real for all $n$. Is it true $f^{(n)}(x_0)$ is real for ...
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### Intuitively explaining the difference between a combination and permutation

I'm having a hard time trying to determine when to use combination and when to use permutation with a problem. Can someone offer a clear and concise explanation or general rules to follow so I don't ...
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### Does $\lim_{n\rightarrow \infty} \int_X f_n - \int_X f\gt 0$ implies that convergence of $f_n$ to $f$ a.e. fails?

I've come across this problem as a part of another proof that I'm writing and I want to know if this is a right conclusion: Let $X$ be a finite measure space and $\{f_n\}$ be a sequence of ...
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### How to determine a in r - in a function of relations

I'm pretty stuck on the following question $f$ on $\mathbb{R}$ given by $xfy\Leftrightarrow (y(2x-3)-3x=y(x^2-2x)-5x^3)$ is a function. Let $g$ be the restriction of $f$ to $\mathbb{Z}^+$, implying ...
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### What is the cardinality of a transcendence basis of $\mathbb{C}$ over $\mathbb{Q}$?

What is the cardinality of a transcendence basis of $\mathbb{C}$ over $\mathbb{Q}$? Is it that of the continuum? Proof?
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### Calculating the Fundamental group of $\Bbb R P^2$

The fundamental group of $\Bbb R P^2$ is $\Bbb Z \times \Bbb Z$. I cannot understand why though, since $\Bbb RP^2$ is a disc with a Möbius strip and the disc is contractible so wouldn't it have ...
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### How do we show that this modular function is one-to-one?

$$f(x) = {(x^2+5)}\bmod{9}$$ $${(x^2 + 5)} \bmod {9} = (x^2 + 5)\bmod 9$$ $$(x^2 + 5) = (x^2 + 5)$$ $$x^2 = x^2$$ $$x = x$$ Is this the correct way to do this? I have no idea how to manipulate the ...
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### Find all Laurent series of the form…

Find all Laurent series of the form $\sum_{-\infty} ^{\infty} a_n$ for the function $f(z)= \frac{z^2}{(1-z)^2(1+z)}$ There are a lot of problems similar to this. What are all the forms? I need to ...
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### Finding connected components of two given spaces

Suppose that in the Cartesian plane $\mathbb{R}^2$ we let $X$ denote the union of all lines through the origin with rational slope. Would that make $X$ connected, since all such lines are connected ...
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### Smooth Monotone $\mathbb{R}^3$ curve with constant (nontrivial) curvature

So I was trying to construct a closed curve in $\mathbb{R}^3$ with constant positive curvature and non-trivial torsion. To do this I tried to glue two helices together in a smooth way with a curve ...
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### Help finding coterminal angles?

I'm trying to find an angle between 0 and 2π that is coterminal with -4π/3 in terms of pi. How do I go about doing this?
### Prove that if $5$ divides $a^2$, then $5$ divides $a$
Ok so my teacher said we can use this sentence: If $a$ is not a multiple of $5$, then $a^2$ is not a multiple of $5$ neither. to prove this sentence: If $a^2$ is a multiple of $5$, then $a$ itself is ...