# All Questions

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### To simplify $\sum\limits_{k|(2m, 2n), k\nmid m} \varphi(k)$

Consider the Euler phi-function $\varphi(n)$ of $n\in \mathbb N$ as the cardinality of $\{1\leq r\leq n: (r, n)=1\}$. I am willing to simplify $$S:=\sum\limits_{k|(2m, 2n), k\nmid m} \varphi(k).$$ ...
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### Parametrization of a curve and by arclength

I am working on differential geometry and am having quite the blast, it's mathematics so what else can you have? Enough about that, I am working on the curves and we have various parametrizations and ...
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### Does permuting indices of a cocycle leave the Cech cohomology class the same?

Consider a $X$ a space, $\{U_i\}_i$ an open covering, and $\mathcal{F}$ a sheaf on $X$. Consider $(c_{i_0,\ldots,i_n})$ a Cech $n$-cocycle. It is theorem that every Cech $n$-cocyle is cohomologous to ...
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### How to prove that there is a bijection from the set Hom$(G_1\oplus G_2, G)$ to Hom$(G_1, G)\times$ Hom$(G_2, G)$?

Willing to solve the following problem but got stuck. Need your help. We denote the collection of all group homomorphisms from $G_1\oplus G_2$ to $G$ as $\text{Hom}(G_1\oplus G_2, G)$. Similarly ...
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### Question in Probability Theory

Lets say that I have 3 sets $A,B,C\subseteq E$ and I know $|E|,|A|,|B|,|C|,|\overline{A}|,|A\cap B|,|B\cap C|,|C\cap A|,|\overline{A}\cap B|,|\overline{A}\cap C|$ and $|A\cap B\cap C|$. Is there any ...
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### Show that $(2^n-1)^{1/n}$ is irrational

How to show that $(2^n-1)^{1/n}$ is irrational for all integer $n\ge 2$? If $(2^n-1)^{1/n}=q\in\Bbb Q$ then $q^n=2^n-1$ which doesn't seem right, but I don't get how to prove it.
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### Integer solution for $x^ay^b-z^ct^d=1$.

P1 Find all nonnegative integer number $(a,b,c,d)$ such that $2^a5^b-3^c11^d=1$. P2 Find all nonnegative integer number $(a,b,c,d)$ such that $2^a3^b-5^c7^d=1$. I am looking for problems ...
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### Showing $P=(o,m) \not\in 2E_m(\mathbb{Q})$ for an elliptic curve $E_m$

I have been having trouble with this question. Let $m\in \mathbb{Z}$ with $m > 0$ and define $E_m : y^2 = x^3 −x+m^2$ Then $E_m$ is an elliptic curve Determine the group sturcture of ...
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### Universal property of kernel of a homomorphism

I am in the beginning of Category Theory. Let $\theta\colon G\rightarrow K$ be a group homomorphism, and $\ker\theta$ denote the kernel of this homomorphism. Let $\varepsilon\colon G\rightarrow K$ ...
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### Radial & Cross-Radial Acceleration: A problem

A particle moves along $r=Ae^{\mu\theta}$ where $\theta=Bt$, prove that its acceleration is proportional to $r$ and makes a constant angle with the radius vector. Approach: $\dot{\theta}=B$ then ...
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### What is the expansion in power series of ${x \over \sin x}$

How can I expand in power series the following function $${x \over \sin x}$$ I know $$\sin x = x - {x^3 \over 3!} + {x^5 \over 5!} -+ \cdots$$ but substitution doesn't give me a hint to continue. ...
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### Calculus: $f◦(g◦h)=f◦g+f◦h$ - Indirect reasoning

"Is it true that $f◦(g◦h)=f◦g+f◦h$?" I think I should use indirect reasoning to prove this but I don't know where to being. Any help?
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### Multiplying $r$ times the function $f(x)=a\sqrt x+b$

I have the next function: $f(x)=a\sqrt{x}+b$ over the domain $\left[ 0, n-1 \right]$, where $n$ is a natural number. For any given $r$ inside the domain (also a natural number), I need to find a ...
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### Can someone solve this? [on hold]

If $$8432: 7 \\ 9213: 0 \\ 5144: 16 \\ 9064: 9 \\$$ Then $$7103: ?$$
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### Permutation formula is wrong for this question.

The number of permutations of n distinct objects taken r at a time is nPr. You have four letters, a, b, c, d. Consider the number of permutations that are possible by taking two letters at a time ...
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### Total number of possible binary operations .

If there are n elements in a set the number of binary operations that can be defined are 2n, am I right or wrong ?
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### Sum of (only certain) prime reciprocicals

It is well known that $$\sum_{p\ is\ prime}\frac1{p}$$ diverges. Is there a simple proof that $$\sum_{p\equiv 1\pmod 4}\frac1{p}$$ and $$\sum_{p\equiv 3\pmod 4}\frac1{p}$$ also diverge? (p ...
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### Probability of getting black or white

If I have a bag of black and white marbles, say 10 black and 10 white, I have a 50% chance of either getting black or white. What if the bag were infinite? Can one talk about probability when ...
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### ZFC,unprovability of existence of a countable model,Skolem construction and paradox

The well-known Skolem construction yields a countable model of ZFC,elemetarily equivalent to the universe of sets $V$.Why this construction is not a proof of existence of models of ZFC,as such proofs ...
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### Trinomial Equation

$$x^n + x^m = 1$$Given $n,m$ are two positive distinct integers, prove that $x$ is irrational number which can be expressed only in terms of $n,m$ (Assume $n$ is greater than $m$)
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### Is there a notation for saying that a function is defined on some subset of a set?

Let $X,Y$ be sets and let $f$ assign to every $x \in X$ some unique $y \in Y$. Then we may write $f: X \to Y$. This notation has the advantage that $f(X)$ need not be $Y$ but must be "within" $Y$. But ...
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### Is the topological entropy of a continuous map $T\colon X\to X$ zero if $X$ is a finite compact topological space?

Let $X$ be a finite compact topological space and $T\colon X\to X$ continuous. As the title already suggests, I am wondering if the topological entropy of $T$, denoted by $h(X,T)$, then is $0$. As ...
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### Which multivariate Gaussian has the highest expected norm?

Let $X$ be an $n$-dimensional Gaussian vector with zero mean and covariance matrix $K$ given by: $$K_{ij} = \begin{cases} p_i(1-p_i) & i=j \\ -p_ip_j & i\neq j\end{cases}~,$$ where ...
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### Does anybody know a proof that the bases of two or more numbers with the same exponent can be multiplied together? (examples in the main text body).

I.e. how can we simply assume that $5^{1/2}\times 6^{1/2} = 30^{1/2}$ (i.e. all we have done is multiplied the bases and applied the exponent to the answer). In more general terms, how can we assume ...
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### Is anyone familiar of a term called “Injected Matrix”?

Is anyone familiar of a term called "Injected Matrix"? I have been asked about it by one of my former students. He is taking a course in stochastic models and encountered the term there. Thank you!
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### Big Oh symbol in Taylor expansions

Consider a remainder of some Taylor series: \begin{align}\frac{Mx^6}{C} + \frac{M'x^8}{C'} + \frac{M'' x^{10}}{C''} + ...\end{align} I want to replace this with $\mathcal{O}(x^\alpha)$ for the best ...
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### Probability problem possibly based on principle of inclusion exclusion

The problem reads as follows: Probabilities that Rajesh passes in Physics, Math and Chemistry are $p$, $m$ and $c$ respectively. Of these subjects, Rajesh has $75%$ chance of passing in at least ...
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### Finding the Number of Subfields of the Splitting Field of $x^{35}-1$ over $\mathbb{F}_8$

Let $E$ be the splitting field of $x^{35}-1$ over the field $\mathbb{F}_8$. Determine $|E|$ and the number of subfields of $E$. Attempt: I am confident that I computed $|E|$ correctly, but I am ...
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### How to find $n$th term of two sequences both dependent on each other?

Given two sequences , $$\large a_{n+1}+b_{n+1}=2\sqrt{2}(a_{n})+2\sqrt{6}(b_{n})$$ $$\large a_{n+1}-b_{n+1}=2\sqrt{2}(b_{n})-2\sqrt{6}(a_{n})$$ Given $\large a_{0}$ and $\large b_{0}$ , How can i find ...