# All Questions

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### Find quotient space on $\mathbb{N}$

On $\mathbb{N}$ is given equivalence relation R with $nRm \iff 4|n-m$. Topology on $\mathbb{N}$ is defined with $\tau=\{\emptyset\}\cup\{U\subseteq\mathbb{N}|n\in U \wedge m|n \implies m\in U\}$. I ...
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### Automorphism of $A[t]/(t^m)$

Let $A$ be a commutative ring and $t$ an indeterminate over $A$. If $f$ is an automorphism of the ring $A[t]/(t^m)$ satisfying $f(x)\equiv x\pmod{(t)}$ for each $x\in A[t]/(t^m)$ with $m$ a positive ...
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### Show that the projection map is Orientation preserving iff n is even

My question is that Orient the unit sphere $S^n$ in $\Bbb R^{n+1}$ as the boundary of the closed unit ball. Let $U$ be the upper hemisphere $U =${$x∈S^n |x^{n+1} >0$}. It is a coordinate chart on ...
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### Second-order ODE with substitution

I’m struggling with this question: Use the substitution $y(t) = z(t)\,e^{-t}$ to transform the ordinary differential equation $$\frac{d^2 y}{dt^2} + 2\,\frac{dy}{dt} + y = t^2 e^{-t}$$ into an ...
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### Embeddings of $GL(n-1,q)$ into $GL(n,q)$

Let $n>2$ and $q$ be a prime power. I want to find all the embeddings of $GL(n-1,q)$ into $GL(n,q)$. An embedding I first thought of is as following. Let $V$ be an $n$-dimensional vector space over ...
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### On the primality of integers of the form $p^2+k$

I am not able to find an answer to the following question: For which positive even integers $k$ is the integer $$p^2+k$$ prime, where $p$ is a prime number $\gt5$?
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### Econometrics Simultaneous equation Indirect Least Squares and Two Stage Least Squares

I still can't figure out this problem. PLEASE HELP! (1) $F_t = a_1 + a_2.C_t + a_3.P_t + e_t$ (2) $P_t = b_1 + b_2.F_t + b_3.S_t + b_4.I_t + u_t$
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### Consider $x^4 \pmod {pq}$, with $p = q = 3 \pmod4$.

Consider $x^4 \pmod {pq}$, with $p = q = 3 \pmod 4$. Would someone explain to me why exactly one of the four square roots of $x^4 \pmod {pq}$ is also a square? This result was given without proof ...
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### The weight of the extended ternary Golay code

I need to show that the extended ternary Golay code with parameters $[12,6]$ is a self dual code and has minimum weight $6$ so i will obtain a $[12,6,6]$-code. I showed the self duality but how can I ...
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### 3D rotation group

It is known that the group $\text{SO}(3)$ of rotation-matrices (matrices $A$ with det(A)=1) are generated from three parameters. This can be expressed by the fact, that any rotation matrix is a ...
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### How do you parametrize the paraboloid to find the flux of $F$ across $S$?

Evaluate the surface integral double integral of $S$ of $F * dS$ for the given vector field $F$ and the oriented surface $S$. In other words, find the flux of $F$ across $S$. For closed surfaces, use ...
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I am studying a paper where it is mentioned that Newton iteration may be used to compute the inverse of $n \times n$, well- conditioned matrix in parallel time $o(\log^2n)$ and that this computation ...
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### Consider a Group $G$ of order $20$ such that $G=Aut$($Z_{25},+) \cong (U_{25}, \cdot)$. Analyze the Sylow subgroups in G.

$G=Aut$($Z_{25},+) \cong (U_{25}, \cdot)$ I know that there is one 5-Sylow subgroup and number of $2-Sylow$ subgroups is either $1$ or $5$. (a) How do I decide whether the number of distinct 2-Sylow ...
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### $(x-a)^2(x-a-1) + p \equiv 0 \bmod{p^3}$

Let p be a prime and a belong to Z. Find all solutions to the equation $$(x-a)^2(x-a-1) + p \equiv 0 \bmod{p^3}$$ I'm having a hard time working with this as such few variables are given. We know p ...
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### Can flux be proportional to $r^2$ in divergence theorem?

The motivation for the divergence being interpreted as the flux of stuff used the following: $$\text{div} F(a) = \lim_{r\to0}\frac{3}{4\pi r^3}\int_{|x-a|=r} F\cdot n dA$$ Without the $r^3$ in the ...
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### Units on a ring

Let $A$ be a ring such that there exists units $u_1, \dots , u_p : u_1 + \dots + u_p=0$ and $p$ a prime. Did $A$ have a subring isomorphic to $\mathbb{Z}/p \mathbb{Z}$ in general Thanks in advance.
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### Proving an Integral domain is a field. [duplicate]

Let $R$ be an integral domain containing a field $F$ as a subring. Show that if $R$ is a finite dimensional vector space over $F$, then $R$ is a field. This is a Ph.D. entrance question, I recently ...
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I am working on some exercises in the book Geometry: A Metric Approach with Models by R.S. Millman. He defines the following map: $$d_S(P,Q):\mathbb R^2\times\mathbb R^2\to\mathbb R\\\ ... 4answers 111 views ### Closed form of a recurrence relation using generating functions It's been awhile since I have done this. The sequence is \displaystyle a_n = a_{n-1} + 5~a_{n-2} with a_{0}=0 and a_{1}=1. I found the generating function to be \displaystyle G(x) = ... 1answer 185 views ### Prove that if Q^tQ = I and A = QR, then \|Ax - b\| = \|Rx - Q^tb\| I have a linear algebra final tomorrow and was practicing a few proofs. I want to make sure this proof is correct. Prove that: If Q^tQ = I and A = QR, then \|Ax - b\| = \|Rx - Q^tb\| ... 2answers 91 views ### Question about derivation in Jordan algebra Let (G,\circ) be a Jordan algebra, then \sigma:G\to G given by$$c\mapsto a\circ(b\circ c)-b\circ(a\circ c),\quad\forall c\in G,$$is a derivation, where a and b are two fixed elements of G. ... 4answers 2k views ### Find \lim\limits_{n \rightarrow \infty}\dfrac{\sin 1+2\sin \frac{1}{2}+\cdots+n\sin \frac{1}{n}}{n} Find$$\lim_{n \rightarrow \infty}\frac{\sin 1+2\sin \frac{1}{2}+\cdots+n\sin \frac{1}{n}}{n}. This is a recent exam question, which I couldn't figure out in the exam. My guess is it doesn't ...
Is it acceptable to write $.001$ rather than $0.001$ when using decimal notation? Are there contexts in which omitting the leading zero is acceptable, and other situations in which it is not?
### Finding Partial Derivative ($n$-dimensional) using implicit differentiation vs explicitly solving
This is a book example (not a homework question) about implicit differentiation on a composite of functions in $n$-dimensional space. But my book explains this example in a very unclear manner. So I ...