# All Questions

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### Is it sensible to define the absolute value of the integral and the derivative with measures in this way?

Is it sensible to define the absolute value of the integral and the derivative with measures in this way? $\mu$ and $\nu$ are measures (functions that take a shape [a set of points] and give the ...
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### Quadratic Approximation Using Chebyshev Economization

For the quadratic approximation of given function, we use following: $\ Q(f) = f(a) + f'(a)*x + f''(a)*(x^2)/2$ In the question, it wants me to find quadratic approximation using chebyshev ...
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### Do local Galois representations always lift?

Suppose $G:=G_F$ is the absolute Galois group of a local (residue char. $\ell$) or global field $F$, and $\bar{\rho}$ a (linear) representation of $G$ on the $\mathbb{F}_q$-module $\mathbb{F}^d_q$, a ...
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### Are every measure on $\mathbb{R}^{n}$ borel and/or regular?

I saw the above question in an exam paper and I am not sure how to even start. The question is true for the case of Lebesgue measure but I am not sure for arbitrary measures. I have tried looking at ...
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### finding general solutions of second order diffrential equation

find the general solution of $$\frac {d^2y}{dx^2} +9y =18$$ I am not sure how to write it in its complementary form because of the roots one being positive and ...
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### Nearest singular matrix

Let the SVD of $A \in \mathbb R^{{n}*{n}}$ be given as $A=\sum_{i=0}^n \sigma_{i}u_{i}v_{i}^{T}$ where $\sigma_{1}\gt \sigma_{2}>{...}>\sigma_{n-1}=\sigma_{n}>0$ Compute a matrix $B$ such ...
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### H prime order, normal subgroup of group G. Prove H in center Z(G).

I am looking at the following question: "Let H be a normal subgroup of prime order p in a finite group G. Suppose that p is the smallest prime that divides the order of G. Prove that H is in the ...
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### Khovanov polynomial twisted unknot. [on hold]

I am reading bar nathans paper about khovanov polynomials and am having a lot of trouble. so if we construct the chain complex we get: $0 \rightarrow V \otimes V \rightarrow V\{1\} \rightarrow 0$ I ...
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### How to prove that $\sin x = x \, _{0}F_{1}(-;\frac{3}{2};-\frac{x^{2}}{4})$?

How to prove that $\sin x = x \, _{0}F_{1}(-;\frac{3}{2};-\frac{x^{2}}{4})$? where $_{0}F_{1}$ is the hypergeometric series?
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### Solution to second order PDE $u_{xy}-xu_x+u=0$

Given second order partial differential equation $u_{xy}-xu_x+u=0$, where $u=u(x,y)$ find the general solution. I tried to use $u(x,y)=v(ξ(x,y),η(x,y))$ substitution to get ...
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### Prove that if $H$ is an abelian subgroup of a group $G$ then $\langle H, Z(G)\rangle$ is abelian.

Prove that if $H$ is an abelian subgroup of a group G then $\langle H, Z(G)\rangle$ is abelian. So I started out trying to show that $H$ is a subset of $Z(G)$ but then I realised that $Z(G)$ is the ...
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### What can you do with a Frobenius Monad? [on hold]

A frobenius monad is an endofunctor that has natural transformations making it both a monad and a comonad. What are the most interesting aspects to Frobenius monads?
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### Without calculating them determine whether $36^2+1$ and $154^2+1$ are prime and find the prime factors if not prime

I know that $36^2+1$ is prime, $154^2+1$ is not, both are equal to 1mod4. The prime divisors of $154^2+1$ should also be of the form 1mod4. Tried showing this by Wilson's theorem however i don't feel ...
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### Surface Normal from Cross Product

Given an equation (in this case $x^2-y^2+z=0$) how would I find the surface normal using a cross product at a certain $(1,2,3)$ point? I know how do it with $grad(f)$ but I presume that isn't what ...
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### What would be the value of the limit $\lim _{x\to \infty} (\frac{3x +1}{3x-1})^{4x}$?

What would be the value of the limit $\lim _{x\to \infty} (\frac{3x +1}{3x-1})^{4x}$? My initial idea was to divide by x in the numerator and denominator. However that would only solve the inner ...
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### Finding the fundamental Pell solution from a system of Pell-like equations

Assume $d$ is a non-square integer, and $r,s,t,w$ are integers, and $n$ and $m$ are integers with $n,m \neq 0,\pm 1$, satisfying the system of Pell-like equations \begin{align} r^2-ds^2 &= m, \\ ...
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### What definition of Reed-Solomon code is correct?

In the discuss with @Evinda we have the contradiction with the definition of Reed-Solomon Codes (not generalized case) over the finite field $\mathbb{F}_q$. We have two papers and ...
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### What is the motivation for normed division algebras?

The famous Hurwitz theorem states that the only normed division algebras are $\mathbb{R}$, $\mathbb{C}$, $\mathbb{H}$ and $\mathbb{O}$. What is some good pedagogical motivation why we should think ...
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### Is the natural logarithm actually unique as a multiplier?

The Wikipedia page on the natural logarithm says: 'Logarithms can be defined to any positive base other than 1, not only e. However, logarithms in other bases differ only by a constant multiplier from ...
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### How can I find the distance between two points within a triangle if I have the distance between each point and each vertex of the triangle?

Title says it all. It would be useful to extend the question to finding the distance if any of the points is outside of the triangle, but I'm trying to figure out the basic problem first.
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### Assume $T$ is a complex operator

Assume $T$ is a complex operator such that $T^{2}=T$. Prove that $Tr(T)$ is a non-negative integer. There is a remark in my book, Suppose the characteristic polynomial $\chi_{T}(x)$ factors intro ...
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### Give a regular expression

Let Σ be {0, 1} Give a regular expression generating words over Σ containing an even number of 1’s or with a length which is multiple of 3. i came up with this solution: ε ( ((0*(10*10*)) + ((0+1) ...
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### Counterexample to Maximum modulus principle [on hold]

How can I show a counterexample to the maximum modulus principle? The Maximum modulus principle states, suppose $f$ is holomorphic and nonconstant in a region $G$. Then $|f|$ does not attain a weak ...
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### Justify the use of a random variable as an estimator

I have a set of independent random variables $U_i\sim \mathcal{N}(\mu,1)$. With them, I create the new set $X_n=\sum\limits_{i=1}^n U_i$. Thus $X_n\sim \mathcal{N}(n\mu,n)$, but they are not ...
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### Is the mixture of Exponential family distributions an Exponential family distribution too?

Consider we have a mixture of multinomials or in a broader sense, a mixture of $f$s where $f$ is an distribution of exponential family type and the membership components are known with the sum of 1. ...
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Newman and Wiegold have studied the AN-groups i.e. the locally nilpotent groups which are not nilpotent but every proper subgroup is nilpotent. I was asking why the notion locally nilpotent was added ...
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### Rationnal canonical form of the matrix $A$

Let the matrix $$A=\begin{bmatrix} 2 & 1 & 2 \\ -2 & -1 & -4 \\\ 1 & 1 & 3 \end{bmatrix}.$$ So far I found the characteristic polynomial ...
Suppose $f$ is in $L_1$ space of $μ$, where $μ$ is the Lebesgue measure. Prove that to each $ϵ>0$, there exists a $δ>0$ so that the Lebesgue integral of the absolute value of $f$ is less than ...
### How to make $S\otimes_R M$ into a left $S[G]$-module
I have a basic question on tensor products which might have been asked before-sorry if this is the case. Let $G$ be a finite group and $R\subset S$ any (unital) ring extension. Let $M$ be a finitely ...