# All Questions

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### How to show that a presheaf kernel of a morphism of sheaves is a sheaf

Let $\varphi:\mathcal{F}\longrightarrow \mathcal{G}$ be a morphism of sheaves. Then the presheaf :$U\mapsto ker(\varphi(U))$ is a sheaf. By the definition of sheaf, it's sufficiently to verify that ...
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### The progression $4n+3$ and primes.

Consider an arithmetic sequence $4n+3$. This sequence contains infinitely many primes and infinitely many composites. It is clear that there cannot be $3$ consecutive primes in the sequence as every ...
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### Higher dimensional Euclidean geometry problem

In my engineering/physics research, I am facing one math problem which I believe should be well established in mathematics... I have a linearly spanned space given by the column vectors of the ...
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### Example Of An Inner Product Space

1). Let $X=\mathcal{L}_2$ be all infinite sequences of complex numbers i.e $(a_1,...,a_n,...)$ with the property that $$\sum_{i=1}^\infty|a_i|^2<\infty$$. If $x$& $y$ are two vectors such that ...
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### Size of Derivation of Lie algebra

If $L$ is a Lie algebra in ${\rm gl}\ (n,{\bf C})$ then $${\rm Der}\ (L)\subset {\rm gl}\ (n,{\bf C})$$ (If $L$ is semisimple then $L = {\rm ad}\ L ={\rm Der}\ L$) Is this true ? Thank you.
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### minimal polynomial of roots of irreducible polynomial

Let $f\in \mathbb{Z}[x]$ be irreducible over $\mathbb{Q}[x]$, the highest degree coefficient of $f$ is $1$. Let $\omega\in \mathbb{C}$ such that $f(\omega)=0$. Can we obtain that the minimal ...
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### Tips for writing proofs

When writing formal proofs in abstract and linear algebra, what kind of jargon is useful for conveying solutions effectively to others? In general, how should one go about structuring a formal proof ...
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### Question on Trigeometry

I'm doing this question and I found out why a, b, and c can't be the answer. What about d, and e? I don't understand them. P.S the right answer is d :) Thank you very much.
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### Doubt about Proposition 2.39 in Dana Williams' crossed product book

You can see the proposition in a google books preview here. First and foremost, my question is: Question: Am I correct to interpret Proposition 2.39 as setting up a bijective correspondence ...
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### How to formalize “conditional random variables”

I've been using "conditional random variables" as a notation aid with some good success in problem solving. But I've heard people claim that one shouldn't define conditional random variables. By a ...
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### How to prove the demicountinuity of nonlinear operators?

Define a nonlinear operator $\mathbf{J}(\mathbf{x}):~\mathbb{R}^3 \rightarrow \mathbb{R}^3$ as $$\mathbf{J}(\mathbf{x}):= |\mathbf{x}|^{-\alpha}\mathbf{x},~0<\alpha<1.$$ How to prove that ...
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### Intuition for the Universal Chord Theorem

So the Universal Chord theorem is a statement and proof that; The numbers of the form $r = \displaystyle \frac{1}{n} \ \ n \ge 1$ are the only numbers such that for any continuous function ...
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### How to justify what solving method to use

How do you determine what solving method to use, for example what is the reason you solve by the quadratic formula, or by factoring, or by completing the square, or by taking a square root.
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### $\mathbb Z_p[T]/(T^a,p^b)\cong\mathbb Z_p[[T]]/(T^a,p^b)$

Let $a,b\in \mathbb N,$ then $$\mathbb Z_p[T]/(T^a,p^b)\cong\mathbb Z_p[[T]]/(T^a,p^b)$$ 1.What is this isomorphism ? 2.How to prove that $|\mathbb Z_p[[T]]/(T,p)^t|=p^{t(t+1)/2}$ Now ...
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### Singular Solutions of this Equation?

How would I find the singular solutions of this equation: y = $ce^{x^2}$ + $ce^{\sin x}$ (where $c$ is a constant). It should be $x^2$ if anyone gets confused by the first part of the equation. ...
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### Finding the area of a region?

I sketched out the graph to get this figure but I can't seem to find the area of the shaded region... would one Y = 4 and the other Y = 8?
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### How to derive the chances of getting the Jackpot in NY Mega Millions with the given answer?

The chances is 1 in 258,890,850: ...
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### Bob has 22 stamps. This is 6 times as many stamps as Joe has. Bob made a model to help him compare the number of stamps they have. [on hold]

N____N____N_____N____N_____N___N__N____ I do not understand why there are 7 boxes at the top. I have tried everyone that I can to figure it out but I come up ...
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### How to measure intensity of attractors

Given a dynamic system with attractors, is there any way to measure the intensity of attractors? I mean intensity by the faster one point far away from the attractor moves to it, the higher intensity ...
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### How to show $|\int_ae^{iz^2}dz|\le\frac{\pi(1-e^{-R^2})}{4R}<\frac{\pi}{4R}$?

Let $a:[0,\frac{\pi}{4}]\mapsto\Bbb C, a(t)=Re^{it}(R>0)$ a curve. Show that $$\left|\int_ae^{iz^2}dz\right|\le\frac{\pi(1-e^{-R^2})}{4R}<\frac{\pi}{4R}$$ Can you help me to solve this? will ...