# All Questions

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### Find point given a line and two angles

Let's say I have two points $p_1=(x_1, y_1)$ and $p_2=(x_2, y_2)$, which are given as two points of a triangle $T$. And let's say I know the angles of $T$ at $p_1$ and $p_2$. How do I find the third ...
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### Use of the Reciprocal Fibonacci constant?

The Reciprocal Fibonacci constant ($\psi$) is defined as $$\psi=\sum_{k=1}^{\infty} \frac{1}{F_k}$$ where $F_{k}$ is the $k^{th}$ Fibonacci number. The irrationality of $\psi$ has been proven. ...
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### Identifying Asymptotes of a Hyperbola

basic hyperbolic functionHow would I find the vertical and horizontal asymptotes of a $y = \frac{1}{x}$ function algebraically? For example, $y = -\frac{2}{x+3}-1$ (as you would type into a ...
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### Is Banach-Alaoglu equivalent to AC?

The Banach-Alaoglu theorem is well-known. It states that the closed unit ball in the dual space of a normed space is $\text{wk}^*$-compact. The proof relies heavily on Tychonoff's theorem. As I have ...
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### Greek sigma symbol correct representation?

I wish to display to people that a number is a total. I have decided to use the greek symbol for sum: Σ Do I prefix or suffix numbers with this symbol? Given ...
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### Basis to a manifold by coordinate balls

I was trying to show that "Every manifold has a basis of coordinate balls". My approach was like this: Given a manifold $M$, it's well-known that for every point $\mathbf{x}\in M$ there exists a ...
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### Fourier Transform of a frequency linearly modulated signal

I'm working on an oscillating signal whose trend can be modelled as a frequency linearly varying function. An example may be as follows: $$\Gamma(t)=\sin(2\pi\nu(t)t)$$ with $$\nu(t)=\nu_0 + at$$ ...
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### Evaluating $\int y^4(1-y)^3 dy$ using integration by parts

Here is the function which could easily be solved using expansion method but how could I solve it using integration by parts $$\int y^4(1-y)^3 dy$$ The problem is, when I apply integration by parts ...
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### Are questions of convergence important in real life?

In the real world, do we ever need to worry about convergence and what not? I am not talking about whether recursive functions and such terminate, but convergence in analysis. It seems like the ...
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### Area of a Spherical Triangle from Side Lengths

I am currently working on a proof involving finding bounds for the f-vector of a simplicial $3$-complex given an $n$-element point set in $\mathbb{E}^3$, and (for a reason I won't explain) am needing ...
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### Finite subgroup

In the following problem, the only parts I didn't understand were c) and e). The remaining I did. Please, help me! Let $A,B$ be subgroups of $G$ that normalize each other. Assume that the set ...
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### Find region of integration where triple integral has maximum value

I have to find the region $E$ where $$\iiint_E (1-x^2 - 2y^2 - 3z^2) dV$$ has maximum value, but I'm not sure how to start. I was thinking of getting the derivative of the integral and then ...
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### Two Equivalent Notions of Algebraic Simple Extension (Proof)

When reading texts about field extensions I've come across the following two definitions for the simple extension $K(\alpha)$ where $\alpha$ some algebraic number over $K$. They are (1) $K(\alpha)$ ...
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### Limits problem in Integration

please look at the following question, Let $X$ denote the diameter of an armored electric cable and $Y$ denote the diameter of the ceramic mold that makes the cable. Both $X$ and $Y$ are scaled so ...
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### Finding Matrix Representation

Problem: Let T: $\mathbb{R}^3\rightarrow\mathbb{R}^3$ be a linear map given by $$T\left[ \begin{matrix} x\\y\\ z\end{matrix} \right]= \left[ \begin{matrix} 3x-y\\z-x\\z-y\\\end{matrix} \right]$$ ...
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### Prove that $\mathrm{Aut}(L(\zeta_n)/K(\zeta_n))$ is a subgroup of $\mathrm{Aut}(L/K)$

Let $L/K$ be a Galois extension of fields and let $\zeta_n$ be a primitive $n$th root of unity in some field extension of $L$, where $n$ is not divisible by the characteristic. Prove that ...
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### A dynamic programming problem with continuous states and observations

I have a dynamic programming expressed in the following Bellman backup equation form, $$V(\boldsymbol{\theta},T)=\max_{i \in N} \mathbb{E} \left[ x_i + V(\boldsymbol{\theta}_{x_i}, T-1) \right]$$ ...
218 views

### Weak derivatives

How can I prove that $|\nabla u|=|\nabla|u||$ when $u$ is regular enough for example Lipschitz or $W^{1,1}_{loc}$. Other question is about the pointwise derivative when $f:[0,1]\to R$ is BV is that ...
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### Dealing with connectness and compactness of matrices.

Consider the set of all $n\times n$ matrices with real entries, considered as the space $\mathbb{R^{n^2}}$ What can we say about connectedness and compactness of the following sets? The set of all ...
761 views

### Formula for the number of latin squares of size $n$?

Is there a "easy to compute" formula for the number of Latin Squares or the number of reduced Latin squares of size $n$?
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### Coordinate change for metrics

I am rather confused by the idea of "geodesic polar coordinates", so I hope someone would kindly explain it to me. As far as my understanding goes, given a Riemannian metric ...
620 views

### $f$ is integrable, prove $F(x) = \int_{-\infty}^x f(t) dt$ is uniformly continuous.

I am not sure how to do this. I can prove it if I know $f$ is bounded, but otherwise I am stuck. $f$ is integrable, prove $F(x) = \int_{-\infty}^x f(t) dt$ is uniformly continuous.
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### Implications of given solutions

This has been solved! Thanks to everyone who read and thought about it Suppose lines of the form $(x_0,y)$ and $(x,y_0)$ for any given $x_0,y_0\in \mathbb R$ are solutions to the system of ...