# All Questions

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### Curve Fitting of Nautical Tabular Data

I have sets of data describing hull forms. The data is in (X, Y, Z) format except that the Y and Z coordinates have been rounded to either 1/16" or 1/32" of an inch. X is always a multiple ...
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### Simpler proof of van Kampen's theorem?

I've been trying to understand the proof of van Kampen's theorem in Hatcher's Algebraic Topology, and I'm a bit confused why it's so long and complicated. Intuitively, the theorem seems obvious to me. ...
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### Solving PDE using singular perturbation theory

I am attempting to solve a partial differential equation, namely the Eikonal equation in two dimensions, perturbatively, because the scenario I am solving presents itself in a perturbative manner. The ...
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### Contradicting the non-existence of a linear map $T: \Bbb R^5 \to \Bbb R^5$ and the Fundamental Theorem of Linear Algebra (from Axler Exercise 3.B(5))

I am asked to prove there does not exist a linear map $T:\Bbb R^5 \to \Bbb R^5$ such that $\operatorname{range}(T) = \operatorname{null}(T)$. I think I understand the proof whereby the Fundamental ...
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### Calculate $\lim_{x \to 0} f(x)$

Let $f:\mathbb{R}\to \mathbb{R}$ be a function. Suppose $\lim_{x \to 0} \frac{f(x)}{x} = 0$. Calculate $\lim_{x \to 0} f(x)$. According to the answer key, $\lim_{x \to 0} f(x) = 0$. I see $f(x)=x^2$ ...
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### Irreducibility of $X\times_k\overline{k}$, $X\times_k k_s$ and $X\times_k K$ (Hartshorne 3.15 (a))

I'm working on Hartshorne's exercise II.3.15 (a), namely: Let $X$ be a scheme of finite type over a field $k$ (not necessarily algebraically closed). Show that the following three conditions are ...
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### Topologies/Norms on Real Closed Fields

Let $\mathbf{R}$ be a real closed field containing $\Bbb{R}$, and let $\mathbf{C} := \mathbf{R}[i]$. Is there a natural norm or topology on $\mathbf{R}$ or $\mathbf{C}$?
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### Matrix is not positive definite eventhough it should be by construction

Let $A$ be $d\times d$ matrix, $b\in\mathbb R^d$, $X$ a $d\times n$ matrix, $W$ an $n\times n$ positive definite diagonal matrix with the diagonal elements summing to one, $1_n$ a vector of $n$ ones ...
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### Find $x^n+y^n+z^n$ general solution

If we know $$x+y+z=1$$$$x^2+y^2+z^2=2$$$$x^3+y^3+z^3=3$$ Is it possible to calculate the general solution for $a_n=x^n+y^n+z^n$? I know $a_5=6$ but the way to get it is more an algorithm than an ...
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### Why is $I = \left \{ \sum_{i=0}^n a_i x^i | a_0 \in 2 \mathbb{Z} \right \} \subseteq \mathbb{Z}[x]$ not a principal ideal?

Why is $I = \left \{ \sum_{i=0}^n a_i x^i | a_0 \in 2 \mathbb{Z} \right \} \subseteq \mathbb{Z}[x]$ not a principal ideal? We saw it as a short example for a non-principal ideal in a linear algebra ...
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### Use the arithmetic mean-geometric mean inequality to establish that $-x < n < m$ implies $(1+x/n)^{n} \leq (1 + x/m)^{m}$

Use the arithmetic mean-geometric mean inequality to establish the following results: (a) If $nt > -1$, then $(1-t)^{n} \geq 1 - nt$ (b) If $-x < n < m$, then $(1+x/n)^{n} \leq (1 + x/m)^{m}$...
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### Find the volume of the part of the ball $\rho \le 7$ that lies between the cones $\phi = \frac \pi 6$ and $\phi = \frac \pi 3$.

My question is simple, when setting up the bounds for this problem, I believed that the bounds would be from $\frac \pi 3$ to $\frac \pi 6$. however its wrong and its actually the other way around, ...
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### Show $l^p$ is not complete with the $q$ norm

I know the question has been asked here, but I do not understand the solution (Are $\ell_p$ spaces complete under the $q$-norm?) I came up with my own solution and was wondering if it is correct. ...
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### How to calculate $f_3(3)$ and $f_4(4)$ in the fast growing hierarchy?

I read some articles about the fast growing hierarchy (and saw some vids), and I wonder how to calculate: $f_3(3)$ And especially $f_4(4)$ I know that there are huge numbers, but I wonder if someone ...
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### Finding k-order statistics among unknowns

Let there be $x^2$ horses, at a time we can compare $x$ of them and establish the order of speed between them. For what is the smallest comparison you can find the $x+1$ speed horse? I will be glad to ...
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### Kernel of $(I-A)^2$ where $A$ has unique real eigenvalue $1$

Assume I have a real square matrix $A$ with a simple eigenvalue $1$ and corresponding eigenvector $v$. Then $v$ should span the kernel of the matrix $I-A$, where $I$ is the identity matrix. Now I am ...
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### How can I integrate an exponential function with transposed and inverted elements?

Suppose that I want to integrate $$\int exp[-\frac{1}{2}((a-b)-y_Lw_L)^T\times \Sigma^{-1} \times ((a-b)-y_Lw_L)]dy_L$$ How can I expand the product inside the exponential function to integrate this? ...
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### Why is it true that if the integral is finite then $\lim_{s\to\infty} \int_{s}^{\infty} f(x) dx=0$?

I saw it whilst reading my lecture notes that Let $f(x)$ be integrable such that $\int_{-\infty}^{\infty} f(x) dx<\infty$ then it means that $\lim_{s\to\infty} \int_{s}^{\infty} f(x) dx=0.$ I can'...
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### How to find the length of a wire making a spherical spiral?

Suppose you have a Christmas ball, which has a decorative lining around it so that it forms a spherical spiral around it. I want to find how long that decorative lining can be given some parameters. ...
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### Fibonacci and tossing coins

Consider the following scheme starting with a sequence $\sigma_0 = \langle 1,1,\dots,1\rangle$ of length $k$, successively followed by sequences $\sigma_i$ of the same length but shifted by one to the ...
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### L'Hopital's rule conditions

I have seen easy geometrical argument why L'Hopital's rule ($\lim_{x \to a} \frac{f(x)}{g(x)} = \lim_{x \to a} \frac{f'(x)}{g'(x)}$) works (local linearization). But, I still don't understand this: ...
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### Prove that if $f$ is bounded above and $d = \sup{f}$, then $\lim_{x \to b^{-}}f(x) = d$ - Thought process behind solution

Let $f:[a,b) \to \mathbb{R}$ be a strictly monotone increasing continuous function on a half closed interval $[a,b)$, and let $d$ be a real number. Prove that if $f$ is bounded above and $d = \sup{f}$,...
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### Localization of a Weyl-Algebra module

My aim is to understand how to describe the localization of modules over the Weyl algebra. I want to be able to do simple examples by hand. I wrote the following code in Macaulay2. ...
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### Euler Approximation of General ODE

I have a slightly general question that might be related to the Euler Approximation of ODE solution, could someone provide some thoughts on this or share some source could potentially helpful? ...
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### Proof of Hoffman & Kruskal's theorem on Unimodularity and Integrality.

I'm reading the following proof. Thm. (Unimodularity and Integrality, Hoffman & Kruskal ): Let $A \in \mathbb{R}^{m \times n}, \operatorname{rank} A=n .$ The following are equivalent: a) $A$...
I'd like to know how fast the number of permutations grows on an $n\times n\times n$ Rubik's Cube as $n$ increases. I'm well aware of the $\frac{3^88!2^{12}12!}{12}$ calculation for the permutations ...
Let $X$ be a normed vector space over a field $\mathbb{K}=\mathbb{R}$ or $\mathbb{C}$, so there is a Banach space $\widehat{X}$ and an isometric mapping $u:X \to \widehat{X}$ such that \$\overline{u(X)}...