# 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 ...
0answers
7 views

### 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. ...
0answers
5 views

### 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 ...
1answer
12 views

### 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 ...
2answers
18 views

### 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 $f(x)=x^2$ ...
0answers
5 views

### 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 ...
0answers
7 views

### 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 $\mathbf{C}$?
0answers
9 views

### 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 ...
0answers
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1answer
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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 ...
0answers
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1answer
21 views

### 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 ...
1answer
14 views

### 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 $(1+x/n)^{n} \leq (1 + x/m)^{m}$...
0answers
12 views

### 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, ...
1answer
16 views

### 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. ...
1answer
15 views

### 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 ...
0answers
6 views

### 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 ...
1answer
25 views

### 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 ...
1answer
15 views

### 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? ...
2answers
25 views

### 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'...
1answer
14 views

### 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. ...
1answer
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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 ...
2answers
38 views

### 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: ...
0answers
19 views

### 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 $d = \sup{f}$,...
0answers
14 views

### 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. ...
0answers
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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? ...
0answers
19 views

### Proof of Hoffman & Kruskal's theorem on Unimodularity and Integrality.

I'm reading the following proof. Thm. (Unimodularity and Integrality, Hoffman & Kruskal [1956]): Let $A \in \mathbb{R}^{m \times n}, \operatorname{rank} A=n .$ The following are equivalent: a) $A$...
0answers
20 views

### Rate of Growth of Permutations of Rubik's Cubes

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 ...
1answer
19 views

### The completion of a normed vector space.

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)}...

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