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2 views

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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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. ...
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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 ...
1
vote
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 ...
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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 $f(x)=x^2$ ...
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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 ...
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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 $\mathbf{C}$?
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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 ...
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0answers
8 views

Triangle problem with a simpler solution

Problem: In the triangle $\mathit{ABC}$ the angle $A$ is 60°. The “interior” circle has center $O$. If $|\mathit{OB}|=8$, $|\mathit{OC}|=7$, how long is $\mathit{OA}$? “Solution”: Let the radius be $...
2
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1answer
16 views

What does it really mean for a model to be pointwise definable?

(Note: I'm only an amateur in logic, so I'm sorry for any weird terminology or notation, or excessive tedious details. Most of what I know is from Kunen's Foundations of Mathematics.) I'm trying to ...
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0answers
7 views

Can these matrices be multiplied in $\mathcal O(n^2)$ time?

Given an $n\times n$ orthogonal matrix $Q$ and a diagonal matrix $D$, what is the cost of multiplying $$QDQ^\top$$ Is there any way this can be accomplished in $\mathcal O(n^2)$ time?
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1answer
10 views

Verify $\tau=\{A \subseteq \mathbb{R}| |\mathbb{R}\setminus A| \leq |\mathbb{N}| \}$ is a topology

Consider the set $\tau=\{A \subseteq \mathbb{R}| |\mathbb{R}\setminus A| \leq |\mathbb{N}| \}$ Verify it is a topology obviously $\mathbb{R} \in \tau$ since $ |\mathbb{R}\setminus \mathbb{R}|=0 \...
2
votes
1answer
24 views

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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0answers
6 views

Closed-Form solution for nested integrals of this polynomial?

I was wondering whether there is a closed-form solution for this (nested) integral: $$ \int_{-1}^{1}\int_{t_{0}}^{1}\int_{t_{1}}^{1}...\int_{t_{a-2}}^{1}\prod_{\begin{array}{c} i<j\\ j=\{0,..,a-1\}\...
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0answers
4 views

How do I find the angle necessary to traverse distance X?

Look at the image, given Radius, length of segment L, and distance X how do I solve for the angle of turn necessary to have segment L pass the distance X (L> X).Circle segment theta question
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0answers
25 views

Proof that $A\mapsto A^3$ is one to one where $A$ is a symmetric matrix

Let $ S_n$ denote the set of real symmetric matrices and $f: S_n\to S_n$, $A\mapsto A^3$. I want to prove that $f$ is a bijective function. Using spectral theorem I manage to prove that $f$ is onto. ...
0
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1answer
10 views

If $\lim_{x \to b^{-}} f(x) = d$ then the image of $f$ is the half closed interval $[f(a),d)$ - Proof feedback

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 $\lim_{x \to b^{-}}f(x) = d$ then the ...
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0answers
10 views

how to use the chain rule in a multivariable function?

Problem: Let the function $f(x,y)=(x^2+y^2)sin(x)$ where $x=r^2e^s$ and $y=rs$ Using the chain rule compute $\frac{\partial f}{\partial r}$ and $\frac{\partial f}{\partial s}$ and then compute $\frac{...
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0answers
10 views

Closed Form Solution to System of Equations

Let $ \mathbf{b} \in \mathbb{R}_+^n$, $\mathbf{V} \in \mathbb{R}_+^{n \times m}$ with $\mathbf{V} \mathbf{1}_m = \mathbf{1}_n$ where $\mathbf{1}_n$ is the vector of ones of size $n$. I have the ...
0
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0answers
7 views

Resolving an integral equal to the exponential generating function involving the Riemann zeta function

It is well-known that $$-\gamma-\psi\left(1-x\right)=\sum_{n=1}^{\infty}\zeta\left(n+1\right)x^{n}$$ Using the OGF to EGF integral transformation, then $$\frac{1}{2\pi}\int_{-\pi}^\pi (-\gamma-\psi(1-...
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0answers
8 views

Given a derivative in some direction, calculate the normal derivative

Suppose $S(u,v,w=0)$ is a surface in $\mathbb R^3$, $\vec p(u,v)$ is a unit vector field, and $f(u,v,w)$ is some smooth enough scalar-valued function. The directional (“oblique”) derivative $\nabla_{\...
0
votes
1answer
28 views

Show that $\sum_{k=1}^\infty \frac{i}{k(k+1)}$ converges. Find its sum.

Show that $\sum_{k=1}^\infty \frac{i}{k(k+1)}$ converges. Find its sum. The presence of the $i$ throws me off. Our professor taught us to find $a$ (the first term in the sequence) and $z$ (the ...
1
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0answers
31 views

invertible matrices are positive definite?

A inverse matrix $B^{-1}$ is it automatically positive definite? Invertible matrices have full rank, and so, nonzero eigenvalues, which in turn implies nonzero determinant (as the product of ...
1
vote
1answer
32 views

How to prove a map is well-defined.

How do I show that $\times_{alg}\colon H_pC_*\otimes H_qD_*\to H_{p+q}(C_*\otimes D_*)$ defined by $\times_{alg}([z]\otimes[w])=[z\otimes w]$ is well-defined? I say that we suppose $[z]\in H_pC_*=\ker\...
0
votes
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 ...
0
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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 $(1+x/n)^{n} \leq (1 + x/m)^{m}$...
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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, ...
0
votes
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. ...
0
votes
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 ...
0
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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 ...
0
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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 ...
0
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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? ...
0
votes
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'...
0
votes
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. ...
1
vote
1answer
30 views

Ratio Test intuitive Idea

My attempt to understand the proof is as follows: If $R=\lim\sup\limits_{n\to\infty}\left|\dfrac{a_{n+1}}{a_n}\right|<1$ implies that for $\epsilon$ such that $0<\epsilon<1-R$ we can find an $...
0
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0answers
6 views

How does the sweep line algorithm check for intersection using vector cross product?

I am trying my best to understand the sweep-line algorithm to find line intersections. I have understood most of the intuition except how it is calculating the intersection between 2 line segments ...
1
vote
0answers
8 views

Tensor product and fiber product of modules

The tensor product of modules commutes with finite products, i.e. if $A,B,C$ are $R$-modules, then we have $$A\otimes_R(B\times C)\cong (A\otimes_R B) \times (A\otimes_R C).$$ My question is whether ...
1
vote
0answers
12 views

Is it allowed to take the total derivative of an infinitesimal, and is it equal to zero?

For instance, I start with this relation: $$ s^2=x^2+y^2 $$ Taking the total derivative on each side, I get: $$ 2sds=2xdx+2ydy $$ Can I take the total derivative a second like this: $$ d[sds]=d[xdx]+d[...
0
votes
1answer
27 views

Show that if $z_n\rightarrow z_0$, then $|z_n|\rightarrow |z_0|$.

Show that if $z_n\rightarrow z_0$, then $|z_n|\rightarrow |z_0|$. So far I've started the proof: Let $\epsilon>0$ be given. Suppose $z_n\rightarrow z_0$. Then there exists $n_0\in\mathbb{N}$ such ...
-1
votes
0answers
19 views

Lemma on primitive divisors given $p$ divides $Q_n$, $Q_m$

Let $p$ be prime. For $n\ne 1,2,6$, p divides $Q_n, Q_m$ for some $0<m<n$. Then $p^2$ does not divide $Q_n$ and $n=p^r z(p)$ for some $r$. Hi, I was reading some lecture notes on primitive ...
2
votes
1answer
31 views

Solvability by radicals

I study the book "Introduction To Field Theory" by Iain Adamson (https://archive.org/details/IntroductionToFieldTheory), and struggle with the Theorem 26.5. on page 166: Let $F$ be a field ...
0
votes
1answer
26 views

Prove that every set and subset with the cofinite topology is compact

Prove that every set with the cofinite topology is compact as well as every subset Solution. Let $X$ be a nonempty set with the cofinite topology and let $ \mathscr{U}$ be an open cover of $ X $. Let $...
3
votes
1answer
42 views

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 ...
3
votes
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: ...
0
votes
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 $d = \sup{f}$,...
0
votes
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. ...
0
votes
0answers
10 views

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? ...
0
votes
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) $A$...
2
votes
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 ...
0
votes
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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