# All Questions

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### Is the length of one cathetus in a right triangle with equal catheti 2?

Look at your watch now. Now try to remember that time. By the time you have finished this sentence that time has probably increased by a couple seconds. Now you may be asking yourself "is this ...
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### Find the “surface vertices” of a collection of points.

I am currently doing some experiments in order to simulate liquids. I have a collection of 3D points that interact with each other to form a body of water. I would like to form a mesh from these ...
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### Wouldn't each addition take time $O(n)$?

I am going over the asymptotic runtime of regular matrix multiplication. Here is a lecture slide I am referencing(too much to type out, shown below), from Algorithms Everything makes sense up ...
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### Integration in complex measure

Let $v$ be a complex measure in $(X,M)$. Then $L^{1}(v)=L^{1}(|v|)$. I have made: $L^1(v)\subset L^1(|v|)$?. Let $g\in L^1(v)$ As $v<<|v|$ and $|v|$ is finite measure, then for chain rule, ...
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### Expressing the Solution to a System of Differential Equations

My professor wrote the solution to a system as $$X = C_1 \begin{bmatrix}1 \\2 \end{bmatrix} e^{\lambda_1t} + C_2 \begin{bmatrix}3 \\4 \end{bmatrix} e^{\lambda_2t}$$ Where the column vectors are the ...
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### How do I interpret this operation?

This question has to do with operations and exploring their characteristics. I have just learned how to extract info from an operations table (what is the identity, inverse, etc.), but this question ...
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### Seeking Recommendation on Pre-Calculus Textbooks!

S.E. advisers, I wrote this email because I am seeking a recommendation on selecting the pre-calculus textbooks. I have been studying the real analysis and number theory, and I felt that I need to ...
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### seperable differential equation question

b) $(2xy^3)dx + (3x2y^2 + y^4)dy = 0$ c) $(2xy^3)dx + (3x2y^2 + y^2)dy = 0$ I know that $c$ is a separable differential equation but $b$ is not. Why? The only difference is the power of the ...
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### Roll one ellipse on another: Locus of center ever a circle?

Let $E_1$ be an ellipse fixed in the plane. Let $E_2$ be a second, possibly different ellipse, which rolls around without slippage outside $E_1$, touching perimeter-to-perimeter. Let $c_2(t)$ be the ...
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### Formula for $r+2r^2+3r^3+…+nr^n$

Is there a formula to get $r+2r^2+3r^3+\dots+nr^n$ provided that $|r|<1$? This seems like the geometric "sum" $r+r^2+\dots+r^n$ so I guess that we have to use some kind of trick to get it, but I ...
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### Is this an instance of the base-rate fallacy?

The following line of probability reasoning is supposedly fallacious, and is an instance of the base-rate fallacy. The argument is that $(1)-(3)$ don't give us enough reason to conclude that $(C)$. ...
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### What is 3^43 mod 33?

I just took math final and one of the question was Find $3^{43}\bmod{33}$. So, I used Euler's function; $\phi(33)=20$. $3^{20}\equiv 1\pmod{\!33}$ By using this fact, I got $27$. One thing ...
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### Stiefel-Whitney Numbers of $\mathbb{R}P^2\times \mathbb{R}P^2$

I'd like to calculate the Stiefel-Whitney numbers of $\mathbb{R}P^2\times\mathbb{R}P^2,$ but don't know how to. My first instinct was to say that the tangent bundle is isomorphic to the product of ...
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### If K $\subset \mathbb{R}$ is compact prove that {fn} converges uniformly to f on K.

Suppose that we have a sequence of functions{f$_{n}$} that converges uniformly to a function f on any (a,b) $\subset \mathbb{R}$. If K $\subset \mathbb{R}$ is compact prove that {f$_{n}$} converges ...
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### Is the only way to go physics is a very goodeth?

Here's a qestion. If the Physics and I chemistry is the best of all the all is and what is this the your phone and is a great way for of to a the first place time half the things time people are just ...
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### Start to Proof of bernoulli polynomials and sums

I need help starting this proof: For all integers k,l,m>=0 and not all equal to 0, (3.7) It says that that comparing the above equation (3.7) with the one discussed earlier in the paper(3.6) (shown ...
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### Probability Density Question Involving an Integral Equation (from Karlin & Taylor's A First Course on Stochastic Processes)

The random variables X and Y have the following properties: X is positive, i.e., $P\{X > 0\} = 1$, with continuous density function $f_X(x)$, and $Y\mid X$ has a uniform distribution on $\{0,X\}$. ...
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### Question on $\mathbb{K}$ notation

In a lot of paper and book $\mathbb{K}$ means $\mathbb{R}$ or $\mathbb{C}$. I know that $\mathbb{R}$ comes from the word real, and $\mathbb{C}$ from the word complex. But what about $\mathbb{K}$? ...
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### Does $\chi(g^{-1})=\overline{\chi (g)}$ hold for infinite groups

Let $\chi$ be the character of some representation $\rho:G \to GL(M)$ over $\mathbb C$. Suppose $G$ is a group, then $\forall g \in G$ of finite order $n$, $\chi(g^{-1})=\overline{\chi (g)}$ ...
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### Vertex invariants based on finding minimal combined shortest paths

A possible vertex invariant for a vertex v is v's smallest n-neighbourhood consisting of the induced subgraph rooted in v of all vertices n edges away from v. Question 1: I'm wondering if this ...
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### Proving De Morgan's law with the minus sign

So I know how to prove De Morgan's Law in this form: $A\cap (B\cup C)^{c}$, what I'm trying to do for practice is prove it in the slightly different notation: $A- (B\cup C)$ I get everything except I ...
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### What does $d\log\left(\frac{y}{x}\right)$ mean mathematically?

I am used to seeing derivatives written as $$\frac{df}{dx}.$$ But my economics professor keeps using notation like $$d\log\left(\frac{y}{x}\right)$$ and I have no idea what this means. What does ...
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### Prove that if $\{1^5,2^5,\ldots, (pq)^5\}$ is a complete residue system mod $pq$, then $\{1^5,2^5,\ldots,p^5\}$ is too, mod $p$

$p,q\ge 2$ are coprime positive integers. Prove that if $\{1^5,2^5,\ldots, (pq)^5\}$ is a complete residue system mod $pq$, then $\{1^5,2^5,\ldots,p^5\}$ is a complete residue system mod $p$. ...
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### Find the maximum of a |cos(z)|

How do you find the maximum of the complex function $|\cos{z}|$ on $[0,2\pi]\times[0,2\pi]$. I believe I'm to use the maximum modulus principle, since the function is entire. I'm just having problems ...
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### Subset of a normal space

Given X a normal space, and a subset $A \subset X$ not closed. Does it imply A is not normal? I understand it does not, Can someone provide me a counterexample?
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### Finding potential of a given vector field

I am trying to solve the following problem: Let $\textbf{F}=f(r) (x,y,z)$ where $r=(x^{2}+y^{2}+z^{2})^{1/2}$. Find an expression for a potential for $\textbf{F}$. Find an expression also for ...