# All Questions

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### How do I solve the following recurrence?

Solve the recurrence $$X_n =\begin{cases} n & 0 \leq n < m\\ X_{n-m} + 1 & n \geq m.\end{cases}$$ So I've started with several base cases, but since the answer depends on $n$'s relation ...
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### Pushout in Set example 2.6.2.2. in David Spivak's Book

Trying to understand Example 2.6.2.2. on page 50 of David I. Spivak's book "Category Theory for Scientists," Spivak gives $X$ as the interval $[0..1]=\{x\in\mathbb{R}|\,0\le x\le 1\}$, $Y$ as the ...
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### If X is a countable KC-space, then every infinite $D\subset X$ contains an infinite subset with only a finite number of accumulation points (in X)

a topological space is called KC- space if every compact set is closed. a topological space is called US, if every convergent sequence has unique limit. generally, KC- space imply US - space. ...
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### What is a noetherian category?

What is a noetherian category? I'm a little bit familiar with category theory, but I've no idea what this could be. Do you know what it is good for or examples?
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### Probability and Statistics 2

John invites 12 friends to a dinner party, half of which are men. Exactly one man and one woman are bringing desserts. If one person from this group is selected at random, what is the probability that ...
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### Integral of Bessel functions

Does anybody know if there is an analytical solution to the following integral of Bessel functions: $$\int J_m^*(kx) \, J_m(kx) \, x \,dx,$$ where $m$ is integer and the problem is that $k$ may be ...
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### Prove: $\int_{-\infty}^{\infty} \frac{\tan^{-1}e^{-\pi x}}{\cosh\frac{3x\zeta(2)}{20}} dx = 10$

I want to solve the following integral: $$\int_{-\infty}^{\infty} \frac{\tan^{-1}e^{-\pi x}}{\cosh\frac{3x\zeta(2)}{20}} dx = 10$$ Have not tried it yet, but it may be tough. All I know is that ζ(2)...
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### Diameter of a circle inscribing a regular pentagon

In his book, The Irrationals, Julian Havil presents a method used by classical Arab scholars for finding the diameter of a circle inscribing a regular pentagon - and then asks the reader to prove the ...
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### Proof for the symmetric difference?

I want to prove that: a) $A \Delta B = \emptyset \Leftrightarrow A = B$ Prove for $\Leftarrow$ Then: $A \Delta B = (A$ \ $A) \cup (A$\ $A) = \emptyset$ Prove for $\Rightarrow$ Then(proof by ...
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### The smallest value of $|a|$ such that the lines $x = a+m$, $y = -2$ and $y = mx$ are concurrent

Question If the line $x = a+m$, $y = -2$ and $y = mx$ are concurrent, the least value of $|a|$ is (A) $\sqrt{2}$ (B) $2\sqrt{2}$ (C) $2\sqrt{3}$ (D) $3\sqrt{2}$ Solution Since the ...
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### Self-teaching myself math from pre-calc and beyond.

Going to be starting grade 12 (pre-calculus) shortly and looking to get ahead. I would like to try some more rigorous stuff on my own and have a couple questions. Ideally I would like to be prepared ...
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### Finding collinearity among variables

I have been reading about how (multi)-collinearity among predictor variables can be determined by looking at the condition number, or smallest eigenvalue, of the covariance matrix. My question is, if ...
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### Hard integral that standard CAS get totally wrong

How to solve the following integral: $$\int_{-\infty }^{\infty }\exp \left ( i\left ( ax^3+bx^2 \right ) \right )dx$$ Standard CAS seem to get it totally wrong, see: http://www.walkingrandomly.com/?...
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### Why $\int_{0}^{+\infty} \frac{\sin(x)}{x} \: dx = \int_{0}^{+\infty} \frac{\sin^{2}(x)}{x^{2}} \: dx$?

In an exercise, I saw the following equality : $$\int_{0}^{+\infty} \frac{\sin(x)}{x} \: dx = \int_{0}^{+\infty} \frac{\sin^{2}(x)}{x^{2}} \: dx$$ At first, I was surprised by this equality. It is ...
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### About zeros of vector fields in compact surfaces

I'm studying compact surfaces and in particular the relationship between zeros of vector fields defined on them and Euler characteristic of the surface herself. Let be $S$ a compact (smooth) surface ...
635 views

### What's the importance of the trig angle formulas?

What's the importance of the trig angle formulas, like the sum and difference formulas, the double angle formula, and the half angle formula? I understand that they help us calculate some trig ratios ...
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### Lagrange multiplier on inequalities

Let's say we have an inequality, then we change it into a function so for example we have: $f(x,y) = x^2-xy-2$ Also we have the constraint $g(x,y) = x+y - 1 = 0$ Using Lagrange multiplier, I get ...
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### Polynomials of roots. Sum and product of roots

If A and B are roots of the equation $px^2 +qx +r=0$ find in terms of $p$,$q$ and $r$ a) $$\frac{1}{A} + \frac{1}{B}$$ Hi, im new to this topic. This is not homework. im just curious of how to do ...
370 views

### A conjecture about vector space

Let $V$ be a $(r+1)$-dimensional vector space, and $p$ be a positive integer and $1\leq p\leq r-1$. Let $$X=\{v_1,\cdots,v_{2r+1-p}\}\subseteq V$$ be a finite set containing $(2r+1-p)$ different ...
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### Convergence in measure of products

Let $\mu$ be a measure on $(X,\mathcal A)$ and let $f, f_1, f_2,\dots$ and $g, g_1, g_2,\dots$be real valued $\mathcal A$- measureable functions on $X$. Show that if $\mu$ is finite, $(f_n)$ ...
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### What is the standard notation to represent the set of primes?

I have seen $\mathbb Z_p$. Are there others, perhaps $\mathbb N_p$? or the set of natural numbers where the totient of $n$ equal $n - 1$ ? $$\{n \in \mathbb N \mid n\ge2,\phi (n) = n-1\}$$
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### Cohomology calculation for maps to the 2-sphere.

Let $Y^3$ be a closed 3-manifold and $f\colon Y\to \operatorname{SO}(3)$, $g\colon Y\to S^2$ be smooth maps. Define $g'\colon Y\to S^2$ be the following composition: Y\xrightarrow{...
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### How can the lyapunov exponents for the Mandelbrot Set be computed?

I am trying to find a way to calculate the Lyapunov exponents of the Mandelbrot set. There are some very nice diagrams that you can find on Flickr of a plot of the Lyapunov exponents of the Mandelbrot ...
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### Every locally compact, second countable Hausdorff space has a countable basis of open sets with compact closure

Let $X$ be a locally compact, second countable Hausdorff space. I want to prove that it has a countable basis of opens with compact closure, and that this basis can be extracted as a subset of any ...
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### Differentiable function (independent of three tetrameters)

Designate all triple of real numbers $A,B,C$ such that function $$f(x)= \begin{cases} \frac{e^{4x} + Ax + B}{x^2} \quad x \neq 0 \\ C \quad \quad \quad \quad x=0 \end{cases}$$ is differentiable in \$...