# All Questions

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### Even natural numbers are sums of two primes with twins or of two primes without twins

I seems to be very few even numbers that can't be written as a sum of two primes with twins or as a sum of two primes without twins. That is, suppose that $\mathbb P'$ is the set of the primes not ...
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### Stirling's formula problem [on hold]

Use stirling's formula to find: $$\displaystyle \lim_{n\to \infty}\dfrac {\ln(n!)}{n\ln(n)}$$ .
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### Is this metric space normable?

Given the function $\rho:\mathbb{R}^n\to\mathbb{R}_+$ defined by $$\rho(x)=\log\left(\frac{\sum\left|x_i\right| e^{\left|x_i\right|}}{W\left(\sum\left|x_i\right| e^{\left|x_i\right|}\right)}\right)$$ ...
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### Question about arc-connected property in a continuum

Suppose $X$ is metric, compact, connected, and $p\in X$. An arc is a copy of $[0,1]$. Is it possible that every two points in $X\setminus \{p\}$ can be joined by an arc, but there is no arc in $X$ ...
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### Conditional probability using set notation [on hold]

Got this wrong on a quiz and i don't have the answers. Need to figure this out for a test coming up. \begin{align} P(A) &= 0.75 \\ P(B\mid A) &= 0.9 \\ P(B\mid A^c) &= 0.8 \\ P(C\mid A\...
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### '2nd order' Picard Iteration

I'm self-studying differential equations using MIT's publicly available materials. One of the problem set exercises deals with what I'm calling a second order Picard Iteration. To be explicit, we ...
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### A tricky integral - $\int_0^1 \sqrt{\frac{1}{(1-t^2)^2}-\frac{(n+1)^2t^{2n}}{(1-t^{2n+2})^2}}dt$

Evaluate: $$\int_0^1 \sqrt{\frac{1}{(1-t^2)^2}-\frac{(n+1)^2t^{2n}}{(1-t^{2n+2})^2}}dt$$ Where $n$ is any positive integer. Introduction: This integral came up while studying the distribution ...
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### Are there any examples of higher order ireducible linear differential operators?

Given a monic, linear differential operator $L = D^n + f_{n-1}(x)D^{n-1} + \dots + f_1(x) + f_0(x)$, say $f_0, \dots, f_{n-1}$ analytic for simplicity's sake, we say that $L$ is irreducible if there ...
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### Evaluate the following: [on hold]

$$1) \cos 2 \theta + \cos 2 \phi$$ and $$2) \sin(\theta + \phi)$$ If $$\sin\theta + \sin\phi = a$$ and $$\cos \theta + \cos\phi = b$$ please provide a detailed solution. I could not ...
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### The Jeep Problem and Nash's Friends

The classical jeep problem is the following. A jeep can carry a maximum load of fuel of 1 gallon, and it travels $l$ miles with $l$ gallons of fuel. The jeep moves along a straight line, and is ...
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### Solve for and Plot the Relationship Between Mean and Standard Deviation of a Normal Distribution Conditional on Satisfaction of A System of Equations

I am trying to use Mathematica, R, or Matlab to solve for (since it cannot seem to be solved analytically) and plot the relationship between mean and standard deviation of a normal distribution ...
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### Is the relation $R$ on $\Bbb N$ given by $(a,b)\in R\iff a\mid b$ an equivalence relation?

$R \subset \Bbb N \times \Bbb N$ Is this an equivalence relation? $$R=\{(a,b)\in \Bbb N\times \Bbb N\,:\,a\mid b\}$$ I would argue that it is reflexive because $a\mid a$, but it is not symmetric ...
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### For which values of t does a matrix not have eigenvalues

I need help solving this problem "For which values of real parameter t does the matrix: \begin{bmatrix} π^2t^2 & 36\\ -36 & 0 \\ \end{bmatrix} NOT have real eigenvalues. Thank you.
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### What is the definition of real numbers? [duplicate]

I only know that the rational and irrational are together called real numbers. Rational numbers can be written in the form $\frac {p}{ q}$ ,where $p \in \mathbb Z$ and $q \in \mathbb Z$. Are there ...
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### Homolorphic vector bundles on almost complex manifolds

Let $M$ be a real manifold with complex structure $J$, making $M$ into an almost complex manifold. I know that the complexification $T_{\textbf{C}}M = TM\otimes \textbf{C}$ of the tangent bundle $TM$ ...
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### Minimum vertex cover exact algorithm analysis

An exact algorithm to find a minimum vertex cover in a simple, undirected graph would be based on the following recursive idea: "either a vertex v is in the minimum cover, or all of its neighbors are"....