# All Questions

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### Examples of compactly supported exact differential forms

I am having some trouble finding any examples of compactly supported exact differential forms on $\mathbb{R}^n$. I have found $e^{\frac{1}{x^2 -1}}$ when taken to be zero everywhere except on the ...
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### $A\cup \text{Fr}A$ is a manifold

I'm dealing with the following exercise from Spivak's "Calculus on Manifolds": Let $A\subset \mathbb R^n$ be an open set such that the (topological) boundary of $A$, $\text{Fr}A$ is an $n-1$ ...
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### Geometric progression question. Year less?

Here is the math question. A 100m cliff erodes by 2/7 of its height each year. (a) What will the height of the cliff be after 10 years? This is how I worked out the question. ...
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### How to show two integers are coprime in a linear combination?

Show that $a$ and $b$ are relatively prime if and only if 1 is expressible as a linear combination of $a$ and $b$. Using that fact, show that any two consecutive integers are relatively prime. ...
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### Show there is no intersection between sequence of balls on an open set.

Is there a way to show that a finite sequence of balls, in an open set $U ⊆ C$, with a small enough radius's such that there is no intersection between any of the balls? Intuitively it makes sense to ...
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### Let $X$ and $Y$ be iid real-valued random variables. Show $P[|X-Y| \le 2] \le 3P[|X-Y| \le 1]$.

Found this question in The Probabilistic Method and tried for hours to prove it, but I'm not getting anywhere. Can anyone walk me through it?
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### Convergence of $\sum_{i \leq n} X_i/n$

I have a question like this: Let $(X_n)$ be an i.i.d sequence of random variables with values in {-1,1}, and define $Y_n:= \sum_{i \leq n} X_i/n$. Show that $(Y_n)$ converges almost surely and in ...
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### Fermat's Little theorem (Num Theory)

How can I compute 31^(1209)mod 101 using Fermat's Little Theorem?
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### $G \subset G(F(x,y)/F)$ where$G = <\sigma>$, $\sigma(x) = y$, $\sigma(y)=x$. Describe the subfield of F(x,y) consisting of elements fixed by $\sigma$

Let $F$ be a field and let $E = F(x,y)$, where $x$ and $y$ are indeterminates. Let $G \subset G(E/F)$ be the subgroup defined by $G = <\sigma>$, where $\sigma(x) = y$ and ...
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### The problem about the gödel proposition V in 1931 paper?

Proposition V says that every recursive function R can find a relation symbol in system P such that: R(v1, v2....,vn) -> prove(subst( r(u1,u2,...,u3), (z(v1),z(v2),...,z(vn))) ~R(v1, v2....,vn) ...
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### How can I calculate O - Notation of an algebraic expression?

Explain the process of obtaining O - Notation for a given algebraic expression. For example 5n^2+n^(3/2)
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### Outer measure of a nested sequence of non-measurable sets

Let $\bigcup_{n=1}^\infty E_n=E$ and $E_{n} \subseteq E_{n+1}$ then $\lim\limits_{n\mapsto \infty} \mu^*(E_n) = \mu^*(E)$ even if each $E_n$ is a non-measurable set, where $\mu^*$ is outer ...
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### turning a certain chebychev polynomial-like expression into a hypergeometric form

Can the following expression be represented in terms of hypergeometric function $$\sqrt{3}\sin(\arcsin(7/25)/3)-\cos(\arcsin(7/25)/3)$$ It looks similar to the one presented on this site
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### Find the most general bilinear transformation that maps $|z-1|=1$ to $\Re(f(z)) = 1$.

Find the most general bilinear transformation that maps $|z-1|=1$ to $\Re(f(z)) = 1$. Attempt Assuming that $$2 \mapsto 1 \\ 0 \mapsto \infty \\ 1+i \mapsto 1+i$$ we have the bilinear ...
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### Still not clear why longest path problem is NP-hard but shortest path is not?

I've heard/read many times that shortest path problem is P, but longest path problem is NP-hard. But I have a problem with this: we say longest path problem is NP-hard because of graphs with positive ...
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### How do you isolate and solve for n in a Sigma notation equation?

"isolate and solve for n:" $$P(n)= \sum_{i=0}^n = (n^2+n)$$ If the above equation is a function of P, how would the equation be stated as a function of N? P.S. what should I be tagging this ...
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### Why cannot the power of $i$ be negative?

In a question, The penultimate step of the result was $(i)^n = 1$, and it required to figure least value of n. I checked the $-4$ option, but it said the answer was $4$. Why so? The complete question ...
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### Showing uniform convergence to origin in 3rd quadrant for $x(t)=\frac{1}{\frac{1}{x_0}-t}$ as $t\ \rightarrow \infty$

I want to show that for the system $\dot{x}=x^2, \dot{y}=y^2$,any solutions starting in the 3rd quadrant not including 0, converge uniformly to the origin. For an initial point $(x_0,y_0)$, (note both ...
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### Converting higher-order ODE to first order ODE

given $y''' + 2y'' -5y' = 2y + 5y^3$ convert to a system of first order equations. My question is do we need to make substitutions for $y$ and $y^3$ or are we only concerned with the derivatives, if ...
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### This seems simple but, how do i prove A isn't in C?

$A\in B$ and $B\in C$, Is it posible to prove $A$ isn't in $C$? Sorry for simplistic exercise, but been wondering this for days now... The question in the book says: Can you deduce that $A\in C$ ? ...
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### Show that if $X$ is a bounded a subset of $\mathbb R$ then the closure of $\bar X$ is also bounded.

Show that if $X$ is a bounded subset of $\mathbb R$ then the closure of $\bar X$ is also bounded. My attempt I can show that if $A$ is bounded then so is $\bar A$. So if we have have limit point of ...
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### Show that the $C_n \geq 4^{n-1}/2^{n}$ where $C_n$ is the Catalan number

I write $C_n=\frac{1}{n+1} {2n\choose n}$ and try to prove this claim by induction. But it didn't quite work out. Any idea how to do this without much computation?
Why is it that when $AD = BC$, this equation becomes a horizontal line? $$y = \frac {Ax+B}{Cx+D}$$ For any other values where $AD$ isn't equal to $BC$ it is a rational function.