Intereting Posts

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Show $au_x+bu_y=f(x,y)$ gives $u(x,y)=(a^2+b^2)^{-\frac{1}{2}}\int_{L}fds +g(bx-ay)$ if $a\neq 0$.
Mathematical Induction divisibility $8\mid 3^{2n}-1$
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Notion of a distribution as acting on tangent spaces
Use $\sum_{n=1}^\infty \frac{1}{n^4} = \frac{\pi^4}{90}$ to compute $\sum_{n=1}^\infty \frac{(-1)^n}{n^4}$
Is the boundary of a boundary a subset of the boundary?
Infinite Series $\sum_{n=1}^\infty\frac{H_n}{n^22^n}$

Exercise 1.2.8 (Part 1), p.8, from *Categories for Types* by Roy L. Crole

**Definition:** Let $X$ be a preordered set and $A \subseteq X$. A *join* of $A$, if such exists, is a least element in the set of upper bounds for $A$. A *meet* of $A$, if such exists, is a greatest element in the set of lower bounds for $A$.

**Exercise:** Make sure you understand the definition of meet and join in a preorder $X$. Think of some simple finite preordered sets in which meets and joins do not exist.

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Let $X = {a, b}$. Define a preorder on $X$ as $a \le a$ and $b \le b$. Now suppose that $\vee X$ is a join of $X$. Then $\vee X$ is an upperbound of $X$. So $a \le \vee X$ and $b \le \vee X$. So $\vee X = a$ and $\vee X = b$, which is a contradiction. Therefor, $\vee X$ does not exist. A similar proof will show that $X$ does not have a meet, either.

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