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## Reversing the translation of the Pillow Image Library Rotation

I’m a bit of newbie at this, trying to rotate an image in Python Pillow without changing the position of the centre of the rotated image. OR by the pillow rotate looks of things… returning the centre back to it’s original spin location. In Pillow (Image.py) there is a function that rotates an image. This […]

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## The interval [a,b] $\subset$ $\mathbb{R}$ and interval [0,1] $\subset$ $\mathbb{R}$ are homeomorphic in their standard topologies. [closed]

Problem: The interval $[a,b]$ $\subset$ $\mathbb{R}$ and interval $[0,1]$ $\subset$ $\mathbb{R}$ are homeomorphic in their standard topologies. Definition of Homeomorphism: We call the underlying bijection of a topological equivalence $h : X \to Y$ a homeomorphism. We say X and Y are homeomorphic. This is how I showed it. Let us define a function $f:[a,b]$$\to$$[0,1]$ […]

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## Implicating convex combinations through inequalities

Given an inequality e.g. $a<b<c$, why does this implicate that $b$ is a convex combination of $a,c$, meaning that there exist two numbers $x,y \in (0,1):x+y=1$ so that $b=xa+yc$?

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## If a function is bounded on a set does this mean ut is constant?

Let $H$ be the upper half plane and let $S$ be the union of the first and third quadrants. (a) Suppose that an entire function $f$ is bounded on $H$. Does it follow that $f$ is constant? (b) Suppose that an entire function f is bounded on $S$. Does it follow that $f$ is constant? […]

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## Finding the joint distribution of two dependent random variables given conditions

Suppose $X \sim U(0,1)$ and $Y|X=x \sim U(0,1-x)$. Find the joint distribution of $(X,Y)$ given that $X\le \frac{1}{2}$, $Y\le \frac{1}{2}.$ My attempt: The pdf of $X$ is $f_{X}(x) = 1$ for $x \in (0,1)$ and the pdf of $Y|X=x$ is $f_{Y|X}(y|x) = \frac{1}{1-x}$ for some $x\in (0,1)$ and $y\in (0,1-x)$. Hence the (unconditional) joint density […]