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Solved formatting problems (#985)
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contents/IFS/IFS.md

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@@ -20,8 +20,8 @@ To begin the discussion of Iterated Function Systems (IFSs), we will first discu
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<img class="center" src="res/IFS_triangle_1.png" alt="Sierpinsky Triangle Chaos Game" style="width:100%">
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This image is clearly a set of triangles embedded in a larger triangle in such a way that it can be continually cut into three identical pieces and still retain its internal structure.
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This idea is known as self-similarity {{"self-similar" | cite }}, and it is usually the first aspect of fractals to catch an audience's attention.
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In fact, there are plenty of uses of fractals and their mathematical underpinnings, such as estimating the coastline of Britain {{ "mandelbrot1967long" | cite}}, identifying fingerprints {{ "jampour2010new" | cite }}, and image compression {{ "fractal-compression" | cite }}{{ "saupe1994review" | cite }}.
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This idea is known as self-similarity {{ "self-similar" | cite }}, and it is usually the first aspect of fractals to catch an audience's attention.
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In fact, there are plenty of uses of fractals and their mathematical underpinnings, such as estimating the coastline of Britain {{ "mandelbrot1967long" | cite }}, identifying fingerprints {{ "jampour2010new" | cite }}, and image compression {{ "fractal-compression" | cite }}{{ "saupe1994review" | cite }}.
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In many more rigorous definitions, a fractal can be described as any system that has a non-integer Hausdorff dimension {{ "3b1bfractal" | cite }}{{ "hausdorff" | cite }}{{ "gneiting2012estimators" | cite }}.
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Though this is an incredibly interesting concept, the discussion of this chapter will instead focus on methods to generate fractal patterns through iterated function systems.
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@@ -41,7 +41,7 @@ f_3(P) &= \frac{P + C}{2}\\
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$$
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Each function will read in a particular location in space (here, $$P \in \mathbb{R}^2$$) and output a new location that is the midpoint between the input location and $$A$$, $$B$$, or $$C$$ for $$f_1$$, $$f_2$$, and $$f_3$$ respectively.
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The union of all of these functions (the set of all possible functions available for use) is often notated as the _Hutchinson operator_ {{ "hutchinson-operator" | cite }}{{ "hutchinson1981fractals" | cite}}, and for this case it would look like this:
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The union of all of these functions (the set of all possible functions available for use) is often notated as the _Hutchinson operator_ {{ "hutchinson-operator" | cite }}{{ "hutchinson1981fractals" | cite }}, and for this case it would look like this:
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$$
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H(P) = \bigcup_{i=1}^3f_i(P)

contents/stacks_and_queues/stacks_and_queues.md

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@@ -19,10 +19,9 @@ Here is a simple implementation of a stack:
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{% sample lang="ts" %}
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[import, lang:"typescript"](code/typescript/stack.ts)
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{% sample lang="java" %}
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[import, lang:"java"](code/java/Stack.java)
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[import, lang:"java"](code/java/StackTest.java)
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{% sample lang = "cpp"%}
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[import, lang:"cpp"](code/cpp/stack.cpp)
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[import, lang:"java"](code/java/StackTest.java)
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{% sample lang="rust" %}
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[import, lang:"rust"](code/rust/Stack.rs)
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{% endmethod %}
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{% sample lang="ts" %}
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[import, lang:"typescript"](code/typescript/queue.ts)
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{% sample lang="java" %}
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[import, lang:"java" ](code/java/Queue.java)
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[import, lang:"java" ](code/java/QueueTest.java)
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{% sample lang = "cpp"%}
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[import, lang:"cpp"](code/cpp/queue.cpp)
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[import, lang:"java" ](code/java/QueueTest.java)
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{% sample lang="rust" %}
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[import, lang:"rust" ](code/rust/Queue.rs)
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{% endmethod %}

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