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6 changes: 3 additions & 3 deletions contents/IFS/IFS.md
Original file line number Diff line number Diff line change
Expand Up @@ -20,8 +20,8 @@ To begin the discussion of Iterated Function Systems (IFSs), we will first discu
<img class="center" src="res/IFS_triangle_1.png" alt="Sierpinsky Triangle Chaos Game" style="width:100%">

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.
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.
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 }}.
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.
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 }}.
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 }}.
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.

Expand All @@ -41,7 +41,7 @@ f_3(P) &= \frac{P + C}{2}\\
$$

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.
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:
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:

$$
H(P) = \bigcup_{i=1}^3f_i(P)
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6 changes: 2 additions & 4 deletions contents/stacks_and_queues/stacks_and_queues.md
Original file line number Diff line number Diff line change
Expand Up @@ -19,10 +19,9 @@ Here is a simple implementation of a stack:
{% sample lang="ts" %}
[import, lang:"typescript"](code/typescript/stack.ts)
{% sample lang="java" %}
[import, lang:"java"](code/java/Stack.java)
[import, lang:"java"](code/java/StackTest.java)
{% sample lang = "cpp"%}
[import, lang:"cpp"](code/cpp/stack.cpp)
[import, lang:"java"](code/java/StackTest.java)
{% sample lang="rust" %}
[import, lang:"rust"](code/rust/Stack.rs)
{% endmethod %}
Expand All @@ -32,10 +31,9 @@ Here is a simple implementation of a queue:
{% sample lang="ts" %}
[import, lang:"typescript"](code/typescript/queue.ts)
{% sample lang="java" %}
[import, lang:"java" ](code/java/Queue.java)
[import, lang:"java" ](code/java/QueueTest.java)
{% sample lang = "cpp"%}
[import, lang:"cpp"](code/cpp/queue.cpp)
[import, lang:"java" ](code/java/QueueTest.java)
{% sample lang="rust" %}
[import, lang:"rust" ](code/rust/Queue.rs)
{% endmethod %}
Expand Down