Re-formatted some blog posts using new formatting script.

5 years ago · c6c704e992
--- a/blogContent/posts/hardware/2018-rochester-maker-faire.md
+++ b/blogContent/posts/hardware/2018-rochester-maker-faire.md
@ -1,78 +1,102 @@
 ![Rochester Maker Faire RITlug table.](media/makerFaire/20181117_091159.jpg)
 # Introduction
 A few weeks ago I presented my [musical floppy drives](https://jrtechs.net/projects/musical-floppy-drive-build-log)
 at the [Rochester Maker Faire](https://rochester.makerfaire.com/) with [RITlug](https://ritlug.com/). 
 Wow, that sentence had a ton of links-- you should check them out. This post is a 
 quick recap of my experience at the Maker Faire and a project update for my musical floppy drive project. For those of you who 
 don't know, [Maker Faires](https://makerfaire.com/) are community gatherings where people "celebrate arts, crafts, 
 engineering, science projects and the Do-It-Yourself (DIY) mindset".
 I would like to give a huge thanks to [Christian](https://ctmartin.me/)
 for providing me with some of these pictures and [RITlug](https://ritlug.com/)
 for giving me the opportunity to present at the Maker Faire. 
 A few weeks ago I presented my [musical floppy
 drives](https://jrtechs.net/projects/musical-floppy-drive-build-log)
 at the [Rochester Maker Faire](https://rochester.makerfaire.com/) with
 [RITlug](https://ritlug.com/).  Wow, that sentence had a ton of
 links-- you should check them out. This post is a  quick recap of my
 experience at the Maker Faire and a project update for my musical
 floppy drive project. For those of you who  don't know, [Maker
 Faires](https://makerfaire.com/) are community gatherings where people
 "celebrate arts, crafts,  engineering, science projects and the
 Do-It-Yourself (DIY) mindset". 
 I would like to give a huge thanks to
 [Christian](https://ctmartin.me/)  for providing me with some of these
 pictures and [RITlug](https://ritlug.com/) for giving me the
 opportunity to present at the Maker Faire.  
 # Project Background
 Musical floppy drives are floppy drives wired to an Arduino to play music. I initially started with two floppy drives;
 however, that grew to 8 drives very fast. 
 Musical floppy drives are floppy drives wired to an Arduino to play
 music. I initially started with two floppy drives; however, that grew
 to 8 drives very fast.  
 <youtube src="Y5msiFa54Ug" />
 I currently have 10 floppy drives stacked together in a tower like structure to make it easier to take to places
 like the Maker Faire and [Imagine RIT](https://www.rit.edu/imagine/). 
 I currently have 10 floppy drives stacked together in a tower like
 structure to make it easier to take to places like the Maker Faire and
 [Imagine RIT](https://www.rit.edu/imagine/).  
 ![Rochester Maker Faire RITlug table.](media/makerFaire/20181117_092417.jpg)
 # Future Project Upgrades
 Although the 10 floppy drives are quite loud in a small room, people typically found it difficult to hear at the
 Maker Faire. This was largely due to the noisiness of the hundreds of people at the convention center. To fix this
 in the future I want to bring a small amplifier for the floppy drives. 
 Although the 10 floppy drives are quite loud in a small room, people
 typically found it difficult to hear at the Maker Faire. This was
 largely due to the noisiness of the hundreds of people at the
 convention center. To fix this in the future I want to bring a small
 amplifier for the floppy drives.  
 The second improvement which I plan on making is creating some form of web application where people can 
 request music for the floppy drives to play. The potential down side of this is that it is often difficult connecting
 to the internet at conventions.
 The second improvement which I plan on making is creating some form of
 web application where people can  request music for the floppy drives
 to play. The potential down side of this is that it is often difficult
 connecting to the internet at conventions. 
 I have a ton of old IDE hard drives that I plan on incorporating into my musical orchestra. The tricky part of that
 upgrade would be to somehow incorporate those into my current floppy drive structure so that it is easy to take places.
 My current plan is to have the hard drives completely separate from the floppy drives and run it on its own Arduino board. 
 I have a ton of old IDE hard drives that I plan on incorporating into
 my musical orchestra. The tricky part of that upgrade would be to
 somehow incorporate those into my current floppy drive structure so
 that it is easy to take places.  My current plan is to have the hard
 drives completely separate from the floppy drives and run it on its
 own Arduino board.  
 # Cool Projects at the Maker Faire
 Although there were a ton of really cool projects at the Maker Faire, I did not have the time explore everything 
 and take a lot of pictures. However, I want to share some of the really cool projects that I saw at the Maker Faire. 
 Although there were a ton of really cool projects at the Maker Faire,
 I did not have the time explore everything  and take a lot of
 pictures. However, I want to share some of the really cool projects
 that I saw at the Maker Faire.  
 ## HatChan
 ![HatChan](media/makerFaire/20181117_095844.jpg)
 This is a project that one person is calling "HatChan". This is a hat equipped with a ton of LED lights, and 
 a fully functioning wireless access point. When you connect to his access point, the captcha portal takes you
 to a 4chan esk site where you can upload pictures -- mostly memes.
 This is a project that one person is calling "HatChan". This is a hat
 equipped with a ton of LED lights, and  a fully functioning wireless
 access point. When you connect to his access point, the captcha portal
 takes you to a 4chan esk site where you can upload pictures -- mostly
 memes. 
 ## 8-bit Computer
 ![Rochester Maker Faire](media/makerFaire/20181117_120948.jpg)
 This one person brought a home made 8-bit computer to the Maker Faire. What I find the most impressive
 about this project was that the person who made it was only a sophomore in high school. 
 This one person brought a home made 8-bit computer to the Maker Faire.
 What I find the most impressive about this project was that the person
 who made it was only a sophomore in high school.  
 ## Foss@Magic
 ![Rochester Maker Faire](media/makerFaire/MVIMG_20181117_151211.jpg)
 *Photo provided by [Christian](https://ctmartin.me/)*
 FOSS@Magic is a organization at RIT which aims to promote
 free and open source software-- i'm also a part of this group. This year the FOSS group demoed their colorful MAGIC letters. On 
 a tablet beneath the letters, there is a web site open where you can select the color of each individual
 RGB letter. 
 FOSS@Magic is a organization at RIT which aims to promote free and
 open source software-- i'm also a part of this group. This year the
 FOSS group demoed their colorful MAGIC letters. On  a tablet beneath
 the letters, there is a web site open where you can select the color
 of each individual RGB letter.  
 ## RITlug
 ![Rochester Maker Faire](media/makerFaire/MVIMG_20181117_151247.jpg)
 *Photo provided by [Christian](https://ctmartin.me/)*
 The RIT Linux Users Group (RITlug) is a student-led organization aimed to promote and teach students about Linux and open source.
 This year we presented my Floppy Drive project, a Raspberry Pi [Lakka](https://www.raspberrypi.org/magpi/easy-retro-gaming-lakka/) gaming set up, a
 [Magic Mirror](https://github.com/MichMich/MagicMirror), and [Tiger OS](https://ritlug.com/tigeros).
 The RIT Linux Users Group (RITlug) is a student-led organization aimed
 to promote and teach students about Linux and open source. This year
 we presented my Floppy Drive project, a Raspberry Pi
 [Lakka](https://www.raspberrypi.org/magpi/easy-retro-gaming-lakka/)
 gaming set up, a [Magic
 Mirror](https://github.com/MichMich/MagicMirror), and [Tiger
 OS](https://ritlug.com/tigeros). 
--- a/blogContent/posts/java/gremlin-in-10-minutes.md
+++ b/blogContent/posts/java/gremlin-in-10-minutes.md
@ -1,44 +1,57 @@
 # What is Gremlin?
 Gremlin is a graph traversal language: think of Gremlin as the SQL for graph databases. Gremlin is not
 a graph database server, it is a language; but, there is a Gremlin Server and a Gremlin Console available for 
 interacting with graph databases. It is possible to use Gremlin on large database platforms 
 like [Titan](https://www.digitalocean.com/community/tutorials/how-to-set-up-the-titan-graph-database-with-cassandra-and-elasticsearch-on-ubuntu-16-04)
 and [HBase](https://docs.janusgraph.org/latest/hbase.html). 
 Gremlin is a graph traversal language: think of Gremlin as the SQL for
 graph databases. Gremlin is not a graph database server, it is a
 language; but, there is a Gremlin Server and a Gremlin Console
 available for  interacting with graph databases. It is possible to use
 Gremlin on large database platforms  like
 [Titan](https://www.digitalocean.com/community/tutorials/how-to-set-up-the-titan-graph-database-with-cassandra-and-elasticsearch-on-ubuntu-16-04)
 and [HBase](https://docs.janusgraph.org/latest/hbase.html).  
 # Graph Data Base Basics
 A graph database is based on graph theory. A graph is composed of nodes, edges, and properties. A key
 object/component in a graph database is stored as a node. Nodes are connected via edges representing
 relationships. For example, you may represent people as nodes and have edges representing friendships. 
 You can assign properties to both nodes and edges. A person (node) may have the properties of age and name, 
 where a friendship (edge) may have a start date property.
 A graph database is based on graph theory. A graph is composed of
 nodes, edges, and properties. A key object/component in a graph
 database is stored as a node. Nodes are connected via edges
 representing relationships. For example, you may represent people as
 nodes and have edges representing friendships.  You can assign
 properties to both nodes and edges. A person (node) may have the
 properties of age and name,  where a friendship (edge) may have a
 start date property. 
 ## Why Graph Databases?
 Graph databases are great for modeling data where the value lies in the shape of the graph. Graph databases
 also allow to to model more complex relationships which would be difficult to model in a normal table-based
 database.
 Graph databases are great for modeling data where the value lies in
 the shape of the graph. Graph databases also allow to to model more
 complex relationships which would be difficult to model in a normal
 table-based database. 
 ## Gremlin Installation
 Download and extract the following:
 Download and extract the following: 
 - [Gremlin Console](https://www.apache.org/dyn/closer.lua/tinkerpop/3.3.3/apache-tinkerpop-gremlin-console-3.3.3-bin.zip)
 - [Gremlin Server](https://www.apache.org/dyn/closer.lua/tinkerpop/3.3.3/apache-tinkerpop-gremlin-server-3.3.3-bin.zip)
 Start the Gremlin server by running it with the start script in the bin folder. As a prerequisite for running gremlin, you
 must have Java installed on your computer.
 ```
 Start the Gremlin server by running it with the start script in the
 bin folder. As a prerequisite for running gremlin, you must have Java
 installed on your computer. 
 ```bash
 ./gremlin-server.sh
 ```
 Start the Gremlin console by running the gremlin.sh or gremlin.bat script in the bin folder.
 ```
 Start the Gremlin console by running the gremlin.sh or gremlin.bat
 script in the bin folder. 
 ```bash
 ./gremlin.sh
 ```
 Now you need to instantiate a new graph on the server to use. To to that, execute the following commands in 
 the Gremlin console.
 Now you need to instantiate a new graph on the server to use. To to
 that, execute the following commands in  the Gremlin console. 
 ```java
 #Creates a empty graph
 gremlin> graph = EmptyGraph.instance()
@ -56,26 +69,32 @@ gremlin> g = graph.traversal().withRemote(DriverRemoteConnection.using(cluster,
 # Gremlin Syntax
 Now that you have your gremlin server and console set up, you are ready to start executing Gremlin queries.
 Now that you have your gremlin server and console set up, you are
 ready to start executing Gremlin queries. 
 ## Adding a Vertex
 In Gremlin nodes are referred to as "Vertexes". To add a node/vertex to the graph, you simply use the 
 command addV() on your graph traversal source. For consistency, most people
 use "g" as their default graph traversal source. To append properties to your your vertex, you add a series of
 ".property('property_name', 'property_value')" strings to the add vertex query. 
 In Gremlin nodes are referred to as "Vertexes". To add a node/vertex
 to the graph, you simply use the  command addV() on your graph
 traversal source. For consistency, most people use "g" as their
 default graph traversal source. To append properties to your your
 vertex, you add a series of ".property('property_name',
 'property_value')" strings to the add vertex query.  
 EX: 
 EX:
 ```java
 g.addV('student').property('name', 'Jeffery').property('GPA', 4.0);
 ```
 ## Updating a Property
 Unlike SQL, you are not limited to a specific schema in a graph database. If you want to add or change 
 a property on a vertex or edge, you simply use the property command again.
 The "g.V(1)" in the following example refers to a specific vertex with the primary id of 1-- the graph database auto assigns these ids.
 You can replace "g.V(1)" with a command to select a specific vertex or edge. 
 Unlike SQL, you are not limited to a specific schema in a graph
 database. If you want to add or change  a property on a vertex or
 edge, you simply use the property command again. The "g.V(1)" in the
 following example refers to a specific vertex with the primary id of
 1-- the graph database auto assigns these ids. You can replace
 "g.V(1)" with a command to select a specific vertex or edge.  
 ```java
 g.V(1).property('name', 'Jeffery R');
@ -83,29 +102,38 @@ g.V(1).property('name', 'Jeffery R');
 ## Selection
 Selecting nodes and edges is the most complicated part of Gremlin. The concept is not particularly hard, but, there
 are dozens of ways to do graph traversals and selections. I will cover the most common aways to traverse a graph.
 Selecting nodes and edges is the most complicated part of Gremlin. The
 concept is not particularly hard, but, there are dozens of ways to do
 graph traversals and selections. I will cover the most common aways to
 traverse a graph. 
 This example will select all vertexes which have the label "student".
 The ".valueMap()" command appended to the end of the query  makes
 Gremlin return a map of all the objects it returns with their
 properties. 
 This example will select all vertexes which have the label "student". The ".valueMap()" command appended to the end of the query 
 makes Gremlin return a map of all the objects it returns with their properties.
 ```java
 g.V().hasLabel('student').valueMap();
 ```
 In this following example, instead of returning a ValueMap of values, we are just returning the names of the students
 in the graph.
 In this following example, instead of returning a ValueMap of values,
 we are just returning the names of the students in the graph. 
 ```java
 g.V().hasLabel('student').values('name');
 ```
 This example will return the GPA of the student with the name "Jeffery R".
 This example will return the GPA of the student with the name "Jeffery
 R". 
 ```java
 g.V().hasLabel('student').has('name', 'Jeffery R').values('gpa');
 ```
 This command will return all the students in order of their GPA.
 This command will return all the students in order of their GPA. 
 ```java
 g.V().hasLabel('student').order().by('gpa', decr).value('name')
 ```
@ -113,10 +141,12 @@ g.V().hasLabel('student').order().by('gpa', decr).value('name')
 ## Adding Edges
 The easiest way (my opinion) to add edges in Gremlin is by
 using aliasing. In this example we select two nodes and assign them a name: in this case it is "a", and "b".
 After we have selected two edges, we can add an edge to them using the "addE()" command. The syntax of this is
 nice because we know that "a" is friends with "b"-- it is easy to tell the direction of the edge. 
 The easiest way (my opinion) to add edges in Gremlin is by using
 aliasing. In this example we select two nodes and assign them a name:
 in this case it is "a", and "b". After we have selected two edges, we
 can add an edge to them using the "addE()" command. The syntax of this
 is nice because we know that "a" is friends with "b"-- it is easy to
 tell the direction of the edge.  
 ```java
 g.V(0).as('a').V(1).as('b').addE('knows')
@ -126,12 +156,15 @@ g.V(0).as('a').V(1).as('b').addE('knows')
 # Using Gremlin with Java
 Now that you know the basic syntax of Gremlin, you are ready to use it somewhere other than the Gremlin console. If you
 are trying to use Gremlin with Java, there is a great Maven dependency for TinkerPop and Gremlin. If you want to quickly 
 connect to your Gremlin server with Java, make sure your server is set up exactly as it was before this tutorial started discussing
 Gremlin syntax. 
 Now that you know the basic syntax of Gremlin, you are ready to use it
 somewhere other than the Gremlin console. If you are trying to use
 Gremlin with Java, there is a great Maven dependency for TinkerPop and
 Gremlin. If you want to quickly  connect to your Gremlin server with
 Java, make sure your server is set up exactly as it was before this
 tutorial started discussing Gremlin syntax.  
 ## Maven dependency for Java:
 ```html
 <!-- https://mvnrepository.com/artifact/com.tinkerpop/gremlin-core -->
 <dependency>
@ -154,8 +187,10 @@ Gremlin syntax.
 </dependency>
 ```
 It is helpful to wrap everything relating to the graph database connection into a single Java class. This is roughly
 the code that I usually use to interact with a Gremlin Server-- anybody is free to use it. 
 It is helpful to wrap everything relating to the graph database
 connection into a single Java class. This is roughly the code that I
 usually use to interact with a Gremlin Server-- anybody is free to use
 it.  
 ```java
 public class GraphConnection
@ -188,6 +223,8 @@ public class GraphConnection
 ```
 ## Basic GraphConnection.java Usage:
 ```java
 RemoteConnection con = new RemoteConnection()
 String query = "g.V().hasLabel('player')" +
@ -202,6 +239,8 @@ this.con.queryGraph(query);
 ```
 ## Overly complex usage with a lambda statement
 ```java
 /**
 * Fetches the list of a player's friends.
@ -212,11 +251,9 @@ this.con.queryGraph(query);
 private List<Player> getFriendsFromGraph(String id)
 {
    List<Player> friends = new ArrayList<>();
    String query = "g.V().hasLabel('player')" +
            ".has('id', '" + id + "')" +
            ".both().valueMap()";
    this.con.queryGraph(query).stream().forEach(r ->
            friends.add(new Player(
                    ((ArrayList) (((HashMap<String, Object>) (r.getObject()))
@ -228,21 +265,22 @@ private List getFriendsFromGraph(String id)
 }
 ```
 The most important thing to do while playing around with Gremlin in Java is to keep an eye on the 
 return type. From experience, I can say that it is often easier to return the vertex from your 
 query rather than returning the valueMap.
 The most important thing to do while playing around with Gremlin in
 Java is to keep an eye on the  return type. From experience, I can say
 that it is often easier to return the vertex from your  query rather
 than returning the valueMap. 
 Without returning the valueMap in the query, you can directly access
 the vertex in the result rather than doing some voodoo witchcraft and
 casting between ArrayLists and HashMaps. 
 Without returning the valueMap in the query, you can directly access the vertex
 in the result rather than doing some voodoo witchcraft and casting between ArrayLists and HashMaps.
 The previous example could be re-written as this: 
 The previous example could be re-written as this:
 ```java
 List<Player> friends = new ArrayList<>();
 String query = "g.V().hasLabel('player')" +
        ".has('id', '" + id + "')" +
        ".both()";
 for(Result r: this.con.queryGraph(query))
 {
    friends.add(new Player(r.getVertex("name").value().toString), 
@ -250,14 +288,15 @@ for(Result r: this.con.queryGraph(query))
 }
 ```
 You now know enough about Gremlin to be dangerous with it. Yay! If you want to do more than basic things with Gremlin,
 I highly suggest that you look at the tutorial [SQL 2 Gremlin](http://sql2gremlin.com/). 
 If you plan on deploying this to production, it is recommended that you use HBase for a persistent back end storage
 server.
 You now know enough about Gremlin to be dangerous with it. Yay! If you
 want to do more than basic things with Gremlin, I highly suggest that
 you look at the tutorial [SQL 2 Gremlin](http://sql2gremlin.com/).  If
 you plan on deploying this to production, it is recommended that you
 use HBase for a persistent back end storage server. 
 # Resources
 - [SQL 2 Gremlin](http://sql2gremlin.com/)
 - [Practical Gremlin](http://kelvinlawrence.net/book/Gremlin-Graph-Guide.html)
 - [Apache TinkerPop](http://tinkerpop.apache.org/)
 - [Steam Friends Graph (Personal Gremlin Project)](https://github.com/jrtechs/SteamFriendsGraph)
 - [Steam Friends Graph (Personal Gremlin Project)](https://github.com/jrtechs/SteamFriendsGraph)
--- a/blogContent/posts/java/top-three-recommended-java-ides.md
+++ b/blogContent/posts/java/top-three-recommended-java-ides.md
@ -1,45 +1,48 @@
 An IDE (Integrated Development Environment) is a single program that wraps
 everything you need to program with. An IDE usually consists of a code editor,
 debugger, and a compiler. A good IDE can make you more productive. Without a IDE
 you would be stuck writing all of your apps in notepad and compiling via command
 line – which is not bad for learning how to program. A good IDE has syntax
 highlighting which makes the code easier to read and error highlighting which
 makes finding that missing semi colon easier. Here is a list of three IDEs that
 work well for me.
 An IDE (Integrated Development Environment) is a single program that
 wraps everything you need to program with. An IDE usually consists of
 a code editor, debugger, and a compiler. A good IDE can make you more
 productive. Without a IDE you would be stuck writing all of your apps
 in notepad and compiling via command line – which is not bad for
 learning how to program. A good IDE has syntax highlighting which
 makes the code easier to read and error highlighting which makes
 finding that missing semi colon easier. Here is a list of three IDEs
 that work well for me. 
 # NetBeans
 NetBeans is the most productive IDE for large java projects. NetBeans is great
 at detecting errors in your code. You can also have multiple projects open in
 the same window which is great for anyone who is constantly flipping between
 projects – even if they are in different coding languages. This is a must have
 program for everyone looking to develop Java. This program also supports PHP,
 JavaScript and C/C++.
 NetBeans is the most productive IDE for large java projects. NetBeans
 is great at detecting errors in your code. You can also have multiple
 projects open in the same window which is great for anyone who is
 constantly flipping between projects – even if they are in different
 coding languages. This is a must have program for everyone looking to
 develop Java. This program also supports PHP, JavaScript and C/C++. 
 ![NetBeans IDE](media/2a20326e77c3f67aa3753401ba4bc612.png)
 # Eclipse
 Eclipse is a very popular programming IDE; however, it is harder to use than
 NetBeans. The largest benefit of Eclipse is that it has tons of templates for
 you to use. Eclipse also has a better GUI creation tool than NetBeans has.
 Eclipse is a very popular programming IDE; however, it is harder to
 use than NetBeans. The largest benefit of Eclipse is that it has tons
 of templates for you to use. Eclipse also has a better GUI creation
 tool than NetBeans has. 
 ![Eclipse](media/f5cb2682e219549cb07acd1d01c2ccb1.png)
 # JCreator
 I want to warn you that this is a old IDE. You hate yourself if you decide to
 use this program. The only reason I am putting this on my list is because I
 think it is great for teaching people how to program – it’s what my high school
 used. Unlike Eclipse and NetBeans, it gives you very little help when writing
 your code. There is no autocompletes or dropdowns when you are accessing members
 of an object. This does not even give you red highlights when you have a syntax
 error. What it instead shows you is the errors that you would receive in the
 command line when you compile this project. This is great for AP Computer
 Science students who have to memorize/learn what certain errors mean.
 I want to warn you that this is a old IDE. You hate yourself if you
 decide to use this program. The only reason I am putting this on my
 list is because I think it is great for teaching people how to program
 – it’s what my high school used. Unlike Eclipse and NetBeans, it gives
 you very little help when writing your code. There is no autocompletes
 or dropdowns when you are accessing members of an object. This does
 not even give you red highlights when you have a syntax error. What it
 instead shows you is the errors that you would receive in the command
 line when you compile this project. This is great for AP Computer
 Science students who have to memorize/learn what certain errors mean. 
 ![JCreator](media/c1e712bdead908ebd72921ec2d1cbd5e.png)
@ -48,11 +51,11 @@ Science students who have to memorize/learn what certain errors mean.
 \*Update 2018
 I have converted to a full time Intellij user since writing this article in
 2015. This program is simply fantastic in every aspect – expect ram usage. If
 you have never heard of Intellij or any of the other JetBrains products I would
 highly recommend you check them out. The main reason I switched was because
 Intellij could scale nicely to 4k and NetBeans did not. The user experience
 overall feels nice.
 I have converted to a full time Intellij user since writing this
 article in 2015. This program is simply fantastic in every aspect –
 expect ram usage. If you have never heard of Intellij or any of the
 other JetBrains products I would highly recommend you check them out.
 The main reason I switched was because Intellij could scale nicely to
 4k and NetBeans did not. The user experience overall feels nice. 
 ![Intellij](media/2a96900bae1e8946a1df243d8318e69f.png)
--- a/blogContent/posts/other/college-cookbook.md
+++ b/blogContent/posts/other/college-cookbook.md
@ -1,21 +1,24 @@
 I occasionally cooked during High School, however, college was my first real experience cooking on 
 a daily basis. Since I have lived to tell the tail, I want to share some of the recipes that
 I created/modified which got me through my first two years of college.
 I occasionally cooked during High School, however, college was my
 first real experience cooking on  a daily basis. Since I have lived to
 tell the tail, I want to share some of the recipes that I
 created/modified which got me through my first two years of college. 
 # Level 1: Cooking in the Dorms
 Being stuck in a dorm room you are limited to only being able to use the microwave.
 Desperate times calls for depraved cooking solutions.
 Being stuck in a dorm room you are limited to only being able to use
 the microwave. Desperate times calls for depraved cooking solutions. 
 ## Ramen Pizza
 ![Romen Pizza Final Result](media/cooking/lvl1/romenPizza.jpg)
 This recipe surprised me as to how it turned out.
 You start making this by microwaving ramen noodles for about a third of the instructed time in a bowl of water.
 When you place the noodles in the bowl, make sure to not break them apart. 
 After cooking, take the romen noodles out of the bowl, dry, and place on a plate.
 Next, you slather the ramen noodles with your desired pizza toppings and place in the microwave for around 2 minutes. 
 This recipe surprised me as to how it turned out. You start making
 this by microwaving ramen noodles for about a third of the instructed
 time in a bowl of water. When you place the noodles in the bowl, make
 sure to not break them apart.  After cooking, take the romen noodles
 out of the bowl, dry, and place on a plate. Next, you slather the
 ramen noodles with your desired pizza toppings and place in the
 microwave for around 2 minutes.  
 ### Ingredients
@ -28,19 +31,22 @@ Next, you slather the ramen noodles with your desired pizza toppings and place i
 ![Eggs in a Mug Result](media/cooking/lvl1/eggs.jpg)
 Life hack: you can cook scrambled eggs in a microwave using a coffee mug.
 The resulting eggs are abnormally fluffy and taste a little strange, but, it makes for a quick meal.
 It just so happens that you can also cook hot dogs in the microwave.
 Life hack: you can cook scrambled eggs in a microwave using a coffee
 mug. The resulting eggs are abnormally fluffy and taste a little
 strange, but, it makes for a quick meal. It just so happens that you
 can also cook hot dogs in the microwave. 
 When I stumbled upon this recipe, I clung to it since it was fast, cheep, 
 and most of the dinning places on campus were closed by the time I got out of work. 
 To make this, you crack and mix two eggs in a coffee cup. 
 You then microwave the coffee mug, mixing the eggs every 30 seconds.
 Half way through cooking the eggs, you add a diced hotdog and some shredded cheese.
 Once your eggs have all cooked, you can add a healthy dosage of salsa.
 Voila, you have an omelet which can compete with whatever they are serving at the dinning hall.
 When I stumbled upon this recipe, I clung to it since it was fast,
 cheep,  and most of the dinning places on campus were closed by the
 time I got out of work.  To make this, you crack and mix two eggs in a
 coffee cup.  You then microwave the coffee mug, mixing the eggs every
 30 seconds. Half way through cooking the eggs, you add a diced hotdog
 and some shredded cheese. Once your eggs have all cooked, you can add
 a healthy dosage of salsa. Voila, you have an omelet which can compete
 with whatever they are serving at the dinning hall. 
 ## Ingredients
 - 1-2 Eggs
 - 1 Hotdog
 - Cheese
@ -48,19 +54,24 @@ Voila, you have an omelet which can compete with whatever they are serving at th
 # Level 2: Cooking With no Time
 Time is *very* scarce as a college student, I found the following recipes to be fast and easy.
 These recipes are very forgiving in the sense that you can change the directions and ingredients and still end up with something resembling food. 
 Time is *very* scarce as a college student, I found the following
 recipes to be fast and easy. These recipes are very forgiving in the
 sense that you can change the directions and ingredients and still end
 up with something resembling food.  
 ## Chili
 ![Chili with rice and cheese](media/cooking/lvl2/chili.jpg)
 It is *very* hard to make a bad chili. 
 All you essentially have to do is gather/prep all the ingredients and plop them in a crock pot and cook for 6-8 hours on low.
 You don't have to cook this in a crock pop, however, it takes more time to prepare on the stove since you have to monitor it while it cooks for an hour on medium heat.
 I usually saute the onions while I cook the ground beef so that it cooks in the grease of the beef. 
 The average prep time for this recipe is around 20-30 minutes. 
 However, if you made this vegetarian you can probably cut the prep time down to a mere 10 minutes. 
 It is *very* hard to make a bad chili.  All you essentially have to do
 is gather/prep all the ingredients and plop them in a crock pot and
 cook for 6-8 hours on low. You don't have to cook this in a crock pop,
 however, it takes more time to prepare on the stove since you have to
 monitor it while it cooks for an hour on medium heat. I usually saute
 the onions while I cook the ground beef so that it cooks in the grease
 of the beef.  The average prep time for this recipe is around 20-30
 minutes.  However, if you made this vegetarian you can probably cut
 the prep time down to a mere 10 minutes.  
 ### Ingredients
@ -86,12 +97,15 @@ However, if you made this vegetarian you can probably cut the prep time down to
 ![Poor mans Chicken Parmesan Serving Picture](media/cooking/lvl2/chickenParm.jpg)
 I call this poor man's chicken parmesan only because it uses chicken patties, this actually tastes really good.
 To prepare this you slather chicken patties with pasta sauce and cheese and then cook normally in the oven.
 While the chicken patties bake, you can cook your normal pasta noodles and sauce.
 You serve the pasta noodles with the sauce topped with the cheesy chicken pattie.
 I call this poor man's chicken parmesan only because it uses chicken
 patties, this actually tastes really good. To prepare this you slather
 chicken patties with pasta sauce and cheese and then cook normally in
 the oven. While the chicken patties bake, you can cook your normal
 pasta noodles and sauce. You serve the pasta noodles with the sauce
 topped with the cheesy chicken pattie. 
 ### Ingredients
 - Chicken Patties
 - Pasta Sauce
 - Pasta Noodles
@ -101,16 +115,21 @@ You serve the pasta noodles with the sauce topped with the cheesy chicken pattie
 ![Chicken Soup Served with Mozzarella Cheese](media/cooking/lvl2/chickenSoup.jpg)
 This is an easy recipe which has *so* many combinations.
 I usually make some form of chicken soup, the rest of the ingredients are entirely dependent on what I have in the kitchen.
 You will have to play around with my recommenced ingredient list to find the perfect combination for you, however, I have not found a combination which did not work with me-- yet. 
 Similar to chili, you just put all the ingredients in a crock pot and let simmer on low for 6-8 hours.
 I would recommend putting in spices like garlic and basil during the last hour of cooking to keep their flavors sharp.
 You want to add the salt and pepper to the soup in the beginning.
 This is an easy recipe which has *so* many combinations. I usually
 make some form of chicken soup, the rest of the ingredients are
 entirely dependent on what I have in the kitchen. You will have to
 play around with my recommenced ingredient list to find the perfect
 combination for you, however, I have not found a combination which did
 not work with me-- yet.  Similar to chili, you just put all the
 ingredients in a crock pot and let simmer on low for 6-8 hours. I
 would recommend putting in spices like garlic and basil during the
 last hour of cooking to keep their flavors sharp. You want to add the
 salt and pepper to the soup in the beginning. 
 ![Chicken Soup with a red sauce](media/cooking/lvl2/redSoup.jpg)
 ### Ingredients
 - Bag of Frozen Chicken Strips
 - 3-4 Diced Potatoes
 - Chicken Stock
@ -135,14 +154,17 @@ You want to add the salt and pepper to the soup in the beginning.
 ![Potato Casserole](media/cooking/lvl2/cheesePotatoes.jpg)
 For this recipe you cut all the vegetables and place them in a large baking pan with the chicken strips.
 If you want, you can also mix in some seasonings like salt, garlic, pepper, and basil. 
 You then then mix everything together with some canola or vegetable oil.
 Cook this casserole in a 350 degrease oven for around 30 minutes.
 Briefly take this out of the oven and top with cheese and ranch dressing and place back in oven for another 10-15 minutes.
 If you are really cheap, you can make this with hot dogs rather than pre-cooked chicken strips. 
 For this recipe you cut all the vegetables and place them in a large
 baking pan with the chicken strips. If you want, you can also mix in
 some seasonings like salt, garlic, pepper, and basil.  You then then
 mix everything together with some canola or vegetable oil. Cook this
 casserole in a 350 degrease oven for around 30 minutes. Briefly take
 this out of the oven and top with cheese and ranch dressing and place
 back in oven for another 10-15 minutes. If you are really cheap, you
 can make this with hot dogs rather than pre-cooked chicken strips.  
 ### Ingredients
 - 4-6 Peeled and Cubed Potatoes
 - 1-2 Diced Onions
 - Bag of Chicken Strips
@ -156,20 +178,25 @@ If you are really cheap, you can make this with hot dogs rather than pre-cooked
 # Level 3 Cooking
 Once you master the art of sauteing onions and making soups, you are ready to advance to level three.
 Nothing I will present is incredibly difficult, but, there is more room for mistakes and takes longer to prepare. 
 Once you master the art of sauteing onions and making soups, you are
 ready to advance to level three. Nothing I will present is incredibly
 difficult, but, there is more room for mistakes and takes longer to
 prepare.  
 ## Fried Rice
 ![Fried Rice Final Result](media/cooking/lvl3/friedRice.jpg)
 The challenging part of this recipe is getting the correct timing.
 To finish this dish in 40 minutes, you want to cook 2 cups of rice, brown ground beef with onions and cook 3 scrambled eggs all simultaneously.
 You then combine all these ingredients in a large fry pan with vegetables and soy sauce.
 If you want to prevent the dish from turning out sticky, you should use left over rice. 
 Or, cook the rice first and then let if cool in the fridge for a few hours.
 The challenging part of this recipe is getting the correct timing. To
 finish this dish in 40 minutes, you want to cook 2 cups of rice, brown
 ground beef with onions and cook 3 scrambled eggs all simultaneously.
 You then combine all these ingredients in a large fry pan with
 vegetables and soy sauce. If you want to prevent the dish from turning
 out sticky, you should use left over rice.  Or, cook the rice first
 and then let if cool in the fridge for a few hours. 
 ### Ingredients
 - 1-2 Cups of Rice
 - 1/2 lbs of Ground Beef
 - 1 Onion
@ -180,17 +207,22 @@ Or, cook the rice first and then let if cool in the fridge for a few hours.
 ![Chicken Right Out of Oven](media/cooking/lvl3/ovenChicken.jpg)
 Cooking raw chicken is definitely not on the syllabus for most college students.
 When cooking the chicken I would strongly recommend using a meat thermometer to make sure that the chicken reaches 165 degrease.
 Either buy chicken breasts or cut a whole chicken-- use the left over chicken spine and wings for chicken stock.
 On a baking pan lather the chicken in the marinara sauce that you just mixed (ingredients below) and sprinkle some salt and pepper on the chicken.
 Let the chicken cook for around 30 minutes in a oven that is 400 degrease (until the chicken reaches 165).
 After the chicken comes out of the oven, sear it on a ripping hot skillet with some more of the marinara sauce.
 Cooking the chicken this way will keep the chicken juicy and the skin crispy. 
 Cooking raw chicken is definitely not on the syllabus for most college
 students. When cooking the chicken I would strongly recommend using a
 meat thermometer to make sure that the chicken reaches 165 degrease.
 Either buy chicken breasts or cut a whole chicken-- use the left over
 chicken spine and wings for chicken stock. On a baking pan lather the
 chicken in the marinara sauce that you just mixed (ingredients below)
 and sprinkle some salt and pepper on the chicken. Let the chicken cook
 for around 30 minutes in a oven that is 400 degrease (until the
 chicken reaches 165). After the chicken comes out of the oven, sear it
 on a ripping hot skillet with some more of the marinara sauce. Cooking
 the chicken this way will keep the chicken juicy and the skin crispy. 
 ![Chicken Searing](media/cooking/lvl3/chickenSear.jpg)
 ### Marinara Sauce Ingredients
 - 4 TBSP Oil
 - 3 TBSP Honey
 - Minced Garlic
@ -201,17 +233,22 @@ Cooking the chicken this way will keep the chicken juicy and the skin crispy.
 ![Ingredients for Stock](media/cooking/lvl3/stockIngredients.jpg)
 If you happen to cook with a whole chicken and end up with some chicken bones, you are in the perfect position to make chicken stock.
 Essentially all you need to do is put your chicken bones, onions, carrots, and basil in a crock pot cook for 8-12 hours on low. 
 Periodically you should skim off the fat that surfaces to the top. 
 At the end of the 12 hours you use a fine mesh skimmer to filter out all the chicken and vegetables.
 If you want a darker flavor to the stock, roast the chicken parts and vegetables in the oven before adding them to the crock pot. 
 If you happen to cook with a whole chicken and end up with some
 chicken bones, you are in the perfect position to make chicken stock.
 Essentially all you need to do is put your chicken bones, onions,
 carrots, and basil in a crock pot cook for 8-12 hours on low. 
 Periodically you should skim off the fat that surfaces to the top.  At
 the end of the 12 hours you use a fine mesh skimmer to filter out all
 the chicken and vegetables. If you want a darker flavor to the stock,
 roast the chicken parts and vegetables in the oven before adding them
 to the crock pot.  
 ![Cooking Chicken Stock](media/cooking/lvl3/cookingBroth.jpg)
 ![Stock Final Result](media/cooking/lvl3/ChickenBroth.jpg)
 ### Ingredients
 - Chicken Bones
 - 1-2 Quartered Onions
 - 2-3 Carrots
@ -226,16 +263,19 @@ If you want a darker flavor to the stock, roast the chicken parts and vegetables
 ![Chicken and Rice Plate Picture](media/cooking/lvl3/chickenAndRice.jpg)
 My version of chicken and rice may not be orthodox. 
 This recipe came into existence because I needed something to make which shared ingredients with the other dishes that I frequently make.
 My version of chicken and rice may not be orthodox.  This recipe came
 into existence because I needed something to make which shared
 ingredients with the other dishes that I frequently make. 
 Essentially, you make rice and put a thick red sauce with chicken, onions, peppers over it.
 You prepare the rice like normal.
 In a large sauce pan you saute onions and peppers.
 Once the onions start to brown, you add diced tomatoes, corn, minced garlic, basil, brown sugar, frozen chicken strips and mustard.
 Once the chicken has thawed, you are ready to serve this dish.
 Essentially, you make rice and put a thick red sauce with chicken,
 onions, peppers over it. You prepare the rice like normal. In a large
 sauce pan you saute onions and peppers. Once the onions start to
 brown, you add diced tomatoes, corn, minced garlic, basil, brown
 sugar, frozen chicken strips and mustard. Once the chicken has thawed,
 you are ready to serve this dish. 
 ### Ingredients
 - 2 Cups Rice
 - Bag of Grilled Chicken Strips
 - 1-2 Onions
@ -245,4 +285,4 @@ Once the chicken has thawed, you are ready to serve this dish.
 - 1 Can Corn
 - Minced Garlic
 - Brown Sugar
 - Basil
 - Basil
--- a/blogContent/posts/programming/cs-theory-exam-2-review.md
+++ b/blogContent/posts/programming/cs-theory-exam-2-review.md
@ -1,19 +1,26 @@
 This a very high level review post that I am making for myself and other people reviewing CS Theory.
 If you want to lean more about content in this blog post I recommend cracking open a text book-- I know, gross.
 This hastily thrown together post will cover how to solve typical problems relating to topics covered by my second CS Theory exam.
 This a very high level review post that I am making for myself and
 other people reviewing CS Theory. If you want to lean more about
 content in this blog post I recommend cracking open a text book-- I
 know, gross. This hastily thrown together post will cover how to solve
 typical problems relating to topics covered by my second CS Theory
 exam. 
 # Myhill-Nerode Theorem
 ## Definition
 L is regular if and only if it has a finite index. The index is the maximum number of elements that are pairwise distinguishable.
 Two strings are said to be pairwise distinguishable if you can append something to both of the strings and it makes one string
 accepted by the language and the other string rejected. 
 The size of an index set X equals the number of equivalence classes it has and the minimum number of states required to represent it
 using a DFA. Each element in the language is accepted by only one equivalence class.
 L is regular if and only if it has a finite index. The index is the
 maximum number of elements that are pairwise distinguishable. Two
 strings are said to be pairwise distinguishable if you can append
 something to both of the strings and it makes one string accepted by
 the language and the other string rejected.  The size of an index set
 X equals the number of equivalence classes it has and the minimum
 number of states required to represent it using a DFA. Each element in
 the language is accepted by only one equivalence class. 
 ## Problem Approach
 Prove that language L is regular.
 Prove that language L is regular. 
 1) Define a set X which is infinite in size - this doesn't have to be in the language.
@ -23,13 +30,13 @@ the other is not in the language.
 ## Example
 Prove the following language is non-regular:
 Prove the following language is non-regular: 
 $$
 L={ww^r | w \in {0,1}^*}
 $$
 answer:
 answer: 
 1)
@ -37,13 +44,14 @@ $$
 X = {(01)^i | i \geq 0}
 $$
 Pick any 2 elements of X and show pairwise distinguishable
 Pick any 2 elements of X and show pairwise distinguishable 
 $$
 x = (01)^i, y = (01)^j  | i \neq j
 $$
 suppose we pick
 suppose we pick 
 $$
 z = (10)^i\\
 xz \in L\\
@ -53,78 +61,91 @@ $$
 # DFA minimization algorithm
 Types of Problems:
 Types of Problems: 
 - Prove DFA is minimal
 - Minimize the DFA
 The argument for DFA minimization comes from the Myhill-Nerode theorem. Given
 a DFA, if you can form a set of strings which represent each state and they are all 
 pairwise distinguishable, then the DFA is minimal with that many states.
 The argument for DFA minimization comes from the Myhill-Nerode
 theorem. Given a DFA, if you can form a set of strings which represent
 each state and they are all  pairwise distinguishable, then the DFA is
 minimal with that many states. 
 ## Prove DFA is minimal
 For these types of problems you construct a table and show that each state is pairwise distinguishable.
 To show pairwise distinguishably you have to show that there exists a string where if appended to one element 
 makes it accepted by the language but pushes the other string out of the language.
 For these types of problems you construct a table and show that each
 state is pairwise distinguishable. To show pairwise distinguishably
 you have to show that there exists a string where if appended to one
 element  makes it accepted by the language but pushes the other string
 out of the language. 
 ### Example
 Prove the following DFA is minimal.
 Prove the following DFA is minimal. 
 ![DFA Example](media/CSTHEORY/DFAMinimalProof.png)
 Find a set of strings which represent the minimal path to each state in the DFA.
 Find a set of strings which represent the minimal path to each state
 in the DFA. 
 $$
 X = \{\epsilon, b, bb, ba\}
 $$
 Show that each state is pairwise distinguishable.
 Show that each state is pairwise distinguishable. 
 ![DFA Example](media/CSTHEORY/DFAMinimalTable.png)
 ## Minimize the DFA
 To use the concept of being indistinguishable to minimize a DFA, you can use a table to keep track which
 states are distinguishable from each other. The states which are not indistinguishable can 
 be combined. To solve one of these problems you start by creating a table which compares each of the 
 states in the DFA. You then go through and mark the states which are indistinguishable -- start with 
 the ones with different accepting statuses. Then you continue marking off states where if you transition with
 a symbol on the DFA you are distinguishable and the other state is non-distinguishable according to the table.
 To use the concept of being indistinguishable to minimize a DFA, you
 can use a table to keep track which states are distinguishable from
 each other. The states which are not indistinguishable can  be
 combined. To solve one of these problems you start by creating a table
 which compares each of the  states in the DFA. You then go through and
 mark the states which are indistinguishable -- start with  the ones
 with different accepting statuses. Then you continue marking off
 states where if you transition with a symbol on the DFA you are
 distinguishable and the other state is non-distinguishable according
 to the table. 
 ### Example
 Minify the Following DFA:
 Minify the Following DFA: 
 ![DFA Example](media/CSTHEORY/DFAMinification.png)
 After marking the states with different accepting criteria as being distinguishable you get this table:
 After marking the states with different accepting criteria as being
 distinguishable you get this table: 
 ![Half Complete Table](media/CSTHEORY/MinificationTable.svg)
 After looping through all pairs and marking them on the table if there exists symbol which results in one state 
 to be distinguishable and one to be indistinguishable you get this table:
 After looping through all pairs and marking them on the table if there
 exists symbol which results in one state  to be distinguishable and
 one to be indistinguishable you get this table: 
 ![Min TableTable](media/CSTHEORY/MinTable2.svg)
 According to the table you are able to combine {D, A, B}, {C, F}, and {E, G}.
 According to the table you are able to combine {D, A, B}, {C, F}, and
 {E, G}. 
 Minimal DFA:
 Minimal DFA: 
 ![Min DFA](media/CSTHEORY/MinimalDFA.svg)
 # Pumping lemma for regular languages
 The pumping lemma cannot prove that a language is regular, however, you can use it
 to show that some languages are non-regular. This theory gets at the idea that if
 a regular language is long enough/infinite, it will have a state somewhere which is
 repeated on the path that accepts the string.
 The pumping lemma cannot prove that a language is regular, however,
 you can use it to show that some languages are non-regular. This
 theory gets at the idea that if a regular language is long
 enough/infinite, it will have a state somewhere which is repeated on
 the path that accepts the string. 
 The accepted strings can be divided into three parts:
 The accepted strings can be divided into three parts: 
 - Symbols leading up to the loop
 - Symbols which complete a loop and come back to start of loop
@ -134,7 +155,7 @@ The accepted strings can be divided into three parts:
 To Show that a language L is not regular using pumping lemma:
 To Show that a language L is not regular using pumping lemma: 
 - Use a proof by contradiction
 - Assume L is regular
@ -148,52 +169,53 @@ To Show that a language L is not regular using pumping lemma:
 ## Example
 Show that the following language is non-regular.
 Show that the following language is non-regular. 
 $$
 {0^n1^n | n \geq 0}
 $$
 Proof by contradiction
 Proof by contradiction 
 Assume that L is regular.
 Assume that L is regular. 
 Let p be the pumping length associated with L
 Let p be the pumping length associated with L 
 $$
 S = o^p1^p
 $$
 S is valid since
 S is valid since 
 $$
 |s| \geq p, S \in L
 $$
 For any valid decomposition
 For any valid decomposition 
 S = xyz
 S = xyz 
 such that |xy| <= p and |y| > 0
 such that |xy| <= p and |y| > 0 
 Consider:
 Consider: 
 $$
 xy^2z
 $$
 By the pumping lemma this should be in the language but it is not. Therefore our assumption that the
 language is regular is false. 
 By the pumping lemma this should be in the language but it is not.
 Therefore our assumption that the language is regular is false.  
 ![Min DFA](media/CSTHEORY/PumpingLemmaExample.svg)
 # Context-free grammars, closure properties for CFLs
 The context-free grammars are a super-set of the regular languages. This means that CFG's can represent 
 some non-regular languages and every regular language. Contest-free Languages are defined by Context-Free Grammars and accepted using
 Push-down Automata machines.
 The context-free grammars are a super-set of the regular languages.
 This means that CFG's can represent  some non-regular languages and
 every regular language. Contest-free Languages are defined by
 Context-Free Grammars and accepted using Push-down Automata machines. 
 Context Free Grammars are Represented using:
 Context Free Grammars are Represented using: 
 - **Terminals** = Set of symbols in that language
 - **Variables** = Set of symbols representing categories
@ -202,7 +224,7 @@ Context Free Grammars are Represented using:
 ## Example 1
 Grammar G:
 Grammar G: 
 $$
 A \rightarrow 0A1 \\
@ -211,7 +233,7 @@ B \rightarrow \# \\
 $$
 This grammar describes the following language:
 This grammar describes the following language: 
 $$
 L = \{0^k\#1^k | k \geq 0\}
@ -219,7 +241,7 @@ $$
 ## Example 2
 Give CFG for non-Palindromes
 Give CFG for non-Palindromes 
 $$
 S \rightarrow aXb | bXa | aSa | bSb | ab | ba \\
@ -227,13 +249,14 @@ X \rightarrow aX | bX | a | b \\
 $$
 In this example, the S rule recursively applies itself until something that is not a palindrome is added.
 Once you exit the S state, you can finish by appending anything to the middle of the string. 
 In this example, the S rule recursively applies itself until something
 that is not a palindrome is added. Once you exit the S state, you can
 finish by appending anything to the middle of the string.  
 ## Example 3
 Give CFG for the following language:
 Give CFG for the following language: 
 $$
 \{a^ib^jc^kd^l | i+k = j + l\}
@ -248,28 +271,32 @@ $$
 ## Closure
 CFLs are closed under union, concatenation, and Kleene star.
 CFLs are closed under union, concatenation, and Kleene star. 
 - Union: Create a new starting variable which goes to either branch.
 -Kleene Star: We can repeatedly concat the derivations of the string. However, we also need to make sure that epsilon occurs in the string. 
 - Concatenation: From start variable we force the concatenation of two variables representing each state sub CFG
 # Parse trees, ambiguity
 Parse Trees are simply graphical means to illustrate a deviation of a string from a grammar.
 The root of the tree will be the start variable, interior nodes are other variables. Leaf nodes are 
 terminal symbols. 
 Parse Trees are simply graphical means to illustrate a deviation of a
 string from a grammar. The root of the tree will be the start
 variable, interior nodes are other variables. Leaf nodes are  terminal
 symbols.  
 A CFG is said to be ambiguous if there is at least one string with two or more distinct derivations.
 Leftmost and rightmost derivations are not ambiguous. 
 A CFG is said to be ambiguous if there is at least one string with two
 or more distinct derivations. Leftmost and rightmost derivations are
 not ambiguous.  
 To remove ambiguity try to force order or break CFG into cases.
 To remove ambiguity try to force order or break CFG into cases. 
 # Chomsky Normal Form
 Useful form for CFGs since they allow you to easily identify if a string is in a language.
 Useful form for CFGs since they allow you to easily identify if a
 string is in a language. 
 Form:
 Form: 
 $$
 A \rightarrow BC\\
@ -277,7 +304,7 @@ A \rightarrow a\\
 S \rightarrow \epsilon
 $$
 Convert CFG to CNF (Chomsky Normal Form).   
 Convert CFG to CNF (Chomsky Normal Form).    
 - Add new start variable   
@ -289,7 +316,7 @@ Convert CFG to CNF (Chomsky Normal Form).
 ## Example
 Convert the following CFG to a CNF
 Convert the following CFG to a CNF 
 $$
 S \rightarrow ASA | aB\\
@ -297,7 +324,7 @@ A \rightarrow B | S\\
 B \rightarrow b | \epsilon
 $$
 Step 1: Add new start variable.
 Step 1: Add new start variable. 
 $$
 S_0 \rightarrow S\\
@ -306,7 +333,7 @@ A \rightarrow B | S\\
 B \rightarrow b | \epsilon
 $$
 Step 2: Remove epsilon rules.
 Step 2: Remove epsilon rules. 
 $$
 S_0 \rightarrow S\\
@ -323,7 +350,7 @@ A \rightarrow B | S\\
 B \rightarrow b
 $$
 Step 3: Remove unit rules
 Step 3: Remove unit rules 
 (Remove A -> B)
@ -354,12 +381,14 @@ $$
 # Push-Down automata
 These are NFAs but they also have a stack. This allows us to solve any problem which
 can be represented with a CFG.
 These are NFAs but they also have a stack. This allows us to solve any
 problem which can be represented with a CFG. 
 The stack has it's own alphabet. The dollar symbol typically represents the empty stack.
 The stack has it's own alphabet. The dollar symbol typically
 represents the empty stack. 
 With each transition you can examine the stack, push to the stack and move to a new state.
 With each transition you can examine the stack, push to the stack and
 move to a new state. 
 ## Example
@ -373,16 +402,18 @@ $$
 # Construction to convert CFG to a PDA
 Basic idea: use stack to hold progressive derivations of a string using rules of grammar.
 Basic idea: use stack to hold progressive derivations of a string
 using rules of grammar. 
 ## Example
 Convert the following CFG to a PDA:
 Convert the following CFG to a PDA: 
 $$
 S \rightarrow aTb | b \\
 T \rightarrow Ta | \epsilon
 $$
 ![](media/CSTHEORY/PDAConstruction.svg)
--- a/blogContent/posts/programming/everything-fibonacci.md
+++ b/blogContent/posts/programming/everything-fibonacci.md
@ -1,19 +1,19 @@
 If you have ever taken a computer science class you probably
 know what the fibonacci sequence is and how to calculate it.
 For those who don't know: [Fibonacci](https://en.wikipedia.org/wiki/Fibonacci)
 is a sequence of numbers starting with 0,1 whose next number is the sum
 of the two previous numbers. After having multiple of my CS classes
 give lectures and homeworks on the Fibonacci sequence; I decided
 to write a blog post going over
 the 4 main ways of calculating the nth term of the Fibonacci sequence.
 In addition to providing the python code for calculating the nth perm of the sequence, a proof for their validity
 and an analysis of their time complexities both mathematically and empirically will
 be examined.
 If you have ever taken a computer science class you probably know what
 the fibonacci sequence is and how to calculate it. For those who don't
 know: [Fibonacci](https://en.wikipedia.org/wiki/Fibonacci) is a
 sequence of numbers starting with 0,1 whose next number is the sum of
 the two previous numbers. After having multiple of my CS classes give
 lectures and homeworks on the Fibonacci sequence; I decided  to write
 a blog post going over the 4 main ways of calculating the nth term of
 the Fibonacci sequence. In addition to providing the python code for
 calculating the nth perm of the sequence, a proof for their validity
 and an analysis of their time complexities both mathematically and
 empirically will be examined. 
 # Slow Recursive Definition
 By the definition of the Fibonacci sequence, it is the most natural to write it as
 a recursive definition.
 By the definition of the Fibonacci sequence, it is the most natural to
 write it as a recursive definition. 
 ```Python
 def fib(n):
@ -24,17 +24,19 @@ def fib(n):
 ##Time Complexity
 Observing that each call has two recursive calls we can place an upper bound on this 
 function as $O(2^n)$. However, if we solve this recurrence we can place a tight bound for time complexity.
 Observing that each call has two recursive calls we can place an upper
 bound on this  function as $O(2^n)$. However, if we solve this
 recurrence we can place a tight bound for time complexity. 
 We can write a recurrence for the number of times fib is called:
 We can write a recurrence for the number of times fib is called: 
 $$
    F(1) = 1\\
    F(n) = F(n-1) + F(n-2)\\
 $$
 Next, we replace F(n) with $a^n$ since we want to find rate of exponential growth.
 Next, we replace F(n) with $a^n$ since we want to find rate of
 exponential growth. 
 $$
    a^n = a^{n-1} + a^{n-2}\\
@ -43,25 +45,28 @@ $$
    a = \frac{1 \pm sqrt(5)}{2}\\
 $$
 From this calculation we can conclude that F(n) $\in \Theta 1.681^n$. We don't have to worry about 
 the negative root since it would not be asymptotically relevant by the definition of $\Theta$. 
 From this calculation we can conclude that F(n) $\in \Theta 1.681^n$.
 We don't have to worry about  the negative root since it would not be
 asymptotically relevant by the definition of $\Theta$.  
 ## Measured Performance
 Here is a graph of the actual performance that I observed from this recursive definition of Fibonacci.
 Here is a graph of the actual performance that I observed from this
 recursive definition of Fibonacci. 
 ![Recursive Definition](media/fibonacci/RecursiveDefinition.png)
 # Accumulation Solution
 The problem with the previous recursive solution is that you had to recalculate certain 
 terms of fibonacci a ton of times. A summation variable would help us avoid this problem.
 You could write this solution using a simple loop or dynamic programming
 , however, I chose to use recursion to demonstrate that it's recursion which made the first
 problem slow. 
 The problem with the previous recursive solution is that you had to
 recalculate certain  terms of fibonacci a ton of times. A summation
 variable would help us avoid this problem. You could write this
 solution using a simple loop or dynamic programming , however, I chose
 to use recursion to demonstrate that it's recursion which made the
 first problem slow.  
 ```Python
@ -70,23 +75,26 @@ def fibHelper(n, a, b):
        return a
    elif n == 1:
        return b
    return fibHelper(n-1, b, a+b)
    return fibHelper(n-1, b, a+b)        
 ```
 ```python
 def fibIterative(n):
    return fibHelper(n, 0, 1)
 ```
 In this code example, fibHelper is a method which accumulates the previous two terms. 
 The fibIterative is a wrapper method which sets the two initial terms equal to 0 and 1 
 representing the fibonacci sequence. At first it may not be obvious that fibIterative(n)
 is equivalent to fib(n). To demonstrate that these two are in fact equivalent, I broke this 
 into two inductive proofs. 
 In this code example, fibHelper is a method which accumulates the
 previous two terms.  The fibIterative is a wrapper method which sets
 the two initial terms equal to 0 and 1  representing the fibonacci
 sequence. At first it may not be obvious that fibIterative(n) is
 equivalent to fib(n). To demonstrate that these two are in fact
 equivalent, I broke this  into two inductive proofs.  
 ## Proof for Fib Helper
 **Lemma:** For any n $\epsilon$ N if n $>$ 1 then 
           $fibHelper(n, a, b) = fibHelper(n - 1, a, b) + fibHelper(n - 2, a, b)$.
 **Proof via Induction**
 **Base Case**: n = 2:
@ -99,7 +107,7 @@ $$
 **Inductive Step:**
 Assume proposition is true for all n and show n+1 follows.
 Assume proposition is true for all n and show n+1 follows. 
 $$
    RHS=fibHelper(n+1;a,b)\\
@ -114,7 +122,7 @@ $\Box$
 ## Proof That fibIterative = Fib
 **Lemma:** For any n $\in$ N, $fib(n)$ = $fibIterative(n, 0, 1)$
 **Proof via Strong Induction**
 **Base Case**: n = 0:
@ -131,7 +139,7 @@ $$
 **Inductive Step:**
 Assume proposition is true for all n and show n+1 follows.
 Assume proposition is true for all n and show n+1 follows. 
 $$
    fib(n+1) = fib(n) + fib(n-1)\\
@ -144,9 +152,10 @@ $\Box$
 ## Time Complexity
 Suppose that we wish to solve for the time complexity in terms of the number of additions needed to be
 computed. Based on fibHelper we can see that it performs one addition every recursive call. 
 We can now form a recurrence for time complexity. 
 Suppose that we wish to solve for the time complexity in terms of the
 number of additions needed to be computed. Based on fibHelper we can
 see that it performs one addition every recursive call.  We can now
 form a recurrence for time complexity.  
 $$
    T(0) = 0\\
@ -155,12 +164,12 @@ $$
    T(n) = n-1\\
 $$
 From this recurrence we can say that fibHelper $\in \Theta(n)$.
 From this recurrence we can say that fibHelper $\in \Theta(n)$. 
 ## Measured Performance
 Notice how much faster this solution is compared to the original recursive solution for
 Fibonacci.
 Notice how much faster this solution is compared to the original
 recursive solution for Fibonacci. 
 ![Iterative Performance](media/fibonacci/Iterative.png)
@ -168,8 +177,9 @@ Fibonacci.
 # Matrix Solution
 We can actually get better than linear for performance for Fibonacci while still using
 recursion. However, to do so we need to know this fact:
 We can actually get better than linear for performance for Fibonacci
 while still using recursion. However, to do so we need to know this
 fact: 
 $$  
 \begin{bmatrix}
@ -183,10 +193,12 @@ F_n & F{n-1}
 $$
 Without any tricks, raising a matrix to a power n times would not get
 us better than linear performance. However, if we use the [Exponentiation by Squaring](https://en.wikipedia.org/wiki/Exponentiation_by_squaring)
 method, we can expect to see logarithmic time. Since two spots in the matrix are always equal,
 I represented the matrix as an array with only three elements to reduce the space and 
 computations required. 
 us better than linear performance. However, if we use the
 [Exponentiation by
 Squaring](https://en.wikipedia.org/wiki/Exponentiation_by_squaring)
 method, we can expect to see logarithmic time. Since two spots in the
 matrix are always equal, I represented the matrix as an array with
 only three elements to reduce the space and  computations required.  
 ```Python
@ -196,8 +208,9 @@ def multiply(a,b):
    product[1] = a[0]*b[1] + a[1]*b[2]
    product[2] = a[1]*b[1] + a[2]*b[2]
    return product
 ```
 ```python
 def power(l, k):
    if k == 1:
        return l
@ -206,8 +219,9 @@ def power(l, k):
        return multiply(temp, temp)
    else:
        return multiply(l, multiply(temp, temp))
 ```
 ```python          
 def fibPower(n):
    l = [1,1,0]
    return power(l, n)[1]
@ -216,15 +230,19 @@ def fibPower(n):
 ## Time Complexity
 For this algorithm, lets solve for the time complexity as the number of additions and multiplications required.
 For this algorithm, lets solve for the time complexity as the number
 of additions and multiplications required. 
 Since we are always multiplying two 2x2 matrices, that operation is constant time.
 Since we are always multiplying two 2x2 matrices, that operation is
 constant time. 
 $$
    T_{multiply} = 9
 $$
 Solving for the time complexity of fib power is slightly more complicated.
 Solving for the time complexity of fib power is slightly more
 complicated. 
 $$
    T_{power}(1) = 0\\
    T_{power}(n) = T(\left\lfloor\dfrac{n}{2}\right\rfloor) + T_{multiply}\\
@ -234,7 +252,8 @@ $$
    T_{power}(n) =  T(\left\lfloor\dfrac{n}{2^k}\right\rfloor) + 9k\\
 $$
 let $k=k_0$ such that $\left\lfloor\dfrac{n}{2^{k_0}}\right\rfloor = 1$
 let $k=k_0$ such that $\left\lfloor\dfrac{n}{2^{k_0}}\right\rfloor =
 1$ 
 $$    
    \left\lfloor\dfrac{n}{2^{k_0}}\right\rfloor = 1 \rightarrow 1 \leq \frac{n}{2^{k_0}} < 2\\
@ -246,13 +265,14 @@ $$
    T_{fibPower}(n) = T_{power}(n)\\
 $$
 Now we can state that $fibPower(n) \in \Theta(log(n))$.
 Now we can state that $fibPower(n) \in \Theta(log(n))$. 
 ## Inductive Proof for Matrix Method
 I would like to now prove that this matrix identity is valid since it is not at first obvious.
 I would like to now prove that this matrix identity is valid since it
 is not at first obvious. 
 **Lemma:** For any n $\epsilon$ N if n $>$ 0 then 
           $$  
@ -266,7 +286,7 @@ I would like to now prove that this matrix identity is valid since it is not at
           \end{bmatrix}^n
           $$
 Let
 Let 
 $$
 A=
@ -317,26 +337,29 @@ $\Box$
 ![FibPower Performance](media/fibonacci/FibPower.png)
 As expected by our mathematical calculations, the algorithm appears to be running in
 logarithmic time. 
 As expected by our mathematical calculations, the algorithm appears to
 be running in logarithmic time.  
 ## Measured Performance With Large Numbers
 ![FibPower Performance](media/fibonacci/FibPowerBigPicture.png)
 When calculating the fibonacci term for extremely large numbers despite having a polynomial
 time complexity, the space required to compute each Fibonacci term grows exponentially. Since our
 performance is only pseudo-polynomial we see a degrade in our performance when calculating 
 large terms of the fibonacci sequence.
 When calculating the fibonacci term for extremely large numbers
 despite having a polynomial time complexity, the space required to
 compute each Fibonacci term grows exponentially. Since our performance
 is only pseudo-polynomial we see a degrade in our performance when
 calculating  large terms of the fibonacci sequence. 
 The one amazing thing to point out here is that despite calculating the 10,000 term of Fibonacci,
 this algorithm is nearly 400 times faster than the recursive algorithm when calculating 
 the 30th term of Fibonacci.
 The one amazing thing to point out here is that despite calculating
 the 10,000 term of Fibonacci, this algorithm is nearly 400 times
 faster than the recursive algorithm when calculating  the 30th term of
 Fibonacci. 
 # Closed Form Definition
 It is actually possible to calculate Fibonacci in constant time using Binet's Formula.
 It is actually possible to calculate Fibonacci in constant time using
 Binet's Formula. 
 $$
    F_n = \frac{(\frac{1+\sqrt{5}}{2})^n-(\frac{1-\sqrt{5}}{2})^n}{\sqrt{5}}
@ -351,8 +374,9 @@ def fibClosedFormula(n):
 ## Derivation of Binet's Formula
 Similar to when we were calculating the time complexity of the basic recursive definition
 , we want to start by finding the two roots of the equation in terms of exponents.
 Similar to when we were calculating the time complexity of the basic
 recursive definition , we want to start by finding the two roots of
 the equation in terms of exponents. 
 $$
    a^n = a^{n-1} + a^{n-2}\\
@ -362,15 +386,15 @@ $$
    a = \frac{1 \pm sqrt(5)}{2}\\
 $$
 Since there are two roots to the equation, the solution of $F_n$ is going to be 
 a linear combination of the two roots.
 Since there are two roots to the equation, the solution of $F_n$ is
 going to be  a linear combination of the two roots. 
 $$
    F_n = c_1(\frac{1 + \sqrt{5}}{2})^n + c_2(\frac{1 - \sqrt{5}}{2})^n
 $$
 Fact: $F_1$ = 1
 Fact: $F_1$ = 1 
 $$
    F_1 = 1\\
@ -378,8 +402,7 @@ $$
    = \frac{c_1}{2} + \frac{c_2}{2} + \frac{c_1\sqrt{5}}{2} - \frac{c_2\sqrt{5}}{2}\\
 $$
 Let $c_1 = \frac{1}{\sqrt{5}}$,
 Let $c_2 = \frac{-1}{\sqrt{5}}$
 Let $c_1 = \frac{1}{\sqrt{5}}$, Let $c_2 = \frac{-1}{\sqrt{5}}$ 
 $$
    F_n = \frac{1}{\sqrt(5)}((\frac{1+\sqrt{5}}{2})^n-(\frac{1-\sqrt{5}}{2})^n)\\
@ -388,8 +411,9 @@ $$
 ## Time Complexity
 Since we managed to find the closed form of the fibonacci sequence we can expect to see constant performance.
 Since we managed to find the closed form of the fibonacci sequence we
 can expect to see constant performance. 
 ## Measured Performance
 ![FibPower Performance](media/fibonacci/ConstantTimeComplexity.png)
 ![FibPower Performance](media/fibonacci/ConstantTimeComplexity.png)