Finding a function's inverse with Wolfram|Alpha

To check my answers when calculating the inverse of functions, I've found this handy widget from Wolfram|Alpha which does the job:

A Function Transformations reference sheet

Since I never miss a change to procrastinate, I decided to spend some time compiling a "cheat sheet" for function transformations.

I've included the following 8 transformations:

1. A constant added to or subtracted from a function shifts its graph vertically.
2. A constant added to or subtracted from a function's input shifts its graph horizontally.
   
3. Multiplying a function by a constant stretches or squishes the function vertically.
4. Multiplying a function's input by a constant stretches or squishes the function horizontally.
   
5. Multiplying a function by a negative number causes its graph to reflect over the x-axis.
6. Multiplying a function's input by a negative number causes its graph to reflect over the y-axis.
   
7. Taking the absolute value of a function moves all the points on its graph above the x-axis.
8. Taking the absolute value of a function's input causes the left side of the graph to clone the right side.

A State Manager in JavaScript

The following is an implementation for a state manager written in JavaScript. More than one state can be "on" at one moment but mutual exclusivity rules can be set up to allow certain states to not be "on" at the same time i.e. if one of the states is switched on while the other is already on, the latter will be switched off since they can't be both on at the same time.

It also supports two hooks which can be used if you want to be notified when states are changed.

It's implemented with a fluent interface to allow for method-chaining.

The source is here: https://github.com/dreasgrech/statemanager-js

I've set up a simple demo to show how it can be used. You can view it here: http://dreasgrech.com/upload/statemanager-js/demo.html

Basic Usage

var state = stateManager();
state.turnOn("state 1", "state 2");

state.on("state 1"); // true
state.on("state 2"); // true
state.on("state 3"); // false

state.toggle("state 2"); // "state 2" is now off
state.on("state 2"); // false

toggle returns the new state after it's been toggled.

Mutual Exclusivity

var state = stateManager();

state.mutex("state 1", "state 3"); // "state 1" and "state 2" now cannot be both on at the same time
state.mutex("state 2", "state 3"); // "state 2" and "state 3" now cannot be both on at the same time

state.turnOn("state 1");
state.turnOn("state 3"); // "state 1" will now be turned off since 1 and 3 are mutually exclusive
state.turnOn("state 1"); // turning "state 1" on, but switching "state 3" off for the same reason
state.turnOn("state 2"); // if "state 3" was on at this point, it would have been switched off

Hooks

var state = stateManager();

state.onStateChanged(function (state, value) { // The global event; fires every time a state changes
  console.log(state + " is now " + value);
});

state.onStateChanged("state 1", function (value) { // The event that will only fire when "state 1" changes
  console.log("[Custom hook] state 1 has been changed.  New value: " + value); 
});

state.turnOn("state 1");
state.turnOn("state 3", "state 2"); 
state.toggle("state 1"); 
state.turnOn("state 2"); 

Output:

state 1 is now 1
[Custom hook] state 1 has been changed. New value: 1
state 3 is now 1
state 2 is now 1
state 1 is now 0
[Custom hook] state 1 has been changed. New value: 0

Note that the onStateChanged event fires only when a state changes. For example, if you turnOn("state 2") while "state 2" is already on, the event will not fire. The same goes for when a state is off and you try to turnOff.

Function composition in JavaScript

Function composition is the application of a result of a function as input to another function. Say we have the following three functions:

Composing functions f, g and h respectively means that we pass the output from function h as the input to function g and then passing the output from g as the input to function f. Mathematically, it would be represented as follows:

That's essentially the same as:

---

Functions in JavaScript are first class citizens. This means that they can be stored in variables and data structures, passed as an argument to functions and also returned from functions. This means that we can easily implement function composition in JavaScript with a very trivial function:

Function.prototype.compose || (Function.prototype.compose = function (f) {
    var g = this;
    return function () {
        return g(f.apply(null, arguments));
    };
});

Here's an example of how we can use it by composing three functions that represent the three polynomials mentioned at the start of the post:

var f = function (x) {
    return x*x - x + 5;
}, g = function (x) {
    return x*x + 3*x - 2;
}, h = function (x) {
    return x*x*x - 2*(x*x) + x -1;
};

Composing them together would then be:

var composed = f.compose(g).compose(h);

The result of the composed function is now the same as the result of f(g(h(x))):

composed(x) === f(g(h(x))); // true

---

If we had a function f and its inverse f^-1 and we compose them, we should get the original input x:

var f = function (x) {
    return (x + 2) / 3 - x;
}, f_inverse = function (x) {
    return (2 - 3*x) / 2;
};

(f.compose(f_inverse)(x) || f_inverse.compose(f)(x) || x) === x;  // true

JavaScript helpers: Iterating an array while skipping select elements

 
Array.prototype.iterate || (Array.prototype.iterate = function (callback, ignore) {
  var i = 0, j = this.length;
    for (; i < j; ++i) {
        if (ignore && ignore(this[i], i)) {
          continue;
        } 
        
        callback(this[i], i);
    }
});

Usage

Here's how we can use it to iterate over a couple of numbers, skipping over the even ones:

 
var arr = [1, 2, 3, 4, 5, 6, 7, 8, 9, 10];

arr.iterate(function (el) {
  console.log(el);
}, function (el) {
  return !(el % 2);
});

Output:

1
3
5
7
9

Here's another example, this time involving objects:

 
var queen = function (reference) {
    return {
        ref: reference
    };
}, arr = [queen("Aragon"), queen("Boleyn"), queen("Seymour"), queen("Cleves"), queen("Howard"), queen("Parr")];

arr.iterate(function (el, i) {
  console.log(i, el.ref);  
}, function (q) {
  return q.ref === "Boleyn"  || q.ref === "Howard";
});

Output (the queens who died with their heads on):

0 Aragon
2 Seymour
3 Cleves
5 Parr

If no ignore function is passed as the second parameter of the iterate method, no elements are skipped:

 
var arr = ["Mr. Albert Show", "Fusion Orchestra", "Fuzzy Duck", "Minotaurus"];

arr.iterate(function (el, i) {
  console.log(el);  
});

Output

Mr. Albert Show
Fusion Orchestra
Fuzzy Duck
Minotaurus

Finding a YouTube video title by scraping

I was recently writing a tool in C# and I needed a way to get a YouTube video title. I could have used the YouTube API but I wasn't bothered about setting everything up and going through the documentation, so I decided to scrape the title off the video page itself.

For this, I first wrote a very simple way to execute a GET request and return the response (Removed all forms of error checking for brevity):

public static string ExecuteGETRequest(string url)
{
    var request = (HttpWebRequest)WebRequest.Create(url);

    using (var response = (HttpWebResponse)request.GetResponse())
    {
        var reader = new StreamReader(response.GetResponseStream());

        return reader.ReadToEnd();
    }
}

And here's the method that does the task at hand:

public static string GetYouTubeVideoTitle(string youtubeLinkUrl)
{
    string response = ExecuteGETRequest(youtubeLinkUrl),
             title = response.Substring(response.IndexOf("<title>\n") + 8);

    title = title.Substring(0, title.IndexOf("\n"));
    return title.Trim();
}

Usage

var title = GetYouTubeVideoTitle("http://www.youtube.com/watch?v=TY9jN6hm3N0"); // "Wicked Lady - Ship of Ghosts (Part 1 of 2) 1972"

Be wary of using this method for extracting titles because it could potentially break in the future due to changes in how YouTube renders the page with HTML; but it may be useful for quick scripts.

Checking for link validity

If you want to make sure the given link actually points to a YouTube video, you can use something like this:

public static bool IsValidYoutubeLink(string youtubeLink)
{
    // Regular expression from http://www.regexlib.com/REDetails.aspx?regexp_id=2569
    return Regex.IsMatch(youtubeLink, @"^http://\w{0,3}.?youtube+\.\w{2,3}/watch\?v=[\w-]{11}");
}

Generating a list of links for a user's YouTube uploads

This is a very simple application I just wrote to generate a list of all of the video upload links given a YouTube user.

Download

Since this was such a small project, I didn't see fit to have a github repository for it so I just uploaded it to my hosting server.

Download the exe (release) file from: http://dreasgrech.com/upload/youtubeuseruploads/YouTubeUserUploads.exe

And the source: http://dreasgrech.com/upload/youtubeuseruploads/YouTubeUserUploads-source.zip
It won't be the best code you'll ever see, but it does the job for the task at hand.

Usage

YouTubeUserUploads.exe <username> [file]

If [file] is not given, a file links.txt will be created in the current directory.

Demo

Here's an example run using my YouTube username:

C:\YouTubeUserUploads.exe agnt666 uploads.txt
Username: agnt666
Saving links to: uploads.txt

Processing...
Finished writing 3 links

That would create (or overwrite) a text file called links.txt and write the links to my three currently uploaded videos at the time of writing:

http://www.youtube.com/watch?v=rsXnEpWyeEg
http://www.youtube.com/watch?v=sxBiD7u7nx8
http://www.youtube.com/watch?v=Ri2SHGZaSqs

Screenshots