Discrete Maths, Lecture-15 : Generating Functions

These are my notes from 15th lecture of discrete mathematics course taught at Arsdigita university by Shai Simonson. Other posts relating to this are here.

This lecture covers Generating Functions(a way to represent sequences), their usage in solving some counting problems and recurrance equations.

Here are my notes...

Discrete Maths, Lecture-14 - Discrete Probability

These are my notes from 14th lecture of discrete mathematics course taught at Arsdigita university by Shai Simonson. Other posts relating to this are here.

This lecture introduces basic discrete probability, mainly the two things..

1. P(A) = #of favorable cases where A happen / total # of cases

2. Probability of A, given B has happened

P(A|B) = P(A /\ B) / P(B)

Here are my notes...

Discrete Maths, ProblemSet#5

These are my solutions to 5th problem set of discrete mathematics course taught at Arsdigita university by Shai Simonson. Other posts relating to this are here.

Doing this problem set makes me realize that One needs to continually practice solving counting problems to learn permutation-combination properly by heart, it really becomes confusing pretty quickly for not-so-trivial cases.

Anyway, Here are my solutions...

anonymous class in scala

Scala also supports java style syntax of defining anonymous classes. Take a look at this code snippet from Actor companion object definition in Actor.scala from scala actors library.

  /* Notice the argument of actor method, its a
   * no-arg(not empty arg list)
   * function that returns Unit */
  def actor(body: => Unit): Actor = {
    /* Here java style syntax of anonymous classes is being used.
     * we're creating an instance not of Actor class but a anonymous
     * subclass that is implementing a method act() which is declared
     * in trait Reactor(also its evident that its not mandatory to use
     * keyword override when implementing something from the trait)
     * and overriding scheduler. */
    val a = new Actor {
      def act() = body
      override final val scheduler: IScheduler = parentScheduler
    }
    /* also see that actor is started right here. */
    a.start()
    a
  }

type param in method name in scala

Type parameters can be specified with methods also like in the example below...
scala> def A[a](b:a){ println(b) }
A: [a](a)Unit

scala> A[String]("hi")
hi
scala> A[String](2)
:6: error: type mismatch;
found : Int(2)
required: String
A[String](2)
^
They're being used in receive method in Actor class, lets take a look at the signature of receive
def receive[R](f: PartialFunction[Any, R]):R
The advantage is that we automatically get the type checking done to be sure that return type of receive is gonna be same as the second type parameter of the PartialFunction.

Discrete Maths, Lecture-13 : Counting-III

These are my notes from 13th lecture of discrete mathematics course taught at Arsdigita university by Shai Simonson. Other posts relating to this are here.

This lecture covers following topics..
1. Inclusion-Exclusion principle (a generalization of sum rule using same theorem from set theory)
2. Pigeonhole Principle
3. Derrangement Problem

Here are my notes...

Discrete Maths, Lecture-12 : Counting-II

These are my notes from 12th lecture of discrete mathematics course taught at Arsdigita university by Shai Simonson. Other posts relating to this are here.

This lecture covers generalized permutation and combinations (that basically means repetition is allowed). Though there are "formulae" presented for each case but more emphasis is(and should be) given to how we can think and solve the problems of generalized permutation and combinations using the already covered 5 principles and simple permutation/combination. The intention of this lecture is not to give you the formula but to teach you how you reach to them.

Here are my notes...