Appendix A: Mathematical Preliminaries

Some students with limited mathematical background may benefit from a brief introduction to some common mathematical notation frequently used in this text.

An efficient way to write a sum of many variables is with the sigma notation [latex]\sum[/latex]. Consider adding up the values [latex]x_{1} + x_{2} + x_{3} + x_{4} + x_{5}[/latex]. This can be compactly represented as:

[latex]\sum_{k=1}^{5} x_{k} = x_{1} + x_{2} + x_{3} + x_{4} + x_{5}[/latex]

Similarly, if we want to represent a product of many variables, we can use the following notation:

[latex]\prod_{k=1}^{n} x_k = x_{1} \times x_{2} \times ... \times x_{n}[/latex]

An example of such a product function is “factorial”, [latex]n![/latex]. This quantity is the produce of the integers from [latex]1[/latex] to [latex]n[/latex], and represents the number of possible arrangements of [latex]n[/latex] distinct objects.

[latex]{n! = \prod_{k=1}^{n} k}[/latex]

Logarithms are an important function to know in bioinformatics as they are commonly used in scoring systems. Logarithms are powerful because they have the following useful algebraic property (among others):

[latex]\log( A B ) = \log(A) + \log(B)[/latex]

We can combine the properties of our sums and products and logs, with the following equation:

[latex]\log \left( \prod_{k=1}^{n} x_{k} \right) = \sum_{k=1}^{n} \log \left( x_{k} \right) .[/latex]

Another very useful mathematical construct is the Kronecker Delta function, and can be used for counting. It is defined as follows:

[latex]\begin{aligned} \delta_{a,b} = \begin{cases} 1 & \mbox{if } a = b\\ 0 & \mbox{if } a \ne b\\ \end{cases} \end{aligned} \label{indicatorFunction}[/latex]