Scalar multiplication is the process of multiplying a scalar to an element in a set.

constituents

• A set \(V\)
• Some \(\lambda \in \mathbb{F}\)
• Each \(v \in V\)

requirements

scalar multiplication is defined by a function that results in \(\lambda v \in V\) (maps back to the space!) to each \(\lambda \in \mathbb{F}\) and each \(v \in V\).

additional information

See also scalar multiplication in \(\mathbb{F}^n\).

scheduling is the tool to figure out which thread can run. Because threads exist in different thread states:

1. running
2. blockde - waiting for an event like disk, network, etc.
3. ready - able to run, but not on CPU yet

a scheduler needs to do the task of ordering ready threads to run, moving running threads to ready when it has ran for enough time. Possible pathways:

1. ready => running
2. blocked => running
3. blocked => ready => running

You can’t go from ready to blocked because you have to do something to know you are blocked.

The second moment of area is a value which—given an origin—describes how point masses are distributed around that origin. (i.e. a number for how point masses are distributed). It is in units \(m^{4}\).

Take, for instance, the following picture:

We have defined an origin at \((0,0)\) of the figure above. Furthermore, we have some \(\rho_{i}\) which is the distance from that origin to each of the infinitesimal areas \(\dd{A}\).

Then, the second moment of area is defined as:

the trick

Here is a pretty ubiquitous trick to solve differential equations of the second order differential equations. It is used to change a second order differential equation to a First-Order Differential Equations.

If you have a differential equation of the shape:

\begin{equation} x^{’’} = f(x,x’) \end{equation}

that, the second derivative is strictly a function between the first derivative value and the current value.

We are going to define a notation \(x’ = v\), which makes sense.