A mathematical model is the representation of a relationship or state or phenomenon in a mathematical form using control variables.
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12 - 6 cloud yes or no?
Mathematical logic is a branch of mathematics which brings together formal logic and mathematics. Mathematical logic entails formal systems for defining the basics and then using the deductive power of logic to develop a system of formal proofs.
Probability means - The likelihood that a particular event will occur.
Interpolation in general is a way to determine intermediate values from a set of coordinates. Linear interpolation would be to fit a single linear function to the entire set of coordinates. Piecewise linear interpolation would then be to determine intermediate values from the set of coordinates by fitting linear functions between each set of coordinates. Connect-the-dots so to speak.
There are no units given so it is not possible to determine whether the linear expansion is in response to heat, strain or some other force.