an integer (\(\mathbb{Z}\)) is the natural numbers, zero, and negative numbers: …,-4,-3,-2,-1,0,1,2,2,3

representing integers

• what are the limitations of computational arithmetic
• how to perform efficient arithmetic
• how to encode data more compactly and efficiently

See also computer number system

The integrating factor \(\rho(x)\) is a value that helps undo the product rule. For which:

\begin{equation} log(\rho(x)) = \int P(x)dx \end{equation}

for some function \(P(x)\).

Separating the \(\rho(x)\) out, we have therefore:

\begin{equation} e^{\int P dx} = \rho(x) \end{equation}

Why is this helpful and undoes the product rule? This is because of a very interesting property of how \(\rho(x)\) behaves.

Goal

We are going to solve the inter-temporal choice problem, for ten time stamps, and perform some numerical optimization of the results

Main Methods

We do this by solving backwards. We will create a variable \(k\) to measure asset, and \(k_{t}\) the remaining asset at time \(t\).

Let us first declare the function for power utility. \(k\) is our asset holding, \(\gamma\) our relative margin of risk, and \(U\) the power utility.

The interaction of multiple agents/decision makers causes additional uncertainty

We have prover \(P\) and randomized verifier \(V\). The \(V\) asks \(P\) for membership statements, and \(P\) responds with statements. These proofs can be used to prove membership in very powerful languages.