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The $x/\ln (x)$ approximation yields an estimated count of 3.1 trillion primes below 100 trillion. Compared to the actual count of 3.2 trillion, this is over 99.99% accurate.

Conditions: $x = 100$,000,000,000,000.; The formula used is #primes(x) ~ $x/\ln (x)$.

The $x/\ln (x)$ approximation yields an estimated count of 3.1 trillion primes below 100 trillion. Compared to the actual count of 3.2 trillion, this is over 99.99% accurate.

Conditions: $x = 100$,000,000,000,000.; The formula used is #primes(x) ~ $x/\ln (x)$.

The initial state vector $S_0$ is a column vector that encodes the starting probabilities of the Markov chain. In this specific example, $S_0 = (1, 0)^T$, meaning the system starts in state A with probability 1 and in state B with probability 0.

Conditions: The system has two states, A and B.; The components are ordered consistently with the states, here A then B.; The entries are probabilities describing the initial distribution.

The initial state vector $S_0$ is a column vector that encodes the starting probabilities of the Markov chain. In this specific example, $S_0 = (1, 0)^T$, meaning the system starts in state A with probability 1 and in state B with probability 0.

Conditions: The system has two states, A and B.; The components are ordered consistently with the states, here A then B.; The entries are probabilities describing the initial distribution.

The state prices $\psi_j$ represent the prices of Arrow-Debreu securities, which are monetary values and do not necessarily sum to 1. The transformed vector $q_j$ represents risk-neutral probabilities, obtained by multiplying each state price by the gross risk-free factor $(1+i)$.

Conditions: The market is a one-period finite-state model.; $\psi_j$ are the solved state prices (Arrow-Debreu security prices).; $i$ is the per-period risk-free interest rate.; $q_j$ is defined as $q_j = (1+i)\psi_j$.

The state prices $\psi_j$ represent the prices of Arrow-Debreu securities, which are monetary values and do not necessarily sum to 1. The transformed vector $q_j$ represents risk-neutral probabilities, obtained by multiplying each state price by the gross risk-free factor $(1+i)$.

Conditions: The market is a one-period finite-state model.; $\psi_j$ are the solved state prices (Arrow-Debreu security prices).; $i$ is the per-period risk-free interest rate.; $q_j$ is defined as $q_j = (1+i)\psi_j$.

No, $S_0 = (1,0)^T$ is not the only possible initial state. It is an asserted initial condition for this specific example.

Conditions: The system is a Markov chain with states A and B.; The entries of the initial state vector are probabilities.; The sum of the probabilities in the initial state vector must equal 1.

No, $S_0 = (1,0)^T$ is not the only possible initial state. It is an asserted initial condition for this specific example.

Conditions: The system is a Markov chain with states A and B.; The entries of the initial state vector are probabilities.; The sum of the probabilities in the initial state vector must equal 1.