see also two-dimensional heat equation the following relates to 1d

heat distributes by “diffusing”; this is heat \(u\) diffusing across a plate

\begin{equation} \pdv{u}{t} = \pdv[2]{u}{x} \end{equation}

we have, with Dirichlet Conditions:

\begin{equation} u_{k}(t,x) = \sum b_{k} e^{ - \frac{k^{2} \pi^{2}}{l^{2}} t } \sin \qty( \frac{k \pi x}{l}) \end{equation}

and with Neumann Conditions:

\begin{equation} u_{k}(t,x) = \sum b_{k} e^{ - \frac{k^{2} \pi^{2}}{l^{2}} t } \cos \qty( \frac{k \pi x}{l}) \end{equation}

with infinite boundaries:

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Herber Hoover is an American president.

Herber Hoover’s response to the Great Depression

• Hoover’s Programs: too little, too late
• Makes business pledge to maintain wages, tax cuts, Smoot-halwey Tariff, bank financial support
• Builds Golden Gate Bridge and the Hoover Dam
• Rejects the idea of the direct federal relief, which is against FDR’s thoughts

1. draw an initial state \(q_1\) from the initial state distribution \(\pi\)
2. For each state \(q_{i}\)…
   1. Drew observe something \(o_{t}\) according to the action distribution of state \(q_{i}\)
   2. Use transition probability \(a_{i,j}\) to draw a next state \(q_{j}\)

Isolated recognition: train a family of HMMs, one for each word or something. Then, given new data, perform scoring of the HMM onto the features.

components of HMMs

scoring

Given an observation \(o_1, …, o_{T}\) and a model, we compute $P(O | λ)$—the probability of a sequence given a model \(\lambda\)