Write a seminar paper. Give a detailed and scientifically sound description of the given topic: literature research, mathematics, algorithms, classification, applications, examples, and a sample program, if it applies.

Long short$-$term memory $($LSTM$)$:

Give a detailed description including a self$-$elaborated example demonstrating the working principles. As minimum requirements, the aspects of RNN$,$ the vanishing gradient problem, the exploding gradient problem, the common LSTM unit, and application examples should be comprised.

more than $20$ pages WITH all the points