I recently came across an excellent survey article, “Morse Theory Indomitable“, by Raoul Bott. It starts with the basic history of Morse functions, and covers the additions of Smale and Witten, and the connections to symplectic reduction. Though, even moreso than being a clear and concise overview of some beautiful mathematics, it is all liberally dosed with personal anecdotes from the life of someone who lived throughout virtually the entire story. It was a throughly enjoyable read, even though I had seen almost all the contained math before.

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April 23, 2019 at 1:49 pm |

indeed appear these these that inncase of other situations for example periodic functions observed prof dr mircea orasanu and prof drd horia orasanu and so can be exposed followed bur ,must expose more many In mathematics, a periodic function is a function that repeats its values in regular intervals or periods. The most important examples are the trigonometric functions, which repeat over intervals of 2π radians. Periodic functions are used throughout science to describe oscillations, waves, and other phenomena that exhibit periodicity. Any function that is not periodic is called aperiodic. A function f is said to be periodic if, for some nonzero constant P, it is the case that

f ( x + P ) = f ( x ) {\displaystyle f(x+P)=f(x)} {\displaystyle f(x+P)=f(x)}

for all values of x in the domain.For example, the sine function is periodic with period 2 π {\displaystyle 2\pi } 2\pi , since