Noetherian rings have applications in physics, and a modified definition of them can be viewed as a generalization of vector spaces. He was the first African American to obtain a PHD in mathematics. His Ph.D. at this time was remarkable, as no place or institution was a friend to Negroes. Indeed, there were just 28 Ph.D.'s in Mathematics awarded in all the country in 1925, but 31 black men were lynched that year. Difference equations are equations in terms of the derivatives of functions.
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Difference equations should be used in situations where we have data given in terms of differentiable functions. Difference equations are very useful in real life since they can be used in situations of discrete data that prevents us from taking the derivative. Instead, we can take the divided difference representing the slope of the secant line, which gives an approximation to the derivative.