Category: Problems
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Graph Lagrangians
Lets review the concept of Lagrangians. Suppose we wish to optimize a function \(f(x_1, x_2, \dots, x_n\) under the boundary condition \(g(x_1, x_2, \dots, x_n = 0\). From calculus, we know that without the boundary condition, the min/max of \(f(x)\) occurs when the partial derivatives vanish. However, the extremas within the boundary might have nonvanishing…
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Integer Valued Functions
In this post, I wish to introduce two elementary number theory problems and their solutions. The solutions require no mathematical prerequisites other than basic understanding of logic and integer valued functions. The typical structure of these problems are as follows. A functional equation is given, along with the domain and the range of the function.…