 # Matthew Sotoudeh - Research & Publications

Here I am collecting a list of proofs that I encounter (or write) and particularly enjoy for one reason or another. Mostly for my own reference, but someone else may enjoy them! Most are not very difficult, but present ideas from an angle slightly different than is traditional. I hope to eventually type up rigorous LaTeX-typeset proofs of each of these and post them here, but that may have to wait a while...

• The completely-visual proof of the log-sum rule. (Exercise from Donald Knuth's The Art of Computer Programming)
• A \cup C = B \cup C and A \cap C = B \cap C implies A = B. (From my discrete mathematics course)
• A special case of the Chinese Remainder Theorem, which can be modeled by and proved from a patterned, sub-divided rectangle. (Used in a problem from Knuth's TAoCP)
• The algebraic rule (a+b)^2 = a^2 + 2ab + b^2 visually with a physical square diagram. (From my evolution and ecology Biology course)
• p^2 + pq = p for alleles in a Harvey-Weinberg equilibrium population. (Evolution-ecology course)
• Three different explanations for the formula n(n+1)/2 for determining the number of possible genotypes given n alleles. (Evolution-ecology course)
• Path independence of conservative vector fields, building off of the famous Escher piece. (Vector analysis course, Hofstadter's Godel, Escher, Bach)
• Generalization of the Jacobian of higher-dimensional spherical coordinate systems. (Vector analysis course TA)
• The Lame proof of the worst-case complexity of the Euclidean algorithm, using the Fibonacci series. (Discrete mathematics course)
• "Godel for Goldilocks" by Prof. Gusfield.