Volume I, Issue 3
A Biological Application of the Calculus of Variations
— In this paper, we introduce the calculus of variations and derive the general Euler-Lagrange equations for functionals that depend on functions of one variable. Although the calculus of variations has traditionally been applied to problems in mechanics, we apply the variational approach to a problem in biology by means of minimal surfaces. We introduce the idea of using space curves to model protein structure and lastly, we analyze the free energy associated with these space curves by deriving two Euler-Lagrange equations dependent on curvature.
Graphings and Unimodularity
—We extend the concept of the law of a finite graph to graphings, which are, in general, infinite graphs whose vertices are equipped with the structure of a probability space. By doing this, we obtain a vast array of new unimodular measures. Furthermore, we work out in full detail a proof of a known result, which states that weak limits preserve unimodularity.