Search

In addition to being intrinsically interesting, calculus of variations constitutes an area that has been extensively considered in both theoretic and applied areas, underlying a large number of structures and phenomena, including the stationary action principle in physics. Basically, given a functional expressing some measurement to be optimized (minimized or maximized), as well as respective boundary conditions, calculus of variations allows the determination of the respective optimal solution, be it a function, a curve, surface, as well as a large range of mathematical structures. The present work aims at developing an introduction to variational calculus which could be hopefully accessible to readers with some background in multivariate calculus. After defining the problem, and presenting some brief historic remarks, we proceed to present a motivational example of calculus of variations respective to a specific type of variation function, namely second-order polynomials adhering to the boundary conditions. In addition to illustrating several concepts related to calculus of variations, this approach also allowed the visualization of the aimed functional behavior as a function of the variation parameter. Next, the Euler-Lagrange equation for functions of one variable was informally derived, and then illustrates its application respectively to the minimum arc-length and soap surface minimization. The interesting possibility of numerical approaches to calculus of variations is also briefly described and illustrated respectively to Euler's direct method. Appendices providing a brief presentation of the chain rule, integration by parts, multivariate derivatives, as well as the multivariate, multiconstraint Lagrange multipliers method are also provided for quick reference.

Complex networks have certain properties that distinguish them from their respective uniform or regular counterparts. One of these properties is the variation of topological properties along different hierarchical levels. In this work, we study how networks that are constructed by repeatedly incorporating a given motif exhibit this property. A motif is henceforth understood as a small subgraph with a reference node where the incorporation respectively occurs. We generate fractal networks using different motifs and observe how their topology changes depending along the growth stages. Two regimes are respectively identified: transient and equilibrium. The former is characterized by significant topological changes that depend on the motif topology, while the equilibrium regime shows more stable, parallel trajectories. A more systematic analysis revealed that the betweenness centrality and the average shortest path lengths were the main topological properties that change along the network growth., Comment: 15 pages, 8 figures