Jean Carlo Moraes

Let q(G) be the minimum number of distinct eigenvalues taken over all real symmetric matrices whose underlying graph is G. Using a linear time algorithm that diagonalizes any symmetric matrix associated to a unicyclic graph, we investigate the maximum eigenvalue multiplicity for these graphs. As an application, we determine the value of q(G) for some family of tadpole graphs, which are unicyclic graphs formed by adding an edge between a cycle and a path.

Recently, a new model was developed in evolutionary game dynamics, extending the standard replicator equation to a finite set of players interacting on some network of connections. Players are modeled as the subpopulations of a multipopulation game, connected on an arbitrary graph, allowed to change their strategies over time and to adopt mixed strategies. A relation between the stability of the mixed equilibrium with the topology of the network was proved to depend on the eigenvalues of the graph's adjacency matrix multiplied by a scalar. The present paper studies the pure (strict) Nash equilibrium, (S)NE, of these games and their connection to the network. Specifically, we present necessary and sufficient conditions for a pure steady state in a coordination or anti-coordination game to be a (strict) Nash equilibrium. A significant application of the model concerns finding the conditions for the onset of pure cooperative or pure defective behavior of players. In this direction, our main result provides conditions in an anti-coordination game such that a pure steady state of the system is a Nash equilibrium if and only if the set of players that cooperate or the set of players that defect are independent dominating sets.

In football, many people argue that in knock-out competitions the team that plays the second game in its stadium would have an advantage, a greater probability of victory in the final outcome of a two leg knock-out match. The purpose of this study is to verify the veracity of this statement using data from the Brazil Cup. We find evidence that the ability spread between the teams participating in a match is the main factor that explains the qualification of one of the teams for the next round. Until 2018, there were three criteria for break the tie in the Brazil Cup, they were used in the respectively order: goal difference, away goals rule, penalty shootout (there is no extra-time in any play-off of the championship). It is estimated that 36% of the matches end tied and need a criterion to determine the winner. Of these, 51% use the goal difference decision, 29% use the away goals rule and 20% penalty shoot-out. When considering the championship in general there is evidence that the home team wins the match in approximately 63% of the matches, a significant advantage. However, in the confrontations that were decided by the away goals rule or the penalty shoot-out, the home team wins percentage is 20% lower, indicating that these criteria level the odds of both teams.

We extend the definitions of dyadic paraproduct and t-Haar multipliers to dyadic operators that depend on the complexity (m,n), for m and n positive integers. We will use the ideas developed by Nazarov and Volberg to prove that the weighted L^2(w)-norm of a paraproduct with complexity (m,n) associated to a function b\in BMO, depends linearly on the A_2-characteristic of the weight w, linearly on the BMO-norm of b, and polynomially in the complexity. This argument provides a new proof of the linear bound for the dyadic paraproduct (the one with complexity (0,0)). Also we prove that the L^2-norm of a t-Haar multiplier for any t and weight w depends on the square root of the C_{2t}-characteristic of w times the square root of the A_2-characteristic of w^{2t} and polynomially in the complexity (m,n), recovering a result of Beznosova for the (0,0)-complexity case.