José Luis Palacios

A Markov Chain Approach to Baseball

Most earlier mathematical studies of baseball required particular models for advancing runners based on a small set of offensive possibilities. Other efforts considered only teams with players of identical ability. We introduce a Markov chain method that considers teams made up of players with different abilities and which is not restricted to a given model for runner advancement. Our method is limited only by the available data and can use any reasonable deterministic model for runner advancement when sufficiently detailed data are not available. Furthermore, our approach may be adapted to include the effects of pitching and defensive ability in a straightforward way. We apply our method to find optimal batting orders, run distributions per half inning and per game, and the expected number of games a team should win. We also describe the application of our method to test whether a particular trade would benefit a team.

Resistance distance in graphs and random walks

We study the resistance distance on connected undirected graphs, linking this concept to the fruitful area of random walks on graphs. We provide two short proofs of a general lower bound for the resistance, or Kirchhoff index, of graphs on N vertices, as well as an upper bound and a general formula to compute it exactly, whose complexity is that of inverting an N×N matrix. We argue that the formulas for the resistance in the case of the Platonic solids can be generalized to all distance-transitive graphs.

Closed‐form formulas for Kirchhoff index

We find closed-form expressions for the resistance, or Kirchhoff index, of certain connected graphs using Fosters theorems, random walks, and the superposition principle.

Foster's Formulas via Probability and the Kirchhoff Index

AbstractBuilding on a probabilistic proof of Foster’s first formula given by Tetali (1994), we prove an elementary identity for the expected hitting times of an ergodic N-state Markov chain which yields as a corollary Foster’s second formula for electrical networks, namely

Two Applications of Urn Processes The Fringe Analysis of Search Trees and The Simulation of Quasi-Stationary Distributions of Markov Chains

\sum {R_{ij} } \frac{{C_{iv} C_{vj} }}{{C_v }} = N - 2,

Bounds for the Kirchhoff index via majorization techniques

where Rij is the effective resistance, as computed by means of Ohm’s law, measured across the endpoints of two adjacent edges (i,v) and (i,v), Civ and Civ are the conductances of these edges, Cv is the sum of all conductances emanating from the common vertex v, and the sum on the left hand side of (1) is taken over all adjacent edges. We show how to extend Foster’s first and second formulas. As an application, we show how to use a “third formula” to compute the Kirchhoff index of a class of graphs with diameter 3.

On the Kirchhoff index of graphs with diameter 2

We give sufficient conditions under whichfor a sequence of nonnegative random variables and [alpha]>0.

New bounds of degree-based topological indices for some classes of c -cyclic graphs

From an urn containing colored balls draw one ball and replace a random number of differently-colored balls, with the distribution of the added balls depending only on the color of the ball drawn. Under mild regularity conditions, the proportions of different colors will converge to deterministic limits. Two applications of this standard result are described. The average efficiency of binary trees, 2-3 trees and other structures for information storage has been studied with diverse techniques. One such technique for estimation is called fringe analysis, and this turns out to involve urn processes. The other topic is the simulation of quasi-stationary distributions of Markov chains. These arise when the chain is conditioned to avoid a taboo set. Here urn processes are used to prove an analog of the classical result on convergence of empirical distributions to the stationary distribution.