Generalizations of the Matching Polynomial to the Multivariate Independence Polynomial

We generalize two main theorems of matching polynomials of undirected simple graphs, namely, real-rootedness and the Heilmann-Lieb root bound. Viewing the matching polynomial of a graph $G$ as the independence polynomial of the line graph of $G$, we determine conditions for the extension of these theorems to the independence polynomial of any graph. In particular, we show that a stability-like property of the multivariate independence polynomial characterizes claw-freeness. Finally, we give and extend multivariate versions of Godsil's theorems on the divisibility of matching polynomials of trees related to $G$.Comment: This research was supported by NSF Grant CCF-155375, 20 page

An overview of probabilistic and time series models in finance

In this paper, we partially review probabilistic and time series models in finance. Both discrete and continuous .time models are described. The characterization of the No- Arbitrage paradigm is extensively studied in several financial market contexts. As the probabilistic models become more and more complex to be realistic, the Econometrics needed to estimate them are more difficult. Consequently, there is still much research to be done on the link between probabilistic and time series models.Modeling Text Databases.- An Overview of Probabilistic and Time Series Models in Finance.- Stereological Estimation of the Rose of Directions.- Approximations for Multiple Scan Statistics.- Krawtchouk Polynomials and Krawtchouk Matrices.- An Elementary Rigorous Introduction to Exact Sampling.- On the Different Extensions of the Ergodic Theorem of Information Theory.- Dynamic Stochastic Models for Indexes and Thesauri.- Stability and Optimal Control.- Statistical Distances Based on Euclidean Graphs.- Implied Volatility.- On the Increments of the Brownian Sheet.- Compound Poisson Approximation.- Penalized Model Selection for Ill-posed Linear Problems.- The Arov-Grossman Model.- Recent Results in Geometric Analysis.- Dependence or Independence of the Sample Mean.- Optimal Stopping Problems for Time-Homogeneous Diffusions.- Criticality in Epidemics.- Acknowledgments.- Reference.- Index