It was argued by Schramm and Smirnov that the critical site percolation exploration path on the triangular lattice converges in distribution to the trace of chordal SLE6. We provide here a detailed proof, which relies on Smirnov’s theorem that crossing probabilities have a conformally invariant scaling limit (given by Cardy’s formula). The version of convergence to SLE6 that we prove suffices f...

Using probabilistic ideas, we prove that if μ is a mean porous measure on R, then the packing dimension of μ is strictly smaller than n. Moreover, we give an explicit bound for the packing dimension, which is asymptotically sharp in the case of small porosity. This result was stated in [D. B. BELIAEV and S. K. SMIRNOV, “On dimension of porous measures”, Math. Ann. 323 (2002) 123-141], but the p...

In statistics, two-sample tests are used to determine whether two samples have been drawn from the same population. A widely used test as such is the Kolmogorov-Smirnov two-sample test. There are other distribution-free tests which might be applied in similar occasions. In this article, we describe a two-sample omnibus test introduced by Epps and Singleton, which has – albeit being distribution...

The Kolmogorov-Smirnov (K-S) test is widely used as a goodness-of-fit test. This thesis consists of two parts to describe ways to improve the classical K-S test in both 1-dimensional and 2-dimensional data. The first part is about how to improve the accuracy of the classical K-S goodness-of-fit test in 1-dimensional data. We replace the p-values estimated by the asymptotic distribution with nea...

In this paper we propose an improvement of the Kolmogorov-Smirnov test for normality. In the current implementation of the Kolmogorov-Smirnov test, a sample is compared with a normal distribution where the sample mean and the sample variance are used as parameters of the distribution. We propose to select the mean and variance of the normal distribution that provide the closest fit to the data....

In this paper we define a discrete subset family of a topological space and basis sigma locally finite and sigma discrete. First, we prove an auxiliary fact for discrete family and sigma locally finite and sigma discrete basis. We also show the necessary condition for the Nagata Smirnov theorem: every metrizable space is T3 and has a sigma locally finite basis. Also, we define a sufficient cond...

This list is a short overview of some of my results. One can nd a more detailed description in my Research Statement or at 1. Simple associative algebras with nite Z-grading. Here, R is a simple algebra with a nite Z-grading over a commutative ring. A description of all such algebras was given 18]. As a corollary, it was proved that every (not necessarily nite dimensional) simple Lie algebra of...

Explicit formulae for asymptotic expansions of Feynman diagrams in typical limits of momenta and masses with external legs on the mass shell are presented. E-mail: [email protected]

In symbolic data analysis, the values of the variables can be, among others, intervals. The algebraic structure of these variables leads us to adapt dissimilarity ones to be able to study them. The Kolmogorov-Smirnov criterion is used as a test selection metric for decision tree induction. For this criterion, the values taken by the explanatory variables have to be ordered. We have been interes...