Christopher D. Sinclair

An application of machine learning to network intrusion detection

Differentiating anomalous network activity from normal network traffic is difficult and tedious. A human analyst must search through vast amounts of data to find anomalous sequences of network connections. To support the analysts job, we built an application which enhances domain knowledge with machine learning techniques to create rules for an intrusion detection expert system. We employ genetic algorithms and decision trees to automatically generate rules for classifying network connections. This paper describes the machine learning methodology and the applications employing this methodology.

The Ginibre Ensemble of Real Random Matrices and its Scaling Limits

We give a closed form for the correlation functions of ensembles of a class of asymmetric real matrices in terms of the Pfaffian of an antisymmetric matrix formed from a 2 × 2 matrix kernel associated to the ensemble. We apply this result to the real Ginibre ensemble and compute the bulk and edge scaling limits of its correlation functions as the size of the matrices becomes large.

Extremal laws for the real Ginibre ensemble

The real Ginibre ensemble refers to the family of n n matrices in which each entry is an independent Gaussian random variable of mean zero and variance one. Our main result is that the appropriately scaled spectral radius converges in law to a Gumbel distribution as n!1. This fact has been known to hold in the complex and quaternion analogues of the ensemble for some time, with simpler proofs. Along the way we establish a new form for the limit law of the largest real eigenvalue.

The range of multiplicative functions on ℂ[x], ℝ[x] and ℤ[x]

Mahlers measure is generalized to create the class of {\it multiplicative distance functions}. These functions measure the complexity of polynomials based on the location of their zeros in the complex plane. Following work of S.-J. Chern and J. Vaaler in \cite{chern-vaaler}, we associate to each multiplicative distance function two families of analytic functions which encode information about its range on \C[x] and \R[x]. These {\it moment functions} are Mellin transforms of distribution functions associated to the multiplicative distance function and demonstrate a great deal of arithmetic structure. For instance, we show that the moment function associated to Mahlers measure restricted to real reciprocal polynomials of degree 2N has an analytic continuation to rational functions with rational coefficients, simple poles at integers between -N and N, and a zero of multiplicity 2N at the origin. This discovery leads to asymptotic estimates for the number of reciprocal integer polynomials of fixed degree with Mahler measure less than T as

Correlation Functions for β=1 Ensembles of Matrices of Odd Size

T \to \infty

The distribution of Mahler's measures of reciprocal polynomials

. To explain the structure of this moment functions we show that the real moment functions of a multiplicative distance function can be written as Pfaffians of antisymmetric matrices formed from a skew-symmetric bilinear form associated to the multiplicative distance function.

Universality for ensembles of matrices with potential theoretic weights on domains with smooth boundary

Using the method of Tracy and Widom we rederive the correlation functions for β=1 Hermitian and real asymmetric ensembles of N×N matrices with N odd.

Conjugate Reciprocal Polynomials with all Roots on the Unit Circle

We study the distribution of Mahler’s measures of reciprocal polynomials with complex coefficients and bounded even degree. We discover that the distribution function associated to Mahler’s measure restricted to monic reciprocal polynomials is a reciprocal (or antireciprocal) Laurent polynomial on [1, ∞) and identically zero on [0, 1). Moreover, the coefficients of this Laurent polynomial are rational numbers times a power of π . We are led to this discovery by the computation of the Mellin transform of the distribution function. This Mellin transform is an even (or odd) rational function with poles at small integers and residues that are rational numbers times a power of π . We also use this Mellin transform to show that the volume of the set of reciprocal polynomials with complex coefficients, bounded degree, and Mahler’s measure less than or equal to one is a rational number times a power of π .

A Generalized Plasma and Interpolation Between Classical Random Matrix Ensembles

Abstract We investigate a two-dimensional statistical model of N charged particles interacting via logarithmic repulsion in the presence of an oppositely charged compact region K whose charge density is determined by its equilibrium potential at an inverse temperature corresponding to β = 2 . When the charge on the region, s , is greater than N , the particles accumulate in a neighborhood of the boundary of K , and form a determinantal point process on the complex plane. We investigate the scaling limit, as N → ∞ , of the associated kernel in the neighborhood of a point on the boundary under the assumption that the boundary is sufficiently smooth. We find that the limiting kernel depends on the limiting value of N / s , and prove universality for these kernels. That is, we show that, the scaled kernel in a neighborhood of a point ζ ∈ ∂ K can be succinctly expressed in terms of the scaled kernel for the closed unit disk, and the exterior conformal map which carries the complement of K to the complement of the closed unit disk. When N / s → 0 we recover the universal kernel discovered by Lubinsky (2010) in [13] .

The partition function of multicomponent log-gases

AbstractWe study the geometry, topology and Lebesgue measure of the set of monic conju-gate reciprocal polynomials of fixed degree with all roots on the unit circle. The set ofsuch polynomials of degree N is naturally associated to a subset of R N−1 . We calculatethe volume of this set, prove the set is homeomorphic to the N − 1 ball and that itsisometry group is isomorphic to the dihedral group of order 2N. 1 Introduction Let N be a positive integer and suppose f(x) is a polynomial in C[x] of degree N. If fsatisfies the identity,(1.1) f(x) = x N f(1/x),then fis said to be conjugate reciprocal, or simply CR. Furthermore, if fis given byf(x) = x N +X Nn=1 c n x N−n .then (1.1) implies that c N = 1,c N−n = c n for 1 ≤ n≤ N− 1 and, if αis a zero of fthen so too is 1/α. The purpose of this manuscript is to study the set of CR polynomialswith all roots on the unit circle. The interplay between the symmetry condition on thecoefficients and the symmetry of the roots allows for a number of interesting theoremsabout the geometry, topology and Lebesgue measure of this set.CR polynomials have various names in the literature including reciprocal, self-reciprocaland self-inversive (though we reserve the term reciprocal for polynomials which satisfy anidentity akin to (1.1) except without both instances of complex conjugation).Thecondition onthe coefficientsofa conjugatereciprocalpolynomialallowsusto identifythe set of CR polynomials with R