arXiv subjects
Publications and source records attributed to Steven Finch.
We collect various facts related loosely to random Gaussian quadrilaterals in the plane. For example, a side of a degenerate quadrilateral (one point inside three others) has a density that is non-Rayleigh.
The joint distribution of two off-diagonal Wishart matrix elements was useful in recent work on geometric probability [Finch 2010]. Not finding such formulas in the literature, we report these here.
Supplements to Mehta & Normand (1997) are given, with regard to integrals involving Euclidean distances between n+1 random points in d-dimensional space, each visited once.
Explicit area expressions are known for a special case, due to Tao & Wu (1987), and lead to calculation of integrals in applied probability.
Let T be a random triangle in a disk D of radius R (meaning that vertices are independent and uniform in D). We determine the bivariate density for two arbitrary sides a,b of T. In particular, we compute that E(a*b)=(0.837...)*R^2, which implies that Var(perimeter)=(0.649...)*R^2. No closed-form expression for either coefficient is known. The Catalan numbers also arise here.
Consider the product of (1-p^(-s))^(-4) over all primes p=1 mod 5. We evaluate its residue at s=1 and compare with the corresponding Mertens constant of Languasco & Zaccagnini. We also count primitive quintic Dirichlet characters mod n and determine their average number as n->infty.
Let f(n) denote the number of odd entries in the nth row of Pascal's binomial triangle. We study "average dispersion" and "typical dispersion" of f(n) -- the latter involves computing a generalized Lyapunov exponent -- and then turn to numerical analysis of higher dimensional examples.
The nth row of Pascal's trinomial triangle gives coefficients of (1+x+x^2)^n. Let g(n) denote the number of such coefficients that are odd. We review Moshe's algorithm for evaluating asymptotics of g(n) -- this involves computing the Lyapunov exponent for certain 2x2 random matrix products -- and then analyze further examples with more terms and higher powers of x.
What are the asymptotic moments of coefficients obtained when expanding prod_{m=1}^infty (1-q^m)^k in series? A few examples are given, as well as a new multiplicative representation for coefficients when k=10 and k=14.
We study a specific convex maximization problem in n-dimensional space. The conjectured solution is proved to be a vertex of the polyhedral feasible region, but only a partial proof of local maximality is known. Integer sequences with interesting patterns arise in the analysis, owing to the number theoretic origin of the problem.
We study a specific convex maximization problem in the space of continuous functions defined on a semi-infinite interval. An unexplained connection to the discrete version of this problem is investigated.
Let F be an infinite field. We prove that the right zero divisors of a three-dimensional associative F-algebra A must form the union of at most finitely many linear subspaces of A. The proof is elementary and written with students as the intended audience.
In this paper we describe an architecture and functionality of main components of a workbench for an acquisition of domain knowledge from large text corpora. The workbench supports an incremental process of corpus analysis starting from a rough automatic extraction and organization of lexico-semantic regularities and ending with a computer supported analysis of extracted data and a semi-automatic refinement of obtained hypotheses. For doing this the workbench employs methods from computational linguistics, information retrieval and knowledge engineering. Although the workbench is currently under implementation some of its components are already implemented and their performance is illustrated with samples from engineering for a medical domain.