On semiring complexity of Schur polynomials

Semiring complexity is the version of arithmetic circuit complexity that allows only two operations: addition and multiplication. We show that semiring complexity of a Schur polynomial {s_\lambda(x_1,\dots,x_k)} labeled by a partition {\lambda=(\lambda_1\ge\lambda_2\ge\cdots)} is bounded by {O(\log(\lambda_1))} provided the number of variables $k$ is fixed

Data-Discriminants of Likelihood Equations

Maximum likelihood estimation (MLE) is a fundamental computational problem in statistics. The problem is to maximize the likelihood function with respect to given data on a statistical model. An algebraic approach to this problem is to solve a very structured parameterized polynomial system called likelihood equations. For general choices of data, the number of complex solutions to the likelihood equations is finite and called the ML-degree of the model. The only solutions to the likelihood equations that are statistically meaningful are the real/positive solutions. However, the number of real/positive solutions is not characterized by the ML-degree. We use discriminants to classify data according to the number of real/positive solutions of the likelihood equations. We call these discriminants data-discriminants (DD). We develop a probabilistic algorithm for computing DDs. Experimental results show that, for the benchmarks we have tried, the probabilistic algorithm is more efficient than the standard elimination algorithm. Based on the computational results, we discuss the real root classification problem for the 3 by 3 symmetric matrix~model.Comment: 2 table

Fast Multivariate Power Series Multiplication in Characteristic Zero

Let k be a field of characteristic zero. We present a fast algorithm formultiplying multivariate power series over k truncated in total degree. Upto logarithmic factors, its complexity is optimal, i.e. linear in the numberof coeffcients of the series.Keywords. Multivariate power series, fast multiplication, complexity.Sociedad Argentina de Informática e Investigación Operativ

Analytical solution of the tooling/workpiece contact interface shape during a flow forming operation

Flow forming involves complicated tooling/workpiece interactions. Purely analytical models of the tool contact area are difficult to formulate, resulting in numerical approaches that are case-specific. Provided are the details of an analytical model that describes the steady-state tooling/workpiece contact area allowing for easy modification of the dominant geometric variables. The assumptions made in formulating this analytical model are validated with experimental results attained from physical modelling. The analysis procedure can be extended to other rotary forming operations such as metal spinning, shear forming, thread rolling and crankshaft fillet rolling.Comment: 28 pages, 11 figure

Uniform Bounds on the Number of Rational Points of a Family of Curves of Genus 2

We exhibit a genus{2 curve C de ned over Q(T ) which admits two independent morphisms to a rank{1 elliptic curve de ned over Q(T ). We describe completely the set of Q(T ){rational points of the curve C and obtain a uniform bound for the number of Q{rational points of a rational specialization C t of the curve C for a certain (possibly in nite) set of values t 2 Q. Furthermore, for this set of values t 2 Q we describe completely the set of Q{rational points of the curve C t . Finally we show how these results can be strengthened assuming a height conjecture of S. Lang

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