A variation on bisecting the binomial coefficients

In this paper, we present an algorithm which allows us to search for all the bisections for the binomial coefficients $\{\binom{n}{k} \}_{k=0,...,n}$ and include a table with the results for all $n\le 154$. Connections with previous work on this topic is included. We conjecture that the probability of having only trivial solutions is $5/6$. \end{abstract}Comment: 14 pages, four tables, two figure

"Rotterdam econometrics": publications of the econometric institute 1956-2005

This paper contains a list of all publications over the period 1956-2005, as reported in the Rotterdam Econometric Institute Reprint series during 1957-2005

On approximate polynomial identity testing and real root finding

In this thesis we study the following three topics, which share a connection through the (arithmetic) circuit complexity of polynomials. 1. Rank of symbolic matrices. 2. Computation of real roots of real sparse polynomials. 3. Complexity of symmetric polynomials. We start with studying the commutative and non-commutative rank of symbolic matrices with linear forms as their entries. Here we show a deterministic polynomial time approximation scheme (PTAS) for computing the commutative rank. Prior to this work, deterministic polynomial time algorithms were known only for computing a 1/2-approximation of the commutative rank. We give two distinct proofs that our algorithm is a PTAS. We also give a min-max characterization of commutative and non-commutative ranks. Thereafter we direct our attention to computation of roots of uni-variate polynomial equations. It is known that solving a system of polynomial equations reduces to solving a uni-variate polynomial equation. We describe a polynomial time algorithm for (n,k,\tau)-nomials which computes approximations of all the real roots (even though it may also compute approximations of some complex roots). Moreover, we also show that the roots of integer trinomials are well-separated. Finally, we study the complexity of symmetric polynomials. It is known that symmetric Boolean functions are easy to compute. In contrast, we show that the assumption VP \neq VNP implies that there exist hard symmetric polynomials. To prove this result, we use an algebraic analogue of the classical Newton iteration.In dieser Dissertation untersuchen wir die folgenden drei Themen, welche durch die (arithmetische) Schaltkreiskomplexität von Polynomen miteinander verbunden sind: 1. der Rang von symbolischen Matrizen, 2. die Berechnung von reellen Nullstellen von dünnbesetzten (“sparse”) Polynomen mit reellen Koeffizienten, 3. die Komplexität von symmetrischen Polynomen. Wir untersuchen zunächst den kommutativen und nicht-kommutativen Rang von Matrizen, deren Einträge aus Linearformen bestehen. Hier beweisen wir die Existenz eines deterministischem Polynomialzeit-Approximationsschemas (PTAS) für die Berechnung des kommutative Ranges. Zuvor waren polynomielle Algorithmen nur für die Berechnung einer 1/2-Approximation des kommutativen Ranges bekannt. Wir geben zwei unterschiedliche Beweise für den Fakt, dass unser Algorithmus tatsächlich ein PTAS ist. Zusätzlich geben wir eine min-max Charakterisierung des kommutativen und nicht-kommutativen Ranges. Anschließend lenken wir unsere Aufmerksamkeit auf die Berechnung von Nullstellen von univariaten polynomiellen Gleichungen. Es ist bekannt, dass das Lösen eines polynomiellem Gleichungssystems auf das Lösen eines univariaten Polynoms zurückgeführt werden kann. Wir geben einen Polynomialzeit-Algorithmus für (n, k, \tau)-Nome, welcher Abschätzungen für alle reellen Nullstellen berechnet (in manchen Fallen auch Abschätzungen von komplexen Nullstellen). Zusätzlich beweisen wir, dass Nullstellen von ganzzahligen Trinomen stets weit voneinander entfernt sind. Schließlich untersuchen wir die Komplexität von symmetrischen Polynomen. Es ist bereits bekannt, dass sich symmetrische Boolesche Funktionen leicht berechnen lassen. Im Gegensatz dazu zeigen wir, dass die Annahme VP \neq VNP bedeutet, dass auch harte symmetrische Polynome existieren. Um dies zu beweisen benutzen wir ein algebraisches Analog zum klassischen Newton-Verfahren

Rendiconti dell'Istituto di Matematica dell'Università di Trieste. An International Journal of Mathematics. Vol. 44 (2012)

Rendiconti dell’Istituto di Matematica dell’Università di Trieste was founded in 1969 by Arno Predonzan, with the aim of publishing original research articles in all fields of mathematics and has been the first Italian mathematical journal to be published also on-line. The access to the electronic version of the journal is free. All published articles are available on-line. The journal can be obtained by subscription, or by reciprocity with other similar journals. Currently more than 100 exchange agreements with mathematics departments and institutes around the world have been entered in

Natural Parameterization

The objective of this project has been to develop an approach for imitating physical objects with an underlying stochastic variation. The key assumption is that a set of “natural parameters” can be extracted by a new subdivision algorithm so they reflect what is called the object’s “geometric DNA”. A case study on one hundred wheat grain crosssections (Triticum aestivum) showed that it was possible to extract thirty-six such parameters and to reuse them for Monte Carlo simulation of “new” stochastic phantoms which possessthe same stochastic behavior as the “original” cross-sections