Computing necessary integrability conditions for planar parametrized homogeneous potentials

Let V\in\mathbb{Q}(i)(\a_1,\dots,\a_n)(\q_1,\q_2) be a rationally parametrized planar homogeneous potential of homogeneity degree $k\neq -2, 0, 2$. We design an algorithm that computes polynomial \emph{necessary} conditions on the parameters (\a_1,\dots,\a_n) such that the dynamical system associated to the potential $V$ is integrable. These conditions originate from those of the Morales-Ramis-Sim\'o integrability criterion near all Darboux points. The implementation of the algorithm allows to treat applications that were out of reach before, for instance concerning the non-integrability of polynomial potentials up to degree $9$. Another striking application is the first complete proof of the non-integrability of the \emph{collinear three body problem}.Comment: ISSAC'14 - International Symposium on Symbolic and Algebraic Computation (2014

Parametric shortest-path algorithms via tropical geometry

We study parameterized versions of classical algorithms for computing shortest-path trees. This is most easily expressed in terms of tropical geometry. Applications include shortest paths in traffic networks with variable link travel times.Comment: 24 pages and 8 figure

ADAM: Analysis of Discrete Models of Biological Systems Using Computer Algebra

Abstract Background Many biological systems are modeled qualitatively with discrete models, such as probabilistic Boolean networks, logical models, Petri nets, and agent-based models, to gain a better understanding of them. The computational complexity to analyze the complete dynamics of these models grows exponentially in the number of variables, which impedes working with complex models. There exist software tools to analyze discrete models, but they either lack the algorithmic functionality to analyze complex models deterministically or they are inaccessible to many users as they require understanding the underlying algorithm and implementation, do not have a graphical user interface, or are hard to install. Efficient analysis methods that are accessible to modelers and easy to use are needed. Results We propose a method for efficiently identifying attractors and introduce the web-based tool Analysis of Dynamic Algebraic Models (ADAM), which provides this and other analysis methods for discrete models. ADAM converts several discrete model types automatically into polynomial dynamical systems and analyzes their dynamics using tools from computer algebra. Specifically, we propose a method to identify attractors of a discrete model that is equivalent to solving a system of polynomial equations, a long-studied problem in computer algebra. Based on extensive experimentation with both discrete models arising in systems biology and randomly generated networks, we found that the algebraic algorithms presented in this manuscript are fast for systems with the structure maintained by most biological systems, namely sparseness and robustness. For a large set of published complex discrete models, ADAM identified the attractors in less than one second. Conclusions Discrete modeling techniques are a useful tool for analyzing complex biological systems and there is a need in the biological community for accessible efficient analysis tools. ADAM provides analysis methods based on mathematical algorithms as a web-based tool for several different input formats, and it makes analysis of complex models accessible to a larger community, as it is platform independent as a web-service and does not require understanding of the underlying mathematics

Combinatorial Resultants in the Algebraic Rigidity Matroid

Motivated by a rigidity-theoretic perspective on the Localization Problem in 2D, we develop an algorithm for computing circuit polynomials in the algebraic rigidity matroid CMn associated to the Cayley-Menger ideal for n points in 2D. We introduce combinatorial resultants, a new operation on graphs that captures properties of the Sylvester resultant of two polynomials in the algebraic rigidity matroid. We show that every rigidity circuit has a construction tree from K4 graphs based on this operation. Our algorithm performs an algebraic elimination guided by the construction tree, and uses classical resultants, factorization and ideal membership. To demonstrate its effectiveness, we implemented our algorithm in Mathematica: it took less than 15 seconds on an example where a Gröbner Basis calculation took 5 days and 6 hrs

A Computer Algebra System for R: Macaulay2 and the m2r Package

Algebraic methods have a long history in statistics. Apart from the ubiquitous applications of linear algebra, the most visible manifestations of modern algebra in statistics are found in the young field of algebraic statistics, which brings tools from commutative algebra and algebraic geometry to bear on statistical problems. Now over two decades old, algebraic statistics has applications in a wide range of theoretical and applied statistical domains. Nevertheless, algebraic statistical methods are still not mainstream, mostly due to a lack of easy off-the-shelf implementations. In this article we debut m2r, an R package that connects R to Macaulay2 through a persistent back-end socket connection running locally or on a cloud server. Topics range from basic use of m2r to applications and design philosophy

Interlacing Ehrhart Polynomials of Reflexive Polytopes

It was observed by Bump et al. that Ehrhart polynomials in a special family exhibit properties similar to the Riemann {\zeta} function. The construction was generalized by Matsui et al. to a larger family of reflexive polytopes coming from graphs. We prove several conjectures confirming when such polynomials have zeros on a certain line in the complex plane. Our main new method is to prove a stronger property called interlacing

Combinatorial resultants in the algebraic rigidity matroid

Motivated by a rigidity-theoretic perspective on the Localization Problem in 2D, we develop an algorithm for computing circuit polynomials in the algebraic rigidity matroid CMn associated to the Cayley-Menger ideal for n points in 2D. We introduce combinatorial resultants, a new operation on graphs that captures properties of the Sylvester resultant of two polynomials in the algebraic rigidity matroid. We show that every rigidity circuit has a construction tree from K4 graphs based on this operation. Our algorithm performs an algebraic elimination guided by the construction tree, and uses classical resultants, factorization and ideal membership. To demonstrate its effectiveness, we implemented our algorithm in Mathematica: it took less than 15 seconds on an example where a Gröbner Basis calculation took 5 days and 6 hrs

Essays on strategic trading

This dissertation discusses various aspects of strategic trading using both analytical modeling and numerical methods. Strategic trading, in short, encompasses models of trading, most notably models of optimal execution and portfolio selection, in which one seeks to rigorously consider various---both explicit and implicit---costs stemming from the act of trading itself. The strategic trading approach, rooted in the market microstructure literature, contrasts with many classical finance models in which markets are assumed to be frictionless and traders can, for the most part, take prices as given. Introducing trading costs to dynamic models of financial markets tend to complicate matters. First, the objectives of the traders become more nuanced since now overtrading leads to poor outcomes due to increased trading costs. Second, when trades affect prices and there are multiple traders in the market, the traders start to behave in a more calculated fashion, taking into account both their own objectives and the perceived actions of others. Acknowledging this strategic behavior is especially important when the traders are asymmetrically informed. These new features allow the models discussed to better reflect aspects real-world trading, for instance, intraday trading patterns, and enable one to ask and answer new questions, for instance, related to the interactions between different traders. To efficiently analyze the models put forth, numerical methods must be utilized. This is, as is to be expected, the price one must pay from added complexity. However, it also opens an opportunity to have a closer look at the numerical approaches themselves. This opportunity is capitalized on and various new and novel computational procedures influenced by the growing field of numerical real algebraic geometry are introduced and employed. These procedures are utilizable beyond the scope of this dissertation and enable one to sharpen the analysis of dynamic equilibrium models.Tämä väitöskirja käsittelee strategista kaupankäyntiä hyödyntäen sekä analyyttisiä että numeerisia menetelmiä. Strategisen kaupankäynnin mallit, erityisesti optimaalinen kauppojen toteutus ja portfolion valinta, pyrkivät tarkasti huomioimaan kaupankäynnistä itsestään aiheutuvat eksplisiittiset ja implisiittiset kustannukset. Tämä erottaa strategisen kaupankäynnin mallit klassisista kitkattomista malleista. Kustannusten huomioiminen rahoitusmarkkinoiden dynaamisessa tarkastelussa monimutkaistaa malleja. Ensinnäkin kaupankävijöiden tavoitteet muuttuvat hienovaraisemmiksi, koska liian aktiivinen kaupankäynti johtaa korkeisiin kaupankäyntikuluihin ja heikkoon tuottoon. Toiseksi oletus siitä, että kaupankävijöiden valitsemat toimet vaikuttavat hintoihin, johtaa pelikäyttäytymiseen silloin, kun markkinoilla on useampia kaupankävijöitä. Pelikäyttäytymisen huomioiminen on ensiarvoisen tärkeää, mikäli informaatio kaupankävijöiden kesken on asymmetristä. Näiden piirteiden johdosta tässä väitöskirjassa käsitellyt mallit mahdollistavat abstrahoitujen rahoitusmarkkinoiden aiempaa täsmällisemmän tarkastelun esimerkiksi päivänsisäisen kaupankäynnin osalta. Tämän lisäksi mallien avulla voidaan löytää vastauksia uusiin kysymyksiin, kuten esimerkiksi siihen, millaisia ovat kaupankävijöiden keskinäiset vuorovaikutussuhteet dynaamisilla markkinoilla. Monimutkaisten mallien analysointiin hyödynnetään numeerisia menetelmiä. Tämä avaa mahdollisuuden näiden menetelmien yksityiskohtaisempaan tarkasteluun, ja tätä mahdollisuutta hyödynnetään pohtimalla laskennallisia ratkaisuja tuoreesta numeerista reaalista algebrallista geometriaa hyödyntävästä näkökulmasta. Väitöskirjassa esitellyt uudet laskennalliset ratkaisut ovat laajalti hyödynnettävissä, ja niiden avulla on mahdollista terävöittää dynaamisten tasapainomallien analysointia