An algebraic characterization of some principal regulated rational cones

AbstractThe aim of this paper is to deal with formal power series over a commutative semiring A. Generalizing Wechler's pushdown automata and pushdown transition matrices yields a characterization of the A-semi-algebraic power series in terms of acceptance by pushdown automata. Principal regulated rational cones generated by cone generators of a certain form are characterized by algebraic systems given in certain matrix form. This yields a characterization of some principal full semi-AFL's in terms of context-free grammars. As an application of the theory, the principal regulated rational cone of one-counter “languages” is considered

Algebraic Statistics

Algebraic Statistics is concerned with the interplay of techniques from commutative algebra, combinatorics, (real) algebraic geometry, and related fields with problems arising in statistics and data science. This workshop was the first at Oberwolfach dedicated to this emerging subject area. The participants highlighted recent achievements in this field, explored exciting new applications, and mapped out future directions for research

Algebraic Systems and Pushdown Automata

The theory of algebraic power series in noncommuting variables, as we un-derstand it today, was initiated in [2] and developed in its early stages by the French school. The main motivation was the interconnection with context-free grammars: the defining equations were made to correspond to context-fre

Decomposition Methods for Nonlinear Optimization and Data Mining

We focus on two central themes in this dissertation. The first one is on decomposing polytopes and polynomials in ways that allow us to perform nonlinear optimization. We start off by explaining important results on decomposing a polytope into special polyhedra. We use these decompositions and develop methods for computing a special class of integrals exactly. Namely, we are interested in computing the exact value of integrals of polynomial functions over convex polyhedra. We present prior work and new extensions of the integration algorithms. Every integration method we present requires that the polynomial has a special form. We explore two special polynomial decomposition algorithms that are useful for integrating polynomial functions. Both polynomial decompositions have strengths and weaknesses, and we experiment with how to practically use them. After developing practical algorithms and efficient software tools for integrating a polynomial over a polytope, we focus on the problem of maximizing a polynomial function over the continuous domain of a polytope. This maximization problem is NP-hard, but we develop approximation methods that run in polynomial time when the dimension is fixed. Moreover, our algorithm for approximating the maximum of a polynomial over a polytope is related to integrating the polynomial over the polytope. We show how the integration methods can be used for optimization. The second central topic in this dissertation is on problems in data science. We first consider a heuristic for mixed-integer linear optimization. We show how many practical mixed-integer linear have a special substructure containing set partition constraints. We then describe a nice data structure for finding feasible zero-one integer solutions to systems of set partition constraints. Finally, we end with an applied project using data science methods in medical research.Comment: PHD Thesis of Brandon Dutr

Über die Maximal Mediated Set Struktur und die Anwendungen Nichtnegativer Circuit Polynome

Certifying the nonnegativity of a polynomial is a significant task both for mathematical and for scientific applications. In general, showing the nonnegativity of a random polynomial is hard. However, for certain classes of polynomials one can find easier conditions that imply their nonnegativity. In this work we investigate both the theoretic and the applied aspects of a special class of polynomials called circuit polynomials. On the theoretical side, we study the relationship of this class of polynomials with another very well studied class called sums of squares using the notion of the maximal mediated set (MMS). We show that MMS is a property of an equivalence class, rather than a property of a single circuit polynomial. With this in mind, we generate a large database of MMS using the software Polymake, and present some statistical and computational observations. On the applied side, we address to the problem of multistationarity in the chemical reaction networks theory by employing a symbolic nonnegativity certification technique via circuit polynomials. The existence of multiple stationary states for a given reaction network with a given starting point is important, as this is closely related to cellular communication in the context of biochemical reaction networks. The existence of multistationarity can be decided by studying the signs of a relevant polynomial whose coefficients are parameterized by the reaction rates. As a case study, we consider the (de)phosphorylation cycle, and use the theory of nonnegative circuit polynomials in order to find a symbolic nonnegativity certificates for the aforementioned polynomial. We provide a method that describes a non-empty open region in the parameter space that enables multistationarity for the (de)phosphorylation cycle. Moreover, we provide an explicit description of such an open region for 2 and 3-site cases.Der Nachweis der Nichtnegativität eines Polynoms ist eine wichtige Aufgabe sowohl für mathematische als auch für wissenschaftliche Anwendungen. Im Allgemeinen ist es schwierig, die Nichtnegativität eines Zufallspolynoms zu zeigen. Für bestimmte Klassen von Polynomen kann man jedoch einfachere Bedingungen finden, die ihre Nichtnegativität implizieren. In dieser Arbeit untersuchen wir sowohl die theoretischen als auch die angewandten Aspekte einer speziellen Klasse von Polynomen, die als circuit Polynome bezeichnet werden. Auf der theoretischen Seite untersuchen wir die Beziehung dieser Klasse von Polynomen mit einer anderen sehr gut untersuchten Klasse namens sums of squares unter Verwendung des Begriffs der maximal mediated set (MMS). Wir zeigen, dass MMS eher eine Eigenschaft einer Äquivalenzklasse als eine Eigenschaft eines circuit polynom ist. Vor diesem Hintergrund erstellen wir mit der Polymake-Software eine große MMS-Datenbank und präsentieren einige statistische und rechnerische Beobachtungen. Auf der angewandten Seite adressieren wir das Problem der Multistationarität in der Theorie chemischer Reaktionsnetzwerke durch die Anwendung einer symbolischen Nichtnegativitäts-Zertifizierungstechnik über circuit Polynome. Die Existenz mehrerer stationärer Zustände für ein gegebenes Reaktionsnetzwerk mit einem gegebenen Startpunkt ist wichtig, da dies eng mit der zellulären Kommunikation im Kontext biochemischer Reaktionsnetzwerke zusammenhängt. Die Existenz von Multistationarität kann durch Studium der Vorzeichen eines relevanten Polynoms entschieden werden, dessen Koeffizienten durch die Reaktionsgeschwindigkeiten parametrisiert werden. Betrachten Sie als Fallbeispiel den (De)Phosphorylierungszyklus und verwenden Sie die Theorie der circuit Polynome, um ein symbolisches Nichtnegativitätszertifikat für das obige Polynom zu finden. Darüber hinaus bieten wir eine explizite Beschreibung einer solchen offenen Region für 2- und 3-Site-Fälle

Instantons, Fluxons and Open Gauge String Theory

We use the exact instanton expansion to illustrate various string characteristics of noncommutative gauge theory in two dimensions. We analyse the spectrum of the model and present some evidence in favour of Hagedorn and fractal behaviours. The decompactification limit of noncommutative torus instantons is shown to map in a very precise way, at both the classical and quantum level, onto fluxon solutions on the noncommutative plane. The weak-coupling singularities of the usual Gross-Taylor string partition function for QCD on the torus are studied in the instanton representation and its double scaling limit, appropriate for the mapping onto noncommutative gauge theory, is shown to be a generating function for the volumes of the principal moduli spaces of holomorphic differentials. The noncommutative deformation of this moduli space geometry is described and appropriate open string interpretations are proposed in terms of the fluxon expansion.Comment: 70 pages, 6 figure

New approach to the stability and control of reaction networks

A new system-theoretic approach for studying the stability and control of chemical reaction networks (CRNs) is proposed, and analyzed. This has direct application to biological applications where biochemical networks suffer from high uncertainty in the kinetic parameters and exact structure of the rate functions. The proposed approach tackles this issue by presenting "structural" results, i.e. results that extract important qualitative information from the structure alone regardless of the specific form of the kinetics which can be arbitrary monotone kinetics, including Mass-Action. The proposed method is based on introducing a class of Lyapunov functions that we call Piecewise Linear in Rates (PWLR) Lyapunov functions. Several algorithms are proposed for the construction of these functions. Subject to mild technical conditions, the existence of these functions can be used to ensure powerful dynamical and algebraic conditions such as Lyapunov stability, asymptotic stability, global asymptotic stability, persistence, uniqueness of equilibria and exponential contraction. This shows that this class of networks is well-behaved and excludes complicated behaviour such as multi-stability, limit cycles and chaos. The class of PWLR functions is then shown to be a subset of larger class of Robust Lyapunov functions (RLFs), which can be interpreted by shifting the analysis to reaction coordinates. In the new coordinates, the problem transforms into finding a common Lyapunov function for a linear parameter varying system. Consequently, dual forms of the PWLR Lyapunov functions are presented, and the interpretation in terms of the variational dynamics and contraction analysis are given. An other class of Piecewise Quadratic in Rates Lyapunov function is also introduced. Relationship with consensus dynamics are also pointed out. Control laws for the stabilization of the proposed class of networks are provided, and the concept of control Lyapunov function is briefly discussed. Finally, the proposed framework is shown to be widely applicable to biochemical networks.Open Acces

Modeling, Analysis, and Optimization Issues for Large Space Structures

Topics concerning the modeling, analysis, and optimization of large space structures are discussed including structure-control interaction, structural and structural dynamics modeling, thermal analysis, testing, and design