49 research outputs found

    Moduli spaces of rational weighted stable curves and tropical geometry

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    We study moduli spaces of rational weighted stable tropical curves, and their connections with the classical Hassett spaces. Given a vector w of weights, the moduli space of tropical w-stable curves can be given the structure of a balanced fan if and only if w has only heavy and light entries. In this case, we can express the moduli space as the Bergman fan of a graphic matroid. Furthermore, we realize the tropical moduli space as a geometric tropicalization, and as a Berkovich skeleton, of the classical moduli space. This builds on previous work of Tevelev, Gibney--Maclagan, and Abramovich--Caporaso--Payne. Finally, we construct the moduli spaces of heavy/light weighted tropical curves as fiber products of unweighted spaces, and explore parallels with the algebraic world.Comment: 26 pages, 8 TikZ figures. v3: Minor changes and corrections. Final version to appear in Forum of Mathematics, Sigm

    The Geometry of Generations

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    We present an intriguing and precise interplay between algebraic geometry and the phenomenology of generations of particles. Using the electroweak sector of the MSSM as a testing ground, we compute the moduli space of vacua as an algebraic variety for multiple generations of Standard Model matter and Higgs doublets. The space is shown to have Calabi–Yau, Grassmannian, and toric signatures, which sensitively depend on the number of generations of leptons, as well as inclusion of Majorana mass terms for right-handed neutrinos. We speculate as to why three generations is special

    Constraint-based Analysis of Substructures of Metabolic Networks

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    Constraint-based methods (CBMs) are promising tools for the analysis of metabolic networks, as they do not require detailed knowledge of the biochemical reactions. Some of these methods only need information about the stoichiometric coefficients of the reactions and their reversibility types, i.e., constraints for steady-state conditions. Nevertheless, CBMs have their own limitations. For example, these methods may be sensitive to missing information in the models. Additionally, they may be slow for the analysis of genome-scale metabolic models. As a result, some studies prefer to consider substructures of networks, instead of complete models. Some other studies have focused on better implementations of the CBMs. In Chapter 2, the sensitivity of flux coupling analysis (FCA) to missing reactions is studied. Genome-scale metabolic reconstructions are comprehensive, yet incomplete, models of real-world metabolic networks. While FCA has proved an appropriate method for analyzing metabolic relationships and for detecting functionally related reactions in such models, little is known about the impact of missing reactions on the accuracy of FCA. Note that having missing reactions is equivalent to deleting reactions, or to deleting columns from the stoichiometric matrix. Based on an alternative characterization of flux coupling relations using elementary flux modes, we study the changes that flux coupling relations may undergo due to missing reactions. In particular, we show that two uncoupled reactions in a metabolic network may be detected as directionally, partially or fully coupled in an incomplete version of the same network. Even a single missing reaction can cause significant changes in flux coupling relations. In case of two consecutive E. coli genome-scale networks, many fully-coupled reaction pairs in the incomplete network become directionally coupled or even uncoupled in the more complete reconstruction. In this context, we found gene expression correlation values being significantly higher for the pairs that remained fully coupled than for the uncoupled or directionally coupled pairs. Our study clearly suggests that FCA results are indeed sensitive to missing reactions. Since the currently available genome-scale metabolic models are incomplete, we advise to use FCA results with care. In Chapter 3, a different, but related problem is considered. Due to the large size of genome-scale metabolic networks, some studies suggest to analyze subsystems, instead of original genome-scale models. Note that analysis of a subsystem is equivalent to deletion of some rows from the stoichiometric matrix, or identically, assuming some internal metabolites to be external. We show mathematically that analysis of a subsystem instead of the original model can lead the flux coupling relations to undergo certain changes. In particular, a pair of (fully, partially or directionally) coupled reactions may be detected as uncoupled in the chosen subsystem. Interestingly, this behavior is the opposite of the flux coupling changes that may happen due to the existence of missing reactions, or equivalently, deletion of reactions. We also show that analysis of organelle subsystems has relatively little influence on the results of FCA, and therefore, many of these subsystems may be studied independent of the rest of the network. In Chapter 4, we introduce a rapid FCA method, which is appropriate for genome-scale networks. Previously, several approaches for FCA have been proposed in the literature, namely flux coupling finder algorithm, FCA based on minimal metabolic behaviors, and FCA based on elementary flux patterns. To the best of our knowledge none of these methods are available as a freely available software. Here, we introduce a new FCA algorithm FFCA (Feasibility-based Flux Coupling Analysis). This method is based on checking the feasibility of a system of linear inequalities. We show on a set of benchmarks that for genome-scale networks FFCA is faster than other existing FCA methods. Using FFCA, flux coupling analysis of genome-scale networks of S. cerevisiae and E. coli can be performed in a few hours on a normal PC. A corresponding software tool is freely available for non-commercial use. In Chapter 5, we introduce a new concept which can be useful in the analysis of fluxes in network substructures. Analysis of elementary modes (EMs) is proven to be a powerful CBM in the study of metabolic networks. However, enumeration of EMs is a hard computational task. Additionally, due to their large numbers, one cannot simply use them as an input for subsequent analyses. One possibility is to restrict the analysis to a subset of interesting reactions, rather than the whole network. However, analysis of an isolated subnetwork can result in finding incorrect EMs, i.e. the ones which are not part of any steady-state flux distribution in the original network. The ideal set of vectors to describe the usage of reactions in a subnetwork would be the set of all EMs projected onto the subset of interesting reactions. Recently, the concept of “elementary flux patterns” (EFPs) has been proposed. Each EFP is a subset of the support (i.e. non-zero elements) of at least one EM. In the present work, we introduce the concept of ProCEMs (Projected Cone Elementary Modes). The ProCEM set can be computed by projecting the flux cone onto the lower-dimensional subspace and enumerating the extreme rays of the projected cone. In contrast to EFPs, ProCEMs are not merely a set of reactions, but from the mathematical point of view they are projected EMs. We additionally prove that the set of EFPs is included in the set of ProCEM supports. Finally, ProCEMs and EFPs are compared in the study of substructures in biological networks

    The Convex Geometry of Linear Inverse Problems

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    In applications throughout science and engineering one is often faced with the challenge of solving an ill-posed inverse problem, where the number of available measurements is smaller than the dimension of the model to be estimated. However in many practical situations of interest, models are constrained structurally so that they only have a few degrees of freedom relative to their ambient dimension. This paper provides a general framework to convert notions of simplicity into convex penalty functions, resulting in convex optimization solutions to linear, underdetermined inverse problems. The class of simple models considered are those formed as the sum of a few atoms from some (possibly infinite) elementary atomic set; examples include well-studied cases such as sparse vectors and low-rank matrices, as well as several others including sums of a few permutations matrices, low-rank tensors, orthogonal matrices, and atomic measures. The convex programming formulation is based on minimizing the norm induced by the convex hull of the atomic set; this norm is referred to as the atomic norm. The facial structure of the atomic norm ball carries a number of favorable properties that are useful for recovering simple models, and an analysis of the underlying convex geometry provides sharp estimates of the number of generic measurements required for exact and robust recovery of models from partial information. These estimates are based on computing the Gaussian widths of tangent cones to the atomic norm ball. When the atomic set has algebraic structure the resulting optimization problems can be solved or approximated via semidefinite programming. The quality of these approximations affects the number of measurements required for recovery. Thus this work extends the catalog of simple models that can be recovered from limited linear information via tractable convex programming

    An Introduction to A Class of Matrix Optimization Problems

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    Ph.DDOCTOR OF PHILOSOPH

    Analysis of the Workspace of Tendon-based Stewart Platforms

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    Tendon-based Stewart platforms are a concept for innovative manipulators where the load to move almost coincides with the payload. After an overview over the state of research and some concepts of kinematics (singularity and redundancy), the thesis discusses aspects of the technically usable workspace (positive tendon forces, limits of tension, singularity, stiffness, collisions between tendens). A representation of the controllablwe workspace by means of polynomial inequalities is developed. Optimal solutions are provided to the problem of finding appropriate force distributions in the tendons. These solutions can be discontinuous in time, but they can be approximated with continuous ones. An algorithm is given for this. From these results, a quality measure for workspace is derived and used to state design rules which help achieving good workspaces. For some systems, sample trajectories are simulated.</p

    Elementary approaches to microbial growth rate maximisation

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    This thesis, called Elementary approaches to microbial growth rate maximisation, reports on a theoretical search for principles underlying single cell growth, in particular for microbial species that are selected for fast growth rates. First, the optimally growing cell is characterised in terms of its elementary modes. We prove an extremum principle: a cell that maximises a metabolic rate uses few Elementary Flux Modes (EFMs, the minimal pathways that support steady-state metabolism). The number of active EFMs is bounded by the number of growth-limiting constraints. Later, this extremum principle is extended in a theory that explicitly accounts for self-fabrication. For this, we had to define the elementary modes that underlie balanced self-fabrication: minimal self-supporting sets of expressed enzymes that we call Elementary Growth Modes (EGMs). It turns out that many of the results for EFMs can be extended to their more general self-fabrication analogue. Where the above extremum principles tell us that few elementary modes are used by a rate-maximising cell, it does not tell us how the cell can find them. Therefore, we also search for an elementary adaptation method. It turns out that stochastic phenotype switching with growth rate dependent switching rates provides an adaptation mechanism that is often competitive with more conventional regulatory-circuitry based mechanisms. The derived theory is applied in two ways. First, the extremum principles are used to review the mathematical fundaments of all optimisation-based explanations of overflow metabolism. Second, a computational tool is presented that enumerates Elementary Conversion Modes. These elementary modes can be computed for larger networks than EFMs and EGMs, and still provide an overview of the metabolic capabilities of an organism
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