12 research outputs found

    Spectral Analysis of Complex Dynamical Systems

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    The spectrum of any differential equation or a system of differential equations is related to several important properties about the problem and its subsequent solution. So much information is held within the spectrum of a problem that there is an entire field devoted to it; spectral analysis. In this thesis, we perform spectral analysis on two separate complex dynamical systems. The vibrations along a continuous string or a string with beads on it are the governed by the continuous or discrete wave equation. We derive a small-vibrations model for multi-connected continuous strings that lie in a plane. We show that lateral vibrations of such strings can be decoupled from their in-plane vibrations. We then study the eigenvalue problem originating from the lateral vibrations. We show that, unlike the well-known one string vibrations case, the eigenvalues in a multi-string vibrating system do not have to be simple. Moreover we prove that the multiplicities of the eigenvalues depend on the symmetry of the model and on the total number of the connected strings [50]. We also apply Nevanlinna functions theory to characterize the spectra and to solve the inverse problem for a discrete multi-string system in a more general setting than it was done in [71],[73], [22], [69]-[72]. We also represent multi-string vibrating systems using a coupling of non-densely defined symmetric operators acting in the infinite dimensional Hilbert space. This coupling is defined by a special set of boundary operators acting in finite dimensional Krein space (the space with indefinite inner product). The main results of this research are published in [50]. The Hypothalamic Pituitary Adrenal (HPA) axis responds to physical and mental challenge to maintain homeostasis in part by controlling the body’s cortisol level. Dysregulation of the HPA axis is implicated in numerous stress-related diseases. For a structured model of the HPA axis that includes the glucocorticoid receptor but does not take into account the system response delay, we first perform rigorous stability analysis of all multi-parametric steady states and secondly, by construction of a Lyapunov functional, we prove nonlinear asymptotic stability for some of multi-parametric steady states. We then take into account the additional effects of the time delay parameter on the stability of the HPA axis system. Finally we prove the existence of periodic solutions for the HPA axis system. The main results of this research are published in [51]

    Operations research: from computational biology to sensor network

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    In this dissertation we discuss the deployment of combinatorial optimization methods for modeling and solve real life problemS, with a particular emphasis to two biological problems arising from a common scenario: the reconstruction of the three-dimensional shape of a biological molecule from Nuclear Magnetic Resonance (NMR) data. The fi rst topic is the 3D assignment pathway problem (APP) for a RNA molecule. We prove that APP is NP-hard, and show a formulation of it based on edge-colored graphs. Taking into account that interactions between consecutive nuclei in the NMR spectrum are diff erent according to the type of residue along the RNA chain, each color in the graph represents a type of interaction. Thus, we can represent the sequence of interactions as the problem of fi nding a longest (hamiltonian) path whose edges follow a given order of colors (i.e., the orderly colored longest path). We introduce three alternative IP formulations of APP obtained with a max flow problem on a directed graph with packing constraints over the partitions, which have been compared among themselves. Since the last two models work on cyclic graphs, for them we proposed an algorithm based on the solution of their relaxation combined with the separation of cycle inequalities in a Branch & Cut scheme. The second topic is the discretizable distance geometry problem (DDGP), which is a formulation on discrete search space of the well-known distance geometry problem (DGP). The DGP consists in seeking the embedding in the space of a undirected graph, given a set of Euclidean distances between certain pairs of vertices. DGP has two important applications: (i) fi nding the three dimensional conformation of a molecule from a subset of interatomic distances, called Molecular Distance Geometry Problem, and (ii) the Sensor Network Localization Problem. We describe a Branch & Prune (BP) algorithm tailored for this problem, and two versions of it solving the DDGP both in protein modeling and in sensor networks localization frameworks. BP is an exact and exhaustive combinatorial algorithm that examines all the valid embeddings of a given weighted graph G=(V,E,d), under the hypothesis of existence of a given order on V. By comparing the two version of BP to well-known algorithms we are able to prove the e fficiency of BP in both contexts, provided that the order imposed on V is maintained

    The Real Topological String on a local Calabi-Yau

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    We study the topological string on local P2 with O-plane and D-brane at its real locus, using three complementary techniques. In the A-model, we refine localization on the moduli space of maps with respect to the torus action preserved by the anti-holomorphic involution. This leads to a computation of open and unoriented Gromov-Witten invariants that can be applied to any toric Calabi-Yau with involution. We then show that the full topological string amplitudes can be reproduced within the topological vertex formalism. We obtain the real topological vertex with trivial fixed leg. Finally, we verify that the same results derive in the B-model from the extended holomorphic anomaly equation, together with appropriate boundary conditions. The expansion at the conifold exhibits a gap structure that belongs to a so far unidentified universality class.Comment: 37 pages, 4 figure

    Constrained Planarity and Augmentation Problems

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    A clustered graph C=(G,T) consists of an undirected graph G and a rooted tree T in which the leaves of T correspond to the vertices of G=(V,E). Each vertex m in T corresponds to a subset of the vertices of the graph called ``cluster''. c-planarity is a natural extension of graph planarity for clustered graphs, and plays an important role in automatic graph drawing. The complexity status of c-planarity testing is unknown. It has been shown by Dahlhaus, Eades, Feng, Cohen that c-planarity can be tested in linear time for c-connected graphs, i.e., graphs in which the cluster induced subgraphs are connected. In the first part of the thesis, we provide a polynomial time algorithms for c-planarity testing of specific planar clustered graphs: Graphs for which - all nodes corresponding to the non-c-connected clusters lie on the same path in T starting at the root of T, or graphs in which for each non-connected cluster its super-cluster and all its siblings in T are connected, - for all clusters m G-G(m) is connected. The algorithms are based on the concepts for the subgraph induced planar connectivity augmentation problem, also presented in this thesis. Furthermore, we give some characterizations of c-planar clustered graphs using minors and dual graphs and introduce a c-planar augmentation method. Parts II deals with edge deletion and bimodal crossing minimization. We prove that the maximum planar subgraph problem remains NP-complete even for non-planar graphs without a minor isomorphic to either K(5) or K(3,3), respectively. Further, we investigate the problem of finding a minimum weighted set of edges whose removal results in a graph without minors that are contractible onto a prespecified set of vertices. Finally, we investigate the problem of drawing a directed graph in two dimensions with a minimal number of crossings such that for every node the incoming and outgoing edges are separated consecutively in the cyclic adjacency lists. It turns out that the planarization method can be adapted such that the number of crossings can be expected to grow only slightly for practical instances

    Aspects of hadron and instatnton physics in lattice quantum field theories

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    Thesis (Ph. D.)--Massachusetts Institute of Technology, Dept. of Physics, 1997.Includes bibliographical references (p. 103-110).by Andrew Pochinsky.Ph.D

    Real Mirror Symmetry and The Real Topological String

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    This thesis is concerned with real mirror symmetry, that is, mirror symmetry for a Calabi-Yau 3-fold background with a D-brane on a special Lagrangian 3-cycle defined by the real locus of an anti-holomorphic involution. More specifically, we will study real mirror symmetry by means of compact 1-parameter Calabi-Yau hypersurfaces in weighted projective space (at tree-level) and non-compact local P2 (at higher genus). For the compact models, we identify mirror pairs of D-brane configurations in weighted projective space, derive the corresponding inhomogeneous Picard-Fuchs equations, and solve for the domainwall tensions as analytic functions over moduli space, thereby collecting evidence for real mirror symmetry at tree-level. A major outcome of this part is the prediction of the number of disk instantons ending on the D-brane for these models. Further, we study real mirror symmetry at higher genus using local P2. For that, we utilize the real topological string, that is, the topological string on a background with O-plane and D-brane on top. In detail, we calculate topological amplitudes using three complementary techniques. In the A-model, we refine localization on the moduli space of maps with respect to the torus action preserved by the anti-holomorphic involution. This leads to a computation of open and unoriented Gromov-Witten invariants that can be applied to any toric Calabi-Yau with involution. We then show that the full topological string amplitudes can be reproduced within the topological vertex formalism. Especially, we obtain the real topological vertex with trivial fixed leg. Finally, we verify that the same results arise in the B-model from the extended holomorphic anomaly equations, together with appropriate boundary conditions, thereby establishing local real mirror symmetry at higher genus. Significant outcomes of this part are the derivation of real Gopakumar-Vafa invariants at high Euler number and degree for local P2 and the discovery of a new kind of gap structure of the closed and unoriented topological amplitudes at the conifold point in moduli space

    The model of Solid Inflation and its outcomes

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    The recent analysis of data collected by the Planck satellite constrains inflationary scenarios through the observational testing of specific primordial bispectra predicted by different inflationary theories. When considering an inflationary scenario, quantum fluctuations of primordial field(s), occurred during inflation and stretched to macroscopic scales, induced fluctuations in the space-time metric. These last fluctuations can be classified as scalar, vector and tensor fluctuations and left a characteristic signature on the Cosmic Microwave Background (CMB). Specifically, the cubic interactions experienced by these fields generated primordial bispectra, which are the Fourier counterparts of three-point expectation values. These primordial bispectra are linked to non-Gaussianities in the statistics of the CMB anisotropies. For this reason, the theoretical study of primordial bispectra predicted by inflationary models is essential for the comprehension of the physics of the very early Universe. The model of Solid Inflation shows differentiating features from other inflationary models such as non-decaying vector perturbations, a mass for gravitational waves and a model-specific scalar bispectrum with non-trivial angular dependence. The aim of this thesis is to integrate the study of essential theoretical elements to augment the range of computed primordial bispectra within the model of Solid Inflation. Specifically, six bispectra involving scalar, vector, tensor perturbations predicted by the model of Solid Inflation will be presented. These bispectra have never appeared in literature and show an enhancement represented by the minimum number of e-foldings at which the CMB modes left the horizon

    Studia Scientiarum Mathematicarum Hungarica

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    Advances in Robot Kinematics : Proceedings of the 15th international conference on Advances in Robot Kinematics

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    International audienceThe motion of mechanisms, kinematics, is one of the most fundamental aspect of robot design, analysis and control but is also relevant to other scientific domains such as biome- chanics, molecular biology, . . . . The series of books on Advances in Robot Kinematics (ARK) report the latest achievement in this field. ARK has a long history as the first book was published in 1991 and since then new issues have been published every 2 years. Each book is the follow-up of a single-track symposium in which the participants exchange their results and opinions in a meeting that bring together the best of world’s researchers and scientists together with young students. Since 1992 the ARK symposia have come under the patronage of the International Federation for the Promotion of Machine Science-IFToMM.This book is the 13th in the series and is the result of peer-review process intended to select the newest and most original achievements in this field. For the first time the articles of this symposium will be published in a green open-access archive to favor free dissemination of the results. However the book will also be o↔ered as a on-demand printed book.The papers proposed in this book show that robot kinematics is an exciting domain with an immense number of research challenges that go well beyond the field of robotics.The last symposium related with this book was organized by the French National Re- search Institute in Computer Science and Control Theory (INRIA) in Grasse, France
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