Computer-Assisted Proofs and Symbolic Computations

We discuss some main points of computer-assisted proofs based on reliable numerical computations. Such so-called self-validating numerical methods in combination with exact symbolic manipulations result in very powerful mathematical software tools. These tools allow proving mathematical statements (existence of a fixed point, of a solution of an ODE, of a zero of a continuous function, of a global minimum within a given range, etc.) using a digital computer. To validate the assertions of the underlying theorems fast finite precision arithmetic is used. The results are absolutely rigorous. To demonstrate the power of reliable symbolic-numeric computations we investigate in some details the verification of very long periodic orbits of chaotic dynamical systems. The verification is done directly in Maple, e.g. using the Maple Power Tool intpakX or, more efficiently, using the C++ class library C-XSC.

Spectroscopy of Electroproduced Light to Medium Mass Lambda Hypernuclei

The E01-011 experiment at Jefferson Laboratory (JLab) studied light-to-medium mass Λ hypernuclei via the AZ + e → [special characters omitted] + e' + K+ electroproduction reaction. Precise measurement of hypernuclear ground state masses and excitation energies provides information about the nature of hyperon-nucleon interactions. Until recently, hypernuclei were studied at accelerator facilities with intense π+ and K- meson beams. The poor quality of these beams limited the resolution of the hypernuclear excitation energy spectra to about 1.5 MeV (FWHM). This resolution is not sufficient for resolving the rich structure observed in the excitation spectra. By using a high quality electron beam and employing a new high resolution spectrometer system, this study aims to improve the resolution to a few hundred keV with an absolute precision of about 100 keV for excitation energies. In this work the high-resolution excitation spectra of [special characters omitted], and [special characters omitted] hypernuclei are presented. In an attempt to emphasize the presence of the core-excited states we introduced a novel likelihood approach to particle identification (PID) to serve as an alternative to the commonly used standard hard-cut PID. The new method resulted in almost identical missing mass spectra as obtained by the standard approach. An energy resolution of approximately 400–500 keV (FWHM) has been achieved, an unprecedented value in hypernuclear reaction spectroscopy. For [special characters omitted] the core-excited configuration has been clearly observed with significant statistics. The embedded Λ hyperon increases the excitation energies of the 11B nuclear core by 0.5–1 MeV. The [special characters omitted] spectrum has been observed with significant statistics for the first time. The ground state is bound deeper by roughly 400 keV than currently predicted by theory. Indication for the core-excited doublet, which is unbound in the core itself, is observed. The measurement of [special characters omitted] provides the first study of a d-shell hypernucleus with sub-MeV resolution. Discrepancies of up to 2 MeV between measured and theoretically predicted binding energies are found. Similar disagreement exists when comparing to the [special characters omitted] mirror hypernucleus. Also the core-excited structure observed between the major s-, p- and d-shell Λ orbits is not consistent with the available theoretical calculations. In conclusion, the discrepancies found in this study will provide valuable input for the further development of theoretical models.

Estabilização de órbitas periódicas instáveis utilizando controle por modos deslizantes com compensação adaptativa

The chaotic behavior has been widely observed in nature, from physical and chemical phenomena to biological systems, present in many engineering applications and found in both simple mechanical oscillators and advanced communication systems. With regard to mechanical systems, the effects of nonlinearities on the dynamic behavior of the system are often of undesirable character, which has motivated the development of compensation strategies. However, it has been recently found that there are situations in which the richness of nonlinear dynamics becomes attractive. Due to their parametric sensitivity, chaotic systems can suffer considerable changes by small variations on the value of their parameters, which is extremely favorable when we want to give greater flexibility to the controlled system. Hence, we analyze in this work the parametric sensitivity of Duffing oscillator, in particular its unstable periodic orbits and Poincar´e section due to changes in nominal value of the parameter that multiplies the cubic term. Since the amount of energy needed to stabilize Unstable Periodic Orbits is minimum, we analyze the control action needed to control and stabilize such orbits which belong to different versions of the Duffing oscillator. For that we will use a smoothed sliding mode controller with an adaptive compensation term based on Fourier series.

The crowded magnetosphere of the post-common-envelope binary QS Virginis

We present high-speed photometry and high-resolution spectroscopy of the eclipsing post-common-envelope binary QS Virginis (QS Vir). Our Ultraviolet and Visual Echelle Spectrograph (UVES) spectra span multiple orbits over more than a year and reveal the presence of several large prominences passing in front of both the M star and its white dwarf companion, allowing us to triangulate their positions. Despite showing small variations on a time-scale of days, they persist for more than a year and may last decades. One large prominence extends almost three stellar radii from the M star. Roche tomography reveals that the M star is heavily spotted and that these spots are long-lived and in relatively fixed locations, preferentially found on the hemisphere facing the white dwarf. We also determine precise binary and physical parameters for the system. We find that the 14 220 ± 350 K white dwarf is relatively massive, 0.782 ± 0.013 M⊙, and has a radius of 0.010 68 ± 0.000 07 R⊙, consistent with evolutionary models. The tidally distorted M star has a mass of 0.382 ± 0.006 M⊙ and a radius of 0.381 ± 0.003 R⊙, also consistent with evolutionary models. We find that the magnesium absorption line from the white dwarf is broader than expected. This could be due to rotation (implying a spin period of only ˜700 s), or due to a weak (˜100 kG) magnetic field, we favour the latter interpretation. Since the M star's radius is still within its Roche lobe and there is no evidence that it is overinflated, we conclude that QS Vir is most likely a pre-cataclysmic binary just about to become semidetached.

The HARPS-N Rocky Planet Search. I. HD 219134 b: A transiting rocky planet in a multi-planet system at 6.5 pc from the Sun

We know now from radial velocity surveys and transit space missions thatplanets only a few times more massive than our Earth are frequent aroundsolar-type stars. Fundamental questions about their formation history,physical properties, internal structure, and atmosphere composition are,however, still to be solved. We present here the detection of a systemof four low-mass planets around the bright (V = 5.5) and close-by (6.5pc) star HD 219134. This is the first result of the Rocky Planet Searchprogramme with HARPS-N on the Telescopio Nazionale Galileo in La Palma.The inner planet orbits the star in 3.0935 ± 0.0003 days, on aquasi-circular orbit with a semi-major axis of 0.0382 ± 0.0003AU. Spitzer observations allowed us to detect the transit of the planetin front of the star making HD 219134 b the nearest known transitingplanet to date. From the amplitude of the radial velocity variation(2.25 ± 0.22 ms-1) and observed depth of the transit(359 ± 38 ppm), the planet mass and radius are estimated to be4.36 ± 0.44 M⊕ and 1.606 ± 0.086R⊕, leading to a mean density of 5.76 ± 1.09 gcm-3, suggesting a rocky composition. One additional planetwith minimum-mass of 2.78 ± 0.65 M⊕ moves on aclose-in, quasi-circular orbit with a period of 6.767 ± 0.004days. The third planet in the system has a period of 46.66 ± 0.08days and a minimum-mass of 8.94 ± 1.13 M⊕, at0.233 ± 0.002 AU from the star. Its eccentricity is 0.46 ±0.11. The period of this planet is close to the rotational period of thestar estimated from variations of activity indicators (42.3 ± 0.1days). The planetary origin of the signal is, however, thepreferredsolution as no indication of variation at the corresponding frequency isobserved for activity-sensitive parameters. Finally, a fourth additionallonger-period planet of mass of 71 M⊕ orbits the starin 1842 days, on an eccentric orbit (e = 0.34 ± 0.17) at adistance of 2.56 AU.The photometric time series and radial velocities used in this work areavailable in electronic form at the CDS via anonymous ftp to http://cdsarc.u-strasbg.fr(ftp://130.79.128.5) or via http://cdsarc.u-strasbg.fr/viz-bin/qcat?J/A+A/584/A72

The perihelion activity of comet 67P/Churyumov-Gerasimenko as seen by robotic telescopes

Around the time of its perihelion passage the observability of 67P/Churyumov-Gerasimenko from Earth was limited to very short windows each morning from any given site, due to the low solar elongation of the comet. The peak in the comet's activity was therefore difficult to observe with conventionally scheduled telescopes, but was possible where service/queue scheduled mode was possible, and with robotic telescopes. We describe the robotic observations that allowed us to measure the total activity of the comet around perihelion, via photometry (dust) and spectroscopy (gas), and compare these results with the measurements at this time by Rosetta's instruments. The peak of activity occurred approximately two weeks after perihelion. The total brightness (dust) largely followed the predictions from Snodgrass et al. 2013, with no significant change in total activity levels from previous apparitions. The CN gas production rate matched previous orbits near perihelion, but appeared to be relatively low later in the year.

Dynamics of coupled modes in two-section semiconductor lasers with saturable absorption

Semiconductor lasers have the potential to address a number of critical applications in advanced telecommunications and signal processing. These include applications that require pulsed output that can be obtained from self-pulsing and mode-locked states of two-section devices with saturable absorption. Many modern applications place stringent performance requirements on the laser source, and a thorough understanding of the physical mechanisms underlying these pulsed modes of operation is therefore highly desirable. In this thesis, we present experimental measurements and numerical simulations of a variety of self-pulsation phenomena in two-section semiconductor lasers with saturable absorption. Our theoretical and numerical results will be based on rate equations for the field intensities and the carrier densities in the two sections of the device, and we establish typical parameter ranges and assess the level of agreement with experiment that can be expected from our models. For each of the physical examples that we consider, our model parameters are consistent with the physical net gain and absorption of the studied devices. Following our introductory chapter, the first system that we consider is a two-section Fabry-Pérot laser. This example serves to introduce our method for obtaining model parameters from the measured material dispersion, and it also allows us to present a detailed discussion of the bifurcation structure that governs the appearance of selfpulsations in two-section devices. In the following two chapters, we present two distinct examples of experimental measurements from dual-mode two-section devices. In each case we have found that single mode self-pulsations evolve into complex coupled dualmode states following a characteristic series of bifurcations. We present optical and mode resolved power spectra as well as a series of characteristic intensity time traces illustrating this progression for each example. Using the results from our study of a twosection Fabry-Pérot device as a guide, we find physically appropriate model parameters that provide qualitative agreement with our experimental results. We highlight the role played by material dispersion and the underlying single mode self-pulsing orbits in determining the observed dynamics, and we use numerical continuation methods to provide a global picture of the governing bifurcation structure. In our concluding chapter we summarise our work, and we discuss how the presented results can inform the development of optimised mode-locked lasers for performance applications in integrated optics.

The SOPHIE search for northern extrasolar planets: X. Detection and characterization of giant planets by the dozen

We present new radial velocity measurements of eight stars that were secured with the spectrograph SOPHIE at the 193 cm telescope of the Haute-Provence Observatory. The measurements allow detecting and characterizing new giant extrasolar planets. The host stars are dwarfs of spectral types between F5 and K0 and magnitudes of between 6.7 and 9.6; the planets have minimum masses Mp sin i of between 0.4 to 3.8 MJup and orbitalperiods of several days to several months. The data allow only single planets to be discovered around the first six stars (HD 143105, HIP 109600, HD 35759, HIP 109384, HD 220842, and HD 12484), but one of them shows the signature of an additional substellar companion in the system. The seventh star, HIP 65407, allows the discovery of two giant planets that orbit just outside the 12:5 resonance in weak mutual interaction. The last star, HD 141399, was already known to host a four-planet system; our additional data and analyses allow new constraints to be set on it. We present Keplerian orbits of all systems, together with dynamical analyses of the two multi-planet systems. HD 143105 is one of the brightest stars known to host a hot Jupiter, which could allow numerous follow-up studies to be conducted even though this is not a transiting system. The giant planets HIP 109600b, HIP 109384b, and HD 141399c are located in the habitable zone of their host star.

Close-in planets around giant stars: lack of hot-Jupiters and prevalence of multiplanetary systems

Extrasolar planets abound in almost any possible configuration. However, until five years ago, there was a lack of planets orbiting closer than 0.5 au to giant or subgiant stars. Since then, recent detections have started to populated this regime by confirming 13 planetary systems. We discuss the properties of these systems in terms of their formation and evolution off the main sequence. Interestingly, we find that 70.0 ± 6.6% of the planets in this regime are inner components of multiplanetary systems. This value is 4.2σ higher than for main-sequence hosts, which we find to be 42.4 ± 0.1%. The properties of the known planets seem to indicate that the closest-in planets (a< 0.06 au) to main-sequence stars are massive (i.e., hot Jupiters) and isolated and that they are subsequently engulfed by their host as it evolves to the red giant branch, leaving only the predominant population of multiplanetary systems in orbits 0.06

Secular and tidal evolution of circumbinary systems

We investigate the secular dynamics of three-body circumbinary systems under the effect of tides. We use the octupolar non-restricted approximation for the orbital interactions, general relativity corrections, the quadrupolar approximation for the spins, and the viscous linear model for tides. We derive the averaged equations of motion in a simplified vectorial formalism, which is suitable to model the long-term evolution of a wide variety of circumbinary systems in very eccentric and inclined orbits. In particular, this vectorial approach can be used to derive constraints for tidal migration, capture in Cassini states, and stellar spin–orbit misalignment. We show that circumbinary planets with initial arbitrary orbital inclination can become coplanar through a secular resonance between the precession of the orbit and the precession of the spin of one of the stars. We also show that circumbinary systems for which the pericenter of the inner orbit is initially in libration present chaotic motion for the spins and for the eccentricity of the outer orbit. Because our model is valid for the non-restricted problem, it can also be applied to any three-body hierarchical system such as star–planet–satellite systems and triple stellar systems.

Neuronal networks with gap junctions: A study of piece-wise linear planar neuron models

The presence of gap junction coupling among neurons of the central nervous systems has been appreciated for some time now. In recent years there has been an upsurge of interest from the mathematical community in understanding the contribution of these direct electrical connections between cells to large-scale brain rhythms. Here we analyze a class of exactly soluble single neuron models, capable of producing realistic action potential shapes, that can be used as the basis for understanding dynamics at the network level. This work focuses on planar piece-wise linear models that can mimic the firing response of several different cell types. Under constant current injection the periodic response and phase response curve (PRC) is calculated in closed form. A simple formula for the stability of a periodic orbit is found using Floquet theory. From the calculated PRC and the periodic orbit a phase interaction function is constructed that allows the investigation of phase-locked network states using the theory of weakly coupled oscillators. For large networks with global gap junction connectivity we develop a theory of strong coupling instabilities of the homogeneous, synchronous and splay state. For a piece-wise linear caricature of the Morris-Lecar model, with oscillations arising from a homoclinic bifurcation, we show that large amplitude oscillations in the mean membrane potential are organized around such unstable orbits.

Instabilities in threshold-diffusion equations with delay

The introduction of delays into ordinary or partial differential equation models is well known to facilitate the production of rich dynamics ranging from periodic solutions through to spatio-temporal chaos. In this paper we consider a class of scalar partial differential equations with a delayed threshold nonlinearity which admits exact solutions for equilibria, periodic orbits and travelling waves. Importantly we show how the spectra of periodic and travelling wave solutions can be determined in terms of the zeros of a complex analytic function. Using this as a computational tool to determine stability we show that delays can have very different effects on threshold systems with negative as opposed to positive feedback. Direct numerical simulations are used to confirm our bifurcation analysis, and to probe some of the rich behaviour possible for mixed feedback.

Relação de alcance em dígrafos transitivos e a propriedade Z

Dissertação (mestrado)—Universidade de Brasília, Instituto de Ciências Exatas, Departamento de Matemática, 2016.

Software for Designing Custom Orbital-Craniofacial Implant Plans

Purpose: Custom cranio-orbital implants have been shown to achieve better performance than their hand-shaped counterparts by restoring skull anatomy more accurately and by reducing surgery time. Designing a custom implant involves reconstructing a model of the patient's skull using their computed tomography (CT) scan. The healthy side of the skull model, contralateral to the damaged region, can then be used to design an implant plan. Designing implants for areas of thin bone, such as the orbits, is challenging due to poor CT resolution of bone structures. This makes preoperative design time-intensive since thin bone structures in CT data must be manually segmented. The objective of this thesis was to research methods to accurately and efficiently design cranio-orbital implant plans, with a focus on the orbits, and to develop software that integrates these methods. Methods: The software consists of modules that use image and surface restoration approaches to enhance both the quality of CT data and the reconstructed model. It enables users to input CT data, and use tools to output a skull model with restored anatomy. The skull model can then be used to design the implant plan. The software was designed using 3D Slicer, an open-source medical visualization platform. It was tested on CT data from thirteen patients. Results: The average time it took to create a skull model with restored anatomy using our software was 0.33 hours ± 0.04 STD. In comparison, the design time of the manual segmentation method took between 3 and 6 hours. To assess the structural accuracy of the reconstructed models, CT data from the thirteen patients was used to compare the models created using our software with those using the manual method. When registering the skull models together, the difference between each set of skulls was found to be 0.4 mm ± 0.16 STD. Conclusions: We have developed a software to design custom cranio-orbital implant plans, with a focus on thin bone structures. The method described decreases design time, and is of similar accuracy to the manual method.

Autonomous formation flying: unified control and collision avoidance methods for close manoeuvring spacecraft

The idea of spacecraft formations, flying in tight configurations with maximum baselines of a few hundred meters in low-Earth orbits, has generated widespread interest over the last several years. Nevertheless, controlling the movement of spacecraft in formation poses difficulties, such as in-orbit high-computing demand and collision avoidance capabilities, which escalate as the number of units in the formation is increased and complicated nonlinear effects are imposed to the dynamics, together with uncertainty which may arise from the lack of knowledge of system parameters. These requirements have led to the need of reliable linear and nonlinear controllers in terms of relative and absolute dynamics. The objective of this thesis is, therefore, to introduce new control methods to allow spacecraft in formation, with circular/elliptical reference orbits, to efficiently execute safe autonomous manoeuvres. These controllers distinguish from the bulk of literature in that they merge guidance laws never applied before to spacecraft formation flying and collision avoidance capacities into a single control strategy. For this purpose, three control schemes are presented: linear optimal regulation, linear optimal estimation and adaptive nonlinear control. In general terms, the proposed control approaches command the dynamical performance of one or several followers with respect to a leader to asymptotically track a time-varying nominal trajectory (TVNT), while the threat of collision between the followers is reduced by repelling accelerations obtained from the collision avoidance scheme during the periods of closest proximity. Linear optimal regulation is achieved through a Riccati-based tracking controller. Within this control strategy, the controller provides guidance and tracking toward a desired TVNT, optimizing fuel consumption by Riccati procedure using a non-infinite cost function defined in terms of the desired TVNT, while repelling accelerations generated from the CAS will ensure evasive actions between the elements of the formation. The relative dynamics model, suitable for circular and eccentric low-Earth reference orbits, is based on the Tschauner and Hempel equations, and includes a control input and a nonlinear term corresponding to the CAS repelling accelerations. Linear optimal estimation is built on the forward-in-time separation principle. This controller encompasses two stages: regulation and estimation. The first stage requires the design of a full state feedback controller using the state vector reconstructed by means of the estimator. The second stage requires the design of an additional dynamical system, the estimator, to obtain the states which cannot be measured in order to approximately reconstruct the full state vector. Then, the separation principle states that an observer built for a known input can also be used to estimate the state of the system and to generate the control input. This allows the design of the observer and the feedback independently, by exploiting the advantages of linear quadratic regulator theory, in order to estimate the states of a dynamical system with model and sensor uncertainty. The relative dynamics is described with the linear system used in the previous controller, with a control input and nonlinearities entering via the repelling accelerations from the CAS during collision avoidance events. Moreover, sensor uncertainty is added to the control process by considering carrier-phase differential GPS (CDGPS) velocity measurement error. An adaptive control law capable of delivering superior closed-loop performance when compared to the certainty-equivalence (CE) adaptive controllers is finally presented. A novel noncertainty-equivalence controller based on the Immersion and Invariance paradigm for close-manoeuvring spacecraft formation flying in both circular and elliptical low-Earth reference orbits is introduced. The proposed control scheme achieves stabilization by immersing the plant dynamics into a target dynamical system (or manifold) that captures the desired dynamical behaviour. They key feature of this methodology is the addition of a new term to the classical certainty-equivalence control approach that, in conjunction with the parameter update law, is designed to achieve adaptive stabilization. This parameter has the ultimate task of shaping the manifold into which the adaptive system is immersed. The performance of the controller is proven stable via a Lyapunov-based analysis and Barbalat’s lemma. In order to evaluate the design of the controllers, test cases based on the physical and orbital features of the Prototype Research Instruments and Space Mission Technology Advancement (PRISMA) are implemented, extending the number of elements in the formation into scenarios with reconfigurations and on-orbit position switching in elliptical low-Earth reference orbits. An extensive analysis and comparison of the performance of the controllers in terms of total Δv and fuel consumption, with and without the effects of the CAS, is presented. These results show that the three proposed controllers allow the followers to asymptotically track the desired nominal trajectory and, additionally, those simulations including CAS show an effective decrease of collision risk during the performance of the manoeuvre.