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This thesis presents a comprehensive approach to enhancing the existing patrimony of SPACEBEL’s physical models for deep space mission simulations. This thesis focuses on two critical areas: the integration of both large and small celestial bodies into mission simulations and the accurate modeling of gravitational fields for small celestial bodies with complex shapes.

The first part addresses the accurate integration of celestial bodies, including both large entities like planets and smaller ones such as asteroids and comets. It focuses on predicting their ephemerides using Chebyshev polynomial interpolation, which provides efficient and precise results over extended mission timescales. Additionally, models were developed for defining and managing reference frames and calculating the spheres of influence (SOI) of celestial bodies, which are crucial for accurate mission simulation and optimising computational time. These models ensure consistent and precise simulations across different reference frames and help define regions where gravitational influences need to be considered. These models have been validated through their application to several well-known missions, including Voyager 2’s journey through the solar system and the Hera mission to the Didymos binary asteroid system. These validations demonstrate the practical utility of the models in enhancing the accuracy of mission simulations, thereby supporting SPACEBEL’s goals in deep space mission simulation software development.

The second part of the thesis explores the precise modeling of gravitational fields for irregularly shaped celestial bodies, such as those in the Didymos-Dimorphos binary system. In this thesis, the polyhedron method was employed. This approach models a celestial body’s shape using polyhedral facets, offering a more accurate representation of its gravitational field. Validation of this approach was carried out by comparing the gravitational field results at the surface of several asteroids with known data. The results show that the polyhedron model provides a more detailed and realistic gravitational field, which is crucial for mission scenarios involving close proximity operations and landings. Additionally, this part of the thesis explores the implications of these gravitational models for spacecraft trajectory propagation in such complex dynamical environments. The findings offer valuable insights into the precision needed for deep space missions simulation, particularly when dealing with the complex gravitational fields of irregular bodies.

Overall, the models developed in this thesis offer valuable advancements for SPACEBEL’s patrimony of physical models, providing enhanced tools for future deep space mission simulation.

Simulation --- Deep Space mission --- Solar system planets --- Polyhedron method --- Irregularly shaped bodies --- Discrete Event Simulation --- SMP2 standard --- Fine Gravitational Field --- Ingénierie, informatique & technologie > Ingénierie aérospatiale

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Heavenly Mathematics traces the rich history of spherical trigonometry, revealing how the cultures of classical Greece, medieval Islam, and the modern West used this forgotten art to chart the heavens and the Earth. Once at the heart of astronomy and ocean-going navigation for two millennia, the discipline was also a mainstay of mathematics education for centuries and taught widely until the 1950's. Glen Van Brummelen explores this exquisite branch of mathematics and its role in ancient astronomy, geography, and cartography; Islamic religious rituals; celestial navigation; polyhedra; stereographic projection; and more. He conveys the sheer beauty of spherical trigonometry, providing readers with a new appreciation of its elegant proofs and often surprising conclusions. Heavenly Mathematics is illustrated throughout with stunning historical images and informative drawings and diagrams. This unique compendium also features easy-to-use appendixes as well as exercises that originally appeared in textbooks from the eighteenth to the early twentieth centuries.

Spherical trigonometry. --- Trigonometry. --- Trig (Trigonometry) --- Geometry --- Mathematics --- Trigonometry, Spherical --- Trigonometry --- Abū 'l-Wafā. --- Abū Mahmūd al-Khujandī. --- Abū Nasr Mansūr ibn 'Alī ibn 'Irāq. --- Abū Sahl al-Kūhī. --- Albert Girard. --- B. M. Brown. --- Cesàro method. --- Christopher Columbus. --- Claudius Ptolemy. --- Earth. --- Elements. --- Georg Rheticus. --- Giuseppe Cesàro. --- Hipparchus of Rhodes. --- Islam. --- Islamic religious rituals. --- John Harrison. --- John Napier. --- Law of Cosines. --- Law of Sines. --- Leonhard Euler. --- Mathematical Collection. --- Mecca. --- Menelaus of Alexandria. --- Menelaus's Theorem. --- Moon. --- Napier's Rules. --- Opus palatinum. --- Planisphere. --- Ptolemy. --- Pythagorean Theorem. --- Rule of Four Quantities. --- Sphaerica. --- Sun. --- acute-angled triangle. --- angle. --- area. --- astrolabe. --- astronomical triangle. --- astronomy. --- cartography. --- celestial motion. --- celestial sphere. --- chronometer. --- classical Greece. --- dead reckoning. --- ecliptic. --- equatorial coordinates. --- geography. --- locality principle. --- logarithms. --- marteloio. --- mathematics. --- method of Saint Hilaire. --- navigation. --- oblique triangle. --- pentagramma mirificum. --- planar Law of Sines. --- plane trigonometry. --- planets. --- polygon. --- polyhedron. --- qibla. --- regular polyhedron. --- right-angled triangle. --- rising time. --- sphere. --- spherical Law of Sines. --- spherical astronomy. --- spherical geometry. --- spherical triangle. --- spherical trigonometry. --- star. --- stars. --- stereographic projection. --- table of sine. --- theorems. --- triangle. --- trigonometric table. --- trigonometry.

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This is the first book to present a complete characterization of Stein-Tomas type Fourier restriction estimates for large classes of smooth hypersurfaces in three dimensions, including all real-analytic hypersurfaces. The range of Lebesgue spaces for which these estimates are valid is described in terms of Newton polyhedra associated to the given surface.Isroil Ikromov and Detlef Müller begin with Elias M. Stein's concept of Fourier restriction and some relations between the decay of the Fourier transform of the surface measure and Stein-Tomas type restriction estimates. Varchenko's ideas relating Fourier decay to associated Newton polyhedra are briefly explained, particularly the concept of adapted coordinates and the notion of height. It turns out that these classical tools essentially suffice already to treat the case where there exist linear adapted coordinates, and thus Ikromov and Müller concentrate on the remaining case. Here the notion of r-height is introduced, which proves to be the right new concept. They then describe decomposition techniques and related stopping time algorithms that allow to partition the given surface into various pieces, which can eventually be handled by means of oscillatory integral estimates. Different interpolation techniques are presented and used, from complex to more recent real methods by Bak and Seeger.Fourier restriction plays an important role in several fields, in particular in real and harmonic analysis, number theory, and PDEs. This book will interest graduate students and researchers working in such fields.

Hypersurfaces. --- Polyhedra. --- Surfaces, Algebraic. --- Fourier analysis. --- Analysis, Fourier --- Mathematical analysis --- Polyhedral figures --- Polyhedrons --- Geometry, Solid --- Shapes --- Algebraic surfaces --- Geometry, Algebraic --- Hyperspace --- Surfaces --- Airy cone. --- Airy-type analysis. --- Airy-type decompositions. --- Fourier decay. --- Fourier integral. --- Fourier restriction estimate. --- Fourier restriction problem. --- Fourier restriction theorem. --- Fourier restriction. --- Fourier transform. --- Greenleaf's restriction. --- Lebesgue spaces. --- LittlewoodАaley decomposition. --- LittlewoodАaley theory. --- Newton polyhedra. --- Newton polyhedral. --- Newton polyhedron. --- SteinДomas-type Fourier restriction. --- auxiliary results. --- complex interpolation. --- dyadic decomposition. --- dyadic decompositions. --- dyadic domain decompositions. --- endpoint estimates. --- endpoint result. --- improved estimates. --- interpolation arguments. --- interpolation theorem. --- invariant description. --- linear coordinates. --- linearly adapted coordinates. --- normalized measures. --- normalized rescale measures. --- one-dimensional oscillatory integrals. --- open cases. --- operator norms. --- phase functions. --- preparatory results. --- principal root jet. --- propositions. --- r-height. --- real interpolation. --- real-analytic hypersurface. --- refined Airy-type analysis. --- restriction estimates. --- restriction. --- smooth hypersurface. --- smooth hypersurfaces. --- spectral localization. --- stopping-time algorithm. --- sublevel type. --- thin sets. --- three dimensions. --- transition domains. --- uniform bounds. --- van der Corput-type estimates.

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The intention of the authors is to examine the relationship between piecewise linear structure and differential structure: a relationship, they assert, that can be understood as a homotopy obstruction theory, and, hence, can be studied by using the traditional techniques of algebraic topology.Thus the book attacks the problem of existence and classification (up to isotopy) of differential structures compatible with a given combinatorial structure on a manifold. The problem is completely "solved" in the sense that it is reduced to standard problems of algebraic topology.The first part of the book is purely geometrical; it proves that every smoothing of the product of a manifold M and an interval is derived from an essentially unique smoothing of M. In the second part this result is used to translate the classification of smoothings into the problem of putting a linear structure on the tangent microbundle of M. This in turn is converted to the homotopy problem of classifying maps from M into a certain space PL/O. The set of equivalence classes of smoothings on M is given a natural abelian group structure.

Algebraic topology --- Piecewise linear topology --- Manifolds (Mathematics) --- Topologie linéaire par morceaux --- Variétés (Mathématiques) --- 515.16 --- PL topology --- Topology --- Geometry, Differential --- Topology of manifolds --- Piecewise linear topology. --- Manifolds (Mathematics). --- 515.16 Topology of manifolds --- Topologie linéaire par morceaux --- Variétés (Mathématiques) --- Affine transformation. --- Approximation. --- Associative property. --- Bijection. --- Bundle map. --- Classification theorem. --- Codimension. --- Coefficient. --- Cohomology. --- Commutative property. --- Computation. --- Convex cone. --- Convolution. --- Corollary. --- Counterexample. --- Diffeomorphism. --- Differentiable function. --- Differentiable manifold. --- Differential structure. --- Dimension. --- Direct proof. --- Division by zero. --- Embedding. --- Empty set. --- Equivalence class. --- Equivalence relation. --- Euclidean space. --- Existential quantification. --- Exponential map (Lie theory). --- Fiber bundle. --- Fibration. --- Functor. --- Grassmannian. --- H-space. --- Homeomorphism. --- Homotopy. --- Integral curve. --- Inverse problem. --- Isomorphism class. --- K0. --- Linearization. --- Manifold. --- Mathematical induction. --- Milnor conjecture. --- Natural transformation. --- Neighbourhood (mathematics). --- Normal bundle. --- Obstruction theory. --- Open set. --- Partition of unity. --- Piecewise linear. --- Polyhedron. --- Reflexive relation. --- Regular map (graph theory). --- Sheaf (mathematics). --- Smoothing. --- Smoothness. --- Special case. --- Submanifold. --- Tangent bundle. --- Tangent vector. --- Theorem. --- Topological manifold. --- Topological space. --- Topology. --- Transition function. --- Transitive relation. --- Vector bundle. --- Vector field. --- Variétés topologiques

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This book offers a self-contained account of the 3-manifold invariants arising from the original Jones polynomial. These are the Witten-Reshetikhin-Turaev and the Turaev-Viro invariants. Starting from the Kauffman bracket model for the Jones polynomial and the diagrammatic Temperley-Lieb algebra, higher-order polynomial invariants of links are constructed and combined to form the 3-manifold invariants. The methods in this book are based on a recoupling theory for the Temperley-Lieb algebra. This recoupling theory is a q-deformation of the SU(2) spin networks of Roger Penrose. The recoupling theory is developed in a purely combinatorial and elementary manner. Calculations are based on a reformulation of the Kirillov-Reshetikhin shadow world, leading to expressions for all the invariants in terms of state summations on 2-cell complexes. Extensive tables of the invariants are included. Manifolds in these tables are recognized by surgery presentations and by means of 3-gems (graph encoded 3-manifolds) in an approach pioneered by Sostenes Lins. The appendices include information about gems, examples of distinct manifolds with the same invariants, and applications to the Turaev-Viro invariant and to the Crane-Yetter invariant of 4-manifolds.

Drie-menigvuldigheden (Topologie) --- Knopentheorie --- Knot theory --- Noeuds [Theorie des ] --- Three-manifolds (Topology) --- Trois-variétés (Topologie) --- Knot theory. --- Algebraic topology --- Invariants --- Mathematics --- Invariants (Mathematics) --- Invariants. --- 3-manifolds (Topology) --- Manifolds, Three dimensional (Topology) --- Three-dimensional manifolds (Topology) --- Low-dimensional topology --- Topological manifolds --- Knots (Topology) --- 3-manifold. --- Addition. --- Algorithm. --- Ambient isotopy. --- Axiom. --- Backslash. --- Barycentric subdivision. --- Bijection. --- Bipartite graph. --- Borromean rings. --- Boundary parallel. --- Bracket polynomial. --- Calculation. --- Canonical form. --- Cartesian product. --- Cobordism. --- Coefficient. --- Combination. --- Commutator. --- Complex conjugate. --- Computation. --- Connected component (graph theory). --- Connected sum. --- Cubic graph. --- Diagram (category theory). --- Dimension. --- Disjoint sets. --- Disjoint union. --- Elaboration. --- Embedding. --- Equation. --- Equivalence class. --- Explicit formula. --- Explicit formulae (L-function). --- Factorial. --- Fundamental group. --- Graph (discrete mathematics). --- Graph embedding. --- Handlebody. --- Homeomorphism. --- Homology (mathematics). --- Identity element. --- Intersection form (4-manifold). --- Inverse function. --- Jones polynomial. --- Kirby calculus. --- Line segment. --- Linear independence. --- Matching (graph theory). --- Mathematical physics. --- Mathematical proof. --- Mathematics. --- Maxima and minima. --- Monograph. --- Natural number. --- Network theory. --- Notation. --- Numerical analysis. --- Orientability. --- Orthogonality. --- Pairing. --- Pairwise. --- Parametrization. --- Parity (mathematics). --- Partition function (mathematics). --- Permutation. --- Poincaré conjecture. --- Polyhedron. --- Quantum group. --- Quantum invariant. --- Recoupling. --- Recursion. --- Reidemeister move. --- Result. --- Roger Penrose. --- Root of unity. --- Scientific notation. --- Sequence. --- Significant figures. --- Simultaneous equations. --- Smoothing. --- Special case. --- Sphere. --- Spin network. --- Summation. --- Symmetric group. --- Tetrahedron. --- The Geometry Center. --- Theorem. --- Theory. --- Three-dimensional space (mathematics). --- Time complexity. --- Tubular neighborhood. --- Two-dimensional space. --- Vector field. --- Vector space. --- Vertex (graph theory). --- Winding number. --- Writhe.