The Polyhedral Geometry of Partially Ordered Sets

Pairs of polyhedra connected by a piecewise-linear bijection appear in different fields of mathematics. The model example of this situation are the order and chain polytopes introduced by Stanley in, whose defining inequalities are given by a finite partially ordered set. The two polytopes have different face lattices, but admit a volume and lattice point preserving piecewise-linear bijection called the transfer map. Other areas like representation theory and enumerative combinatorics provide more examples of pairs of polyhedra that are similar to order and chain polytopes. The goal of this thesis is to analyze this phenomenon and move towards a common theoretical framework describing these polyhedra and their piecewise-linear bijections. A first step in this direction was done by Ardila, Bliem and Salazar, where the authors generalize order and chain polytopes by replacing the defining data with a marked poset. These marked order and chain polytopes still admit a piecewise-linear transfer map and include the Gelfand-Tsetlin and Feigin-Fourier-Littelmann-Vinberg polytopes from representation theory among other examples. We consider more polyhedra associated to marked posets and obtain new results on their face structure and combinatorial interplay. Other examples found in the literature bear resemblance to these marked poset polyhedra but do not admit a description as such. This is our motivation to consider distributive polyhedra, which are characterized by describing networks by Felsner and Knauer analogous to the description of order polytopes by Hasse diagrams. For a subclass of distributive polyhedra we are able to construct a piecewise-linear bijection to another polyhedron related to chain polytopes. We give a description of this transfer map and the defining inequalities of the image in terms of the underlying network

Symmetry in Applied Mathematics

Applied mathematics and symmetry work together as a powerful tool for problem reduction and solving. We are communicating applications in probability theory and statistics (A Test Detecting the Outliers for Continuous Distributions Based on the Cumulative Distribution Function of the Data Being Tested, The Asymmetric Alpha-Power Skew-t Distribution), fractals - geometry and alike (Khovanov Homology of Three-Strand Braid Links, Volume Preserving Maps Between p-Balls, Generation of Julia and Mandelbrot Sets via Fixed Points), supersymmetry - physics, nanostructures -chemistry, taxonomy - biology and alike (A Continuous Coordinate System for the Plane by Triangular Symmetry, One-Dimensional Optimal System for 2D Rotating Ideal Gas, Minimal Energy Configurations of Finite Molecular Arrays, Noether-Like Operators and First Integrals for Generalized Systems of Lane-Emden Equations), algorithms, programs and software analysis (Algorithm for Neutrosophic Soft Sets in Stochastic Multi-Criteria Group Decision Making Based on Prospect Theory, On a Reduced Cost Higher Order Traub-Steffensen-Like Method for Nonlinear Systems, On a Class of Optimal Fourth Order Multiple Root Solvers without Using Derivatives) to specific subjects (Facility Location Problem Approach for Distributed Drones, Parametric Jensen-Shannon Statistical Complexity and Its Applications on Full-Scale Compartment Fire Data). Diverse topics are thus combined to map out the mathematical core of practical problems

LIPIcs, Volume 244, ESA 2022, Complete Volume

LIPIcs, Volume 244, ESA 2022, Complete Volum

Mecánica Discreta para Sistemas Forzados y Ligados

Tesis inédita de la Universidad Complutense de Madrid, Facultad de Ciencias Matemáticas, leída el 10/07/2019Geometric mechanics is a branch of mathematics that studies classical mechanics of particles and fields from the point of view of geometry and its relation to symmetry. One of its most interesting developments was bringing together numerical analysis and geometry by relating what is known as discrete mechanics with numerical integration. This is called geometric integration. In the last 30 years this latter field has exploded with researchfrom the purely theoretical to the strictly applied. Variational integrators are a type of geometric integrators arising naturally from the discretization process of variational principles in mechanics. They display some of the most salient features of the theory, such as symplecticity, preservation of momenta and quasi-preservation of energy. These methods also apply very naturally to optimal control problems, also based on variational principles. Unfortunately, not all mechanical systems of interest admit a variational formulation. Such is the case of forced and nonholonomic mechanical systems. In this thesis we study both of these types of systems and obtain several new results. By geometrizing a new technique of duplication of variables and applying it, we were able to definitely prove the order of integrators for forced systems by using only variational techniques. Furthermore, we could also extend these results to the reduced setting in Lie groups, leading us to a very interesting geometric structure, Poisson groupoids. In addition, we developed new methods to geometrically integrate nonholonomic systems to arbitrary order preserving their constraints exactly. These methods can be seen as nonholonomic extensions of variational methods, and we were able to prove their order, although not through variational means. These methods have a nice geometric interpretation and thanks to their closeness to variational methods, they can be easily generalized to other geometric settings, such as Lie group integration. Finally, we were able to apply these new methods to optimal control problems...La mecánica clásica es un campo tan fundamental para la física como la geometría lo es para las matemáticas. Ambos están interrelacionados y su estudio conjunto así como sus interacciones forman lo que hoy se conoce como la mecánica geométrica [véase, por ejemplo, AM78; Arn89; Hol11a; Hol11 b]. Hoy es bien sabido que el concepto de simetría tiene importantes consecuencias para los sistemas mecánicos. En particular, la evolución de los sistemas mecánicos suele mostrar ciertas propiedades de preservación en forma de cantidades conservadas del movimiento o preservación de estructuras geométricas. Ser capaces de capturar estas propiedades es vital para tener una imagen fiel, tanto en términos cuantitativos como cualitativos, de cara al estudio de estos sistemas. Esto tiene gran importancia en el campo teórico y también el aplicado, como en la ingeniería. La experimentación en laboratorios y la generación de prototipos son procesos costosos y que requieren de tiempo, y para determinad os sistemas pueden no ser siquiera factibles. Con la llegada el ordenador, simular y experimentar con sistemas mecánicos de forma rápida y económica se convirtió en una realidad . Desde sencillas simulaciones balísticas para alumnos de secundaria a simulaciones de dinámica molecular a gran escala; desde la planificación de trayectorias para vehículos autónomos a la estimación de movimientos en robots bípedos; desde costosas simulaciones basadas en modelos físicos para la industria de la animación a la simulación de sólidos rígidos y deformables en tiempo real para la industria del videojuego, el tratamiento numérico de sistemas de complejidad creciente se ha convertido en una necesidad. Naturalmente surgieron nuevos algoritmos capaces de conservar gran parte de las propiedades geométricas de estos sistemas, configurando lo que a hora se conoce como integración geométrica [véase SC94; HLW1O]. En los últimos 20 a 30 años se han dado grandes pasos en esta dirección, con el desarrollo de métodos que conservan energía, métodos simplécticos y multisimplécticos, métodos que preservan el espacio de configuración y más. Aún así, la investigación en esta área está todavía lejos de acabar. Por ejemplo , los sistemas sometidos a fuerzas externas y con ligaduras ofrecen ciertas dificultades que han de ser abordadas, y esta tesis se dedica a explorar estos dos casos ofreciendo nuevos desarrollos y resultados...Fac. de Ciencias MatemáticasTRUEunpu

Problems in Control, Estimation, and Learning in Complex Robotic Systems

In this dissertation, we consider a range of different problems in systems, control, and learning theory and practice. In Part I, we look at problems in control of complex networks. In Chapter 1, we consider the performance analysis of a class of linear noisy dynamical systems. In Chapter 2, we look at the optimal design problems for these networks. In Chapter 3, we consider dynamical networks where interactions between the networks occur randomly in time. And in the last chapter of this part, in Chapter 4, we look at dynamical networks wherein coupling between the subsystems (or agents) changes nonlinearly based on the difference between the state of the subsystems. In Part II, we consider estimation problems wherein we deal with a large body of variables (i.e., at large scale). This part starts with Chapter 5, in which we consider the problem of sampling from a dynamical network in space and time for initial state recovery. In Chapter 6, we consider a similar problem with the difference that the observations instead of point samples become continuous observations that happen in Lebesgue measurable observations. In Chapter 7, we consider an estimation problem in which the location of a robot during the navigation is estimated using the information of a large number of surrounding features and we would like to select the most informative features using an efficient algorithm. In Part III, we look at active perception problems, which are approached using reinforcement learning techniques. This part starts with Chapter 8, in which we tackle the problem of multi-agent reinforcement learning where the agents communicate and classify as a team. In Chapter 9, we consider a single agent version of the same problem, wherein a layered architecture replaces the architectures of the previous chapter. Then, we use reinforcement learning to design the meta-layer (to select goals), action-layer (to select local actions), and perception-layer (to conduct classification)

Theory for the FCC-ee : Report on the 11th FCC-ee Workshop

The Future Circular Collider (FCC) at CERN, a proposed 100-km circular facility with several colliders in succession, culminates with a 100 TeV proton-proton collider. It offers a vast new domain of exploration in particle physics, with orders of magnitude advances in terms of Precision, Sensitivity and Energy. The implementation plan foresees, as a first step, an Electroweak Factory electron-positron collider. This high luminosity facility, operating between 90 and 365 GeV centre-of-mass energy, will study the heavy particles of the Standard Model, Z, W, Higgs, and top with unprecedented accuracy. The Electroweak Factory $e^+e^-$ collider constitutes a real challenge to the theory and to precision calculations, triggering the need for the development of new mathematical methods and software tools. A first workshop in 2018 had focused on the first FCC-ee stage, the Tera-Z, and confronted the theoretical status of precision Standard Model calculations on the Z-boson resonance to the experimental demands. The second workshop in January 2019, which is reported here, extended the scope to the next stages, with the production of W-bosons (FCC-ee-W), the Higgs boson (FCC-ee-H) and top quarks (FCC-ee-tt). In particular, the theoretical precision in the determination of the crucial input parameters, alpha_QED, alpha_QCD, M_W, m_t at the level of FCC-ee requirements is thoroughly discussed. The requirements on Standard Model theory calculations were spelled out, so as to meet the demanding accuracy of the FCC-ee experimental potential. The discussion of innovative methods and tools for multi-loop calculations was deepened. Furthermore, phenomenological analyses beyond the Standard Model were discussed, in particular the effective theory approaches. The reports of 2018 and 2019 serve as white papers of the workshop results and subsequent developments