A graded subring of an inverse limit of polynomial rings

We study the power series ring R= K[[x1,x2,x3,...]]on countably infinitely many variables, over a field K, and two particular K-subalgebras of it: the ring S, which is isomorphic to an inverse limit of the polynomial rings in finitely many variables over K, and the ring R', which is the largest graded subalgebra of R. Of particular interest are the homogeneous, finitely generated ideals in R', among them the generic ideals. The definition of S as an inverse limit yields a set of truncation homomorphisms from S to K[x1,...,xn] which restrict to R'. We have that the truncation of a generic I in R' is a generic ideal in K[x1,...,xn]. It is shown in Initial ideals of Truncated Homogeneous Ideals that the initial ideal of such an ideal converge to the initial ideal of the corresponding ideal in R'. This initial ideal need no longer be finitely generated, but it is always locally finitely generated: this is proved in Gröbner Bases in R'. We show in Reverse lexicographic initial ideals of generic ideals are finitely generated that the initial ideal of a generic ideal in R' is finitely generated. This contrast to the lexicographic term order. If I in R' is a homogeneous, locally finitely generated ideal, and if we write the Hilbert series of the truncated algebras K[x1,...,xn] module the truncation of I as qn(t)/(1-t)n, then we show in Generalized Hilbert Numerators that the qn's converge to a power series in t which we call the generalized Hilbert numerator of the algebra R'/I. In Gröbner bases for non-homogeneous ideals in R' we show that the calculations of Gröbner bases and initial ideals in R' can be done also for some non-homogeneous ideals, namely those which have an associated homogeneous ideal which is locally finitely generated. The fact that S is an inverse limit of polynomial rings, which are naturally endowed with the discrete topology, provides S with a topology which makes it into a complete Hausdorff topological ring. The ring R', with the subspace topology, is dense in R, and the latter ring is the Cauchy completion of the former. In Topological properties of R' we show that with respect to this topology, locally finitely generated ideals in R'are closed.

Discrete Event Systems based Design Patterns for Diagnosability Analysis of Automated Manufacturing Systems

The main goal of this thesis is to facilitate the process of industrial automated systems development applying formal methods to ensure the reliability of systems. A new formulation of distributed diagnosability problem in terms of Discrete Event Systems theory and automata framework is presented, which is then used to enforce the desired property of the system, rather then just verifying it. This approach tackles the state explosion problem with modeling patterns and new algorithms, aimed for verification of diagnosability property in the context of the distributed diagnosability problem. The concepts are validated with a newly developed software tool.

Probabilistic Recursion Theory and Implicit Computational Complexity

In this thesis we provide a characterization of probabilistic computation in itself, from a recursion-theoretical perspective, without reducing it to deterministic computation. More specifically, we show that probabilistic computable functions, i.e., those functions which are computed by Probabilistic Turing Machines (PTM), can be characterized by a natural generalization of Kleene's partial recursive functions which includes, among initial functions, one that returns identity or successor with probability 1/2. We then prove the equi-expressivity of the obtained algebra and the class of functions computed by PTMs. In the the second part of the thesis we investigate the relations existing between our recursion-theoretical framework and sub-recursive classes, in the spirit of Implicit Computational Complexity. More precisely, endowing predicative recurrence with a random base function is proved to lead to a characterization of polynomial-time computable probabilistic functions.

A feasible theory of truth over combinatory algebra

We define an applicative theory of truth TPT which proves totality exactly for the polynomial time computable functions. TPT has natural and simple axioms since nearly all its truth axioms are standard for truth theories over an applicative framework. The only exception is the axiom dealing with the word predicate. The truth predicate can only reflect elementhood in the words for terms that have smaller length than a given word. This makes it possible to achieve the very low proof-theoretic strength. Truth induction can be allowed without any constraints. For these reasons the system TPT has the high expressive power one expects from truth theories. It allows embeddings of feasible systems of explicit mathematics and bounded arithmetic. The proof that the theory TPT is feasible is not easy. It is not possible to apply a standard realisation approach. For this reason we develop a new realisation approach whose realisation functions work on directed acyclic graphs. In this way, we can express and manipulate realisation information more efficiently.

The flexibility matrix o timber composite beams with a discrete connection system

This work presents a method for the analysis of timber composite beams which considers the slip in the connection system, based on assembling the flexibility matrix of the whole structure. This method is based on one proposed by Tommola and Jutila (2001). This paper extends the method to the case of a gap between two pieces with an arbitrary location at the first connector, which notably broadens its practical application. The addition of the gap makes it possible to model a cracked zone in concrete topping, as well as the case in which forming produces the gap. The consideration of induced stresses due to changes in temperature and moisture content is also described, while the concept of equivalent eccentricity is generalized. This method has important advantages in connection with the current European Standard EN 1995-1-1: 2004, as it is able to deal with any type of load, variable section, discrete and non-regular connection systems, a gap between the two pieces, and variations in temperature and moisture content. Although it could be applied to any structural system, it is specially suited for the case of simple supported and continuous beams. Working examples are presented at the end, showing that the arrangement of the connection notably modifies shear force distribution. A first interpretation of the results is made on the basis of the strut and tie theory. The examples prove that the use of EC-5 is unsafe when, as a rule of thumb, the strut or compression field between the support and the first connector is at an angle with the axis of the beam of less than 60º.

A geometric characterization of the upper bound for the span of the jones polynomial

Let D be a link diagram with n crossings, sA and sB be its extreme states and |sAD| (respectively, |sBD|) be the number of simple closed curves that appear when smoothing D according to sA (respectively, sB). We give a general formula for the sum |sAD| + |sBD| for a k-almost alternating diagram D, for any k, characterizing this sum as the number of faces in an appropriate triangulation of an appropriate surface with boundary. When D is dealternator connected, the triangulation is especially simple, yielding |sAD| + |sBD| = n + 2 - 2k. This gives a simple geometric proof of the upper bound of the span of the Jones polynomial for dealternator connected diagrams, a result first obtained by Zhu [On Kauffman brackets, J. Knot Theory Ramifications6(1) (1997) 125–148.]. Another upper bound of the span of the Jones polynomial for dealternator connected and dealternator reduced diagrams, discovered historically first by Adams et al. [Almost alternating links, Topology Appl.46(2) (1992) 151–165.], is obtained as a corollary. As a new application, we prove that the Turaev genus is equal to the number k of dealternator crossings for any dealternator connected diagram

Contributions to trajectory prediction theory and its application to arrival management for air traffic control

La gestión del tráfico aéreo (Air Traffic Management, ATM) está experimentando un cambio de paradigma hacia las denominadas operaciones basadas trayectoria. Bajo dicho paradigma se modifica el papel de los controladores de tráfico aéreo desde una operativa basada su intervención táctica continuada hacia una labor de supervisión a más largo plazo. Esto se apoya en la creciente confianza en las soluciones aportadas por las herramientas automatizadas de soporte a la decisión más modernas. Para dar soporte a este concepto, se precisa una importante inversión para el desarrollo, junto con la adquisición de nuevos equipos en tierra y embarcados, que permitan la sincronización precisa de la visión de la trayectoria, basada en el intercambio de información entre ambos actores. Durante los últimos 30 a 40 años las aerolíneas han generado uno de los menores retornos de la inversión de entre todas las industrias. Sin beneficios tangibles, la industria aérea tiene dificultades para atraer el capital requerido para su modernización, lo que retrasa la implantación de dichas mejoras. Esta tesis tiene como objetivo responder a la pregunta de si las capacidades actualmente instaladas en las aeronaves comerciales se pueden aplicar para lograr la sincronización de la trayectoria con el nivel de calidad requerido. Además, se analiza en ella si, conjuntamente con mejoras en las herramientas de predicción trayectorias instaladas en tierra en para facilitar la gestión de las arribadas, dichas capacidades permiten obtener los beneficios esperados en el marco de las operaciones basadas en trayectoria. Esto podría proporcionar un incentivo para futuras actualizaciones de la aviónica que podrían llevar a mejoras adicionales. El concepto operacional propuesto en esta tesis tiene como objetivo permitir que los aviones sean pilotados de una manera consistente con las técnicas actuales de vuelo optimizado. Se permite a las aeronaves que desciendan en el denominado “modo de ángulo de descenso gestionado” (path-managed mode), que es el preferido por la mayoría de las compañías aéreas, debido a que conlleva un reducido consumo de combustible. El problema de este modo es que en él no se controla de forma activa el tiempo de llegada al punto de interés. En nuestro concepto operacional, la incertidumbre temporal se gestiona en mediante de la medición del tiempo en puntos estratégicamente escogidos a lo largo de la trayectoria de la aeronave, y permitiendo la modificación por el control de tierra de la velocidad de la aeronave. Aunque la base del concepto es la gestión de las ordenes de velocidad que se proporcionan al piloto, para ser capaces de operar con los niveles de equipamiento típicos actualmente, dicho concepto también constituye un marco en el que la aviónica más avanzada (por ejemplo, que permita el control por el FMS del tiempo de llegada) puede integrarse de forma natural, una vez que esta tecnología este instalada. Además de gestionar la incertidumbre temporal a través de la medición en múltiples puntos, se intenta reducir dicha incertidumbre al mínimo mediante la mejora de las herramienta de predicción de la trayectoria en tierra. En esta tesis se presenta una novedosa descomposición del proceso de predicción de trayectorias en dos etapas. Dicha descomposición permite integrar adecuadamente los datos de la trayectoria de referencia calculada por el Flight Management System (FMS), disponibles usando Futuro Sistema de Navegación Aérea (FANS), en el sistema de predicción de trayectorias en tierra. FANS es un equipo presente en los aviones comerciales de fuselaje ancho actualmente en la producción, e incluso algunos aviones de fuselaje estrecho pueden tener instalada avionica FANS. Además de informar automáticamente de la posición de la aeronave, FANS permite proporcionar (parte de) la trayectoria de referencia en poder de los FMS, pero la explotación de esta capacidad para la mejora de la predicción de trayectorias no se ha estudiado en profundidad en el pasado. La predicción en dos etapas proporciona una solución adecuada al problema de sincronización de trayectorias aire-tierra dado que permite la sincronización de las dimensiones controladas por el sistema de guiado utilizando la información de la trayectoria de referencia proporcionada mediante FANS, y también facilita la mejora en la predicción de las dimensiones abiertas restantes usado un modelo del guiado que explota los modelos meteorológicos mejorados disponibles en tierra. Este proceso de predicción de la trayectoria de dos etapas se aplicó a una muestra de 438 vuelos reales que realizaron un descenso continuo (sin intervención del controlador) con destino Melbourne. Dichos vuelos son de aeronaves del modelo Boeing 737-800, si bien la metodología descrita es extrapolable a otros tipos de aeronave. El método propuesto de predicción de trayectorias permite una mejora en la desviación estándar del error de la estimación del tiempo de llegada al punto de interés, que es un 30% menor que la que obtiene el FMS. Dicha trayectoria prevista mejorada se puede utilizar para establecer la secuencia de arribadas y para la asignación de las franjas horarias para cada aterrizaje (slots). Sobre la base del slot asignado, se determina un perfil de velocidades que permita cumplir con dicho slot con un impacto mínimo en la eficiencia del vuelo. En la tesis se propone un nuevo algoritmo que determina las velocidades requeridas sin necesidad de un proceso iterativo de búsqueda sobre el sistema de predicción de trayectorias. El algoritmo se basa en una parametrización inteligente del proceso de predicción de la trayectoria, que permite relacionar el tiempo estimado de llegada con una función polinómica. Resolviendo dicho polinomio para el tiempo de llegada deseado, se obtiene de forma natural el perfil de velocidades optimo para cumplir con dicho tiempo de llegada sin comprometer la eficiencia. El diseño de los sistemas de gestión de arribadas propuesto en esta tesis aprovecha la aviónica y los sistemas de comunicación instalados de un modo mucho más eficiente, proporcionando valor añadido para la industria. Por tanto, la solución es compatible con la transición hacia los sistemas de aviónica avanzados que están desarrollándose actualmente. Los beneficios que se obtengan a lo largo de dicha transición son un incentivo para inversiones subsiguientes en la aviónica y en los sistemas de control de tráfico en tierra. ABSTRACT Air traffic management (ATM) is undergoing a paradigm shift towards trajectory based operations where the role of an air traffic controller evolves from that of continuous intervention towards supervision, as decision making is improved based on increased confidence in the solutions provided by advanced automation. To support this concept, significant investment for the development and acquisition of new equipment is required on the ground as well as in the air, to facilitate the high degree of trajectory synchronisation and information exchange required. Over the past 30-40 years the airline industry has generated one of the lowest returns on invested capital among all industries. Without tangible benefits realised, the airline industry may find it difficult to attract the required investment capital and delay acquiring equipment needed to realise the concept of trajectory based operations. In response to these challenges facing the modernisation of ATM, this thesis aims to answer the question whether existing aircraft capabilities can be applied to achieve sufficient trajectory synchronisation and improvements to ground-based trajectory prediction in support of the arrival management process, to realise some of the benefits envisioned under trajectory based operations, and to provide an incentive for further avionics upgrades. The proposed operational concept aims to permit aircraft to operate in a manner consistent with current optimal aircraft operating techniques. It allows aircraft to descend in the fuel efficient path managed mode as preferred by a majority of airlines, with arrival time not actively controlled by the airborne automation. The temporal uncertainty is managed through metering at strategically chosen points along the aircraft’s trajectory with primary use of speed advisories. While the focus is on speed advisories to support all aircraft and different levels of equipage, the concept also constitutes a framework in which advanced avionics as airborne time-of-arrival control can be integrated once this technology is widely available. In addition to managing temporal uncertainty through metering at multiple points, this temporal uncertainty is minimised by improving the supporting trajectory prediction capability. A novel two-stage trajectory prediction process is presented to adequately integrate aircraft trajectory data available through Future Air Navigation Systems (FANS) into the ground-based trajectory predictor. FANS is standard equipment on any wide-body aircraft in production today, and some single-aisle aircraft are easily capable of being fitted with FANS. In addition to automatic position reporting, FANS provides the ability to provide (part of) the reference trajectory held by the aircraft’s Flight Management System (FMS), but this capability has yet been widely overlooked. The two-stage process provides a ‘best of both world’s’ solution to the air-ground synchronisation problem by synchronising with the FMS reference trajectory those dimensions controlled by the guidance mode, and improving on the prediction of the remaining open dimensions by exploiting the high resolution meteorological forecast available to a ground-based system. The two-stage trajectory prediction process was applied to a sample of 438 FANS-equipped Boeing 737-800 flights into Melbourne conducting a continuous descent free from ATC intervention, and can be extrapolated to other types of aircraft. Trajectories predicted through the two-stage approach provided estimated time of arrivals with a 30% reduction in standard deviation of the error compared to estimated time of arrival calculated by the FMS. This improved predicted trajectory can subsequently be used to set the sequence and allocate landing slots. Based on the allocated landing slot, the proposed system calculates a speed schedule for the aircraft to meet this landing slot at minimal flight efficiency impact. A novel algorithm is presented that determines this speed schedule without requiring an iterative process in which multiple calls to a trajectory predictor need to be made. The algorithm is based on parameterisation of the trajectory prediction process, allowing the estimate time of arrival to be represented by a polynomial function of the speed schedule, providing an analytical solution to the speed schedule required to meet a set arrival time. The arrival management solution proposed in this thesis leverages the use of existing avionics and communications systems resulting in new value for industry for current investment. The solution therefore supports a transition concept from mixed equipage towards advanced avionics currently under development. Benefits realised under this transition may provide an incentive for ongoing investment in avionics.

Discrete linear stability and sensitivity analysis of fluid flows

Este trabajo presenta un método discreto para el cálculo de estabilidad hidrodinámica y análisis de sensibilidad a perturbaciones externas para ecuaciones diferenciales y en particular para las ecuaciones de Navier-Stokes compressible. Se utiliza una aproximación con variable compleja para obtener una precisión analítica en la evaluación de la matriz Jacobiana. Además, mapas de sensibilidad para la sensibilidad a las modificaciones del flujo de base y a una fuerza constante permiten identificar las regiones del campo fluido donde una modificacin (ej. fuerza puntual) tiene un efecto estabilizador del flujo. Se presentan cuatro casos de prueba: (1) un caso analítico para comprobar la derivación discreta, (2) una cavidad cerrada a bajo Reynolds para mostrar la mayor precisión en el cálculo de los valores propios con la aproximación de paso complejo, (3) flujo 2D en un cilindro circular para validar la metodología, y (4) flujo en un cavidad abierta, presentado para validar el método en casos de inestabilidades convectivamente inestables. Los tres últimos casos mencionados (2-4) se resolvieron con las ecuaciones de Navier-Stokes compresibles, utilizando un método Discontinuous Galerkin Spectral Element Method. Se obtuvo una buena concordancia para el caso de validación (3), cuando se comparó el nuevo método con resultados de la literatura. Además, este trabajo muestra que para el cálculo de los modos propios directos y adjuntos, así como para los mapas de sensibilidad, el uso de variables complejas es de suprema importancia para obtener una predicción precisa. El método descrito es aplicado al análisis para la estabilización de la estela generada por un disco actuador, que representa un modelo sencillo para hélices, rotores de helicópteros o turbinas eólicas. Se explora la primera bifurcación del flujo para un disco actuador, y se sugiere que está asociada a una inestabilidad de tipo Kelvin-Helmholtz, cuya estabilidad se controla con en el número de Reynolds y en la resistencia del disco actuador (o fuerza resistente). En primer lugar, se verifica que la disminución de la resistencia del disco tiene un efecto estabilizador parecido a una disminución del Reynolds. En segundo lugar, el análisis hidrodinmico discreto identifica dos regiones para la colocación de una fuerza puntual que controle las inestabilidades, una cerca del disco y otra en una zona aguas abajo. En tercer lugar, se muestra que la inclusión de un forzamiento localizado cerca del actuador produce una estabilización más eficiente que al forzar aguas abajo. El análisis de los campos de flujo controlados confirma que modificando el gradiente de velocidad cerca del actuador es más eficiente para estabilizar la estela. Estos resultados podrían proporcionar nuevas directrices para la estabilización de la estela de turbinas de viento o de marea cuando estén instaladas en un parque eólico y minimizar las interacciones no estacionarias entre turbinas. ABSTRACT A discrete framework for computing the global stability and sensitivity analysis to external perturbations for any set of partial differential equations is presented. In particular, a complex-step approximation is used to achieve near analytical accuracy for the evaluation of the Jacobian matrix. Sensitivity maps for the sensitivity to base flow modifications and to a steady force are computed to identify regions of the flow field where an input could have a stabilising effect. Four test cases are presented: (1) an analytical test case to prove the theory of the discrete framework, (2) a lid-driven cavity at low Reynolds case to show the improved accuracy in the calculation of the eigenvalues when using the complex-step approximation, (3) the 2D flow past a circular cylinder at just below the critical Reynolds number is used to validate the methodology, and finally, (4) the flow past an open cavity is presented to give an example of the discrete method applied to a convectively unstable case. The latter three (2–4) of the aforementioned cases were solved with the 2D compressible Navier–Stokes equations using a Discontinuous Galerkin Spectral Element Method. Good agreement was obtained for the validation test case, (3), with appropriate results in the literature. Furthermore, it is shown that for the calculation of the direct and adjoint eigenmodes and their sensitivity maps to external perturbations, the use of complex variables is paramount for obtaining an accurate prediction. An analysis for stabilising the wake past an actuator disc, which represents a simple model for propellers, helicopter rotors or wind turbines is also presented. We explore the first flow bifurcation for an actuator disc and it suggests that it is associated to a Kelvin- Helmholtz type instability whose stability relies on the Reynolds number and the flow resistance applied through the disc (or actuator forcing). First, we report that decreasing the disc resistance has a similar stabilising effect to an decrease in the Reynolds number. Second, a discrete sensitivity analysis identifies two regions for suitable placement of flow control forcing, one close to the disc and one far downstream where the instability originates. Third, we show that adding a localised forcing close to the actuator provides more stabilisation that forcing far downstream. The analysis of the controlled flow fields, confirms that modifying the velocity gradient close to the actuator is more efficient to stabilise the wake than controlling the sheared flow far downstream. An interesting application of these results is to provide guidelines for stabilising the wake of wind or tidal turbines when placed in an energy farm to minimise unsteady interactions.

A positivity result in the theory of Macdonald polynomials

We outline here a proof that a certain rational function Cn(q, t), which has come to be known as the “q, t-Catalan,” is in fact a polynomial with positive integer coefficients. This has been an open problem since 1994. Because Cn(q, t) evaluates to the Catalan number at t = q = 1, it has also been an open problem to find a pair of statistics a, b on the collection

High-speed polynomial basis multipliers over GF(2(m)) for special pentanomial.

Efficient hardware implementations of arithmetic operations in the Galois field are highly desirable for several applications, such as coding theory, computer algebra and cryptography. Among these operations, multiplication is of special interest because it is considered the most important building block. Therefore, high-speed algorithms and hardware architectures for computing multiplication are highly required. In this paper, bit-parallel polynomial basis multipliers over the binary field GF(2(m)) generated using type II irreducible pentanomials are considered. The multiplier here presented has the lowest time complexity known to date for similar multipliers based on this type of irreducible pentanomials.

A planar geometrical model for representing multidimensional discrete spaces and multiple-valued logic functions /

Bibliography: p. 37-38.

On decomposition of sub-linearised polynomials

We give a detailed exposition of the theory of decompositions of linearised polynomials, using a well-known connection with skew-polynomial rings with zero derivative. It is known that there is a one-to-one correspondence between decompositions of linearised polynomials and sub-linearised polynomials. This correspondence leads to a formula for the number of indecomposable sub-linearised polynomials of given degree over a finite field. We also show how to extend existing factorisation algorithms over skew-polynomial rings to decompose sub-linearised polynomials without asymptotic cost.

Quantum computing and polynomial equations over the finite field Z(2)

What is the computational power of a quantum computer? We show that determining the output of a quantum computation is equivalent to counting the number of solutions to an easily computed set of polynomials defined over the finite field Z(2). This connection allows simple proofs to be given for two known relationships between quantum and classical complexity classes, namely BQP subset of P-#P and BQP subset of PP.

Convergence analysis of a discrete-time single-unit gradient ICA algorithm

We revisit the one-unit gradient ICA algorithm derived from the kurtosis function. By carefully studying properties of the stationary points of the discrete-time one-unit gradient ICA algorithm, with suitable condition on the learning rate, convergence can be proved. The condition on the learning rate helps alleviate the guesswork that accompanies the problem of choosing suitable learning rate in practical computation. These results may be useful to extract independent source signals on-line.

A methodology for analysis and control of discrete event dynamic systems

The rapid developments in computer technology have resulted in a widespread use of discrete event dynamic systems (DEDSs). This type of system is complex because it exhibits properties such as concurrency, conflict and non-determinism. It is therefore important to model and analyse such systems before implementation to ensure safe, deadlock free and optimal operation. This thesis investigates current modelling techniques and describes Petri net theory in more detail. It reviews top down, bottom up and hybrid Petri net synthesis techniques that are used to model large systems and introduces on object oriented methodology to enable modelling of larger and more complex systems. Designs obtained by this methodology are modular, easy to understand and allow re-use of designs. Control is the next logical step in the design process. This thesis reviews recent developments in control DEDSs and investigates the use of Petri nets in the design of supervisory controllers. The scheduling of exclusive use of resources is investigated and an efficient Petri net based scheduling algorithm is designed and a re-configurable controller is proposed. To enable the analysis and control of large and complex DEDSs, an object oriented C++ software tool kit was developed and used to implement a Petri net analysis tool, Petri net scheduling and control algorithms. Finally, the methodology was applied to two industrial DEDSs: a prototype can sorting machine developed by Eurotherm Controls Ltd., and a semiconductor testing plant belonging to SGS Thomson Microelectronics Ltd.