On Queueinginventory Models – product form solution; reservation, Cancellation and common life time

Queueing theory is the mathematical study of ‘queue’ or ‘waiting lines’ where an item from inventory is provided to the customer on completion of service. A typical queueing system consists of a queue and a server. Customers arrive in the system from outside and join the queue in a certain way. The server picks up customers and serves them according to certain service discipline. Customers leave the system immediately after their service is completed. For queueing systems, queue length, waiting time and busy period are of primary interest to applications. The theory permits the derivation and calculation of several performance measures including the average waiting time in the queue or the system, mean queue length, traffic intensity, the expected number waiting or receiving service, mean busy period, distribution of queue length, and the probability of encountering the system in certain states, such as empty, full, having an available server or having to wait a certain time to be served.

The problem of the spreading of a liquid film along a solid surface: A new mathematical formulation

A new mathematical model is proposed for the spreading of a liquid film on a solid surface. The model is based on the standard lubrication approximation for gently sloping films (with the no-slip condition for the fluid at the solid surface) in the major part of the film where it is not too thin. In the remaining and relatively small regions near the contact lines it is assumed that the so-called autonomy principle holds—i.e., given the material components, the external conditions, and the velocity of the contact lines along the surface, the behavior of the fluid is identical for all films. The resulting mathematical model is formulated as a free boundary problem for the classical fourth-order equation for the film thickness. A class of self-similar solutions to this free boundary problem is considered.

Formulation of the standard model in terms of spinor fields

We propose an alternative formulation of the Standard Model which reduces the number of free parameters. In our framework, fermionic fields are assigned to fundamental representations of the Lorentz and the internal symmetry groups, whereas bosonic field variables transform as direct products of fundamental representations of all symmetry groups. This allows us to reduce the number of fundamental symmetries. We formulate the Standard Model by considering the SU(3) and SU(2) symmetry groups as the underlying symmetries of the fundamental interactions. This allows us to suggest a model, for the description of the interactions of the intermediate bosons among themselves and interactions of fermions, that makes use of just two parameters. One parameter characterizes the symmetric phase, whereas the other parameter (the asymmetry parameter) gives the breakdown strength of the symmetries. All coupling strengths of the Standard Model are then derived in terms of these two parameters. In particular, we show that all fermionic electric charges result from symmetry breakdown.

Predictive formulation of the Nambu-Jona-Lasinio model

A novel strategy to handle divergences typical of perturbative calculations is implemented for the Nambu-Jona-Lasinio model and its phenomenological consequences investigated. The central idea of the method is to avoid the critical step involved in the regularization process, namely, the explicit evaluation of divergent integrals. This goal is achieved by assuming a regularization distribution in an implicit way and making use, in intermediary steps, only of very general properties of such regularization. The finite parts are separated from the divergent ones and integrated free from effects of the regularization. The divergent parts are organized in terms of standard objects, which are independent of the ( arbitrary) momenta running in internal lines of loop graphs. Through the analysis of symmetry relations, a set of properties for the divergent objects are identified, which we denominate consistency relations, reducing the number of divergent objects to only a few. The calculational strategy eliminates unphysical dependencies of the arbitrary choices for the routing of internal momenta, leading to ambiguity-free, and symmetry-preserving physical amplitudes. We show that the imposition of scale properties for the basic divergent objects leads to a critical condition for the constituent quark mass such that the remaining arbitrariness is removed. The model becomes predictive in the sense that its phenomenological consequences do not depend on possible choices made in intermediary steps. Numerical results are obtained for physical quantities at the one-loop level for the pion and sigma masses and pion-quark and sigma-quark coupling constants.

Operator formulation of q-deformed dual string model

We present an operator formulation of the q-deformed dual string model amplitude using an infinite set of q-harmonic oscillators. The formalism attains the crossing symmetry and factorization and allows to express the general n-point function as a factorized product of vertices and propagators.

Loop formulation of the supersymmetric nonlinear O(N) sigma model

We derive the fermion loop formulation for the supersymmetric nonlinear O(N) sigma model by performing a hopping expansion using Wilson fermions. In this formulation the fermionic contribution to the partition function becomes a sum over all possible closed non-oriented fermion loop configurations. The interaction between the bosonic and fermionic degrees of freedom is encoded in the constraints arising from the supersymmetry and induces flavour changing fermion loops. For N ≥ 3 this leads to fermion loops which are no longer self-avoiding and hence to a potential sign problem. Since we use Wilson fermions the bare mass needs to be tuned to the chiral point. For N = 2 we determine the critical point and present boson and fermion masses in the critical regime.

Mathematical analysis of a model of chemotaxis arising from morphogenesis

We consider non-negative solution of a chemotaxis system with non constant chemotaxis sensitivity function X. This system appears as a limit case of a model formorphogenesis proposed by Bollenbach et al. (Phys. Rev. E. 75, 2007).Under suitable boundary conditions, modeling the presence of a morphogen source at x=0, we prove the existence of a global and bounded weak solution using an approximation by problems where diffusion is introduced in the ordinary differential equation. Moreover,we prove the convergence of the solution to the unique steady state provided that ? is small and ? is large enough. Numerical simulations both illustrate these results and give rise to further conjectures on the solution behavior that go beyond the rigorously proved statements.

Development of a 3-Dimensional Mathematical Collision Risk Model Based on Recorded Aircraft Trajectories to Estimate the Safety Level in High Density En-Route Airspaces

Como en todos los medios de transporte, la seguridad en los viajes en avión es de primordial importancia. Con los aumentos de tráfico aéreo previstos en Europa para la próxima década, es evidente que el riesgo de accidentes necesita ser evaluado y monitorizado cuidadosamente de forma continúa. La Tesis presente tiene como objetivo el desarrollo de un modelo de riesgo de colisión exhaustivo como método para evaluar el nivel de seguridad en ruta del espacio aéreo europeo, considerando todos los factores de influencia. La mayor limitación en el desarrollo de metodologías y herramientas de monitorización adecuadas para evaluar el nivel de seguridad en espacios de ruta europeos, donde los controladores aéreos monitorizan el tráfico aéreo mediante la vigilancia radar y proporcionan instrucciones tácticas a las aeronaves, reside en la estimación del riesgo operacional. Hoy en día, la estimación del riesgo operacional está basada normalmente en reportes de incidentes proporcionados por el proveedor de servicios de navegación aérea (ANSP). Esta Tesis propone un nuevo e innovador enfoque para evaluar el nivel de seguridad basado exclusivamente en el procesamiento y análisis trazas radar. La metodología propuesta ha sido diseñada para complementar la información recogida en las bases de datos de accidentes e incidentes, mediante la provisión de información robusta de los factores de tráfico aéreo y métricas de seguridad inferidas del análisis automático en profundidad de todos los eventos de proximidad. La metodología 3-D CRM se ha implementado en un prototipo desarrollado en MATLAB © para analizar automáticamente las trazas radar y planes de vuelo registrados por los Sistemas de Procesamiento de Datos Radar (RDP) e identificar y analizar todos los eventos de proximidad (conflictos, conflictos potenciales y colisiones potenciales) en un periodo de tiempo y volumen del espacio aéreo. Actualmente, el prototipo 3-D CRM está siendo adaptado e integrado en la herramienta de monitorización de prestaciones de Aena (PERSEO) para complementar las bases de accidentes e incidentes ATM y mejorar la monitorización y proporcionar evidencias de los niveles de seguridad. ABSTRACT As with all forms of transport, the safety of air travel is of paramount importance. With the projected increases in European air traffic in the next decade and beyond, it is clear that the risk of accidents needs to be assessed and carefully monitored on a continuing basis. The present thesis is aimed at the development of a comprehensive collision risk model as a method of assessing the European en-route risk, due to all causes and across all dimensions within the airspace. The major constraint in developing appropriate monitoring methodologies and tools to assess the level of safety in en-route airspaces where controllers monitor air traffic by means of radar surveillance and provide aircraft with tactical instructions lies in the estimation of the operational risk. The operational risk estimate normally relies on incident reports provided by the air navigation service providers (ANSPs). This thesis proposes a new and innovative approach to assessing aircraft safety level based exclusively upon the process and analysis of radar tracks. The proposed methodology has been designed to complement the information collected in the accident and incident databases, thereby providing robust information on air traffic factors and safety metrics inferred from the in depth assessment of proximate events. The 3-D CRM methodology is implemented in a prototype tool in MATLAB © in order to automatically analyze recorded aircraft tracks and flight plan data from the Radar Data Processing systems (RDP) and identify and analyze all proximate events (conflicts, potential conflicts and potential collisions) within a time span and a given volume of airspace. Currently, the 3D-CRM prototype is been adapted and integrated in AENA’S Performance Monitoring Tool (PERSEO) to complement the information provided by the ATM accident and incident databases and to enhance monitoring and providing evidence of levels of safety.

Mathematical development of stress-strain law for rock block contacts

The fracture behavior of rock block contacts has been studied for many years. Unfortunately, up to now, there is not a rigorous formulation or a solid theoretical foundation to support it. A mathematical development to represent the failure mechanism which occurs in the contacts between rock blocks is presented to evaluate the performance of breaking mechanism of such blocks relating it to the morphology of the contact and mechanical parameters of the material. The examined framework includes the evaluation of the surface roughness of first order in the failure mechanism of the granular particles of large size and the development of a theoretical model describing the morphology of the contact between rock blocks.

Toy model for a relational formulation of quantum theory

In the absence of an external frame of reference-i.e., in background independent theories such as general relativity-physical degrees of freedom must describe relations between systems. Using a simple model, we investigate how such a relational quantum theory naturally arises by promoting reference systems to the status of dynamical entities. Our goal is twofold. First, we demonstrate using elementary quantum theory how any quantum mechanical experiment admits a purely relational description at a fundamental. Second, we describe how the original non-relational theory approximately emerges from the fully relational theory when reference systems become semi-classical. Our technique is motivated by a Bayesian approach to quantum mechanics, and relies on the noiseless subsystem method of quantum information science used to protect quantum states against undesired noise. The relational theory naturally predicts a fundamental decoherence mechanism, so an arrow of time emerges from a time-symmetric theory. Moreover, our model circumvents the problem of the collapse of the wave packet as the probability interpretation is only ever applied to diagonal density operators. Finally, the physical states of the relational theory can be described in terms of spin networks introduced by Penrose as a combinatorial description of geometry, and widely studied in the loop formulation of quantum gravity. Thus, our simple bottom-up approach (starting from the semiclassical limit to derive the fully relational quantum theory) may offer interesting insights on the low energy limit of quantum gravity.