Calculus of variations on time scales and applications to economics

We consider some problems of the calculus of variations on time scales. On the beginning our attention is paid on two inverse extremal problems on arbitrary time scales. Firstly, using the Euler-Lagrange equation and the strengthened Legendre condition, we derive a general form for a variation functional that attains a local minimum at a given point of the vector space. Furthermore, we prove a necessary condition for a dynamic integro-differential equation to be an Euler-Lagrange equation. New and interesting results for the discrete and quantum calculus are obtained as particular cases. Afterwards, we prove Euler-Lagrange type equations and transversality conditions for generalized infinite horizon problems. Next we investigate the composition of a certain scalar function with delta and nabla integrals of a vector valued field. Euler-Lagrange equations in integral form, transversality conditions, and necessary optimality conditions for isoperimetric problems, on an arbitrary time scale, are proved. In the end, two main issues of application of time scales in economic, with interesting results, are presented. In the former case we consider a firm that wants to program its production and investment policies to reach a given production rate and to maximize its future market competitiveness. The model which describes firm activities is studied in two different ways: using classical discretizations; and applying discrete versions of our result on time scales. In the end we compare the cost functional values obtained from those two approaches. The latter problem is more complex and relates to rate of inflation, p, and rate of unemployment, u, which inflict a social loss. Using known relations between p, u, and the expected rate of inflation π, we rewrite the social loss function as a function of π. We present this model in the time scale framework and find an optimal path π that minimizes the total social loss over a given time interval.

Economical and operational optimization of CALB immobilization on MP1000 for amidation of olein fatty acids with ethanolamine

Dissertação de Mestrado , Engenharia Biológica, Faculdade de Engenharia de Recursos Naturais, Universidade do Algarve, 2008

Mathematica in the classroom: new tools for exploring precalculus and differential calculus

Prémio de Melhor Artigo de Jovem Investigador atribuído pela empresa Timberlake, apresentado na 1ª Conferência Nacional sobre Computação Simbólica no Ensino e na Investigação - CSEI2012, que decorreu no IST nos dias 2 e 3 de Abril.

Towards a hydrodynamic operational model of the Algarve coast and St. Vincent Cape

This study attempts to implement a hydrodynamic operational model which can ultimately be used for projecting oil spill dispersal patterns and also sewage, pollution and can also be used in wave forecasting. A two layer nested model was created using MOHID Water, which is powerful ocean modelling software. The first layer (father) is used to impose the boundary conditions for the second layer (son). This was repeated for two different wind dominant regimes, Easterly and Westerly winds respectively. A qualitative comparison was done between measured tidal data and the tidal output. Sea surface temperature was also qualitatively compared with the model’s results. The results from both simulations were analysed and compared to historical literature. The comparison was done at the surface layer, 100 metre depth and at 800m depth. In the surface layer the first simulation generated an upwelling event near Cape St. Vincent and within the Algarve. The second simulation generated a non-upwelling event within which the surface was flow reversed and the warm water mass was along the Algarve coastline and evening turning clockwise around Cape St. Vincent. At the 100 metre depth for both simulations, velocity vortexes were observed near Cape St. Vincent travelling northerly and southerly at various instances. At 800metre depth a strong oceanic flow was observed moving north westerly along the continental shelf.

Development of an operational tool for oil spill forecast: application to oil exposed regions

Tese de doutoramento, Ciências do Mar, da Terra e do Ambiente (Modelação), Faculdade de Ciências e Tecnologia, Universidade do Algarve, 2014

A method for analysing operational complexity in supply chains

This paper proposes a method for analysing the operational complexity in supply chains by using an entropic measure based on information theory. The proposed approach estimates the operational complexity at each stage of the supply chain and analyses the changes between stages. In this paper a stage is identified by the exchange of data and/or material. Through analysis the method identifies the stages where the operational complexity is both generated and propagated (exported, imported, generated or absorbed). Central to the method is the identification of a reference point within the supply chain. This is where the operational complexity is at a local minimum along the data transfer stages. Such a point can be thought of as a ‘sink’ for turbulence generated in the supply chain. Where it exists, it has the merit of stabilising the supply chain by attenuating uncertainty. However, the location of the reference point is also a matter of choice. If the preferred location is other than the current one, this is a trigger for management action. The analysis can help decide appropriate remedial action. More generally, the approach can assist logistics management by highlighting problem areas. An industrial application is presented to demonstrate the applicability of the method.

On a generalized Laguerre operational matrix of fractional integration

A new operationalmatrix of fractional integration of arbitrary order for generalized Laguerre polynomials is derived.The fractional integration is described in the Riemann-Liouville sense.This operational matrix is applied together with generalized Laguerre tau method for solving general linearmultitermfractional differential equations (FDEs).Themethod has the advantage of obtaining the solution in terms of the generalized Laguerre parameter. In addition, only a small dimension of generalized Laguerre operational matrix is needed to obtain a satisfactory result. Illustrative examples reveal that the proposedmethod is very effective and convenient for linear multiterm FDEs on a semi-infinite interval.

On development of fractional calculus during the last fifty years

Fractional calculus generalizes integer order derivatives and integrals. During the last half century a considerable progress took place in this scientific area. This paper addresses the evolution and establishes an assertive measure of the research development.

A fractional approach for the motion planning of redundant and hyper-redundant manipulators

The trajectory planning of redundant robots through the pseudoinverse control leads to undesirable drift in the joint space. This paper presents a new technique to solve the inverse kinematics problem of redundant manipulators, which uses a fractional differential of order α to control the joint positions. Two performance measures are defined to examine the strength and weakness of the proposed method. The positional error index measures the precision of the manipulator's end-effector at the target position. The repeatability performance index is adopted to evaluate if the joint positions are repetitive when the manipulator execute repetitive trajectories in the operational workspace. Redundant and hyper-redundant planar manipulators reveal that it is possible to choose in a large range of possible values of α in order to get repetitive trajectories in the joint space.

Recent history of fractional calculus

This survey intends to report some of the major documents and events in the area of fractional calculus that took place since 1974 up to the present date.

Some pioneers of the applications of fractional calculus

In the last decades fractional calculus (FC) became an area of intensive research and development. This paper goes back and recalls important pioneers that started to apply FC to scientific and engineering problems during the nineteenth and twentieth centuries. Those we present are, in alphabetical order: Niels Abel, Kenneth and Robert Cole, Andrew Gemant, Andrey N. Gerasimov, Oliver Heaviside, Paul Lévy, Rashid Sh. Nigmatullin, Yuri N. Rabotnov, George Scott Blair.

Condition-Based Diagnosis of Mechatronic Systems Using a Fractional Calculus Approach

While fractional calculus (FC) is as old as integer calculus, its application has been mainly restricted to mathematics. However, many real systems are better described using FC equations than with integer models. FC is a suitable tool for describing systems characterised by their fractal nature, long-term memory and chaotic behaviour. It is a promising methodology for failure analysis and modelling, since the behaviour of a failing system depends on factors that increase the model’s complexity. This paper explores the proficiency of FC in modelling complex behaviour by tuning only a few parameters. This work proposes a novel two-step strategy for diagnosis, first modelling common failure conditions and, second, by comparing these models with real machine signals and using the difference to feed a computational classifier. Our proposal is validated using an electrical motor coupled with a mechanical gear reducer.

Pseudo Phase Plane and Fractional Calculus Modelling of Western Global Economic Downturn

This paper applies Pseudo Phase Plane (PPP) and Fractional Calculus (FC) mathematical tools for modeling world economies. A challenging global rivalry among the largest international economies began in the early 1970s, when the post-war prosperity declined. It went on, up to now. If some worrying threatens may exist actually in terms of possible ambitious military aggression, invasion, or hegemony, countries’ PPP relative positions can tell something on the current global peaceful equilibrium. A global political downturn of the USA on global hegemony in favor of Asian partners is possible, but can still be not accomplished in the next decades. If the 1973 oil chock has represented the beginning of a long-run recession, the PPP analysis of the last four decades (1972–2012) does not conclude for other partners’ global dominance (Russian, Brazil, Japan, and Germany) in reaching high degrees of similarity with the most developed world countries. The synergies of the proposed mathematical tools lead to a better understanding of the dynamics underlying world economies and point towards the estimation of future states based on the memory of each time series.

The effect of key process operational conditions on enhanced biological phosphorus removal from wastewater

Enhanced biological phosphorus removal (EBPR) is the most economic and sustainable option used in wastewater treatment plants (WWTPs) for phosphorus removal. In this process it is important to control the competition between polyphosphate accumulating organisms (PAOs) and glycogen accumulating organisms (GAOs), since EBPR deterioration or failure can be related with the proliferation of GAOs over PAOs. This thesis is focused on the effect of operational conditions (volatile fatty acid (VFA) composition, dissolved oxygen (DO) concentration and organic carbon loading) on PAO and GAO metabolism. The knowledge about the effect of these operational conditions on EBPR metabolism is very important, since they represent key factors that impact WWTPs performance and sustainability. Substrate competition between the anaerobic uptake of acetate and propionate (the main VFAs present in WWTPs) was shown in this work to be a relevant factor affecting PAO metabolism, and a metabolic model was developed that successfully describes this effect. Interestingly, the aerobic metabolism of PAOs was not affected by different VFA compositions, since the aerobic kinetic parameters for phosphorus uptake, polyhydroxyalkanoates (PHAs) degradation and glycogen production were relatively independent of acetate or propionate concentration. This is very relevant for WWTPs, since it will simplify the calibration procedure for metabolic models, facilitating their use for full-scale systems. The DO concentration and aerobic hydraulic retention time (HRT) affected the PAO-GAO competition, where low DO levels or lower aerobic HRT was more favourable for PAOs than GAOs. Indeed, the oxygen affinity coefficient was significantly higher for GAOs than PAOs, showing that PAOs were far superior at scavenging for the often limited oxygen levels in WWTPs. The operation of WWTPs with low aeration is of high importance for full-scale systems, since it decreases the energetic costs and can potentially improve WWTP sustainability. Extended periods of low organic carbon load, which are the most common conditions that exist in full-scale WWTPs, also had an impact on PAO and GAO activity. GAOs exhibited a substantially higher biomass decay rate as compared to PAOs under these conditions, which revealed a higher survival capacity for PAOs, representing an advantage for PAOs in EBPR processes. This superior survival capacity of PAOs under conditions more closely resembling a full-scale environment was linked with their ability to maintain a residual level of PHA reserves for longer than GAOs, providing them with an effective energy source for aerobic maintenance processes. Overall, this work shows that each of these key operational conditions play an important role in the PAO-GAO competition and should be considered in WWTP models in order to improve EBPR processes.