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## Über dieses Buch

The focus of this volume is research carried out as part of the program Mathematics of Planet Earth, which provides a platform to showcase the essential role of mathematics in addressing planetary problems and creating a context for mathematicians and applied scientists to foster mathematical and interdisciplinary developments that will be necessary to tackle a myriad of issues and meet future global challenges.

Earth is a planet with dynamic processes in its mantle, oceans and atmosphere creating climate, causing natural disasters and influencing fundamental aspects of life and life-supporting systems. In addition to these natural processes, human activity has increased to the point where it influences the global climate, impacts the ability of the planet to feed itself and threatens the stability of these systems. Issues such as climate change, sustainability, man-made disasters, control of diseases and epidemics, management of resources, risk analysis and global integration have come to the fore.

Written by specialists in several fields of mathematics and applied sciences, this book presents the proceedings of the International Conference and Advanced School Planet Earth, Mathematics of Energy and Climate Change held in Lisbon, Portugal, in March 2013, which was organized by the International Center of Mathematics (CIM) as a partner institution of the international program Mathematics of Planet Earth 2013. The book presents the state of the art in advanced research and ultimate techniques in modeling natural, economical and social phenomena. It constitutes a tool and a framework for researchers and graduate students, both in mathematics and applied sciences.

## Inhaltsverzeichnis

### Max-Stability at Work (or Not): Estimating Return Levels for Daily Rainfall Data

Abstract
When we are dealing with meteorological data, usually one is interested in the analysis of maximal observations and records over time, since these entail negative consequences—risk events. Extreme Value Theory has proved to be a powerful and useful tool to describe situations that may have a significant impact in many application areas, where knowledge of the behavior of the tail of a distribution is of main interest. The classical Gnedenko theorem establishes that there are three type of possible limit max-stable distributions for maxima of blocks of independent and identically distributed (iid) observations. However, for the types of data to which extreme value models are commonly applied, temporal independence is usually an unrealistic assumption and one could ask about the appropriateness of max-stable models. Luckily, stationary and weekly dependent series follow the same distributional limit laws as those of independent series, although with parameters affected by dependence. For rainfall data, we will play with these results, analyzing max-stability at work for rare events estimation and the real impact of “neglecting” iid property.
Maria Isabel Fraga Alves

### Impacts of Vaccination and Behavior Change in the Optimal Intervention Strategy for Controlling the Transmission of Tuberculosis

Abstract
A dynamical model of TB for two age groups that incorporate vaccination of children at birth, behavior change in adult population, treatment of infectious children and adults is formulated and analyzed. Three types of control measures (vaccination, behavior change and anti-TB treatment strategies) are applied with separate rate for children and adults to analyze the solution of the controlled system by using the concept of optimal control theory. It is indicated that vaccination at birth and treatment for both age groups have impact in reducing the value of the reproduction number ($$\mathcal{R}_{o}$$) whereas behavior modification does not have any impact on $$\mathcal{R}_{o}$$. Pontryagin’s Minimum Principle has been used to characterize the optimal level of controls applied on the model. It is shown that the optimal combination strategy of vaccination, behavior change and treatment for the two age groups can help to reduce the disease epidemic with minimum cost of interventions, in shorter possible time.
Temesgen Debas Aweke, Semu Mitiku Kassa

### Modeling of Extremal Earthquakes

Abstract
Natural hazards, such as big earthquakes, affect the lives of thousands of people at all levels. Extreme-value analysis is an area of statistical analysis particularly concerned with the systematic study of extremes, providing an useful insight to fields where extreme values are probable to occur. The characterization of the extreme seismic activity is a fundamental basis for risk investigation and safety evaluation. Here we study large earthquakes in the scope of the Extreme Value Theory. We focus on the tails of the seismic moment distributions and we propose to estimate relevant parameters, like the tail index and high order quantiles using the geometric-type estimators. In this work we combine two approaches, namely an exploratory oriented analysis and an inferential study. The validity of the assumptions required are verified, and both geometric-type and Hill estimators are applied for the tail index and quantile estimation. A comparison between the estimators is performed, and their application to the considered problem is illustrated and discussed in the corresponding context.
Margarida Brito, Laura Cavalcante, Ana Cristina Moreira Freitas

### Detonation Wave Solutions and Linear Stability in a Four Component Gas with Bimolecular Chemical Reaction

Abstract
We consider a four component gas undergoing a bimolecular chemical reaction of type $$A_{1} + A_{2} \rightleftharpoons A_{3} + A_{4}$$, described by the Boltzmann equation (BE) for chemically reactive mixtures. We adopt hard-spheres elastic cross sections and modified line-of-centers reactive cross sections depending on both the activation energy and geometry of the reactive collisions. Then we consider the hydrodynamic limit specified by the reactive Euler equations, in an earlier stage of the chemical reaction, when the gas is far from equilibrium (slow chemical reaction). In particular, the rate of the chemical reaction obtained in this limit shows an explicit dependence on the reaction heat and on the activation energy. Starting from this kinetic setting, we study the dynamics of planar detonation waves for the considered reactive gas and characterize the structure of the steady detonation solution. Then, the problem of the hydrodynamic linear stability of the detonation solution is treated, investigating the response of the steady solution to small rear boundary perturbations. A numerical shooting technique is used to determine the unstable modes in a pertinent parametric space for the considered problem. Numerical simulations are performed for the Hydrogen-Oxygen system and some representative results are presented, regarding the steady detonation wave solution and linear stability.
F. Carvalho, A. W. Silva, A. J. Soares

### Mathematical Aspects of Coagulation-Fragmentation Equations

Abstract
We give an overview of the mathematical literature on the coagulation-like equations, from an analytic deterministic perspective. In Sect. 1 we present the coagulation type equations more commonly encountered in the scientific and mathematical literature and provide a brief historical overview of relevant works. In Sect. 2 we present results about existence and uniqueness of solutions in some of those systems, namely the discrete Smoluchowski and coagulation-fragmentation: we start by a brief description of the function spaces, and then review the results on existence of solutions with a brief description of the main ideas of the proofs. This part closes with the consideration of uniqueness results. In Sects. 3 and 4 we are concerned with several aspects of the solutions behaviour. We pay special attention to the long time convergence to equilibria, self-similar behaviour, and density conservation or lack thereof.
F. P. da Costa

### Resampling-Based Methodologies in Statistics of Extremes: Environmental and Financial Applications

Abstract
Resampling computer intensive methodologies, like the jackknife and the bootstrap are important tools for a reliable semi-parametric estimation of parameters of extreme or even rare events. Among these parameters we mention the extreme value index, ξ, the primary parameter in statistics of extremes. Most of the semi-parametric estimators of this parameter show the same type of behaviour: nice asymptotic properties, but a high variance for small k, the number of upper order statistics used in the estimation, a high bias for large k, and the need for an adequate choice of k. After a brief reference to some estimators of the aforementioned parameter and their asymptotic properties we present an algorithm that deals with an adaptive reliable estimation of ξ. Applications of these methodologies to the analysis of environmental and financial data sets are undertaken.
M. Ivette Gomes, Lígia Henriques-Rodrigues, Fernanda Figueiredo

### On the Optimal Control of Flow Driven Dynamic Systems

Abstract
The objective of this work is to develop a mathematical framework for the modeling, control and optimization of dynamic control systems whose state variable is driven by interacting ODE’s (ordinary differential equations) and solutions of PDE’s (partial differential equations). The ultimate goal is to provide a sound basis for the design and control of new advanced engineering systems arising in many important classes of applications, some of which may encompass, for example, underwater gliders and mechanical fishes. For now, the research effort has been focused in gaining insight by applying necessary conditions of optimality for shear flow driven dynamic control systems which can be easily reduced to problems with ODE dynamics. In this article we present and discuss the problem of minimum time control of a particle advected in a Couette and Poiseuille flows, and solve it by using the maximum principle.
Teresa Grilo, Sílvio M. A. Gama, Fernando Lobo Pereira

### An Overview of Network Bifurcations in the Functionalized Cahn-Hilliard Free Energy

Abstract
The functionalized Cahn-Hilliard (FCH) free energy models interfacial energy in amphiphilic phase-separated mixtures. Its minimizers and quasi-minimizers encompass rich classes of network morphologies with detailed inner layers incorporating bilayers, pore, pearled pore, and micelle type structures. We present an overview of the stability of the network morphologies as well as the competitive evolution of bilayer and pore morphologies under a gradient flow in three space-dimensions.
Noa Kraitzman, Keith Promislow

### The Economics of Ethanol: Use of Indirect Policy Instruments

Abstract
General equilibrium models typically ignore environmental goods because it is assumed that they have zero price. In the United States the Renewable Fuel Standard was introduced to offset the carbon emissions from by burning ethanol. The model in this study extends the standard general equilibrium approach to consider both positive and negative externalities. The negative externality is due to gasoline consumption while the positive externality is from substitution of ethanol for gasoline.
Charles B. Moss, Andrew Schmitz, Troy G. Schmitz

### Geostatistical Analysis in Extremes: An Overview

Abstract
Classical statistics of extremes is very well developed in the univariate context for modeling and estimating parameters of rare events. Whenever rain, snow, storms, hurricanes, earthquakes, and so on, happen the analysis of extremes is of primordial importance. However such rare events often present a temporal aspect, a spatial aspect or both. Classical geostatistics, widely used for spatial data, is mostly based on multivariate normal distribution, inappropriate for modeling tail behavior. The analysis of spatial extreme data, an active research area, lies at the intersection of two statistical domains: extreme value theory and geostatistics. Some statistical tools are already available for the spatial modeling of extremes, including Bayesian hierarchical models, copulas and max-stable random fields. The purpose of this chapter is to present an overview of basic spatial analysis of extremes, in particular reviewing max-stable processes. A real case study of annual maxima of daily rainfall measurements in the North of Portugal is slightly discussed as well the main functions in R environment for doing such analysis.
M. Manuela Neves

### Reducing the Minmax Regret Robust Shortest Path Problem with Finite Multi-scenarios

Abstract
The minmax regret robust shortest path problem is a combinatorial optimization problem that can be defined over networks where costs are assigned to arcs under a given scenario. This model can be continuous or discrete, depending on whether costs vary within intervals or within discrete sets of values. The problem consists in finding a path that minimizes the maximum deviation from the shortest paths over all scenarios. This work focuses on designing tools to reduce the network, in order to make easier the search for an optimum solution. With this purpose, methods to identify useless nodes to be removed and to detect arcs that surely belong to the optimum solution are developed. Two known algorithms for the robust shortest path problem are tested on random networks with and without these preprocessing rules.
Marta M. B. Pascoal, Marisa Resende

### Mathematics of Energy and Climate Change: From the Solar Radiation to the Impacts of Regional Projections

Abstract
This chapter focuses on the natural and anthropogenic drivers of climate change and on the assessment of potential impacts of regional projections for different scenarios of future climate. Internal and external forcing factors of climate change are associated to changes in the most important processes of energy transfer with influence on the energy balance of the climate system. The role of the solar activity, regular variations in the orbital parameters of the Earth and the radiative forcing which comprises the changes in the chemical composition of the atmosphere and the characteristics of the radiative processes that occur in the atmosphere and on the surface of the Earth will be discussed. Recent evidences of climate change and the general characteristics of the climate models used in climate projection will be presented. The chapter ends with results of some case studies of potential impacts of regional climate change projections in Portugal, namely in forest fire regime, extreme precipitation intensity and in the design of storm water drainage infrastructures.
Mário Gonzalez Pereira

### Infinite Horizon Optimal Control for Resources Management in Agriculture

Abstract
This article concerns an optimal control based framework for the optimization of resources in agriculture taking into account the environment sustainability. A decentralized, adaptive, hierarchic architecture to support long term coordinated decision-making strategies is required in order to achieve the common long term desired equilibrium in the environment state, and, at the same time, allow the economic sustainability of a number of distributed farm producers with, possibly conflicting, short term economic goals. The overall coordination is achieved by an adaptive Model Predictive Control structure that, on the one end hand, promotes the long term common good by approximating the solution to an infinite horizon optimal control problem, and, on the other hand, provides agro-chemical indicators to each one of the local farmers. We will emphasize on the importance of optimality results for infinite horizon optimal control problems of the Mayer type depending on the state at the final time while satisfying constraints at both trajectory endpoints.
Fernando Lobo Pereira

### Distributed Reasoning

Abstract
This paper discusses the problem of learning a global model from local information. We consider ubiquitous streaming data sources, such as sensor networks, and discuss efficient learning distributed algorithms. We present the generic framework of distributed sources of data, an illustrative algorithm to monitor the global state of the network using limited communication between peers, and an efficient distributed clustering algorithm.
Pedro Rodrigues, João Gama

### Multiscale Internet Statistics: Unveiling the Hidden Behavior

Abstract
Being able to characterize and predict the behavior of Internet users based only on layer 2 statistics can be very important for network managers and/or network operators. Operators can perform a low level monitoring of the communications at the network entry points, independently of the data encryption level and even without being associated with the network itself. Based on this low level data, it is possible to optimize the access service, offer new security threats detection services and infer the users behavior, which consists of identifying the underlying web application that is responsible by the layer 2 traffic at different time instants and characterize the usage dynamics of the different web applications. Several identification methodologies have been proposed over the years to classify and identify IP applications, each one having its own advantages and drawbacks: port-based analysis, deep packet inspection, behavior-based approaches, learning theory, among others. Although some of them are very efficient when applied to specific scenarios, all approaches fail when only low level statistics are available or under data encryption restrictions. In this work, we propose the use of multiscaling traffic characteristics to differentiate web applications and the use of a Markovian model to characterize the dynamics of user actions over time. By applying the proposed methodology to Wi-Fi layer 2 traffic generated by users accessing different common web services/contents through HTTP (namely social networking, web news and web-mail applications), it was possible to achieve a good prediction of the different users behaviors. The classification results obtained show that the developed multiscaling traffic Markovian model has the potential to efficiently identify, model and predict Internet users behaviors based only on layer 2 traffic statistics.
Paulo Salvador, António Nogueira, Eduardo Rocha

### The Role of Clouds, Aerosols and Galactic Cosmic Rays in Climate Change

Abstract
A review of the role played by clouds, by natural and anthropogenic aerosols and by their interaction, on climate, is presented. The suggestion that galactic cosmic rays may affect the interaction between clouds/aerosols and climate is here discussed in the context of the CLOUD (Cosmics Leaving Outdoor Droplets) experiment at CERN. The experiment has shown that cosmic rays enhance aerosol nucleation and cloud condensation but the effect is too weak to have an impact on climate during a solar cycle or over the last century. The CLOUD experiment has also revealed a nucleation mechanism involving the formation of clusters containing sulphuric acid and oxidized organic molecules.
Filipe Duarte Santos

### Long Time Behaviour and Self-similarity in an Addition Model with Slow Input of Monomers

Abstract
We consider a coagulation equation with constant coefficients and a time dependent power law input of monomers. We discuss the asymptotic behaviour of solutions as t → , and we prove solutions converge to a similarity profile along the non-characteristic direction.
Rafael Sasportes

### Modelling the Fixed Bed Adsorption Dynamics of CO2/CH4 in 13X Zeolite for Biogas Upgrading and CO2 Sequestration

Abstract
The sorption of $$\mathrm{CO}_{2}$$ and $$\mathrm{CH}_{4}$$ in binderless beads of 13X zeolite has been investigated between 313 and 423 K and total pressure up to 0.5 MPa through fixed bed adsorption experiments. Experimental selectivities $$\mathrm{CO}_{2}/\mathrm{CH}_{4}$$ range from 37 at a low pressure of 0.0667 MPa to approximately 5 at the high temperature of 423 K. The breakthrough curves measured show a plateau of pure $$\mathrm{CH}_{4}$$ of approximately 6 min depending of the operating conditions chosen. A mathematical model was developed and tested predicting with good accuracy the behaviour of the fixed bed adsorption experiments being a valuable tool for the design of cyclic adsorption processes for biogas upgrading and $$\mathrm{CO}_{2}$$ capture using 13X zeolite.
José A. C. Silva, Alírio E. Rodrigues

### Detection of Additive Outliers in Poisson INAR(1) Time Series

Abstract
Outlying observations are commonly encountered in the analysis of time series. In this paper a Bayesian approach is employed to detect additive outliers in order one Poisson integer-valued autoregressive time series. The methodology is informative and allows the identification of the observations which require further inspection. The procedure is illustrated with simulated and observed data sets.
Maria Eduarda Silva, Isabel Pereira

### From Ice to Penguins: The Role of Mathematics in Antarctic Research

Abstract
Mathematics underpins all modern Antarctic science as illustrated by numerous activities carried out during the international year “Mathematics for Planet Earth”. Here, we provide examples of some ongoing applications of mathematics in a wide range of Antarctic science disciplines: (1) Feeding and foraging of marine predators; (2) Fisheries management and ecosystem modelling; and (3) Climate change research. Mathematics has allowed the development of diverse models of physical and ecological processes in the Antarctic. It has provided insights into the past dynamics of these systems and allows projections of potential future conditions, which are essential for understanding and managing the effects of fishing and climate change. Highly specific methods and models have been developed to address particular questions in each discipline, from the detailed analyses of remote-sensed predator tracking data to the assessment of the outputs from multiple global climate models. A key issue, that is common to all disciplines, is how to deal with the inherent uncertainty that arises from limited data availability and the assumptions or simplifications that are necessary in the analysis and modeling of interacting processes. With the continued rapid development of satellite-based and remote observation systems (e.g. ocean drifters and automatic weather stations), and of new methods for genetic analyses of biological systems, a step-change is occurring in the magnitude of data available on all components of Antarctic systems. These changes in data availability have already led to the development of new methods and algorithms for their efficient collection, validation, storage and analysis. Further progress will require the development of a wide range of new and innovative mathematical approaches, continuing the trend of world science becoming increasingly international and interdisciplinary.
José C. Xavier, S. L. Hill, M. Belchier, T. J. Bracegirdle, E. J. Murphy, J. Lopes Dias

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