Dynamics, Games and Science

It is recognized that corrupt behavior determines the institutional types of an economic system where an institution is ruled out by economic agents (e.g. officials-public or private) abusing their role to procure gain for themselves (rent-seeking activities) or somebody else. In this vein, we study an evolutionary model of institutional corruption. We show that income inequality and income taxation are the main factors (explanatory variables) for fighting institutional corruption. We conclude with some feasible policies on institutions, beliefs and incentives to combat the corruption.

We review previous results providing sufficient conditions to determine the global dynamics for equivariant maps of the plane with a unique fixed point which is also hyperbolic.

We consider a model, due to Andersen and Nielsen (Econ Lett 118(2):262–264, 2013), concerning the behavior of a risk-averse sports team under uncertainty in demand: the team chooses a value for the price of its ticket, but the ticket demand is stochastic at the moment of decision. For this model, we carry out a decision analysis by studying several comparative-static effects not considered by the authors in their paper. Specifically, we examine the effect of changes in the team’s risk aversion, and also the effect of a variation in the risk of the random demand. Furthermore, we enhance the model by considering a proportional profit tax, and we study the effect of a variation in the tax rate. We derive some conditions under which the sports team finds optimal to reduce the ticket price as a consequence of a rise in the tax rate.

The Robinson-Schensted-Knuth (RSK) correspondence is an important combinatorial bijection between two line arrays of positive integers (or non-negative integer matrices) and pairs of semi-standard Young tableaux (SSYTs). One of its applications, in the theory of Schur polynomials, is a bijective proof of the well known Cauchy identity. An interesting analogue of this bijection was given by Mason, where SSYTs are replaced by semi-skyline augmented fillings (SSAFs), originated in the Haglund-Haiman-Loehr formula for non-symmetric Macdonald polynomials. The latter object SSAF has the advantage of detecting the key of a SSYT which is easily read off from the SSAF shape. Using this analogue, we have previously considered the restriction of RSK correspondence to multisets of cells in a (truncated) staircase. The image is described by a Bruhat order inequality between the keys of the recording and the insertion fillings. This has allowed to derive a (truncated) triangular version of the Cauchy identity, due to Lascoux, where Schur polynomials are replaced by key polynomials or Demazure characters. We now consider the restriction of RSK to a near staircase, in French convention, where the top leftmost and the bottom rightmost cells and also possibly some cells in the diagonal layer are deleted. The image is described by additional Bruhat order inequalities, specified by the cells in the diagonal layer. The bijection is then used to extend the triangular version to near staircases, also a version due to Lascoux, where Demazure characters are now under the action of Demazure operators specified by the cells in the diagonal layer. Our analysis is made in the framework of Fomin’s growth diagrams where a formulation of the Mason’s analogue is given. This is then used to show how to pass from a triangular shape to a near staircase, via the action crystal operators, and how this affects the keys in the image of the RSK.

Using a cross-section database that observes the Portuguese labour market in two different phases of the business cycle, the present paper aims to address the issue of the segmentation of the Portuguese labour market taking into account the heterogeneity resulting from different unemployment characteristics observed along the Portuguese geographical space and applying two optimization clustering methods: the

k

-means and the spectral methods. The

k

-means is a traditional optimisation clustering method applied to cluster data observations. Spectral clustering is an alternative method based on the computation of the dominant eigenvectors of a matrix related with the distance among data points. The results obtained by the two methods are not identical but are very close and show that, apart the economic phase of the cycle, Portugal presents two very different profiles of registered unemployment. One of them can be considered problematic because it presents a higher percentage of unemployed women, long duration unemployed and unemployed with low levels of formal education—these are the groups that present more difficulties in the labour market and for which is more difficult to find a job after losing one. The segmentation of the labour market is a reality and the labour market is not adjusting to the business cycle.

Given a finite group

G

and

$$\nu \in \mathcal{P}(G)$$

, we study the dynamics of the linear automorphisms of a convolution measure algebra over

G

,

$$T_{\nu }(\mu ) =\nu {\ast}\mu$$

. In order to understand and classify the asymptotic behavior of this dynamical system we provide an alternative to classical results, a very direct way to understand convergence of the sequence

$$\{\nu ^{n}\}_{n\in \mathbb{N}}$$

, where

$$\nu ^{n} =\underbrace{\mathop{ \nu {\ast}\ldots {\ast}\nu }}\limits _{n}$$

, through the subgroup generated by its support.

We consider optimal control problems where the dynamical system and the running cost are affected by fast periodic oscillations of the state variables. We show that, under suitable controllability and structure assumptions, it is possible to describe the limiting optimal control problem. The proofs make use of results in the theory of homogenization and singular perturbations of Hamilton-Jacobi equations.

We consider ordinary differential equations on the unit simplex of

$$\mathbb{R}^{n}$$

that naturally occur in population games, models of learning and self reinforced random processes. Generalizing and relying on an idea introduced in Dupuis and Fisher (On the construction of Lyapunov functions for nonlinear Markov processes via relative entropy, 2011), we provide conditions ensuring that these dynamics are gradient like and satisfy a suitable “angle condition”. This is used to prove that omega limit sets and chain transitive sets (under certain smoothness assumptions) consist of equilibria; and that, in the real analytic case, every trajectory converges toward an equilibrium. In the reversible case, the dynamics are shown to be

C

1

close to a gradient vector field. Properties of equilibria -with a special emphasis on potential games—and structural stability questions are also considered.

We derive from first principles the dynamical equations that govern the interaction of small-amplitude water waves with freely floating obstacles in a stratified multilayer fluid. Focusing on two-layer fluids, we present the equations in an easily manageable matrix form, write down conditions for the stability of equilibrium and, by limiting ourselves to time-harmonic motions, recast the problem as a spectral boundary-value problem composed of a differential equation and an algebraic system, coupled through boundary conditions. Proceeding with a suitable variational and operator formulation, we present an elimination scheme that simplifies the system to a linear spectral problem for a self-adjoint operator in a Hilbert space. Under symmetry assumptions on the geometry of the fluid domain, we derive a sufficient condition guaranteeing the existence of trapped modes in a two-layer fluid channel.

Shannon’s switching game is a combinatorial game invented by C. Shannon circa 1955 as a simple model for breakdown repair of the connectivity of a network. The game was completely solved by A. Lehman, shortly after, in what is considered the first application of matroid theory. In the middle 1980s Y. O. Hamidoune and M. Las Vergnas introduced and solved directed versions of the game for graphs considering their generalization to oriented matroids. We do a brief review of the main results and conjectures of the directed case.

This paper presents a method to measure the functional efficiency of futures markets in terms of social welfare using a standard futures market structural model. Employing the concept of social surplus, it can be shown that the error committed when using futures prices to estimate spot prices in the future results in a welfare loss caused by the erroneous allocation of resources. Therefore, the social welfare associated with the presence of futures markets can be measured using a social loss (SL) statistic and its components. The results confirm the consistency and robustness of the method. Finally, several practical uses for the SL statistic are suggested.

This brief survey of nonlinear Leslie models focuses on the fundamental bifurcation that occurs when the extinction equilibrium destabilizes as

R

0

increases through 1. Of particular interest is the bifurcation that occurs when only the oldest age class is reproductive, in which case the Leslie projection matrix is not primitive. This case is distinguished by the invariance of the boundary of the positive cone on which orbits contain temporally synchronized, missing age classes and by the bifurcation of oscillatory attractors, lying on the boundary of the positive cone, in addition to the bifurcation of positive equilibria. The lack of primitivity of the Leslie projection matrix, while seemingly only a mathematically technicality, corresponds to a fundamental life history strategy in population dynamics, namely, semelparity (when individuals have one reproductive event before dying). The study of semelparous Leslie models was historically motivated by the synchronized outbreak cycles of periodical insects, the most famous being the long-lived cicadas (C.

magicicada

spp).

These are the notes of the short course “Stability of quasi-periodic dynamics” given at the Advanced School Planet Earth, Dynamics, Games and Science II held in Lisbon, Portugal, from 28 August to 6 September 2013 and organized by the International Center of Mathematics CIM - Portugal. We review some recent results concerning the stability of non-autonomous linear differential equations with a quasi-periodic forcing.

This paper deals with equilibrium existence for incomplete markets economies with finitely-lived agents and infinitely-lived agents when default is allowed and borrowers have to constitute collateral in terms of durable goods. In the first model, lenders are protected by an exogenous personalized collateral. In the second model, the personalized collateral requirements are endogenously determined by a financial institution whose objective is to minimize the default rate taking into account agent’s default history.

In the present paper, we study forward-forward mean-field games with a power dependence on the measure and subquadratic Hamiltonians. These problems arise in the numerical approximation of stationary mean-field games. We prove the existence of smooth solutions under dimension and growth conditions for the Hamiltonian. To obtain the main result, we combine Sobolev regularity for solutions of the Hamilton-Jacobi equation (using Gagliardo-Nirenberg interpolation) with estimates of polynomial type for solutions of the Fokker-Planck equation.

Consider a typical agency relation involving a capital owner and a manager. The principal (i.e., the capital owner) has a potential budget to assign to investment projects. The effective amount of investment will be a share of the potential level, given the specific form of interaction that will be established between the principal and the agent (i.e., the manager). The budget setting problem originating from this relation is evaluated from the point of view of the manager, who wants to maximize the received budget, in an intertemporal basis. The optimal control problem is subject to a constraint, which indicates how the assigned budget evolves over time. In this constraint, a matching function takes a central role; the arguments of the function are the agent’s effort to absorb new funds and the financial resources the principal has available but has not yet channeled to the manager.

This paper extends the Ramsey-Cass-Koopmans (RCK) model by considering both a non constant number of hours worked by each individual through time and leisure, which includes healthcare and creative activities. With this extension, the seminal RCK model can be used to analyse the economic growth effects arising from governmental policies. In this context, governmental expenditures financed by lump-sum taxes and inefficient expenditures lead to a decrease in the short, medium and long-run economic growth.

This is an article I wrote for Dynamics, Games, and Science. In Dynamics, Game, and Science, one of the most important equilibrium states is a Gibbs state. The deformation of a Gibbs state becomes an important subject in these areas. An appropriate metric on the space of underlying dynamical systems is going to be very helpful in the study of deformation. The Teichmüller metric becomes a natural choice. The Teichmüller metric, just like the hyperbolic metric on the open unit disk, makes the space of underlying dynamical systems a complete space. The Teichmüller metric precisely measures the change of the eigenvalues at all periodic points which are essential data needed to obtain the Gibbs state for a given dynamical system. In this article, I will introduce the Teichmüller metric and, subsequently, a generalization of Gibbs theory which we call geometric Gibbs theory.

The kinematic equations for rolling a sphere on another sphere, subject to non-holonomic constraints of non-slip and non-twist, are known and can be found in [

7

]. Here we present an alternative approach to derive these kinematic equations which is also suitable for describing the rolling of more general manifolds embedded in Euclidean space. This approach consists on rolling each of the manifolds separately on a common affine tangent space and then using the transitive and symmetric properties of rolling maps to derive the kinematic equations of rolling one manifold on the other. We use this approach to derive the kinematic equations for rolling an

n

-dimensional sphere on another one with the same dimension. It is also well known that the sphere rolling on sphere system is controllable, except when the two spheres have equal radii. This is a theoretical result that guarantees the possibility to roll one of the spheres on the other from any initial configuration to any final configuration without violating the non-holonomic constraints. However, from a practical viewpoint it is important to know how this is done. To answer this more applied question, we present a constructive proof of the controllability property, by showing how the forbidden motions can be performed by rolling without slip and twist. This is also illustrated for 2-dimensional spheres.

Consider the shift

σ

acting on the Bernoulli space

$$\varSigma =\{ 1,2,\ldots,n\}^{\mathbb{N}}$$

. We denote

$$\hat{\varSigma }=\{ 1,2,\ldots,n\}^{\mathbb{Z}} =\varSigma \times \varSigma$$

. We analyze several properties of the maximizing probability

μ

∞

, 

A

of a Hölder potential

$$A:\varSigma \rightarrow \mathbb{R}$$

. Associated to

A

(

x

), via the involution kernel,

W

(

x

, 

y

),

$$W:\hat{\varSigma }\rightarrow \mathbb{R}$$

, one can get the dual potential

A

∗

(

y

), where

$$(x,y) \in \hat{\varSigma }$$

. We denote

$$\mu _{\infty,A^{{\ast}}}$$

the maximizing probability for

A

∗

. We would like to consider the transport problem from

μ

∞

, 

A

to

$$\mu _{\infty,A^{{\ast}}}$$

. In this case, it is natural to consider the cost function

c

(

x

, 

y

) = 

I

(

x

) −

W

(

x

, 

y

) +

γ

, where

I

is the deviation function for

μ

∞

, 

A

, as the limit of Gibbs probabilities

μ

β A

for the potential

β A

when

β

 → 

∞

. The value

γ

is a constant which depends on

A

. We could also take

c

 = −

W

above. We denote by

$$\mathcal{K} = \mathcal{K}(\mu _{\infty,A},\mu _{\infty,A^{{\ast}}})$$

the set of probabilities

$$\hat{\eta }(x,y)$$

on

$$\hat{\varSigma }$$

, such that

$$\pi _{x}^{{\ast}}(\hat{\eta }) =\mu _{\infty,A},\,\,\text{and}\,\,\pi _{y}^{{\ast}}(\hat{\eta }) =\mu _{\infty,A^{{\ast}}}\,.$$

We describe the minimal solution

$$\hat{\mu }$$

(which is invariant by the shift on

$$\hat{\varSigma }$$

) of the Transport Problem, that is, the solution of

$$\displaystyle{\inf _{\hat{\eta }\in \mathcal{K}}\int \int c(x,y)\,d\,\hat{\eta } =\, -\,\max _{\hat{\eta }\in \mathcal{K}}\int \int (W(x,y)-\gamma )\,d\,\hat{\eta }.\,}$$

The optimal pair of functions for the Kantorovich Transport dual Problem is (−

V

, −

V

∗

), where we denote the two calibrated sub-actions by

V

and

V

∗

, respectively, for

A

and

A

∗

. We show that the involution kernel

W

is cyclically monotone. In other words, satisfies a twist condition in the support of

$$\hat{\mu }$$

. We analyze the question: is the support of

$$\hat{\mu }$$

a graph? We also investigate the question of finding an explicit expression for the function

$$f:\varSigma \rightarrow \mathbb{R}$$

whose

c

-subderivative determines the graph. We also analyze the same kind of problem for expanding transformations on the circle.

Computer vision problems typically have geometric constraints. When two cameras view a 3D scene from two distinct positions, or a single camera views a 3D scene from two different locations, there are a number of geometric relations between the 3D points and their projections onto the 2D images. These relations lead to constraints between the image points. In particular, the epipolar constraint encodes the relation between correspondences across two images of the same scene. In a calibrated setting, the epipolar constraint is parameterized by essential matrices, which form the Essential Manifold. The reconstruction of a video from several images of a scene can be formulated as an interpolation problem on this manifold. An approach that simplifies the generation of an interpolating curve consists in projecting the problem to a linear manifold where it can be solved easily, and then projecting back the solution on the nonlinear manifold. The projection is realized by rolling the Essential Manifold, without slip and twist, over an affine tangent space. This gives particular relevance to rolling motions in the context of certain computer vision problems. Having this in mind, we derive the kinematic equations for the rolling motions of the Essential Manifold and present explicit solutions when it rolls along geodesics.

The definition and study of pattern avoidance for set partitions, which is an analogue of pattern avoidance for permutations, begun with Klazar. Sagan continued his work by considering set partitions which avoid a single partition of three elements, and Goyt generalized these results by considering partitions which avoid any family of partitions of a 3-element set. In this paper we enumerate and describe set partitions, even set partitions and odd set partitions without singletons which avoid any family of partitions of a 3-element set. The characterizations of these families allow us to conclude that the corresponding sequences are

P

-recursive. We also construct Gray codes for the sets of singletons free partitions that avoid a single partition of three elements.

We consider association schemes with

d

classes and the underlying Bose-Mesner algebra,

$$\mathcal{A}$$

. Then, by taking into account the relationship between the Hadamard and the Kronecker products of matrices and making use of some matrix techniques over the idempotents of the unique basis of minimal orthogonal idempotents of

$$\mathcal{A}$$

, we prove some results over the Krein parameters of an association scheme.

In this paper, a bivariate integer-valued autoregressive model with periodic structure is introduced and studied in some detail. The model can be view as a generalization of the one considered in Pedeli and Karlis (Stat. Model. 11:325–349, 2011). Emphasis is placed on models with periodic bivariate Poisson innovations. Basic probabilistic and statistical properties of the model are discussed as well as parameter estimation and forecasting. The proposed model is applied to a bivariate data series concerning the monthly number of fires in neighbor counties, Aveiro and Coimbra, in Portugal.

In the context of the current Eurozone crisis, the study of the effects of uncertainty in the macrodynamics of employment is a topic of major importance. This paper tackles this challenging question. At a first step a non-ideal relay hysteresis type microeconomic model of employment adjustment with uncertainty is presented. Then, an aggregation mechanism is explicitly considered in order to analyse the aggregate level of employment. Finally, as a new feature, uncertainty is considered endogenously determined by the actual state of the economy. Aggregate time-series built from micro monthly data on a representative sample of Portuguese manufacturing firms is used on a computational implementation of the linear play model of hysteresis. Results illustrate that uncertainty enhances the hysteretic behaviour of employment in small firms, but this effect is not significant for large ones.

Black scabbardfish (

Aphanopus carbo

Lowe, 1839) is a widely distributed species across the Atlantic ocean. In Portuguese mainland waters the existing specimens are immature (not able to reproduce). It is admitted that they have migrated from the West of the British Isles and that they remain in the area for some years, until they attain an adequate size or physiological conditions which allow them to migrate and reproduce elsewhere.The present study aims to model the dynamics of the population of black scabbardfish living in the International Council for the Exploration of the Seas Division IXa, for which disaggregated data are available, although within the context of a larger population. With this purpose, a state-space model is used, which enables the estimation of the unknown abundance (latent process) by exploring its dependency relationship with the observational data on the species fishing landings in that area. The population is partitioned into length groups and the population evolution process is subdivided into biological related subprocesses.The estimation is achieved within a Bayesian paradigm, where all the available biological information is incorporated in the prior distributions of the parameters of the subprocesses. Later, short-term trajectories of the population living in IXa are studied, via simulations that are constructed based on different management scenarios.

The concept of entropy has been applied to such different fields as thermodynamics, cosmology, biology, chemistry, information theory and economics. An interesting application of entropy in the latter field is the existence of a complete ordering of information structures represented by the decrease in entropy, computed à la Shannon, of the agent’s beliefs. In this paper we will apply this entropy ordering to information structures used in experiments assessing the role of communication in coordination games.

This paper begins a new line in the estimate and classification of New household formation in Spain. It starts with the study of the emancipation of young people dependent of their parents and proposes a micro-econometric analysis to find and measure socioeconomic factors that affect the decisions youngsters make when they leave their parent’s home. In the first place a discrete choice model three level nested multinomial logit based in population characteristics is proposed. In order to improve the results avoiding systematic biases and making use of all the information in the data source, the model is replaced by a sequence of three binary logits. The period of study extends from 2008 to 2011 so it will be useful to find evidence of how the economic crisis has affected the current trends of Spanish growing New household formation levels and increasing emigration of young dependents. The gap between Spain and the rest of European countries concerning Emancipation and New household formation levels is reducing since the last nineties but the high level of unemployment in the current crisis has supposed a brake in that trend.

Skin cancer is one of the most common malignancies in humans. Early detection of suspicious skin signs is critical to prevent this kind of malignancy, and various disciplines can play a crucial role in its detection. The lesion border is especially relevant for diagnosis, and provides information on the shape of the lesion, growth path, and growth rate. Digital image processing methods can be used to perform automatic lesion border detection; nonetheless, the presence of artifacts may induce artificial borders, thereby jeopardizing the efficiency of automatic detection algorithms. Artifact removal is a necessary pre-processing step to improve the accuracy quality of the border identification. In this work, we present a method to identify and remove artifacts in dermoscopic images. This pre-processing step enhances the output of the segmentation of the lesion. This process is based on several applications of the Local Normalization, which is a method that increases the local contrast between local pixels, improving the overall quality of the image, especially with non-uniform illumination. The process is scale sensitive and uses a multi-scale approach adaptable to every shape and size of skin lesions.

This chapter concerns dynamics, bifurcations and synchronization properties of von Bertalanffy’s functions, a new class of continuous one-dimensional maps, which was first studied in [22]. This family of unimodal maps is proportional to the right hand side of von Bertalanffy’s growth equation. We provide sufficient conditions for the occurrence of stability, period doubling, chaos and non admissibility of von Bertalanffy’s dynamics. These dynamics are dependent on the variation of the intrinsic growth rate of the individual weight, which is given by

r

 = 

r

(

K

, 

W

∞

), where

K

is von Bertalanffy’s growth rate constant and

W

∞

is the asymptotic weight. A central point of our investigation is the study of bifurcations structure for this class of functions, on the two-dimensional parameter space (

K

, 

W

∞

). Another important approach in this work is the study of synchronization phenomena of von Bertalanffy’s models in some types of networks: paths, grids and lattices. We study the synchronization level when the local dynamics vary and the topology of the network is fixed. This variation is expressed by the Lyapunov exponents, as a function of the intrinsic growth rate

r

. Moreover, we present some results about the evolution of the network synchronizability, as the number of nodes increases, keeping fixed the local dynamics, in some types of networks: paths, grids and lattices. We also discuss the evolution of the network synchronizability as the number of edges increases. To support our results, we present numerical simulations for these types of networks.

In the present survey, we give an overview of some recent developments on examples of differential equations whose flows have heteroclinic cycles and networks; we fit some properties of their nonwandering sets into the classic theory of hyperbolic and pseudo-hyperbolic sets.

Dengue is a vector-borne disease and 40 % of world population is at risk. Dengue transcends international borders and can be found in tropical and subtropical regions around the world, predominantly in urban and semi-urban areas. A model for dengue disease transmission, composed by mutually-exclusive compartments representing the human and vector dynamics, is presented in this study. The data is from Madeira, a Portuguese island, where an unprecedented outbreak was detected on October 2012. The aim of this work is to simulate the repercussions of the control measures in the fight of the disease.

In this work we describe the saturated numerical semigroups, and characterize the SAT system of generators for them. We see how we can arrange them in a tree rooted in

$$\mathbb{N}$$

and describe the sons of any vertex of this tree. Finally, we present an algorithm for computing the set of saturated numerical semigroups of a given genus

In this paper we estimate a small structural model, in order to forecast the key macroeconomic variables of output growth and underlying inflation. In contrast to models with purely statistical foundations, the Bayesian Vector Autoregressive Dynamic Stochastic General Equilibrium (BVAR-DSGE) model, uses the theoretical information of a DSGE model to offset insample overfitting. We compare the forecast performance of BVAR-DSGE model with Minesota VAR and independently estimates DSGE model. The open economy DSGE model of Lubik and Schorfheide (2007) is implemented to provide prior information for the VAR.

The application of compound tests in clinical analysis or acceptance sampling exults in resource savings. Furthermore, quantitative compound tests allow to infer whether the amount of some substance of any individual in the group is greater or lower than a prefixed threshold. However, the use of this type of tests must be done with caution to avoid having a high probability of misclassification. This work uses the weight of the tails of the underlying distribution as a measure of the adequacy of the application of continuous compounds tests.

We consider the problem of financing two productive sectors in an economy through bank loans, when the sectors may experience independent demands for money but when it is desirable for each to maintain an independently determined sequence of prices. An idealized central bank is compared with a collection of commercial banks that generate profits from interest rate spreads and flow those through to a collection of consumer/owners who are also one group of borrowers and lenders in the private economy. We model the private economy as one in which both production functions and consumption preferences for the two goods are independent, and in which one production process experiences a shock in the demand for money arising from an opportunity for risky innovation of its production function. An idealized, profitless central bank can decouple the sectors, but for-profit commercial banks inherently propagate shocks in money demand in one sector into price shocks with a tail of distorted prices in the other sector. The connection of profits with efficiency-reducing propagation of shocks is mechanical in character, in that it does not depend on the particular way profits are used strategically within the banking system. In application, the tension between profits and reserve requirements is essential to enabling but also controlling the distributed perception and evaluation services provided by commercial banks. We regard the inefficiency inherent in the profit system as a source of costs that are paid for distributed perception and control in economies.

We review the optimal control of systems modeling the dynamics of tuberculosis. Time dependent control functions are introduced in the mathematical models, representing strategies for the improvement of the treatment and cure of active infectious and/or latent individuals. Optimal control theory allows then to find the optimal way to implement the strategies, minimizing the number of infectious and/or latent individuals and keeping the cost of implementation as low as possible. An optimal control problem is proposed and solved, illustrating the procedure. Simulations show an effective reduction in the number of infectious individuals.

In the last decade wildfires became a serious problem in Portugal due to socieconomic and climate change trends. In order to analyse wildfire data, we employ beta regression for modelling the proportion of burned wild area, under a Bayesian perspective. Our main goal is to find out fire risk factors that influence the proportion of area burned and what may make a wild area susceptible or resistant to fire. Then, we analyse wildfire data in Portugal during 1990–1994 through Bayesian normal and beta regression models that use Markov chain Monte Carlo methods for estimating quantities of interest.

The subject of

H

-decompositions of graphs was first introduced by Erdős, Goodman and Pósa in 1966. Given graphs

G

and

H

, an

H-decomposition

of

G

is a partition of the edge set of

G

, such that, each part is either a single edge or forms a graph isomorphic to

H

. Let

ϕ

(

n

, 

H

) be the smallest number

ϕ

, such that, any graph

G

with

n

vertices admits an

H

-decomposition with at most

ϕ

parts. The exact computation of

ϕ

(

n

, 

H

) for an arbitrary

H

is still an open problem. In this paper we will survey recent results about

H

-decompositions of graphs and we will also introduce its Ramsey or coloured version together with recent results on this problem.