A cellular automaton model for collective neural dynamics

A stochastic epidemic model for the collective behaviour of a large set of Boolean automata placed upon the sites of a complete graph is revisited. In this paper we study the generalisation of the model to take into account inhibitory neurons. The resulting stochastic cellular automata are completely defined by five parameters: the number of excitatory neurons, N, the number of inhibitory neurons, M, the probabilities of excitation, α, and inhibition, γ, among neurons and the spontaneous transition rate from the firing to the quiescent state, β.We propose that the background of the electroencephalographic signals could be mimicked by the fluctuations in the total number of firing neurons in the excitatory subnetwork. These fluctuations are Gaussian and the mean-square displacement from an initial state displays a strongly subdiffusive behaviour approximately given by , where NA=β/(β+Mγ), τ=2(Nα−β). Comparison with real EEG records exhibits good agreement with these predictions.     

A Cayley theorem for distributive lattices

A Cayley-like representation theorem for distributive lattices is proved. Support of the research of the first author by the Czech Government Research Project MSM 6198959214 is gratefully acknowledged.     

A size-width inequality for distributive lattices

We show that every collection ofw sets such that none contains any other generates at least 3w-2 sets under the operations of taking intersections and unions. In particular, we prove that if the finite distributive lattice ? contains an antichain of sizew, then |?| ≧3w, forw≠1, 2, 3, 6, where the minimal exceptional cases arise from the Boolean algebras ?_n withn=0, 1, 2, 3, 4 atoms.     

A Sahlqvist theorem for distributive modal logic

In this paper we consider distributive modal logic, a setting in which we may add modalities, such as classical types of modalities as well as weak forms of negation, to the fragment of classical propositional logic given by conjunction, disjunction, true, and false. For these logics we define both algebraic semantics, in the form of distributive modal algebras, and relational semantics, in the form of ordered Kripke structures. The main contributions of this paper lie in extending the notion of Sahlqvist axioms to our generalized setting and proving both a correspondence and a canonicity result for distributive modal logics axiomatized by Sahlqvist axioms. Our proof of the correspondence result relies on a reduction to the classical case, but our canonicity proof departs from the traditional style and uses the newly extended algebraic theory of canonical extensions.     

Critical phenomenon for a percolation model

We consider a percolation model which consists of oriented lines placed randomly on the plane. The lines are of random length and at a random angle with respect to the horizontal axis and are placed according to a Poisson point process; the length, angle, and orientation being independent of the underlying Poisson process. We establish a critical behaviour of this model, i.e., percolation occurs for large intensity of the Poisson process and does not occur for smaller intensities. In the special case when the lines are of fixed unit length and are either oriented vertically up or oriented horizontally to the left, with probability p or (1-p), respectively, we obtain a lower bound on the critical intensity of percolation.     

The collective reserving model

This paper sets out a model for analysing claims development data, which we call the collective reserving model (CRM). The model is defined on the individual claim level and it produces separate IBNR and RBNS reserve estimators at the collective level without using any approximations. The CRM is based on ideas from a paper by Verrall, Nielsen and Jessen (VNJ) from 2010 in which a model is proposed that relies on a claim giving rise to a single payment. This is generalised by the CRM to the case of multiple payments per claim. All predictors of outstanding claims payments for the VNJ model are shown to hold for this new model. Moreover, the quasi-Poisson GLM estimation framework will be applicable as well, but without using an approximation. Furthermore, analytical expressions for the variance of the total outstanding claims payments are given, with a subdivision on IBNR and RBNS claims. To quantify the effect of allowing only one payment per claim, the model is related and compared to the VNJ model, in particular by looking at variance inequalities. The double chain ladder (DCL) method is discussed as an estimation method for this new model and it is shown that both the GLM- and DCL-based estimators are consistent in terms of an exposure measure. Lastly, both of these methods are shown to asymptotically reproduce the regular chain ladder reserve estimator when restricting predictions to the lower right triangle without the tail, motivating the chain ladder technique as a large-exposure approximation of this model.     

A collective model for the formation of public opinion: An application to nuclear public acceptance

A collective model is proposed for the formation of public opinion and applied to the problem of public acceptance for nuclear energy in Japan. The selection of attitude for the people to a certain subject is assumed to be made depending on the sociopsychological pressure suffering from the surroundings, and on the quality and quantity of information on that subject released from the newsmedia. The reportorial stance of the newsmedia, on the other hand, is assumed to vary depending on, not only public opinion, but also the resultant of the mutual watch of the newsmedia to each other, concerning the state of information on the subject. Both an individual person and an individual medium are regarded as an interactive but structureless particle, and the interactions between the particles of the people and of media are described by introducing respective potentials of the type of molecular dynamics. Applying this model to the issue of electricity generation by nuclear energy, it was found to well reproduce the observed movement of public opinion in Japan. It also became clear that this sort of physical model is quite useful for understanding the status of socially collective phenomena in which nonlinearity and self-organization are strongly concerned.     

Pricing catastrophe swaps: A contingent claims approach

In this paper, we comprehensively analyze the catastrophe (cat) swap, a financial instrument which has attracted little scholarly attention to date. We begin with a discussion of the typical contract design, the current state of the market, as well as major areas of application. Subsequently, a two-stage contingent claims pricing approach is proposed, which distinguishes between the main risk drivers ex-ante as well as during the loss reestimation phase and additionally incorporates counterparty default risk. Catastrophe occurrence is modeled as a doubly stochastic Poisson process (Cox process) with mean-reverting Ornstein–Uhlenbeck intensity. In addition, we fit various parametric distributions to normalized historical loss data for hurricanes and earthquakes in the US and find the heavy-tailed Burr distribution to be the most adequate representation for loss severities. Applying our pricing model to market quotes for hurricane and earthquake contracts, we derive implied Poisson intensities which are subsequently condensed into a common factor for each peril by means of exploratory factor analysis. Further examining the resulting factor scores, we show that a first order autoregressive process provides a good fit. Hence, its continuous-time limit, the Ornstein–Uhlenbeck process should be well suited to represent the dynamics of the Poisson intensity in a cat swap pricing model.     

Pricing catastrophe swaps: A contingent claims approach

In this paper, we comprehensively analyze the catastrophe (cat) swap, a financial instrument which has attracted little scholarly attention to date. We begin with a discussion of the typical contract design, the current state of the market, as well as major areas of application. Subsequently, a two-stage contingent claims pricing approach is proposed, which distinguishes between the main risk drivers ex-ante as well as during the loss reestimation phase and additionally incorporates counterparty default risk. Catastrophe occurrence is modeled as a doubly stochastic Poisson process (Cox process) with mean-reverting Ornstein-Uhlenbeck intensity. In addition, we fit various parametric distributions to normalized historical loss data for hurricanes and earthquakes in the US and find the heavy-tailed Burr distribution to be the most adequate representation for loss severities. Applying our pricing model to market quotes for hurricane and earthquake contracts, we derive implied Poisson intensities which are subsequently condensed into a common factor for each peril by means of exploratory factor analysis. Further examining the resulting factor scores, we show that a first order autoregressive process provides a good fit. Hence, its continuous-time limit, the Ornstein-Uhlenbeck process should be well suited to represent the dynamics of the Poisson intensity in a cat swap pricing model.     

A generalization of completely distributive lattices

Presented by Bjarni Jónsson.     

Boundary-value problem for density-velocity model of collective motion of organisms

The collective motion of organisms is observed at almost all levels of biological systems. In this paper the density-velocity model of the collective motion of organisms is analyzed. This model consists of a system of nonlinear parabolic equations, a forced Burgers equation for velocity and a mass conservation equation for density. These equations are supplemented with the Neumann boundary conditions for the density and the Dirichlet boundary conditions for the velocity. The existence, uniqueness and regularity of solution for the density-velocity problem is proved in a bounded 1D domain. Moreover, a priori estimates for the solutions are established, and existence of an attractor is proved. Finally, some numerical approximations for asymptotical behavior of the density-velocity model are presented.     

A topological characterization of complete distributive lattices

An ordered compact space is a compact topological space X, endowed with a partially ordered relation, whose graph is a closed set of X × X (cf. [4]). An important subclass of these spaces is that of Priestley spaces, characterized by the following property: for every x, y ? X with x ? y there is an increasing clopen set A (i.e. A is closed and open and such that a ? A, a ? z implies that z?A) which separates x from y, i.e., x ? A and y ? A. It is known (cf. [5, 6]) that there is a dual equivalence between the category Ld01 of distributive lattices with least and greatest element and the category P of Priestley spaces.In this paper we shall prove that a lattice L ? Ld01 is complete if and only if the associated Priestley space X verifies the condition: (E0) D ? X, D is increasing and open implies $\text{D}{}^1$ is increasing clopen (where A¹ denotes the least increasing set which includes A).This result generalizes a well-known characterization of complete Boolean algebras in terms of associated Stone spaces (see [2, Ch. III, Section 4, Lemma 1], for instance).We shall also prove that an ordered compact space that fulfils (E0) is necessarily a Priestley space.