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Many disease fighting strategies have focused on the generation of NK cells, since they constitute the main immune barrier against cancer and intracellular pathogens such as viruses. Therefore, a predictive model for the development of NK cells would constitute a useful tool to test several hypotheses regarding the production of these cells during both physiological and pathological conditions. Here, we present a boolean network model that reproduces experimental results reported on the literature regarding the progressive stages of the development of NK cells in wild-type and mutant backgrounds. The model allows for the simulation of different conditions, including extracellular micro-environment as well as the simulation of genetic alterations. It also describes how NK cell differentiation depends on a molecular regulatory network that controls the specification of lymphoid lineages, such as T and B cells, which share a common progenitor with NKs. Furthermore, the study shows that the structure of the regulatory network strongly determines the stability of the expression patterns against perturbations.

In this study we investigate the dynamics of a nonlinear discrete-time duopoly game, where the players have heterogeneous expectations linear demand and cost functions. Two players with different expectations are considered; one is boundedly rational and the other thinks with adaptive expectations. We show that the model gives more complex chaotic and unpredictable trajectories as a consequence of change in the marginal costs of the players. The chaotic features are justified numerically via computing Lyapunov numbers, sensitive dependence on initial conditions and the box dimension of the chaotic attractor.

Writing the article “Time independent pricing of options in range bound markets” [1], the question in the title came naturally to my mind. It is stated, in the above article, that in certain market conditions the stock price is subjected to an equation that exactly matches a time independent Schrodinger equation. The time independent equation for options valuation is used further to explain a stock market phenomenon that resembles an α particle decay tunneling effect. The transmission coefficient for the stock price tunneling effect it is also deduced. Although, it may not have important impact in quantum physics, the philosophical aspects residing in the use of quantum mechanics for stock market specific are very important.

In this study we investigate the dynamics of a nonlinear discrete-time duopoly game, where the players have heterogeneous expectations linear demand and quadratic cost functions. Two players with different expectations are considered; one is boundedly rational and the other thinks with adaptive expectations. We show that the model gives more complex chaotic and unpredictable trajectories as a consequence of change in the slope of the marginal costs. The chaotic features are justified numerically via computing Lyapunov numbers, sensitive dependence on initial conditions and the box dimension of the chaotic attractor.

The change of the functional graph of a linear discrete dynamical system is described under transformation of the support graph of the system. Namely, the support graph is transformed by adding two dominating vertices. [ABSTRACT FROM AUTHOR]

Let E(X, f) be the Ellis semigroup of a dynamical system (X, f) where X is a compact metric space. We analyze the cardinality of E(X, f) for a compact countable metric space X. A characterization when E(X, f) and E(X,f)∗=E(X,f)\{fn:n∈N} are both finite is given. We show that if the collection of all periods of the periodic points of (X, f) is infinite, then E(X, f) has size 2ℵ0. It is also proved that if (X, f) has a point with a dense orbit and all elements of E(X, f) are continuous, then |E(X,f)|≤|X|. For dynamical systems of the form (ω2+1,f), we show that if there is a point with a dense orbit, then all elements of E(ω2+1,f) are continuous functions. We present several examples of dynamical systems which have a point with a dense orbit. Such systems provide examples where E(ω2+1,f) and ω2+1 are homeomorphic but not algebraically homeomorphic, where ω2+1 is taken with the usual ordinal addition as semigroup operation. [ABSTRACT FROM AUTHOR]

Many disease fighting strategies have focused on the generation of NK cells, since they constitute the main immune barrier against cancer and intracellular pathogens such as viruses. Therefore, a predictive model for the development of NK cells would constitute a useful tool to test several hypotheses regarding the production of these cells during both physiological and pathological conditions. Here, we present a boolean network model that reproduces experimental results reported on the literature regarding the progressive stages of the development of NK cells in wild-type and mutant backgrounds. The model allows for the simulation of different conditions, including extracellular micro-environment as well as the simulation of genetic alterations. It also describes how NK cell differentiation depends on a molecular regulatory network that controls the specification of lymphoid lineages, such as T and B cells, which share a common progenitor with NKs. Furthermore, the study shows that the structure of the regulatory network strongly determines the stability of the expression patterns against perturbations. [ABSTRACT FROM AUTHOR]

Markov map is one example of interval maps where it is a piecewise expanding map and obeys the Markov property. One well-known example of Markov map is the doubling map, a map which has two subintervals with equal partitions. In this paper, we are interested to investigate another type of Markov map, the so-called skewed doubling map. This map is a more generalized map than the doubling map. Thus, the aims of this paper are to find the fixed points as well as the periodic points for the skewed doubling map and to investigate the sensitive dependence on initial conditions of this map. The method considered here is the cobweb diagram. Numerical results suggest that there exist dense of periodic orbits for this map. The sensitivity of this map to initial conditions is also verified where small differences in initial conditions give different behaviour of the orbits in the map. [ABSTRACT FROM AUTHOR]

We present properties of sets of invariant lines for Brouwer homeomorphisms which are not necessarily embeddable in a flow. Using such lines we describe the structure of equivalence classes of the codivergency relation. We also obtain a result concerning the set of regular points. [ABSTRACT FROM AUTHOR]

Many discrete models of biological networks rely exclusively on Boolean variables and many tools and theorems are available for analysis of strictly Boolean models. However, multilevel variables are often required to account for threshold effects, in which knowledge of the Boolean case does not generalise straightforwardly. This motivated the development of conversion methods for multilevel to Boolean models. In particular, Van Ham’s method has been shown to yield a one-to-one, neighbour and regulation preserving dynamics, making it the de facto standard approach to the problem. However, Van Ham’s method has several drawbacks: most notably, it introduces vast regions of “non-admissible” states that have no counterpart in the multilevel, original model. This raises special difficulties for the analysis of interaction between variables and circuit functionality, which is believed to be central to the understanding of dynamic properties of logical models. Here, we propose a new multilevel to Boolean conversion method, with software implementation. Contrary to Van Ham’s, our method doesn’t yield a one-to-one transposition of multilevel trajectories; however, it maps each and every Boolean state to a specific multilevel state, thus getting rid of the non-admissible regions and, at the expense of (apparently) more complicated, “parallel” trajectories. One of the prominent features of our method is that it preserves dynamics and interaction of variables in a certain manner. As a demonstration of the usability of our method, we apply it to construct a new Boolean counter-example to the well-known conjecture that a local negative circuit is necessary to generate sustained oscillations. This result illustrates the general relevance of our method for the study of multilevel logical models. [ABSTRACT FROM AUTHOR]

The game among three competing firms, which is called a triopolistic game, is formulated and studied in this paper. The adopted prices by those firms are derived from a quadratic form of utility that is quasi-concave and has a strict convexity to the origin level curves. The paper is divided into three parts: the first part analyzes a static model on which firms use different strategic variables. The stability condition of the equilibrium points of that model is investigated. In the second part, it is assumed that the three firms may conjecture that their opponents’ current step will remain as it is in the next time step and therefore an instantaneous adjustment model is considered. Schur–Cohn criterion of stability is adopted to detect the stability condition of the equilibrium points of that model. The fractional bounded rationality is introduced in the third part. We introduce in this part a dynamical model of three competing firms whose strategic variables are quantities. As in the previous models, the stability condition of the equilibrium points of the model is studied. The main interesting observation we obtain in this part is that for the firms to stay stable for a long time in the market they should play with fractional bounded rationality rather than the traditional bounded rationality. This important observation is confirmed using numerical simulation for some cases under the same values of parameters. Furthermore, for the models considered in this paper some numerical experiments are carried out to satisfy our theoretical results. Moreover, Karush–Kuhn–Tucker (KKT) proximity measure based on evolutionary algorithm is performed in order to confirm the obtained results. [ABSTRACT FROM AUTHOR]

In this paper, I continue the study of the mathematical models presented in [J. C. Larsen, Models of cancer growth, J. Appl. Math. Comput. 53(1-2) (2015) 613-645] and [J. C. Larsen, The bistability theorem in a model of metastatic cancer, to appear in Appl. Math.]. I shall prove the bistability theorem for the ODE model from [Larsen, 2015]. It is a mass action kinetic system in the variables cancer, GF growth factor and GI growth inhibitor. This theorem says that for some values of the parameters, there exist two positive singular points , of the vector field. Here and is stable and is unstable, see Sec. 2. There is also a discrete model in [Larsen, 2015], it is a linear map () on three-dimensional Euclidean vector space with variables where these variables have the same meaning as in the ODE model above. In [Larsen, 2015], I showed that one can sometimes find affine vector fields on three-dimensional Euclidean vector space whose time one map is . I shall also show this in the present paper in a more general setting than in [Larsen, 2015]. This enables me to find an expression for the rate of change of cancer growth on the coordinate hyperplane in Euclidean vector space. I also present an ODE model of cancer metastasis with variables where is primary cancer and is metastatic cancer and GF, GI are growth factors and growth inhibitors, respectively. [ABSTRACT FROM AUTHOR]