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Motivated by the China's five element philosophy (CFEP), we construct a permuted Volterra quadratic stochastic operator acting on the four-dimensional simplex. This operator (depending on 10 parame...

The aim of this research is to provide a model for optimising the sectoral portfolio selection by using game theory during a general elections phase. We select stocks from Bursa Malaysia and calculate the payoff for each stock and its coalition sectors by averaging returns. The value of the game is at the same time the characteristic function of a multiple-player game which will be applied to obtain the Shapley value using the cooperative game theory approach for finding the optimal increment of the returns. We also compare the Shapley value percentages obtained for both periods of the 13th Malaysia General Election (GE13) and 14th Malaysia General Election (GE14) to indicate the impact of GE14 on investment. This research will lead to optimal portfolio selection before and after an election.

Let $$S^{m-1}$$ be the simplex in $${{\mathbb {R}}}^m$$ , and $$V:S^{m-1}\rightarrow S^{m-1}$$ be a nonlinear mapping then this operator satisfies an ergodic theorem if the limit $$\begin{aligned} \lim _{n\rightarrow \infty }\frac{1}{n}\sum _{k=1}^n V^k(x) \end{aligned}$$ exists for every $$x\in S^{m-1}$$ . It is a well known fact that this ergodicity may fail for Volterra quadratic operators, so it is natural to characterize all non-ergodic operators. However, there is an ongoing problem even in the low dimensional simplexes. In this paper, we solve the mentioned problem within Volterra cubic stochastic operators acting on two-dimensional simplex.

We consider the λ model, a generalization of the Potts model, with spin values {1, 2, 3} on the order-two Cayley tree. We describe the model ground states and prove that translation-invariant Gibb measures exist, which means that a phase transition exists. We establish that two-periodic Gibbs measures exist.

In this paper, we consider the $\lambda$-model with nearest neighbor interactions and with competing Potts interactions on the Cayley tree of order-two. We notice that if $\lambda$-function is taken as a Potts interaction function, then this model contains as a particular case of Potts model with competing interactions on Cayley tree. In this paper, we first describe all ground states of the model. We point out that the Potts model with considered interactions was investigated only numerically, without rigorous (mathematical) proofs. One of the main points of this paper is to propose a measure-theoretical approach for the considered model in more general setting. Furthermore, we find certain conditions for the existence of Gibbs measures corresponding to the model, which allowed to establish the existence of the phase transition., Comment: 23 pages. arXiv admin note: text overlap with arXiv:1704.01933

We give the full descriptions of the dynamical behaviour of the Potts–Bethe mapping of degree four on ℚ5. For a, b ∈ ℚ5, the Potts–Bethe mapping is written as follows fa,b(x)=(ax+bx+a+b−1)4.(1)When |a − 1|5 < |b + 1|5 < 1, there exists a subsystem (𝒥, fa,b) that is topologically conjugate to the chaotic full shift dynamics on four symbols. We also show that for any initial point x∈ ℚ5\(J∪∪j=0∞fa,b−j{ 1−a−b }), the trajectory of the Potts–Bethe mapping converges to a unique attracting fixed point.We give the full descriptions of the dynamical behaviour of the Potts–Bethe mapping of degree four on ℚ5. For a, b ∈ ℚ5, the Potts–Bethe mapping is written as follows fa,b(x)=(ax+bx+a+b−1)4.(1)When |a − 1|5 < |b + 1|5 < 1, there exists a subsystem (𝒥, fa,b) that is topologically conjugate to the chaotic full shift dynamics on four symbols. We also show that for any initial point x∈ ℚ5\(J∪∪j=0∞fa,b−j{ 1−a−b }), the trajectory of the Potts–Bethe mapping converges to a unique attracting fixed point.

In this paper, we consider the cubic stochastic operator (CSO) defined on 1-dimensional simplex, S1. We provide a full description of orthogonal preserving (OP) cubic stochastic operators on the simplex. We provide full description of the fixed points subject to two different parameters for the Volterra OP CSO on the simplex. In the last section we describe the behaviour of the fixed points.

In this paper, we consider the λ-model on Cayley tree for order two with Potts competing nearest-neighbor and prolonged next-nearest neighbor-interactions. We described the construction of the Gibbs measure for the considered model. We proved the existence of the translation-invariant limiting Gibbs measures for the model.

A new combinatorial interpretation of Raney numbers is proposed. We apply this combinatorial interpretation to solve several tree enumeration counting problems. Further a generalized Catalan triangle is introduced and some of its properties are proved.

In this paper we consider the $\lambda$-model on the Cayley tree of order two. We describe periodic and weakly periodic ground states for the considered model., Comment: 20 pages, 1 figure

We investigate an Ising model with two restricted competing interactions (nearest neighbors, and one-level neighbors) on the Cayley tree of order 5. The translation Gibbs measures is considered for this model. Our result of the critical curve shows that the phase transition occurs in this model, further it confirms a particular case of a conjecture.

We showed that one form of generalized Catalan numbers is the solution to the problem of finding different connected component with finite vertices containing a fixed root for the semi-infinite Cayley tree of order 3. We give the formula for the full graph, Cayley tree of order 3 which is derived from the generalized Catalan numbers. Using ratios of Gamma functions, two upper bounds are given for problem defined on semi-infinite Cayley tree of order 3 as well as the full graph.

We have found an exact phase diagram of the Potts model with competing nearest neighbor and next nearest neighbor interactions on the Bethe lattice of order two. The diagram consists of five phases: ferromagnetic, paramagnetic, modulated, antiphase and paramodulated, all meeting at the multicritical point ( T = 0 , p = 1 / 3 ) . We report on a new phase which we denote as paramodulated, found at low temperatures and characterized by zero average magnetization lying inside the modulated phase. Such a phase, inherent in the Potts model has no analogues in the Ising setting.

We consider q-component models on both the integer lattice ℤ2 and a second-order Cayley tree and study phase diagrams of these models.

In this paper we consider an Ising model with three competing restricted interactions on the Cayley tree J2(J3). The translation invariant and periodic Gibbs measures for these models are investigated and the problem of the phase transition in these classes is solved.

We analyse the relation between local two-atom and total multi-atom entanglements in the Dicke system composed of a large number of atoms. We use concurrence as a measure of entanglement between two atoms in the multi-atom system, and the spin squeezing parameter as a measure of entanglement in the whole n-atom system. In addition, the influence of the squeezing phase and bandwidth on entanglement in the steady-state Dicke system is discussed. It is shown that the introduction of a squeezed field leads to a significant enhancement of entanglement between two atoms, and the entanglement increases with increasing degree of squeezing and bandwidth of the incident squeezed field. In the presence of a coherent field the entanglement exhibits a strong dependence on the relative phase between the squeezed and coherent fields, that can jump quite rapidly from unentangled to strongly entangled values when the phase changes from zero to π. We find that the jump of the degree of entanglement is due to a flip of the spin squeezing from one quadrature component of the atomic spin to the other component when the phase changes from zero to π. We also analyse the dependence of the entanglement on the number of atoms and find that, despite the reduction in the degree of entanglement between two atoms, a large entanglement is present in the whole n-atom system and the degree of entanglement increases as the number of atoms increases.

The exact solution of an Ising model with competing restricted interactions on the Cayley tree, and in the absence of an external field is presented. A critical curve is defined where it is possible to get phase transitions above it, and a single Gibbs state is obtained elsewhere.

In the present paper, we consider orthogonal preserving quadratic stochastic operators defined on infinite dimensional simplex. We provide a full descriptions of such kind of operators. Moreover, certain examples are given.

In the paper, we consider the λ-model with spin values {1, 2, 3} on the Cayley tree of order two. We describe ground states of the model.

We found an exact formulation for a finite sub-tree counting problem. Solution to two extremal cases are Catalan Triangle introduced by Shapiro and ballot Catalan triangles. The general solution could be expressed as linear combination of these Catalan triangles.

We investigate an Ising model with two restricted competing interactions (nearest neighbors, and one-level neighbors) on the Cayley tree of order four. We derive a recurrent equation for the Cayley tree of order k. We found an analytic solution for the given interactions in the case of order 4. Our result of the critical curve shows the existence of the phase transition occurs in this model. We also give the calculation of the free energy from the description of Gibbs measure of the given Hamiltonian on Cayley tree of order four.