Programming in Polygon R&D: Explorations with a Spatial Language II

This paper introduces the language associated with a polygon microworld called PolygonR&D, which has the mathematical crispness of Logo and has the discreteness and simplicity of a Turing machine. In this microworld, polygons serve two purposes: as agents (similar to the turtles in Logo), and as data (landmarks in the plane). Programming the spatial behaviour of polygon agents is achieved by a simple variable-free language. Although limited in the number of instructions, the language allows for complex outcomes such as creation of sophisticated tilings, algorithm visualization, simulations, and even numerical computation. The ease of constructing the variety of examples shown in this paper indicates the microworld’s potential in both secondary and post-secondary education. An erratum to this article can be found at     

Spatiotemporal Dynamic Analysis in a Time-space Discrete Brusselator Mode

In this paper, we study the spatiotemporal patterns of a Brusselator model with discrete time-space by using the coupled mapping lattice (CML) model. The existence and stability conditions of the equilibrium point are obtained by using linear stability analysis. Then, applying the center manifold reduction theorem and the bifurcation theory, the parametric conditions of the flip and the Neimark-Sacker bifurcation are described respectively. Under space diffusion, the model admits the Turing instability at stable homogeneous solutions under some certain conditions. Two nonlinear mechanisms, including flip-Turing instability and Neimark-Sacker-Turing instability, are presented. Through numerical simulation, periodic windows, invariant circles, chaotic phenomenon and some interesting spatial patterns are found.     

Isomorphism of lattices of recursively enumerable sets

Let , and for , let be the lattice of subsets of which are recursively enumerable relative to the ``oracle' . Let be , where is the ideal of finite subsets of . It is established that for any , is effectively isomorphic to if and only if , where is the Turing jump of . A consequence is that if , then . A second consequence is that can be effectively embedded into preserving least and greatest elements if and only if .

     

Analysis of ratio-dependent food chain model

In this paper, a food chain model with ratio-dependent functional response is studied under homogeneous Neumann boundary conditions. The large time behavior of all non-negative equilibria in the time-dependent system is investigated, i.e., conditions for the stability at equilibria are found. Moreover, non-constant positive steady-states are studied in terms of diffusion effects, namely, Turing patterns arising from diffusion-driven instability (Turing instability) are demonstrated. The employed methods are comparison principle for parabolic problems and Leray-Schauder Theorem.     

Modulo computably enumerable degrees by cupping partners

Cupping partners of an element in an upper semilattice with a greatest element 1 are those joining the element to 1. We define a congruence relation on such an upper semilattice by considering the elements having the same cupping partners as equivalent. It is interesting that this congruence relation induces a non-dense quotient structure of computably enumerable Turing degrees. Another main interesting phenomenon in this article is that on the computably enumerable degrees, this relation is different from that modulo the noncuppable ideal, though they define a same equivalent class for the computable Turing degree.     

Distinguishing between evidence and its explanations in the steering of atomic clocks

Quantum theory reflects within itself a separation of evidence from explanations. This separation leads to a known proof that: (1) no wave function can be determined uniquely by evidence, and (2) any chosen wave function requires a guess reaching beyond logic to things unforeseeable. Chosen wave functions are encoded into computer-mediated feedback essential to atomic clocks, including clocks that step computers through their phases of computation and clocks in space vehicles that supply evidence of signal propagation explained by hypotheses of spacetimes with metric tensor fields.     

What is a universal computing machine?

A computer is classically formalised as a universal Turing machine or a similar device. However over the years a lot of research has focused on the computational properties of dynamical systems other than Turing machines, such cellular automata, artificial neural networks, mirrors systems, etc.In this paper we propose a unifying formalism derived from a generalisation of Turing’s arguments. Then we review some of universal systems proposed in the literature and show that are particular case of this formalism. Finally, we review some of the attempts to understand the relation between dynamical and computational properties of a system.     

Lattice-valued fuzzy Turing machines: Computing power, universality and efficiency

In this paper we study fuzzy Turing machines with membership degrees in distributive lattices, which we called them lattice-valued fuzzy Turing machines. First we give several formulations of lattice-valued fuzzy Turing machines, including in particular deterministic and non-deterministic lattice-valued fuzzy Turing machines (l-DTMcs and l-NTMs). We then show that l-DTMcs and l-NTMs are not equivalent as the acceptors of fuzzy languages. This contrasts sharply with classical Turing machines. Second, we show that lattice-valued fuzzy Turing machines can recognize n-r.e. sets in the sense of Bedregal and Figueira, the super-computing power of fuzzy Turing machines is established in the lattice-setting. Third, we show that the truth-valued lattice being finite is a necessary and sufficient condition for the existence of a universal lattice-valued fuzzy Turing machine. For an infinite distributive lattice with a compact metric, we also show that a universal fuzzy Turing machine exists in an approximate sense. This means, for any prescribed accuracy, there is a universal machine that can simulate any lattice-valued fuzzy Turing machine on it with the given accuracy. Finally, we introduce the notions of lattice-valued fuzzy polynomial time-bounded computation (lP) and lattice-valued non-deterministic fuzzy polynomial time-bounded computation (lNP), and investigate their connections with P and NP. We claim that lattice-valued fuzzy Turing machines are more efficient than classical Turing machines.     

Spatial Dynamics of a Diffusive Predator-prey Model with Leslie-Gower Functional Response and Strong Allee Effect

In this paper, spatial dynamics of a diffusive predator-prey model with Leslie-Gower functional response and strong Allee effect is studied. Firstly, we obtain the critical condition of Hopf bifurcation and Turing bifurcation of the PDE model. Secondly, taking self-diffusion coefficient of the prey as bi- furcation parameter, the amplitude equations are derived by using multi-scale analysis methods. Finally, numerical simulations are carried out to verify our theoretical results. The simulations show that with the decrease of self- diffusion coefficient of the prey, the preys present three pattern structures: spot pattern, mixed pattern, and stripe pattern. We also observe the transi- tion from spot patterns to stripe patterns of the prey by changing the intrinsic growth rate of the predator. Our results reveal that both diffusion and the intrinsic growth rate play important roles in the spatial distribution of species.     

Effect of herd shape in a diffusive predator-prey model with time delay

In this paper, we deal with the effect of the shape of herd behavior on the interaction between predator and prey. The model analysis was studied in three parts. The first, The analysis of the system in the absence of spatial diffusion and the time delay, where the local stability of the equilibrium states, the existence of Hopf bifurcation have been investigated. For the second part, the spatiotemporal dynamics introduce by self diffusion was determined, where the existence of Hopf bifurcation, Turing driven instability, Turing-Hopf bifurcation point have been proved. Further, the order of Hopf bifurcation points and regions of the stability of the non trivial equilibrium state was given. In the last part of the paper, we studied the delay effect on the stability of the non trivial equilibrium, where we proved that the delay can lead to the instability of interior equilibrium state, and also the existence of Hopf bifurcation. A numerical simulation was carried out to insure the theoretical results.