Narrow Results

In this paper, a discrete version of continuous non-autonomous predator–prey model with delays is investigated. By using Gaines and Mawhin's continuation theorem of coincidence degree theory and the method of Lyapunov function, some sufficient conditions for the existence and globally asymptotically stability of positive periodic solution of difference equations in consideration are established. Finally, some numerical examples are given to verify the theoretical analysis.

The design of circuits which are robust against variations in operating and process conditions is crucial in today's IC industry. In the analog design flow this problem can be tackled during the sizing of a new circuit. However, hardly any methods are available which support the designer to compute such a robust design if discrete parameters should be considered in this design step. Discrete parameters arise predominantly if a layout-friendly sizing should be computed in the sense that, e.g., a manufacturing grid for the transistor lengths and widths is considered or that transistor multipliers are used to allow the layout of a transistor as multifinger or common centroid structure without applying rounding operations to the carefully computed sizing. This paper presents a new Branch-and-Bound based approach which allows the automatic computation of a robust design using classical and realistic worst case analysis. The results of the sizing of three circuits show that the new approach is highly efficient. The robustness of the results computed by the new approach is validated by Monte Carlo analyses.

We show that a discrete dynamical system with delayed arguments that describe the computational performance of a neural network with a loop structure can exhibit the coexistence of huge number of periodic solutions, and we describe the domains of attraction for the stable periodic orbits. This demonstrates a great potential of discrete time delayed neural nets with a loop structure for the purpose of storage and retrieval of periodic patterns.

A discrete nonlinear model is studied and sufficient conditions which guarantee the permanence of the model are obtained. Assuming that the coefficients in the model are periodic, the existence of periodic solutions are obtained. Sufficient conditions are obtained to ensure the global stability of the positive periodic solution by constructing a suitable Lyapunov function.

The global stability of a nonlinear discrete population model of Volterra type is studied. The model incorporates time delays. By linearization of the model at positive solutions and construction of Liapunov functionals, sufficient conditions are obtained to ensure that a positive solution of the model is stable and attracts all positive solutions. An example shows the feasibility of our main results.

In this paper, we deal with a discrete n-species non-autonomous Lotka–Volterra competitive systems with infinite delays and feedback control, obtain sufficient conditions for the permanence of the systems.

We discuss the performance of the Search and Fourier Transform algorithms on a hybrid computer constituted of classical and quantum processors working together. We show that this semi-quantum computer would be an improvement over a pure classical architecture, no matter how few qubits are available and, therefore, it suggests an easier implementable technology than a pure quantum computer with arbitrary number of qubits.

We show that an ultradiscrete analogue of the third Painlevé equation admits discrete Riccati type solutions. We derive these solutions by considering a framework in which the ultradiscretization process arises as a restriction of a non-archimedean valuation over a field. Using this framework we may relax the conditions one requires to apply the ultradiscretization process. We derive a family of transcendental solutions that appear as the non-archimedean field valuation of hypergeometric functions.

In this paper, we consider the discrete Hematopoiesis model with a time delay:

In this paper, we consider a discrete Lotka–Volterra competitive system with the effect of toxic substances and feedback controls. By using the method of discrete Lyapunov function and by developing a new analysis technique, we obtain the sufficient conditions which guarantee that one of the two species will be driven to extinction while the other will be permanent. We improve the corresponding results of Li and Chen [Extinction in two-dimensional discrete Lotka–Volterra competitive system with the effect of toxic substances, Dynam. Contin. Discrete Impuls. Syst. Ser. B Appl. Algorithms 15 (2008) 165–178]. Also, an example together with their numerical simulations shows the feasibility of our main results. It is shown that toxic substances and feedback control variables play an important role in the dynamics of the system.

The Markov brothers’ inequalities in approximation theory concern polynomials p of degree n and assert bounds for the kth derivatives |p^(k)|, 1 ≤ k ≤ n, on [−1, 1], given that |p| ≤ 1 on [−1, 1]. Here we go in the other direction, seeking bounds for |p|, given a bound for |p^(k)|. For the problem to be meaningful, additional restrictions on p must be imposed, for example, p(−1) = p′(−1) = … = p^(k−1)(−1) = 0. The problem then has an easy solution in the continuous case, where the polynomial and their derivatives are considered on the whole interval [−1, 1], but is more challenging, and also of more interest, in the discrete case, where one focuses on the values of p and p^(k) on a given set of n−k +1 distinct points in [−1, 1]. Analytic solutions are presented and their fine structure analyzed by computation.

Beginning with an elementary, oscillatory discrete dynamical system associated with the square root of minus one, we study both the foundations of mathematics and physics. Position and momentum do not commute in our discrete physics. Their commutator is related to the diffusion constant for a Brownian process and to the Heisenberg commutator in quantum mechanics. We take John Wheeler's idea of It from Bit as an essential clue and we rework the structure of that bit to a logical particle that is its own anti-particle, a logical Marjorana particle. This is our key example of the amphibian nature of mathematics and the external world. We show how the dynamical system for the square root of minus one is essentially the dynamics of a distinction whose self-reference leads to both the fusion algebra and the operator algebra for the Majorana Fermion. In the course of this, we develop an iterant algebra that supports all of matrix algebra and we end the essay with a discussion of the Dirac equation based on these principles.