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This paper deals with the existence and multiplicity of nontrivial periodic solutions to the first order nonlinear difference equation with delay of the form△x(n)=-f(x(n-T)) By applying the critical point theory,several sufficient conditions are obtained for the periodic solutions to the equation above under the assumption that f satisfies certain growth conditions both at zero and at infinity.Some examples are given to illustrate the feasibility of the main results.

This paper is to solve several problems on the global attractivity of the zero solution of the nonautonomous difference equation $$x_{n + 1} - x_n + P_n x_{n - k_n } = 0,n \in \mathbb{Z} (0)$$ , where P n is a sequence of nonnegative real numbers, and k n is a sequence of nonnegative integers with n − k n →∞ as n → ∞.

This research focuses on observing and measuring the topography of thin film buckle in micro-nano scale, which is based on focus measure function and Shape From Focus technology, Gaussian interpolation. A PZT stepper motor which is well calibrated is used to change the distance between thin film surface and the microscope lens. Focusing-evaluation-functions elements of each image which is taken at each step are corresponded to the distance values obtained from the PZT stepper motor. Then, focusingevaluation- functions of each element at different steps can be seemed as a function of distance values. The actual distance of each element between the film and the lens, when the element is clearest, can be obtained by the Gaussian interpolation when the element focusing-evaluation-function reaches the peak value. Furthermore, this method is applied successfully to measure the full field topography of straight side buckle and telephone cord buckle.

In this paper the existence of nontrivial periodic solution for second order asymptotically linear difference equation at resonance is obtained. The methods used here are based on combining the minimax methods and the Morse theory, especially the observation on the critical groups.

By the critical point theory, we study the existence and multiplicity of periodic solutions to the following system of delay differential equations: (*) x ′ ( t ) = - f ( x ( t - r ) ) , where x ∈ R n , f ∈ C ( R n , R n ) , and r > 0 is a given constant. A sufficient condition on the existence and multiplicity of periodic solutions for Eq. ( * ) is obtained. To the authors’ knowledge, this is the first time, the existence of periodic solutions to systems of delay differential equations is dealt with by using variational approaches directly.

In this paper, an existence theorem is obtained for periodic solutions of a second-order discrete Hamiltonian system with a change of sign in the potential by the minimax methods in the critical point theory.

At the end of the 20th century, China launched the ‘Grain-for-Green’ Project (GGP) that recommended the abandonment of low-yielding sloping farmland (>15°) prone to soil degradation by erosion, to allow recovery through natural vegetative regeneration. The effect of this policy on soil nitrogen (N) cycling, as fertilization applications are also withdrawn after abandonment, is poorly understood. A space-for-time approach was applied to investigate the responses of nitrogen functional genes (NFGs) in soil profiles (surface to bedrock) associated with progressive vegetative recovery (sloping farmland > recently abandoned sloping farmland > secondary forest > primary forest) at the Puding Karst Critical Zone Observatory in Guizhou province, southwest China. Coincident soil chemical properties (dissolved organic carbon (DOC), nitrate ( N O 3 - -N), ammonium ( N H 4 + -N), available inorganic phosphorus (AP), soil organic carbon (SOC), total nitrogen (TN) and total phosphorus (TP)) were also quantified. We found that the absolute abundance of NFGs significantly varied according to the phase of vegetation recovery, and that concentrations of AP and N O 3 - -N were the best explanatory variables. The external N from fertilizer application promoted the absolute abundances of ammonia-oxidizing archaea (AOA) and ammonia-oxidizing bacteria (AOB) in sloping farmland. The relative abundances of chiA (associated with decomposition) increased with soil depth across all vegetation recovery phases. The relative abundances of chiA and nifH (associated with N fixation) accounted for the largest proportion (58–72%) of the measured NFGs, indicating that active N-acquisition increased along the vegetation recovery gradient. The ratios of (chiA + nifH)/(AOA + AOB) and the sums of (nirK + nirS) were larger in the forest soil than those of sloping farmland and abandoned sloping farmland, implying a greater capacity for N storage potential, though accompanied by increased gas N emission potential, in the karst forest ecosystems. Our results provide a new and comprehensive understanding of soil N cycling potentials at the microscale in degraded and recovering karst ecosystems.

Consider the second-order discrete system where f ∈ C (R × Rm, Rm), f(t + M, Z) = f(t, Z) for any (t, Z) ∈ R × Rm and M is a positive integer. By making use of critical-point theory, the existence of M-periodic solutions of (*) is obtained.

Cognitive radio networks (CRNs) are expected to improve spectrum utilization significantly by allowing secondary users (SUs) to opportunistically access the licensed spectrum of primary users (PUs). In an integrated network consisting of multiple heterogeneous CRNs, SUs with multiple interfaces may have to conduct inter-system or intra-system spectrum handoff due to the arrival of PUs or performance degradation on serving spectrum. In this case, designing an optimal spectrum handoff scheme which offers quality of service (QoS) guarantee and performance enhancement of the SUs is of particular importance. On the other hand, resource allocation strategy on target channel also plays an important role in affecting the transmission performance of handoff SUs. In this paper, we jointly design spectrum handoff and resource allocation strategy for handoff SUs in heterogeneously integrated CRNs. To achieve joint resource management among various CRNs, we propose a joint radio resource management architecture, based on which the proposed spectrum handoff and resource allocation scheme can be conducted. Jointly considering the transmission performance of the handoff SUs, we formulate the total energy efficiency of the SUs and design an optimization problem which maximizes the energy efficiency subject to spectrum handoff, QoS, and power constraints of the SUs. An iterative algorithm is proposed to solve the formulated nonlinear fractional optimization problem. Within each iteration, the optimization problem is transformed equivalently into two subproblems, i.e., power allocation subproblem of each SU-spectrum pair and spectrum handoff subproblem for all the SUs. The two subproblems are solved, respectively, through applying Lagrange dual method and the Kuhn-Munkres (K-M) algorithm. Numerical results demonstrate the efficiency of the proposed algorithm.