Stability and Neimark–Sacker bifurcation of a ratio‐dependence predator–prey model

In this paper, stability and bifurcation of a two‐dimensional ratio‐dependence predator–prey model has been studied in the close first quadrant . It is proved that the model undergoes a period‐doubling bifurcation in a small neighborhood of a boundary equilibrium and moreover, Neimark–Sacker bifurcation occurs at a unique positive equilibrium. We study the Neimark–Sacker bifurcation at unique positive equilibrium by choosing b as a bifurcation parameter. Some numerical simulations are presented to illustrate theocratical results. Copyright © 2017 John Wiley & Sons, Ltd.     

Bogdanov‐Takens bifurcations of codimensions 2 and 3 in a Leslie‐Gower predator‐prey model with Michaelis‐Menten–type prey harvesting

The Bogdanov‐Takens bifurcations of a Leslie‐Gower predator‐prey model with Michaelis‐Menten–type prey harvesting were studied. In the paper “Diff. Equ. Dyn. Syst. 20(2012), 339‐366,” Gupta et al proved that the Leslie‐Gower predator‐prey model with Michaelis‐Menten–type prey harvesting has rich dynamics. Some equilibria of codimension 1 and their bifurcations were discussed. In this paper, we find that the model has an equilibrium of codimensions 2 and 3. We also prove analytically that the model undergoes Bogdanov‐Takens bifurcations (cusp cases) of codimensions 2 and 3. Hence, the model can have 2 limit cycles, coexistence of a stable homoclinic loop and an unstable limit cycle, supercritical and subcritical Hopf bifurcations, and homoclinic bifurcation of codimension 1 as the values of parameters vary. Moreover, several numerical simulations are conducted to illustrate the validity of our results.     

A self‐adaptive intelligence gray prediction model with the optimal fractional order accumulating operator and its application

The self‐adaptive intelligence gray predictive model (SAIGM) has an alterable‐flexible model structure, and it can build a dynamic structure to fit different external environments by adjusting the parameter values of SAIGM. However, the order number of the raw SAIGM model is not optimal, which is an integer. For this, a new SAIGM model with the fractional order accumulating operator (SAIGM_FO) was proposed in this paper. Specifically, the final restored expression of SAIGM_FO was deduced in detail, and the parameter estimation method of SAIGM_FO was studied. After that, the Particle Swarm Optimization algorithm was used to optimize the order number of SAIGM_FO, and some steps were provided. Finally, the SAIGM_FO model was applied to simulate China's electricity consumption from 2001 to 2008 and forecast it during 2009 to 2015, and the mean relative simulation and prediction percentage errors of the new model were only 0.860% and 2.661%, in comparison with the ones obtained from the raw SAIGM model, the GM(1, 1) model with the optimal fractional order accumulating operator and the GM(1, 1) model, which were (1.201%, 5.321%), (1.356%, 3.324%), and (2.013%, 23.944%), respectively. The findings showed both the simulation and the prediction performance of the proposed SAIGM_FO model were the best among the 4 models.     

Eigenvalue confinement and spectral gap for random simplicial complexes

We consider the adjacency operator of the Linial‐Meshulam model for random simplicial complexes on n vertices, where each d‐cell is added independently with probability p to the complete ‐skeleton. Under the assumption , we prove that the spectral gap between the smallest eigenvalues and the remaining eigenvalues is with high probability. This estimate follows from a more general result on eigenvalue confinement. In addition, we prove that the global distribution of the eigenvalues is asymptotically given by the semicircle law. The main ingredient of the proof is a Füredi‐Komlós‐type argument for random simplicial complexes, which may be regarded as sparse random matrix models with dependent entries. © 2017 Wiley Periodicals, Inc. Random Struct. Alg., 51, 506–537, 2017     

Pre-service elementary teachers’ strategies for tiling and relating area units

It has been established that preservice elementary school teachers (PSTs) often employ procedural methods when solving measurement problems without conceptual understanding or flexibility, but a significant gap exists in the literature identifying why. Through the lens of discrete and continuous interpretations of area, this study extends the research base by describing strategies PSTs use to tile a two-dimensional space with varying size tiles and what these strategies imply about PSTs’ conceptions of area measurement. These strategies and implied conceptions enable further discussion on the multiple purposes of the area model as an illustrative measure for mathematics throughout the elementary school curriculum.     

Symmetrization of matrix technique in a finite element context

A finite element Galerkin-based formulation of the mass conservation and momentum equations can require, if convective type terms are retained in the coefficient matrix, a non-symmetric solver. The resulting increase in core storage for efficient utilization of CPU time can be considerable. The current paper advocates a simple symmetrization of matrix technique, at element level which results in a considerable reduction in core requirement. The increase in CPU time required when solving linear systems of equations is considerable. However, for nonlinear systems the penalty can be negligible.     

Beijing Lectures on the Grade Restriction Rule

The authors describe the relationships between categories of B-branes in different phases of the non-Abelian gauged linear sigma model.The relationship is described explicitly for the model proposed by Hori and Tong with non-Abelian gauge group that connects two non-birational Calabi-Yau varieties studied by Rφdland.A grade restriction rule for this model is derived using the hemisphere partition function and it is used to map B-type D-branes between the two Calabi-Yau varieties.     

Multiple level finite element analysis and its applications to tidal current flow in Tokyo Bay

A finite element method for the analysis of a one level and a multiple level current flow is presented. The basic equations can be derived from the three-dimensional Navier-Stokes equations under the shallow water assumptions. The standard finite element method has been introduced using the linear interpolation function based on a triangular finite element. For each level, the finite element subdivisions are not required to be coincident. To integrate the discretized equations numerically in time, an improved two step explicit scheme is employed. The multiple level finite element method is applied to a tidal flow analysis of Tokyo Bay.The multiple level tidal flow analysis is performed at the entrance channel of Tokyo Bay. The density of water is assumed to be constant for each level. The vertical profiles of the numerical velocity are compared with those of the observed velocity. The flow directions and the order of velocity are both well in agreement with the observed data. The tidal flow pattern in Tokyo Bay has been shown to be expressed by the multiple level flow assuming that the density of seawater is levelwise constant.The numerical tidal flow computation of Tokyo Bay carried out using a one level model is compared with observed data. The one level numerical values will be used to specify the boundary conditions for the multiple level analysis. Both numerical and observed results correspond extremely well in this computation. The two dominant circulated residual flows have been computed, and they coincide with the observed facts.     

Spatial exponential fitting scheme for the solution of steady and unsteady diffusion-convection problems in two-dimensions

Using a spatial exponential fitting scheme, the steady diffusion-convection problem in two-dimensions is reduced to a system of linear equations, while the unsteady problem is reduced to a system of ordinary-differential equations that may be solved by a fully-implicit time scheme. Also, a predictor-corrector fitting scheme is presented for solving the unsteady problem. Numerical results for the Raithby steady problem and the unsteady dispersion in a shallow sea problem are shown.     

A model for the economic evaluation of master city plans: a pilot study of Västerås,Sweden

The paper introduces the framework, problems addressed, objective function, types of variables and so on for a model designed to facilitate the economic evaluation of master city plans. The model presented here has been used in a pilot study of the city of Västerås, Sweden. It consists of three main parts, data, results and method. Some conclusions are drawn.