A preconditioner for the h-p version of the finite element method in two dimensions

Summary. A preconditioner, based on a two-level mesh and a two-level orthogonalization, is proposed for the - version of the finite element method for two dimensional elliptic problems in polygonal domains. Its implementation is in parallel on the subdomain level for the linear or bilinear (nodal) modes, and in parallel on the element level for the high order (side and internal) modes. The condition number of the preconditioned linear system is of order , where is the diameter of the -th subdomain, and are the diameter of elements and the maximum polynomial degree used in the subdomain. This result reduces to well-known results for the -version (i.e. ) and the -version (i.e. ) as the special cases of the - version. Received August 15, 1995 / Revised version received November 13, 1995     

Multiple impulse solutions to McKean's caricature of the nerve equation. II. Stability

We study McKean's caricature of a nerve conduction equation where H is the Heaviside function. It is proved that an n-ple impulse solution resembling the superposition of n unstable solitary impulses has at most 2n - 1, and at least n, unstable modes: exactly n unstable modes corresponding to the amplitudes and the rest of them corresponding to the spacings. The n amplitude modes always exist. We prove also that for an n-ple impulse solution resembling the superposition of n stable solitary impulses, there are at most n - 1 unstable modes and all of them are of spacing type.     

基于溢出效应与平台服务的竞争网络平台的销售模式研究

网络平台竞争与发展成为平台运营理论与实践的关键问题,急需解决。本文应用博弈论研究存在溢出效应和网络平台竞争环境下平台服务内生的网络平台销售模式选择,即网络平台商是允许制造商在其网络平台上直销产品的平台销售模式,还是网络平台商先从制造商购买产品再销售给消费者的转销模式。主要研究竞争网络平台的三种结构模型:双转销模式、双平台销售模式和混合模式。研究结果显示:网络平台商的利润随正溢出效应增加而增加,随负溢出效应增加而减少。当网络平台服务敏感程度较高且网络平台竞争较为激烈时,两个网络平台商均选择平台销售模式,形成占优均衡;此时,网络平台商获得最高利润,而且网络零售价和网络平台服务水平最高。     

The influence and selection of subspaces for a posteriori error estimators

Summary. The element residual method for a posteriori error estimation is analyzed for degree finite element approximation on quadrilateral elements. The influence of the choice of subspace used to solve the element residual problem is studied. It is shown that the resulting estimators will be consistent (or asymptotically exact) for all if and only if the mesh is parallel. Moreover, even if the mesh consists of rectangles, then the estimators can be inconsistent when . The results provide concrete guidelines for the selection of a posteriori error estimators and establish the limits of their performance. In particular, the use of the element residual method for high orders of approximation (such as those arising in the - version finite element method) is vindicated. The mechanism behind the rather poor performance of the estimators is traced back to the basic formulation of the residual problem. The investigations reveal a deficiency in the formulation, leading, as it does, to spurious modes in the true solution of the residual problem. The recommended choice of subspaces may be viewed as being sufficient to guarantee that the spurious modes are filtered out from the approximate solution while at the same time retaining a sufficient degree of approximation to represent the true modes. Received February 27, 1995 / Revised version received June 7, 1995     

Chaos in M(atrix) theory

We consider the classical and quantum dynamics in M(atrix) theory. Using a simple ansatz we show that a classical trajectory exhibits a chaotic motion. We argue that the holographic feature of M(atrix) theory is related with the repulsive feature of energy eigenvalues in quantum chaotic system. Chaotic dynamics in N = 2 supersymmetric Yang—Mills theory is also discussed. We demonstrate that after the separation of “slow” and “fast” modes there is a singular contribution from the “slow” modes to the Hamiltonian of the “fast” modes.     

Linear algebra students’ modes of reasoning: Geometric representations

Main goal of our research was to document differences on the types of modes linear algebra students displayed in their responses to the questions of linear independence from two different assignments. In this paper, modes from the second assignment are discussed in detail. Second assignment was administered with the support of graphical representations through an interactive web-module. Additionally, for comparison purposes, we briefly talk about the modes from the first assignment. First assignment was administered with the support of computational devices such as calculators providing the row reduced echelon form (rref) of matrices. Sierpinska’s framework on thinking modes (2000) was considered while qualitatively documenting the aspects of 45 matrix algebra students’ modes of reasoning. Our analysis revealed 17 categories of the modes of reasoning for the second assignment, and 15 categories for the first assignment. In conclusion, the findings of our analysis support the view of the geometric representations not replacing one’s arithmetic or algebraic modes but encouraging students to utilize multiple modes in their reasoning. Specifically, geometric representations in the presence of algebraic and arithmetic modes appear to help learners begin to consider the diverse representational aspects of a concept flexibly.     

How do ecosystems respond to external perturbations?

This paper attempts to study the influence of environmental perturbations on evolutionary modes (dynamical regimes) of model ecosystems. Three model systems are analysed and the relative importance of these modes is depicted in basin boundary structures. It is found that sudden perturbations usually influence these modes significantly. The coexistence of two or more distinct attractors on the same set of parametric values in nonlinear dynamical systems suggests that the environmental forces may alter the dynamical regimes of these systems. Two-dimensional parameter scans suggest that even smooth (periodic or seasonal) perturbations, which bring changes in system parameters and thus decide dynamical modes to be displayed in a particular situation, are able to induce variations in these modes. It should be noted that the nature of changes induced by the two types of perturbations is different. While the former causes predictable changes in the dynamical modes, the latter forces the system to meander among different dynamical regimes aimlessly.The present study indicates that the influences of sudden (unforeseen) perturbations (e.g., forest fire, drought, flood, invasion by exotic species, etc.) on the ecosystem dynamics can be controlled provided the system's dynamic complexity is understood properly. The knowledge of these influences may help us in gaining insight into how ecological disasters can be managed.     

Semilattice modes I: the associated semiring

We examine idempotent, entropic algebras (modes) which have a semilattice term. We are able to show that any variety of semilattice modes has the congruence extension property and is residually small. We refine the proof of residual smallness by showing that any variety of semilattice modes of finite type is residually countable. To each variety of semilattice modes we associate a commutative semiring satisfying 1 +r=1 whose structure determines many of the properties of the variety. This semiring is used to describe subdirectly irreducible members, clones, subvariety lattices, and free spectra of varieties of semilattice modes.Presented by J. Berman.Part of this paper was written while the author was supported by a fellowship from the Alexander von Humboldt Stiftung.     

Active control of flexible systems

Since mechanically flexible systems are distributed-parameter systems, they are infinite-dimensional in theory and, in practice, must be modelled by large-dimensional systems. The fundamental problem of actively controlling very flexible systems is to control a large-dimensional system with a much smaller dimensional controller. For example, a large number of elastic modes may be needed to describe the behavior of a flexible satellite; however, active control of all these modes would be out of the question due to onboard computer limitations and modelling error. Consequently, active control must be restricted to a few critical modes. The effect of the residual (uncontrolled) modes on the closed-loop system is often ignored. In this paper, we consider the class of flexible systems that can be described by a generalized wave equation,u _tt+Au=F, which relates the displacementu(x,t) of a body Θ inn-dimensional space to the applied force distributionF(x,t). The operatorA is a time-invariant symmetric differential operator with a discrete, semibounded spectrum. This class of distributed parameter systems includes vibrating strings, membranes, thin beams, and thin plates. The control force distribution $$F(x,t) = \sum\limits_{i = 1}^M { \delta (x - x_i )f_i (t)} $$ is provided byM point force actuators located at pointsx _i on the body. The displacements (or their velocities) are measured byP point sensorsy _i(t)=u(z _j,t), oru _t(z _j,t),j=1, 2, ...,P, located at various pointsz _j along the body. We obtain feedback control ofN modes of the flexible system and display the controllability and observability conditions required for successful operation. We examine the control and observation spillover due to the residual modes and show that the combined effect of spillover can lead to instabilities in the closed-loop system. We suggest some remedies for spillover, including a straightforward phase-locked loop prefilter, to remove the instability mechanism. To illustrate the concepts of this paper, we present the results of some numerical studies on the active control of a simply supported beam. The beam dynamics are modelled by the Euler-Bernoulli partial differential equation, and the feedback controller is obtained by the above procedures. One actuator and one sensor (at different locations) are used to control three modes of the beam quite effectively. A fourth residual mode is simulated, and the destabilizing effect of control and observation spillover together on this mode is clearly illustrated. Once observation spillover is eliminated (e.g., by prefiltering the sensor outputs), the effect of control spillover alone on this system is negligible.     

Super-high-frequency oscillations in a discontinuous dynamic system with time delay

We study oscillations in the discontinuous dynamic system with time delay . This is a typical model of relay feedback with delay. It is known that stable modes in this system have a bounded oscillation frequency. Here we consider transient processes and obtain the following result: under some restrictions ofF, the average oscillation frequency of any solution becomes finite after a period of time, i.e. super-high-frequency oscillations (with infinite frequency) exist only in a finite time interval. Moreover, we give an effective upper bound on the length of this interval.     

Long Nonlinear Surface Waves in the Presence of a Variable Current

We investigate the eigenvalue problem governing the propagation of long nonlinear surface waves when there is a current beneath the surface, y being the vertical coordinate. The amplitude of such waves evolves according to the KdV equation and it was proved by Burns [ 1 ] that their speed of propagation c is such that there is no critical layer (i.e., c lies outside the range of ). If, however, the critical layer is nonlinear, the result of Burns does not necessarily apply because the phase change of linear theory then vanishes. In this paper, we consider specific velocity profiles and determine c as a function of Froude number for modes with nonlinear critical layers. Such modes do not always exist, the case of the asymptotic suction profile being a notable example. We find, however, that singular modes can be obtained for boundary layer profiles of the Falkner–Skan similarity type, including the Blasius case. These and other examples are treated and we examine singular solutions of the Rayleigh equation to gain insight about the long wave limit of such solutions.     

An applicatin of approximate inertial manifolds to a weakly damped nonlinear schrödinger equation

Nonlinear Galerkin methods (NGMs) based on pproximate inertial manifolds are applied to a weakly dissipative nonlinear Schrödinger equation. The purpose is to capture critical and chaotic behavior with as few modes as possible. Density functions are used on both the energy and instantaneous Lyapunov exponents to determine convergence of a chaotic attractor as the number of modes is increased. The computations presented here indicate a substantial reduction in the number of modes needed for the NGMs, compared to that needed for the traditional Galerkin method.     

Object modeling and path computation for multimodal travel systems

This paper describes a multimodal travel system (MTS) designed to address the needs of a variety of demand-responsive transport. An origin–destination (O–D) trip in transportation network can be accomplished by using multiple modes. In urban network passengers may boarding buses or metros to go from one place to another, and modes as autobus or trains are used by passengers to travel between cities. The work focuses on the network object modeling and multimodal shortest path algorithm. A solution to the problem of long-run planning of transit on multimodal network has been implemented and tested. The work presents the general results found, and the proposed algorithm recognizes the set of constraints related to the time schedule and the sequence of used modes in a O–D trip. The aim is to provide a tool for detecting the facilities of using different travel modes through a transportation network. Routings may include distinct combination of rail, and route. Geographic Information Systems (GIS) were invaluable in the cost-effective construction and maintenance of this work and the subsequent validation of mode sequences and paths selections. Attention is devoted to the multimodal path operator as well as to the use of GIS-transit planning.     

考虑网络平台服务的电子零售商运营模式选择

针对供应商竞争情形下拥有网络平台的电子零售商运营模式选择问题,考虑网络平台服务对网络平台绩效的影响,研究电子零售商运营模式决策与供应商销售模式选择。本文构建了网络平台服务内生的供应商竞争模型,识别电子零售商选择转销模式或平台模式运营的条件,揭示实现帕累托改进的收益分享契约特征。研究结果显示:当产品间竞争不激烈且服务增值成本指数相对较高时,若收益分享比例小于某临界值,电子零售商偏好平台模式运营,且平台模式中网络平台服务水平高而零售价格低;当产品间竞争较激烈时,电子零售商总偏好转销模式运营;当产品市场规模不高且收益分享比例在某区域时,电子零售商和供应商都偏好平台模式。     

Exact algorithms for multi-criteria multi-modal shortest path with transfer delaying and arriving time-window in urban transit network

This paper investigates the solution algorithms for the multi-criteria multi-modal shortest path problem (M-SPP), which belongs to the set of problems known as NP-hard, in urban transit network (UTN). The related M-SPP is one of the important and practical problems in several fields such as urban transportation system and freight transportation. The UTN is composed of multiple modes (e.g., automobile, bus, subway, light rail, pedestrian and so on). To get their destination, the passengers can alternate between different modes. As a special demand, the time-window is usually associated with the M-SPP. Because of the service time-limit of modes, the available modes at a stop are varied with the time. So the optimal M-SPP with arriving time-window cannot be simply obtained by finding the optimal M-SPP firstly and then reversely deducing the leaving time-window of the origin according to the arriving time-window of destination. In this paper, the M-SPP with arriving time-window is firstly proposed. To solve the multi-criteria M-SPPs (MM-SPP) with transfer delaying, an improved exact label correcting algorithm (LCA) is designed and, to solve the proposed MM-SPPs with both of transfer delaying and arriving time-window, an exact reverse LCA is designed. Finally, some computing examples are given to test the effectiveness of the designed algorithms.     

Multimodal admittance method in waveguides and singularity behavior at high frequencies

This work presents a multimodal method for the propagation in a waveguide with varying height and its relation to trapped modes or quasi-trapped modes. The coupled mode equations are obtained by projecting the Helmholtz equation on the local transverse modes. To solve this problem we integrate the Riccati equation governing the admittance matrix (Dirichlet-to-Neumann operator). For many propagating modes, i.e. at high frequencies, the numerical integration of the Riccati equation shows that the rule is that this matrix has quasi-singularities associated to quasi-trapped modes.     

Propagation of localized bending deformations in microtubules

A continuum medium model is proposed describing a microtubule (MT) as an elastic rod. When the MT is subjected to a constant bending force, the dynamics of the angular deviation, with respect to the MT's rectilinear configuration, is governed by a Sine-Gordon equation. Particular analytical solutions are kink and anti-kink bending modes which may propagate at various speeds along the MT's length. Kinetic energies of these modes compare with thermal and ATP hydrolysis energies. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Criteria for trapped modes in a cranked channel with fixed and freely floating bodies

Trapped modes in the linearized water wave problem are localized free oscillations in an unbounded fluid with a free surface. For sometime, it has been known that certain structures, fixed or freely floating, can support such modes. In this paper, we consider the problem on a channel, which consists of a finite part and two cylindrical outlets into infinity. The finite (bounded) part may contain some submerged and/or surface-piercing bodies. Since the ordinary scattering matrix can by no means contribute any information on trapped modes, we introduce the fictitious scattering operator and present a criterion for the existence of trapped modes. The criterion states that the number of trapped modes is the difference of the multiplicities of the eigenvalue 1 of the fictitious scattering operator and the eigenvalue ?i of the scattering matrix.