Comment on "Identification of low order manifolds: validating the algorithm of Maas and Pope" [Chaos 9, 108-123 (1999)

It is claimed by Rhodes, Morari, and Wiggins [Chaos 9, 108-123 (1999)] that the projection algorithm of Maas and Pope [Combust. Flame 88, 239-264 (1992)] identifies the slow invariant manifold of a system of ordinary differential equations with time-scale separation. A transformation to Fenichel normal form serves as a tool to prove this statement. Furthermore, Rhodes, Morari, and Wiggins [Chaos 9, 108-123 (1999)] conjectured that away from a slow manifold, the criterion of Maas and Pope will never be fulfilled. We present two examples that refute the assertions of Rhodes, Morari, and Wiggins. In the first example, the algorithm of Maas and Pope leads to a manifold that is not invariant but close to a slow invariant manifold. The claim of Rhodes, Morari, and Wiggins that the Maas and Pope projection algorithm is invariant under a coordinate transformation to Fenichel normal form is shown to be not correct in this case. In the second example, the projection algorithm of Maas and Pope leads to a manifold that lies in a region where no slow manifold exists at all. This rejects the conjecture of Rhodes, Morari, and Wiggins mentioned above.     

Second order tangent bundles of infinite dimensional manifolds

The second order tangent bundle T²M of a smooth manifold M consists of the equivalent classes of curves on M that agree up to their acceleration. It is known [Analele Stiintifice ale Universitatii Al. I. Cuza 28 (1982) 63] that in the case of a finite n-dimensional manifold M, T²M becomes a vector bundle over M if and only if M is endowed with a linear connection. Here we extend this result to M modeled on an arbitrarily chosen Banach space and more generally to those Fréchet manifolds which can be obtained as projective limits of Banach manifolds. The result may have application in the study of infinite dimensional dynamical systems.     

A geometrical version of the higher order Hamilton formalism in fibred manifolds

It has been clarified recently that an r-th order Lagrangian on a fibred manifold Y → X does not determine a unique Poincaré-Cartan form provided dim X > 1 and r > 2, [1], [4], [6], [9], [10]. To make this fact more transparent, we introduced a new operation generalizing the formal exterior differentiation, [6]. In the present paper we deduce in such a way that a unique Poincaré-Cartan form can be determined by means of a simple additional structure - a linear symmetric connection Г on the base manifold X (or, more generally, by a convenient splitting S). Then we present a suitable geometric definition of a regular r-th order Lagrangian on Y and we prove that any our Poincaré-Cartan form can be used in a geometrical version of the higher order Hamilton formalism.     

Impulsive control functional differential systems of fractional order: stability with respect to manifolds

In this paper, we consider nonlinear impulsive functional differential systems involving Caputo fractional derivatives. Using vector multivariable piecewise continuous Lyapunov functions and a new vector fractional comparison principle, stability criteria with respect to a manifold are established. Some examples are presented in order to illustrate our theoretical findings.     

On low order moments of the homodyned-K distribution

Fractional low order moments have been reported as beneficial for sampling computations using the K distribution. However, it has been recently pointed out that this it not the case for the homodyned-K distribution for a tissue discrimination problem. In this paper we show that such an statement is not fully justified. To that end, we follow a standard pattern recognition procedure both to determine class separability measures and to classify data with several classifiers. We conclude that the optimum order of the moments is intimately linked to the specific statistical properties of the tissues to be discriminated. Some ideas on how to choose the optimum order are discussed.     

Doped spin liquid: Luttinger sum rule and low temperature order

We analyze a model of two-leg Hubbard ladders weakly coupled by interladder tunneling. At half filling a semimetallic state with small Fermi pockets is induced beyond a threshold tunneling strength. The sign changes in the single electron Green's function relevant for the Luttinger sum rule now take place at surfaces with both zeros and infinities with important consequences for the interpretation of angle-resolved photoemission spectroscopy experiments. Residual interactions between electron and holelike quasiparticles cause a transition to long range order at low temperatures. The theory can be extended to small doping leading to superconducting order.     

一种二维不稳定流形的新算法及其应用

提出了连续时间系统二维(不)稳定流形的一种新数值算法,不但可以快速地求得流形的直观图像,而且能够准确地获取流形上各点的位置、时间、轨道距离等丰富的信息,从而有利于人们从几何上去研究系统的全局行为,如边界特征、演化过程、奇异环等等.本算法首先通过解初值问题求出均匀分布的相邻轨道,然后连接这些轨道既得流形面.Lorenz系统原点的稳定流形的计算表明本算法快速有效.此外,通过试着寻找异宿轨道,还研究了一个三维神经网络中的混沌产生机理.     

Almost product manifolds as the low energy geometry of Dirichlet branes

The European Physical Journal C - Any candidate theory of quantum gravity must address the breakdown of the classical smooth manifold picture of space-time at distances comparable to the Planck...     

连续时间系统二维不稳定流形的异构算法

非线性系统的二维流形通常具有复杂几何结构和丰富动力学信息,因此在流形计算与可视化时存在大量的不可避免的数值计算.因此,如何高效地完成这些计算就成了关键问题.鉴于当今计算机的异构发展趋势(包含多核CPU和通用GPU),本文在兼顾精度和通用性的基础上,提出了适用于新一代计算平台的快速流形计算方法.本算法将计算任务分为轨道延伸和三角形生成两部分,前者运算量大而单一适合GPU完成,后者运算量小而复杂适合CPU执行.通过对Lorenz系统原点稳定流形的计算,表明本算法能充分发挥异构平台的综合性能,可大幅度提高计算速 关键词： 不稳定流形 流形计算 异构计算 Lorenz系统     

激光靶耦合问题的二阶精度重映算法研究

 任意拉格朗日欧拉方法被认为是解决2维流体力学大变形数值模拟的有效途径之一。以激光黑腔靶耦合问题为背景,结合质量流的物理意义,提出了一种二阶守恒物理量重映的算法,并利用1维激光靶耦合程序对方法及其可行性进行了考察,结合物理问题调整网格,用较少的网格得到了与较多网格拉氏计算同样的结果。为算法在2维LARED-H程序中的应用奠定了基础。     

激光靶耦合问题的二阶精度重映算法研究

任意拉格朗日欧拉方法被认为是解决2维流体力学大变形数值模拟的有效途径之一。以激光黑腔靶耦合问题为背景,结合质量流的物理意义,提出了一种二阶守恒物理量重映的算法,并利用1维激光靶耦合程序对方法及其可行性进行了考察,结合物理问题调整网格,用较少的网格得到了与较多网格拉氏计算同样的结果。为算法在2维LARED-H程序中的应用奠定了基础。     

Emergence of novel phenomena on the border of low dimensional spin and charge order

Proximity to magnetic order as well as low dimensionality are both beneficial to superconductivity at elevated temperatures. Materials on the border of magnetism display a wide range of novel and potentially useful phenomena: high T_cs, heavy fermions, coexistence of magnetism and superconductivity and giant magnetoresistance. Low dimensionality is linked to enhanced fluctuations and, in the case of heavy fermions, has been experimentally shown to be beneficial for the fluctuations that are responsible for the rich abundance of novel emergent phases. This experimental strategy motivated us to explore 2D insulating magnets with a view to investigate phase evolution across metal-insulator and magnetic-non-magnetic boundaries. This has been a fruitful venture with totally novel results different to our expectations. We present results from 2 distinct systems. The MPS₃ family are highly anisotropic in both their crystal and magnetic structures. FePS₃ in particular is a model insulating honeycomb antiferromagnet. We find that the application of pressure to FePS₃ induces an insulator to metal transition. The second system, Cs₂CuCl₄, is a highly-frustrated quantum spin liquid at low temperature. The competition of the 3 relevant exchange couplings is delicately balanced. It has been shown to become antiferromagnetic at very low temperatures (~1 K). We have found that the application of pressure for 3 days or more followed by a return to ambient pressure stabilises a totally distinct magnetic ground state.     

Effects of low order atmosphere-turbulence aberrations on the vortex carrying Gaussian beam

The normalized intensity distributions at the focal plane of a vortex carrying Gaussian beam propagating in weak/middle turbulent atmosphere channel with z-tilt aberration, defocus aberration, astigmatism aberration or total turbulent aberration are discussed by numerical calculation. Our results show that the effect of z-tilt aberration on the intensity distribution of optical vortex beam is main effect of total turbulent aberration. In weak turbulent region, the effects of defocus and astigmatism aberration on the intensity distribution of a vortex carrying Gaussian beam can be ignored. In middle turbulent region, the effect of z-tilt aberration is still the most significant, but the effects of defocus and astigmatism aberrations, specially the effect of astigmatism on the center dark core of the intensity distribution at focal plane, can no longer be disregarded. Our results also show that for three low order aberrations, the beam with values of the topological charge, the beam have larger beam-radius and undergo smaller effects of z-tilt turbulent aberration on the doughnut distribution. For defocus aberration, the radius of the center dull of beam intensity increases with the values of topological charges increasing. And for astigmatism aberration, the beam with odd number topological charges, the center dull speck of beam intensity becomes to the bright speck. But for even number topological charges, the center dull speck of beam intensity maintains the dull one.     

验证感生电场存在的实验

针对磁场变化时激发产生的感生电场很难演示的问题,设计了演示感生电场存在的实验. 该实验将不同功能的仪器组合起来使用,通过亥姆霍兹线圈的交变磁场对带电粒子的作用效果,直观说明感生电场的存在.     

High order approximate low frequency theory of elastic anisotropic lining and coating

A problem of the dynamic behavior of an elastic layer coupled to one or two thick elastic solids is considered. All the materials may possess a general anisotropy and the layer is assumed to be thin enough with respect to the characteristic wavelength. Introducing the asymptotic power series with respect to the thickness-over-wavelength ratio for the main quantities and using the asymptotic integration method the displacements and stresses on the layer surfaces are related. Thus, the so-called impedance boundary conditions (IBC) are deduced for three cases--for a coated substrate with given displacements or with given stresses on the surface and for two substrates with a layer in between. In contrast to previous papers these IBC are obtained for the most general situation with the asymptotic accuracy up to the sixth order, uniform with respect to the representation of the displacements and stresses. Presented theory can be used for studying the surface and interface phenomena as well as for calculating fields and spectra of layered solids. The results are validated numerically and compared with those of other authors.     

Efficiency of the microcanonical over-relaxation algorithm for vector spins analyzing first and second order transitions

We simulate vectorial spin systems solely with the microcanonical over-relaxation algorithm where the temperature is calculated by a formula of Nurdin and Schotte. We show that this procedure is the most efficient local algorithm besides the nonlocal cluster algorithm not only for first order transitions but also for second order ones. A comparison is made with the Metropolis, heat bath, multicanonical and the Creutzs demon algorithms. We study, using these algorithms, the frustrated J1-J2 model on a cubic lattice for XY, Heisenberg and O(4) spins. These models have a breakdown of symmetry Z3 SO(N)/SO(N-1) for the number N = 2,3,4 of spin components leading to transitions of first order. We show that they are strongly first order. Then, to test the over-relaxation update for second order transitions, we study a ferromagnet on a cubic lattice and a frustrated antiferromagnet on a stacked triangular lattice. We finally point out the advantages and the flaws of the over-relaxation procedure.