THE STRESS SUBSPACE OF HYBRID STRESS ELEMENT AND THE DIAGONALIZATION METHOD FOR FLEXIBILITY MATRIX H

The following is proved: 1) The linear independence of assumed stress modes is the necessary and sufficient condition for the nonsingular flexibility matrix; 2) The equivalent assumed stress modes lead to the identical hybrid element. The Hilbert stress subspace of the assumed stress modes is established. So, it is easy to derive the equivalent orthogonal normal stress modes by Schmidt's method. Because of the resulting diagonal flexibility matrix, the identical hybrid element is free from the complex matrix inversion so that the hybrid efficiency, is improved greatly. The numerical examples show that the method is effective.

DIAGONALIZATION METHOD FOR A SINGULARLY PERTURBED VECTOR ROBIN PROBLEM

In this paper the method and technique of the diagonalization are employed to transform a vector second-order nonlinear system into two first-order approximate diagonalized systems. The existence and the asymptotic behavior of the solutions are obtained for a vector second-order nonlinear Robin problem of singular perturbation type.

Zeno and the Wrong Understanding of Motion—A Philosophical-Mathematical Inquiry into the Concept of Finitude as a Peculiarity of Infinity

In contrast to the solutions of applied mathematics to Zeno’s paradoxes, I focus on the concept of motion and show that, by distinguishing two different forms of motion, Zeno’s apparent paradoxes are not paradoxical at all. Zeno’s paradoxes indirectly prove that distances are not composed of extensionless points and, in general, that a higher dimension cannot be completely composed of lower ones. Conversely, lower dimensions can be understood as special cases of higher dimensions. To illustrate this approach, I consider Cantor’s only apparent proof that the real numbers are uncountable. However, his widely accepted indirect proof has the disadvantage that it depends on whether there is another way to make the real numbers countable. Cantor rightly assumes that there can be no smallest number between 0 and 1, and therefore no beginning of counting. For this reason he arbitrarily lists the real numbers in order to show with his diagonal method that this list can never be complete. The situation is different if we start with the largest number between 0 and 1 (0.999…) and use the method of an inverted triangle, which can be understood as a special fractal form. Here we can construct a vertical and a horizontal stratification with which it is actually possible to construct all real numbers between 0 and 1 without exception. Each column is infinite, and each number in that column is the starting point of a new triangle, while each row is finite. Even in a simple sine curve, we experience finiteness with respect to the y-axis and infinity with respect to the x-axis. The first parts of this article show that Zeno’s assumptions contradict the concept of motion as such, so it is not surprising that this misconstruction leads to contradictions. In the last part, I discuss Cantor’s diagonal method and explain the method of an inverted triangle that is internally structured like a fractal by repeating this inverted triangle at each column. The consequence is that we encounter two very different methods of counting. Vertically it is continuous, horizontally it is discrete. While Frege, Tarski, Cantor, Gödel and the Vienna Circle tried to derive the higher dimension from the lower, a procedure that always leads to new contradictions and antinomies (Tarski, Russell), I take the opposite approach here, in which I derive the lower dimension from the higher. This perspective seems to fail because Tarski, Russell, Wittgenstein, and especially the Vienna Circle have shown that the completeness of the absolute itself is logically contradictory. For this reason, we agree with Hegel in assuming that we can never fully comprehend the Absolute, but only its particular manifestations—otherwise we would be putting ourselves in the place of the Absolute, or even God. Nevertheless, we can understand the Absolute in its particular expressions, as I will show with the modest example of the triangle proof of the combined horizontal and vertical countability of the real numbers, which I developed in rejection of Cantor’s diagonal proof. .

Energy Spectrum for a Short-Range 1/r Singular Potential with a Non-Orbital Barrier Using the Asymptotic Iteration Method

Using the asymptotic iteration method, we obtain the S-wave solution for a short-range three-parameter central potential with 1/r singularity and with a non-orbital barrier. To the best of our knowledge, this is the first attempt at calculating the energy spectrum for this potential, which was introduced by H. Bahlouli and A. D. Alhaidari and for which they obtained the “potential parameter spectrum”. Our results are also independently verified using a direct method of diagonalizing the Hamiltonian matrix in the J-matrix basis.

Surface states in crystals with low-index surfaces

For most of the conventional crystals with low-index surfaces, the hopping between the nearest neighbor （1NN） crystal planes （CPs） is dominant and the ones from the nNN （2 〈 n 〈 ∞） CPs are relatively weak, considered as small perturbations. The recent theoretical analysisIll has demonstrated the absence of surface states at the level of the hopping approximation between the INN CPs when the original infinite crystal has the geometric reflection symmetry （GRS） for each CP. Meanwhile, based on the perturbation theory, it has also been shown that small perturbations from the hopping between the nNN （2 〈 n 〈 ∞） CPs and surface relaxation have no impact on the above conclusion. However, for the crystals with strong intrinsic spin-orbit coupling （SOC）, the dominant terms of intrinsic SOC associate with two INN bond hoppings. Thus SOC will significantly contribute the hoppings from the INN and/or 2NN CPs except the ones within each CP. Here, we will study the effect of the hopping between the 2NN CPs on the surface states in model crystals with three different type structures （Type I： “……P-P-P-P……”, Type II： “……-P-Q-P-Q……” and Type III：“……P=Q-P=Q……” where P and Q indicate CPs and the signs “-” and “=” mark the distance between the INN CPs）. In terms of analytical and numerical calculations, we study the behavior of surface states in three types after the symmetric/asymmetric hopping from the 2NN CPs is added. We analytically prove that the symmetric hopping from the 2NN CPs cannot induce surface states in Type I when each CP has only one electron mode. The numerical calculations also provide strong support for the conclusion, even up to 5NN. However, in general, the coupling from the 2NN CPs （symmetric and asymmetric） is favorable to generate surface states except Type I with single electron mode only.

Local structure distortion and spin Hamiltonian parameters for Cr^(3+)-V_(Zn) tetragonal defect centre in Cr^(3+) doped KZnF_3 crystal

The quantitative relationship between the spin Hamiltonian parameters （D, g｜｜ Ag） and the crystal structure parameters for the Cr3＋-Vzη tetragonal defect centre in a Cr3＋ ：KZnF3 crystal is established by using the superposition model. On the above basis, the local structure distortion and the spin Hamiltonian parameter for the Cr3＋-Vzn tetragonal defect centre in the KZnF3 crystal are systematically investigated using the complete diagonalization method. It is found that the Vzn vacancy and the differences in mass, radius and charge between the Cr3＋ and the Zn2＋ ions induce the local lattice distortion of the Cr3＋ centre ions in the KZnF3 crystal. The local lattice distortion is shown to give rise to the tetragonal crystal field, which in turn results in the tetragonal zero-field splitting parameter D and the anisotropic g factor Ag. We find that the ligand F- ion along I001] and the other five F- ions move towards the central Cr3＋ by distances of A1 = 0.0121 nm and A2 = 0.0026 nm, respectively. Our approach takes into account the spin-rbit interaction as well as the spin-spin, spin other-orbit, and orbit-rbit interactions omitted in the previous studies. It is found that for the Cr3＋ ions in the Cr3＋：KZnF3 crystal, although the spin-rbit mechanism is the most important one, the contribution to the spin Hamiltonian parameters from the other three mechanisms, including spin- spin, spin-other-orbit, and orbit-orbit magnetic interactions, is appreciable and should not be omitted, especially for the zero-field splitting （ZFS） parameter D.

Existence of Solution for a Coupled System of Fractional Integro-Differential Equations on an Unbounded Domain

We present the existence of solution for a coupled system of fractional integro-differential equations. The differential operator is taken in the Caputo fractional sense. We combine the diagonalization method with Arzela-Ascoli theorem to show a fixed point theorem of Schauder.

Sliding Mode Control for Singularly Perturbed Systems Using Accurate Reduced Model

In order to deal with unmodeled dynamics in large vehicle systems, which have an ill condition of the state matrix, the use of model order reduction methods is a good approach. This article presents a new construction of the sliding mode controller for singularly perturbed systems. The controller design is based on a linear diagonal transformation of the singularly perturbed model. Furthermore, the use of a single sliding mode controller designed for the slow component of the diagonalized system is investigated. Simulation results indicate the performance improvement of the proposed controllers.

SINGULAR PERTURBATION OF TWO─POINT BOUNDARY VALUE PROBLEM FOR NONLINEAR SYSTEM

In this paper we study the singular perturbation of boundary value problem for second order nonlinear system by the method and the technique of diagonalization. Under the appropriate assumptions we prove the existence of solution and give its asymptotic estimation as ε→0+

NUMERICAL SIMULATION OF WIND-INDUCED MOTION IN SUSPENDED SEDIMENT TRANSPORT

In regions with broad water surfaces such as lakes, reservoirs and coastal areas, the wind stress on the flow motion generates a significant impact. The wind stress is an unsteady force which makes numerical simulation difficult. This paper presents a two-dimensional （2-D） mathematical model of the impact of wind-induced motion on suspended sediment transport at Beijing＇s 13-Ling Reservoir. The model uses the Diagonal Cartesian Method （DCM） with a wetting-drying dynamic boundary to trace variations in the water level. The calculation results have been tested against in situ measurements. The measurements confirm the model＇s accuracy and agreement with the actual situation at the reservoir. The calculations also indicate that wind stress holds the key to suspended sediment transport at Beijing＇s 13-Ling Reservoir, especially when westerly winds prevail.

The Singularly Perturbed Problem of Vector Integro-differential Equations

The singularly perturbed boundary value problem of scalar integro-differential equations has been studied extensively by the differential inequality method . However, it does not seem possible to carry this method over to a corresponding nonlinear vector integro-differential equation. Therefore , for n-dimensional vector integro-differential equations the problem has not been solved fully. Here, we study this nonlinear vector problem and obtain some results. The approach in this paper is to transform the appropriate integro-differential equations into a canonical or diagonalized system of two first-order equations.

On Dissipation and Dispersion Errors Optimization,A-Stability and SSP Properties

In a recent paper(Du and Ekaterinaris,2016)optimization of dissipation and dispersion errors was investigated.A Diagonally Implicit Runge-Kutta(DIRK)scheme was developed by using the relative stability concept,i.e.the ratio of absolute numerical stability function to analytical one.They indicated that their new scheme has many similarities to one of the optimized Strong Stability Preserving(SSP)schemes.They concluded that,for steady state simulations,time integration schemes should have high dissipation and low dispersion.In this note,dissipation and dispersion errors for DIRK schemes are studied further.It is shown that relative stability is not an appropriate criterion for numerical stability analyses.Moreover,within absolute stability analysis,it is shown that there are two important concerns,accuracy and stability limits.It is proved that both A-stability and SSP properties aim at minimizing the dissipation and dispersion errors.While A-stability property attempts to increase the stability limit for large time step sizes and by bounding the error propagations via minimizing the numerical dispersion relation,SSP optimized method aims at increasing the accuracy limits by minimizing the difference between analytical and numerical dispersion relations.Hence,it can be concluded that A-stability property is necessary for calculations under large time-step sizes and,more specifically,for calculation of high diffusion terms.Furthermore,it is shown that the oscillatory behavior,reported by Du and Ekaterinaris(2016),is due to Newton method and the tolerances they set and it is not related to the employed temporal schemes.