Describing Chaos of Continuous Time System Using Bounded Space Curve

The qualitative solutions of dynamical system expressed with nonlinear differential equation can be divided into two categories. One is that the motion of phase point may approach infinite or stable equilibrium point eventually. Neither periodic excited source nor self-excited oscillation exists in such nonlinear dynamic circuits, so its solution cannot be treated as the synthesis of multiharmonic. And the other is that the endless vibration of phase point is limited within certain range, moreover possesses character of sustained oscillation, namely the bounded nonlinear oscillation. It can persistently and repeatedly vibration after dynamic variable entering into steady state;moreover the motion of phase point will not approach infinite at last;system has not stable equilibrium point. The motional trajectory can be described by a bounded space curve. So far, the curve cannot be represented by concretely explicit parametric form in math. It cannot be expressed analytically by human. The chaos is a most universally common form of bounded nonlinear oscillation. A number of chaotic systems, such as Lorenz equation, Chua’s circuit and lossless system in modern times are some examples among thousands of chaotic equations. In this work, basic properties related to the bounded space curve will be comprehensively summarized by analyzing these examples.

THE BESOV AND TRIEBEL-LIZORKIN SPACES WITH HIGH ORDER ON THE LIPSCHITZ CURVES

The Besov spaces B_p^(α,4)(Γ)and Triebel-Lizorkin spaces F_p^(α,4)(Γ)with high order x∈R on a Lipschitz curve Γ are defind,when 1≤p≤∞,1≤q≤∞.To compare to the classical case.a difference characterization of such spaces in the case|x|<1 is given also.

Generalized elastic curves in the Lorentz flat space L^4

In this paper, we study the extremals of the curvature energy actions on non-null curves in the four-dimensional Lorentz-Minkowski space. We derive the motion equations and find three Killing fields along the generalized elastic curves. We also construct a cylindrical coordinate system using these Killing fields and express the generalized elastic curves by means of quadratures.

Conservation laws for energy and momentum in curved spaces

In arbitrary Riemannian 4-spaces, continuity equations are constructed which could be interpreted as conservation laws for the energy and momentum of the gravitational field. Special attention is given to general relativity to obtain, of natural manner, the pseudotensors of Einstein, Landau-Lifshitz, Moeller, Goldberg and Stachel, and also the conservation equations of Komar, Trautman, DuPlessis and Moss.

Evolutions of the Ruled Surfaces via the Evolution of Their Directrix Using Quasi Frame along a Space Curve

In this paper, evolutions of ruled surfaces generated by the quasi normal and quasi binormal vector fields of space curve are presented. These evolutions of the ruled surfaces depend on the evolutions of their directrix using quasi frame along a space curve.

Efficient Flexible M-Tree Bulk Loading Using FastMap and Space-Filling Curves

Many database applications currently deal with objects in a metric space.Examples of such objects include unstructured multimedia objects and points of interest(POIs)in a road network.The M-tree is a dynamic index structure that facilitates an efficient search for objects in a metric space.Studies have been conducted on the bulk loading of large datasets in an M-tree.However,because previous algorithms involve excessive distance computations and disk accesses,they perform poorly in terms of their index construction and search capability.This study proposes two efficient M-tree bulk loading algorithms.Our algorithms minimize the number of distance computations and disk accesses using FastMap and a space-filling curve,thereby significantly improving the index construction and search performance.Our second algorithm is an extension of the first,and it incorporates a partitioning clustering technique and flexible node architecture to further improve the search performance.Through the use of various synthetic and real-world datasets,the experimental results demonstrated that our algorithms improved the index construction performance by up to three orders of magnitude and the search performance by up to 20.3 times over the previous algorithm.

Equivalence of String Classical and Quantum Energy beside Equivalence of Wave Packet and Relativistic Velocity in Eucleadian and Curved Space

Plank quantum and classical string energy relations seem to be uncorrelated. This work correlated them. The relativistic energy-momentum relation has been used together with plank and de Brogglie hypothesis to prove that the wave group velocity is equal to the particle velocity in both ordinary and curved space. The plank energy relation is shown also to be related to the classical energy relation of an oscillating string. Starting from plank energy relation for n photons and performing integration, the expression of classical string energy was obtained. This means that one can treat electromagnetic waves as a collection of continuous photons having frequencies ranging from zero to w. Conversely, starting from classical string energy relation by differentiating it with respect to angular frequency, the plank quantum energy for n photons has been found. This means that the quanta results from separation of electromagnetic waves to single isolated waves. Each wave consists of n photons or quanta.

A SPACE THEORY BASIS FOR CURVE FITTING PROBLEMS

itherto, a precision Concept for curve fitting problems has not been set. By using the theory of functional analysis, the author of this paper established a space theory basis for curve fitting problems. Also given in the paper is the precision concept of the curve fitting problems and the method for constructing the fitting of a curve satisfying given precision requirements.

From Yang-Mills Photon in Curved Spacetime to Dark Energy Density

We start from quantum field theory in curved spacetime to derive a new Einstein-like energy mass relation of the type E=γmc2 where γ=1/22 is a Yang-Mills Lorentzian factor, m is the mass and c is the velocity of light. Although quantum field in curved spacetime is not a complete quantum gravity theory, our prediction here of 95.4545% dark energy missing in the cosmos is almost in complete agreement with the WMAP and supernova measurements. Finally, it is concluded that the WMAP and type 1a supernova 4.5% measured energy is the ordinary energy density of the quantum particle while the 95.5% missing dark energy is the energy density of the quantum wave. Recalling that measurement leads to quantum wave collapse, it follows that dark energy as given by E(D) = mc2 (21/22) cannot be detected using conventional direct measurement although its antigravity effect is manifested through the increasing rather than decreasing speed of cosmic expansion.

Research on a toolpath generation method of NC milling based on space-filling curve

Tool path generated by space-filling curve always turns frequently causing trembling to machine,reducing toollife and affecting workpiece quality. Length and generation time of tool paths are both relatively long. In order to solve these problems,a toolpath generation method of NC milling based on space-filling curve is proposed. First,T-spline surface is regarded as the modeling surface,the grid,which is based on the limited scallop-height,can be got in the parameter space,and the influence value of grid node is determined. Second,a box is defined and planned,and the tool paths are got preliminarily,which is based on minimal spanning tree; Finally,based on an improved chamfering algorithm,the whole tool paths are got. A simulation system is developed for computer simulation,and an experiment is carried out to verify the method. The results of simulation and experiment show that the method is effective and feasible,and length and time of the tool paths are reduced.

一种新的基于factor curve的变形控制方法

本文提出了一种新的基于ｆａｃｔｏｒｃｕｒｖｅ的变形控制方法．Ｂａｒｒ的整体非线性变形和Ｗａｔｔ的时空因子曲线方法都不同程度地依赖于某种解析表达式，不便于进行统一的控制．作者首次对因子曲线进行了分类，提供了一种统一的控制方式，因而非常适合于动画系统．把因子曲线本身作为可变参数的动画控制方法不仅包含了Ｂａｒｒ的ｔａｐｅｒｉｎｇ和ｔｗｉｓｔｉｎｇ操作，而且还包含了Ｗａｔｔ的因子曲线变形控制方法．因而动画师有更多的余地对变形进行有效的控制．

Salinity effects on soil shrinkage characteristic curves of fine-grained geomaterials

Fine-grained clayey soils are prone to substantial volume changes during desiccation in response to the dynamics of their moisture regime,and are of critical importance in several geotechnical and geoenvironmental engineering applications. As such, the complex interactions between the fraction of soil solids and the ionic pore fluid play a critical role in governing such volume changes, and have been the focus in studies dealing with marine geotechnology, mine-tailing ponds, engineered barrier systems, etc.With this in mind, the present investigation evaluates the volume changes and accompanying densification from a saturated slurry state to a constant volume state of a reference fine-grained geomaterial,kaolin, subjected to evaporative dewatering. For this purpose, several parametric studies involving determination of soil shrinkage characteristic curves(SSCCs) of kaolin under the influence of varied salt constituents and concentrations of pore fluid are performed. Furthermore, a critical assessment of SSCCs depicting progressive shrinkage and volume change behaviour of geomaterials is provided, followed by the analysis of experimentally obtained SSCCs of the kaolin to explore the impacts of pore fluid salinity.Moreover, the SSCCs are parameterised with a predictive model and the fitting parameters are used to quantitatively demonstrate the salinity-dependent volume change response of a representative finegrained porous system.

NORMAL CRITERIA FOR A FAMILY OF HOLOMORPHIC CURVES

In this paper,we extend the concept of holomorphic curves sharing hyperplanes and introduce definitions of restricted hyperplanes and partial shared hypersurfaces.Then,we prove several normal criteria of the family of holomorphic curves and holomorphic mappings that concern restricted hyperplanes and partial shared hypersurfaces.These results generalize the Montel-type normal criterion of holomorphic curves.

Type Curves Relating Well Spacing and Heterogeneity to Oil Recovery in a Water Flooded Reservoir—A Case Study

The design of an optimum spacing between oil wells entails both reservoir characterization and economics considerations. High hydrocarbon recovery requires short distances between wells. However, higher well density leads to a greater development cost. Accordingly, determination of an optimum well spacing is primordial in the development of oil fields. As a matter of fact, the identification of optimum well spacing for heterogeneous sandstone reservoirs undergoing waterflooding requires extensive analytical and numerical studies. The intent of this work is therefore to develop type curves as a quick tool in estimating ultimate recovery and reduce excessive reservoir simulation cost in analog reservoirs. These type curves utilize reservoir heterogeneity and well spacing in the estimating of oil recovery. In this work, we investigated numerically the effects of heterogeneity and well spacing on ultimate recovery using Eclipse black oil simulation and PEEP economic software 2015 and 2009 versions, respectively. The study involved a 50-ft thick Middle Eastern reservoir with porosity variability ranging from 0.2 to 0.9. Corresponding average matrix permeabilities of 1, 10 and 100 md were considered. Type curves relating well spacing and heterogeneity to ultimate oil recovery were developed. Type curves and net present value calculations indicated that there is exists an ultimate well spacing for each of the considered matrix permeabilities.

A THEOREM ON THE CONVEXITY OF PLANAR PARAMETRIC CURVES OF DEGREE N

The main result of this paper is a theorem about the convexity of curves of degree n on a plane. As its application ,we obtained a sufficient condition that a space curve of degree n in R^3 has no singularity points and staying points.

Performance Analysis of Curved Track G2T-FSO(Ground-to-Train Free Space Optical)Model under Various Weather Conditions

The demand for broadband data services on high-speed trains is rapidly growing as more people commute between their homes and workplaces.However,current radio frequency(RF)technology cannot adequately meet this demand.In order to address the bandwidth constraint,a technique known as free space optics(FSO)has been proposed.This paper presents a mathematical derivation and formulation of curve track G2T-FSO(Ground-to-train Free Space Optical)model,where the track radius characteristics is 2667 m,divergence angle track is 1.5°for train velocity at V=250 km/h.Multiple transmitter configurations are proposed to maximize coverage range and enhance curve track G2T-FSO link performance under varying weather conditions.The curved track G2T-FSO model was evaluated in terms of received power,signal-to-noise ratio(SNR),bit error rate(BER),and eye diagrams.The results showed maximum coverage lengths of 618,505,365,and 240 m for 4Tx/1Rx,3Tx/1Rx,2Tx/1Rx,and 1Tx/1Rx configurations,respectively.The analyzed results demonstrate that the G2T-FSO link can be effectively implemented under various weather conditions.

The Faber Operator Acting on BMOA,BMO-Teichmüller Space and Chord-arc Curves

After reviewing Grunsky operator and Faber operator acting on Dirichlet space,we discuss the boundedness of Faber operator on BMOA,a new subject which turns out to be closely related to the BMO theory of the universal Teichmüller space.In particular,we show that the Faber operator acts as a bounded operator on BMOA if the symbol conformal map stays nearly to the base point in the BMO-Teichmüller space.Meanwhile,we obtain several results on quasiconformal mappings,BMOTeichm¨uller space and chord-arc curves as well.As by-products,this provides a complex analysis approach to the boundedness of the Cauchy integral acting on BMO functions on a chord-arc curve near to the unit circle in the BMO-Teichmüller space.

STEADY-STATE RESPONSE OF A TIMOSHENKO BEAM ON AN ELASTIC HALF-SPACE UNDER A MOVING LOAD

By introducing the equivalent stiffness of an elastic half-space interacting with a Timoshenko beam, the displacement solution of the beam resting on an elastic half-space subjected to a moving load is presented. Based on the relative relation of wave velocities of the half-space and the beam, four cases with the combination of different parameters of the half-space and the beam, the system of soft beam and hard half-space, the system of sub-soft beam and hard half- space, the system of sub-hard beam and soft half-space, and the system of hard beam and soft half-space are considered. The critical velocities of the moving load are studied using dispersion curves. It is found that critical velocities of the moving load on the Timoshenko beam depend on the relative relation of wave velocities of the half-space and the beam. The Rayleigh wave velocity in the half-space is always a critical velocity and the response of the system will be infinite when the load velocity reaches it. For the system of soft beam and hard half-space, wave velocities of the beam are also critical velocities. Besides the shear wave velocity of the beam, there is an additional minimum critical velocity for the system of sub-soft beam and hard half-space. While for systems of （sub-） hard beams and soft half-space, wave velocities of the beam are no longer critical ones. Comparison with the Euler-Bernoulli beam shows that the critical velocities and response of the two types of beams are much different for the system of （sub-） soft beam and hard half-space but are similar to each other for the system of （sub-） hard beam and soft half space. The largest displacement of the beam is almost at the location of the load and the displacement along the beam is almost symmetrical if the load velocity is smaller than the minimum critical velocity （the shear wave velocity of the beam for the system of soft beam and hard half-space）. The largest displacement of the beam shifts behind the load and the asymmetry of the displacement along the beam increases with the increase of the load velocity due to the damping and wave racliation. The displacement of the beam at the front of the load is very small if the load velocity is larger than the largest wave velocity of the beam and the half space. The results of the present study provide attractive theoretical and practical references for the analysis of ground vibration induced by the high-speed train.

基于改进NSGA-Ⅱ的纤维缠绕落纱点轨迹采样特征权重优化

针对基于空间特征曲线特征函数的纤维缠绕落纱点轨迹采样算法无法自动选择特征权重的问题,建立以特征权重为变量,以得到采样点线性插值生成曲线与原曲线的MAE,RMSE为目标函数的双目标优化模型。提出基于改进NSGA-Ⅱ算法的双目标优化求解方法以优化特征权重。实例验证表明,与传统NSGA-Ⅱ算法相比,改进NSGA-Ⅱ算法求得Pareto解集的MAE,RMSE平均下降了0.002和0.105,算法选取特征权重的MAE,RMSE比特征权重为(0.1,0.3)的MAE,RMSE分别降低了约12.9%和8.5%,比特征权重为(0.9,0.1)的MAE,RMSE分别降低了约20.6%和11.4%,有效地提高了落纱点轨迹采样的精度。

高效前缀约简的三维Hilbert空间填充曲线编解码算法

3维Hilbert空间填充曲线(3D HSFC)的编码和解码效率对空间查询处理、图像处理等领域的应用举足轻重。现有的3维编解码算法独立编解码每一个点,忽略了Hilbert曲线的局部保持特性。为了提高编解码效率,该文设计了高效的3D状态视图,并提出一种新的前缀约简的3D HSFC编码算法(PR-3HE)和前缀约简3D HSFC解码算法(PR-3HD),这两个算法通过公共前缀的定义和识别、公共前缀约简及多种优化技术来最小化需要编码的阶数,从而提高3D HSFC的编解码效率。理论上证明:当编码或解码一个k阶的窗体(窗体内总共含有2k×2k×2k个点)时,PR-3HE平均每个点的编码阶数不超过2,PR-3HD平均解码阶数不超过8/7。相对于传统的基于迭代的方法,编解码时间复杂度从O(k)降低到了O(1)。实验结果表明,该文算法在模拟数据集和真实数据集上的表现显著优于现有算法。