浅谈高考数学试卷上的“任意”

“任意”作为一种特殊的数学语言，近年频频出现在高考试卷上，下面结合例题，从三个方面描述“任意”的含义．

Simulation of extrusion process of complicated aluminium profile and die trial

The arbitrary Lagrangian-Eulerian（ALE） adaptive remeshing technology and the HyperXtrude software of transient finite element simulations were used on analogue simulation of aluminium extrusion processing.The field distributions of strain rate,stress,temperature and velocity of metal flow were obtained.The results are basically consistent with the experiment,which indicates that this method may successfully predict the defects in the actual extrusion process.

Modeling seismic wave propagation within complex structures

Seismic modeling is a useful tool for studying the propagation of seismic waves within complex structures. However, traditional methods of seismic simulation cannot meet the needs for studying seismic wavefields in the complex geological structures found in seismic exploration of the mountainous area in Northwestern China. More powerful techniques of seismic modeling are demanded for this purpose. In this paper, two methods of finite element-finite difference method （FE-FDM） and arbitrary difference precise integration （ADPI） for seismic forward modeling have been developed and implemented to understand the behavior of seismic waves in complex geological subsurface structures and reservoirs. Two case studies show that the FE-FDM and ADPI techniques are well suited to modeling seismic wave propagation in complex geology.

ALE FINITE ELEMENT ANALYSIS OF NONLINEAR SLOSHING PROBLEMS BASED ON F LUID VELOCITY POTENTIAL

Base d on fluid velocity potential, an ALE finite element formulation for the analysi s of nonlinear sloshing problems has been developed. The ALE kinemat ical description is introduced to move the computational mesh independently of f luid motion, and the container fixed noninertial coordinate system is employed to establish the governing equations so that the mesh is needed to be updated in this coordinate system only. This leads to a very simple mesh moving algorithm which makes it easy to trace the motion of the moving boundaries and the free su rface without producing undesirable distortion of the computational mesh. The fi nite element method and finite difference method are used spacewise and timewise , respectively. A numerical example involving either forced horizontal oscillati on or forced pitching oscillation of the fluid filled container is presented to illustrate the effectiveness and the robustness of the method. In additi on, this work can be extended for the fluid structure interaction problems.

Finite-difference numerical modeling with even-order accuracy in two-phase anisotropic media

To improve the accuracy of the conventional finite-difference method, finitedifference numerical modeling methods of any even-order accuracy are recommended. We introduce any even-order accuracy difference schemes of any-order derivatives derived from Taylor series expansion. Then, a finite-difference numerical modeling method with any evenorder accuracy is utilized to simulate seismic wave propagation in two-phase anisotropic media. Results indicate that modeling accuracy improves with the increase of difference accuracy order number. It is essential to find the optimal order number, grid size, and time step to balance modeling precision and computational complexity. Four kinds of waves, static mode in the source point, SV wave cusps, reflection and transmission waves are observed in two-phase anisotropic media through modeling.

An Algorithm for Partitioning Polygons into Convex Parts

An algorithm for partitioning arbitrary simple polygons into a number of convex parts was presented. The concave vertices were determined first, and then they were moved by using the method connecting the concave vertices with the vertices of falling into its region B,so that the primary polygon could be partitioned into two subpolygons. Finally, this method was applied recursively to the subpolygons until all the concave vertices were removed. This algorithm partitions the polygon into O(l) convex parts, its time complexity is max(O(n),O(l 2)) multiplications, where n is the number of vertices of the polygon and l is the number of the concave vertices.

Optimized nonhyperbolic moveout correction equation of long-offset seismic data for TI media with an arbitrary spatial orientation

The conventional long-offset nonhyperbolic moveout equation is derived for the transverse isotropic media with a vertical symmetric axis（VTI）.It cannot be extended to the transverse isotropic media with an arbitrary spatial orientation of symmetry axis（ATI）.In this paper,we optimize a modified long-offset nonhyperbolic moveout equation for ATI media based on the conventional nonhyperbolic moveout equation and the exact analytical solution of the quartic moveout coefficient（A_4） and NMO velocity for ATI media that were derived in our previous work.Compared with the exact traveltimes of the ray-tracing algorithm for anisotropic media,this optimized equation can be used to calculate the traveltime varying with survey line azimuth in arbitrary strong or weak ATI media.It can replace the time-consuming, multi-offset,and multi-azimuth ray tracing method for forward modeling of long-offset reflection traveltimes in ATI media,which is useful to further anisotropic parameter inversion using long-offset nonhyperbolic moveout.

Bounds of Interpolation Error for Arbitrary Narrow Quadrilateral Quasi-Wilson Element

Based on the construction of reference e le ment and bilinear transformation, a quasi-Wilson element for arbitrary narrow q uadrilateral is presented. Using the interpolation Theorem for narrow quadrilate ral isoparametric finite element and related methods, the bounds of interpolatio n error for arbitrary narrow quadrilateral quasi-Wilson element are obtained in case when the condition ρ K/h K≥σ 0】0 is not satisfied, where h K is the diameter of the element K and ρ K is the diameter of an ins cribed circle in K. The interpolation error is O(h2 K) in the L2( K)-norm and O(h K) in the H1(K) -norm provided that the in terpolated function belongs to H2(K).

Methodological Studies on Genomic DNA Extraction and Purification from Plant Drug Materials

This paper reports the rationale and methods of DNA extraction and purification from nine species of Compositae and four commercial drugs of corresponding plant Elephantopus scaber. The comparison of three methods: CsCl gradient, CTAB/CsCl gradient and CTAB miniprep extraction by yield, purity and factors affecting PCR was carried out. In conclusion, CTAB miniprep method provides a rapid, effective, economic approach for isolating genomic DNA for Chinese drug identification by genomic fingerprints.

理性精神引领下的高中数学概念课程教学案例--以“任意角”教学为例

数学理性精神是一种科学精神,它主张尊重数学的规律性,重视数学思想和方法的作用,强调数学与现实世界的联系,提倡从实际问题中抽象出数学模型,用数学语言表达和刻画现实世界。在数学课堂教学中融入理性精神,能够帮助学生更好地掌握数学知识,理性地看待问题。在“任意角”教学中,为了促进学生理性思维的发展,教师可以引入章节内容,帮助学生明晰研究路径;借助生活情境、动画演示、模型展示等形象化教学方式帮助学生温故知新,激发认知冲突;倡导学生自主探究,生成任意角的概念;精析例题,进行变式拓展;做好课堂小结,让学生在学习过程中提高数学推理能力,培养学生的理性精神,发展学生的核心素养。

Three-dimensional arbitrarily anisotropic modeling for time-domain airborne electromagnetic surveys

Electrically anisotropic strata are abundant in nature, so their study can help our data interpretation and our understanding of the processes of geodynamics. However, current data processing generally assumes isotropic conditions when surveying anisotropic structures, which may cause discrepancies between reality and electromagnetic data interpretation. Moreover, the anisotropic interpretation of the time-domain airborne electromagnetic （TDAEM） method is still confined to one dimensional （1D） cases, and the corresponding three-dimensional （3D） numerical simulations are still in development. In this study, we expanded the 3D TDAEM modeling of arbitrarily anisotropic media. First, through coordinate rotation of isotropic conductivity, we obtained the conductivity tensor of an arbitrary anisotropic rock. Next, we incorporated this into Maxwell＇s equations, using a regular hexahedral grid of vector finite elements to subdivide the solution area. A direct solver software package provided the solution for the sparse linear equations that resulted. Analytical solutions were used to verify the accuracy and feasibility of the algorithm. The proven model was then applied to analyze the effects of arbitrary anisotropy in 3D TDAEM via the distribution of responses and amplitude changes, which revealed that different anisotropy situations strongly affected the responses of TDAEM.

CONVERGENCERATESIN THESTRONG LAWSOFASYMPTOTICALLY NEGATIVELY ASSOCIATEDRANDOM FIELDS

In this paper, a notion of negative side ρ \|mixing ( ρ\+- \|mixing) which can be regarded as asymptotic negative association is defined, and some Rosenthal type inequalities for ρ\+- \|mixing random fields are established. The complete convergence and almost sure summability on the convergence rates with respect to the strong law of large numbers are also discussed for ρ\+-\| mixing random fields. The results obtained extend those for negatively associated sequences and ρ\+*\| mixing random fields.

MPC-based path tracking with PID speed control for high-speed autonomous vehicles considering time-optimal travel

In order to track the desired path as fast as possible,a novel autonomous vehicle path tracking based on model predictive control(MPC)and PID speed control was proposed for high-speed automated vehicles considering the constraints of vehicle physical limits,in which a forward-backward integration scheme was introduced to generate a time-optimal speed profile subject to the tire-road friction limit.Moreover,this scheme was further extended along one moving prediction window.In the MPC controller,the prediction model was an 8-degree-of-freedom(DOF)vehicle model,while the plant was a 14-DOF vehicle model.For lateral control,a sequence of optimal wheel steering angles was generated from the MPC controller;for longitudinal control,the total wheel torque was generated from the PID speed controller embedded in the MPC framework.The proposed controller was implemented in MATLAB considering arbitrary curves of continuously varying curvature as the reference trajectory.The simulation test results show that the tracking errors are small for vehicle lateral and longitudinal positions and the tracking performances for trajectory and speed are good using the proposed controller.Additionally,the case of extended implementation in one moving prediction window requires shorter travel time than the case implemented along the entire path.

Probabilistic Teleportation of an Arbitrary Two-Particle State and Its Quantum Circuits

Two simple schemes for probabilistic teleportation of an arbitrary unknown two-particle state using a non-maximally entangled EPR pair and a non-maximally entangled GHZ state as quantum channels are proposed. After receiving Alice＇s Bell state measurement results, Bob performs a collective unitary transformation on his inherent particles without introducing the auxiliary qubit. The original state can be probabilistically teleported. Meanwhile, quantum circuits for realization of successful teleportation are also presented.

Effects of die geometry on non-equal channel lateral extrusion(NECLE) of AZ80 magnesium alloy

Non-equal channel lateral extrusion（NECLE） is a new process that can be used to attain higher grain refinement in comparison with equal channel lateral extrusion（ECLE）. The die design for this process was numerically and experimentally studied. After finding a good correlation between the numerical and experimental results, more comprehensive FE analyses were carried out. Different die geometrical parameters were considered and their effects on the induced plastic strain, stress distribution, velocity field and forming load of the process were investigated. It was found that by this process with a suitable set of die geometrical parameters, higher induced effective strain and more homogeneous strain distribution could be achieved in comparison with ECLE operation.

Interaction of a circular cavity and a beeline crack in right-angle plane impacted by SH-wave

The problem of scattering of SH-wave by a circular cavity and an arbitrary beeline crack in right-angle plane was investigated using the methods of Green's function,complex variables and muti-polar coordinates.Firstly,we constructed a suitable Green's function,which is an essential solution to the displacement field for the elastic right-angle plane possessing a circular cavity while bearing out-of-plane harmonic line source load at arbitrary point.Secondly,based on the method of crack-division,integration for solution was established,then expressions of displacement and stress were obtained while crack and circular cavities were both in existence.Finally,the dynamic stress concentration factor around the circular cavity and the dynamic stress intensity factor at crack tip were discussed to the cases of different parameters in numerical examples.Calculation results show that the crack produces adverse engineering influence on both of the dynamic stress concentration factor and the dynamic stress intensity factor.

A NEW APPROACH OF SIMULTANEOUS FAULTS DIAGNOSIS BASED ON RANDOM SETS AND DSMT

Simultaneous faults often occur in running equipments, in order to solve the problems of the simultaneous faults, a new approach based on random sets and Dezert-Smarandache Theory (DSmT) is proposed in this paper. Firstly, the simultaneous faults' model is built based on the generalized frame of discernment in DSmT. Secondly, according to the unified description of combination rules in evidence reasoning based on random sets, a new combination rule for simultaneous faults diagnosis is proposed. Thirdly, according to the working characteristics and environment of the sensors used to acquire fault characteristic information, a new method to construct basic probability assignment function is pro- posed based on membership. Finally, diagnosis result is obtained by use of the new combination rule combined with decision rules. A case pertaining to the fault diagnosis for a multi-function rotor test-bed is given, and the result shows that the proposed diagnosis approach is feasible and efficient.

A NEW METHOD FOR CONSTRUCTING FUZZY EVIDENCE BASED ON THE NON-CONSONANT RANDOM SET

Natural-language information is often mathematically expressed by fuzzy sets. With the random set theory as a bridge, this kind of information can be transformed into fuzzy evidence in Dempster-Shafer (DS) theory. Then Dempster's combination rule or other combination rules of evi- dence can be used perfectly for fusing natural-language and other information. However, this traditional transformation involves the use of α -cutsets to construct the focal elements which have to be repre- sented as consonant sets. This construction is very inflexible and unreasonable in some practical ap- plications. In this paper, with the desire to overcome this limitation, a method for constructing more general non-consonant focal elements is proposed based on the random set theory. Some examples are given to show the generality and the efficiency of this new method. Finally, we validate that non-consonant constructions provide less degrees of total uncertainty than that of the consonant case in these examples by using the evaluation criterion of total uncertainty.

Probabilistic Teleportation of an Arbitrary Two-Atom State in Cavity QED

We propose a scheme for the teleportation of an arbitrary two-atom state by using two pairs of two-atom nonmaximally entangled states as the quantum channel in cavity QED. It is Shown that no matter whether the arbitrary two-atom pure state to be teleported is entangled or not, our teleportation scheme can always be probabilistically realized. The success prohability of teleportation is determined by the smaller coemcients of the two initially entangled atom pairs.

Auxiliary Equation Method and New Exact Solutions of BKP Equation

In this paper, auxiliary equation method is proposed for constructing more general exact solutions of nonlinear partial differential equation with the aid of symbolic computation. We study the （2＋1）-dimensional BKP equation and get a series of new types of traveling wave solutions. The method used here can be also extended to other nonlinear partial differential equations.