Connecting the 3D DGS Calques3D with the CAS Maple

Many (2D) Dynamic Geometry Systems (DGSs) are able to export numeric coordinates and equations with numeric coefficients to Computer Algebra Systems (CASs). Moreover, different approaches and systems that link (2D) DGSs with CASs, so that symbolic coordinates and equations with symbolic coefficients can be exported from the DGS to the CAS, already exist. Although the 3D DGS Calques3D can export numeric coordinates and equations with numeric coefficients to Maple and Mathematica, it cannot export symbolic coordinates and equations with symbolic coefficients. A connection between the 3D DGS Calques3D and the CAS Maple, that can handle symbolic coordinates and equations with symbolic coefficients, is presented here. Its main interest is to provide a convenient time-saving way to explore problems and directly obtain both algebraic and numeric data when dealing with a 3D extension of “ruler and compass geometry”. This link has not only educational purposes but mathematical ones, like mechanical theorem proving in geometry, geometric discovery (hypotheses completion), geometric loci finding… As far as we know, there is no comparable “symbolic” link in the 3D case, except the prototype 3D-LD (restricted to determining algebraic surfaces as geometric loci).     

Rigid body concept for geometric nonlinear analysis of 3D frames, plates and shells based on the updated Lagrangian formulation

In the nonlinear analysis of elastic structures, the displacement increments generated at each incremental step can be decomposed into two components as the rigid displacements and natural deformations. Based on the updated Lagrangian (UL) formulation, the geometric stiffness matrix [k_g] is derived for a 3D rigid beam element from the virtual work equation using a rigid displacement field. Further, by treating the three-node triangular plate element (TPE) as the composition of three rigid beams lying along the three sides, the [k_g] matrix for the TPE can be assembled from those of the rigid beams. The idea for the UL-type incremental-iterative nonlinear analysis is that if the rigid rotation effects are fully taken into account at each stage of analysis, then the remaining effects of natural deformations can be treated using the small-deformation linearized theory. The present approach is featured by the fact that the formulation is simple, the expressions are explicit, and all kinds of actions are considered in the stiffness matrices. The robustness of the procedure is demonstrated in the solution of several benchmark problems involving the postbuckling response.     

Interactive 3D video editing

We present a generic and versatile framework for interactive editing of 3D video footage. Our framework combines the advantages of conventional 2D video editing with the power of more advanced, depth-enhanced 3D video streams. Our editor takes 3D video as input and writes both 2D or 3D video formats as output. Its underlying core data structure is a novel 4D spatio-temporal representation which we call the video hypervolume. Conceptually, the processing loop comprises three fundamental operators: slicing, selection, and editing. The slicing operator allows users to visualize arbitrary hyperslices from the 4D data set. The selection operator labels subsets of the footage for spatio-temporal editing. This operator includes a 4D graph-cut based algorithm for object selection. The actual editing operators include cut & paste, affine transformations, and compositing with other media, such as images and 2D video. For high-quality rendering, we employ EWA splatting with view-dependent texturing and boundary matting. We demonstrate the applicability of our methods to post-production of 3D video.     

医学图像三维可视化中任意平面切片的研究与实现

对医学图像的三维重建技术进行研究,利用介于面绘制和体绘制之间的切片法来实现三维体数据可视化,并针对断层图像序列所生成三维体数据的切片方法进行了深入讨论。提出一种可操作性很强的切片方法,能较好地实现三维体数据中冠状面、矢状面、特别是任意平面的切片。同时,该方法已经用VC6．0结合VTK（Visualization Toolkit）编程进行实现,效果符合医学诊断要求,并得到了有关应用。     

医学图象体数据场的剖切显示方法

为了给医生提供一些全面、直观和准确的诊断信息 ,讨论了一种对医学图象体数据场进行二维和三维剖切显示的方法 ,其中对于二维剖切显示 ,可首先由用户通过人机交互的方式来给出对体数据场进行任意剖切显示的路径 ;然后运用反求工程的求解思路沿切割路径进行量化和采样来得到沿切割路径上每个采样点在对应图象坐标空间中的位置 ,并由此获得剖切平面上所有采样点的空间位置坐标 ;最后通过对体数据场进行重采样和插值操作来实现对空间体数据场的二维任意剖切显示 .对于三维剖切显示 ,可首先采用与二维剖切操作类似的图象采样和插值操作来获得纹理数据阵列 ,然后结合纹理映射技术 ,将纹理数据映射到三维坐标空间中 ,以实现数据场的三维空间剖切显示     

Estimation of an unknown cartographic projection and its parameters from the map

This article presents a new off-line method for the detection, analysis and estimation of an unknown cartographic projection and its parameters from a map. Several invariants are used to construct the objective function ? that describes the relationship between the 0D, 1D, and 2D entities on the analyzed and reference maps. It is minimized using the Nelder-Mead downhill simplex algorithm. A simplified and computationally cheaper version of the objective function ? involving only 0D elements is also presented. The following parameters are estimated: a map projection type, a map projection aspect given by the meta pole K coordinates [φ _k , λ _k ], a true parallel latitude φ ₀, central meridian longitude λ ₀, a map scale, and a map rotation. Before the analysis, incorrectly drawn elements on the map can be detected and removed using the IRLS. Also introduced is a new method for computing the L ₂ distance between the turning functions Θ₁, Θ₂ of the corresponding faces using dynamic programming. Our approach may be used to improve early map georeferencing; it can also be utilized in studies of national cartographic heritage or land use applications. The results are presented both for the real cartographic data, representing early maps from the David Rumsay Map Collection, and for the synthetic tests.     

Maple procedures for the coupling of angular momenta. IX. Wigner D-functions and rotation matrices

The Wigner D-functions, , are known for their frequent use in quantum mechanics. Defined as the matrix elements of the rotation operator in R³ and parametrized in terms of the three Euler angles α, β, and γ, these functions arise not only in the transformation of tensor components under the rotation of the coordinates, but also as the eigenfunctions of the spherical top. In practice, however, the use of the Wigner D-functions is not always that simple, in particular, if expressions in terms of these and other functions from the theory of angular momentum need to be simplified before some computations can be carried out in detail.To facilitate the manipulation of such Racah expressions, here we present an extension to the Racah program [S. Fritzsche, Comput. Phys. Comm. 103 (1997) 51] in which the properties and the algebraic rules of the Wigner D-functions and reduced rotation matrices are implemented. Care has been taken to combine the standard knowledge about the rotation matrices with the previously implemented rules for the Clebsch-Gordan coefficients, Wigner n−j symbols, and the spherical harmonics. Moreover, the application of the program has been illustrated below by means of three examples.

Program summary

Title of program:RacahCatalogue identifier:ADFv_9_0Program summary URL:http://cpc.cs.qub.ac.uk/summaries/ADFv_9_0Program obtainable from: CPC Program Library, Queen's University of Belfast, N. IrelandCatalogue identifier of previous version: ADFW, ADHW, title RacahJournal reference of previous version(s): S. Fritzsche, Comput. Phys. Comm. 103 (1997) 51; S. Fritzsche, S. Varga, D. Geschke, B. Fricke, Comput. Phys. Comm. 111 (1998) 167; S. Fritzsche, T. Inghoff, M. Tomaselli, Comput. Phys. Comm. 153 (2003) 424.Does the new version supersede the previous one: Yes, in addition to the spherical harmonics and recoupling coefficients, the program now supports also the occurrence of the Wigner rotation matrices in the algebraic expressions to be evaluated.Licensing provisions:NoneComputer for which the program is designed and others on which it is operable: All computers with a license for the computer algebra package Maple [Maple is a registered trademark of Waterloo Maple Inc.]Installations:University of Kassel (Germany)Operating systems under which the program has been tested: Linux 8.2+Program language used:Maple, Release 8 and 9Memory required to execute with typical data:10-50 MBNo. of lines in distributed program, including test data, etc.:52 653No. of bytes in distributed program, including test data, etc.:1 195 346Distribution format:tar.gzipNature of the physical problem: The Wigner D-functions and (reduced) rotation matrices occur very frequently in physical applications. They are known not only as the (infinite) representation of the rotation group but also to obey a number of integral and summation rules, including those for their orthogonality and completeness. Instead of the direct computation of these matrices, therefore, one first often wishes to find algebraic simplifications before the computations can be carried out in practice.Reasons for new version: The Racah program has been found an efficient tool during recent years, in order to evaluate and simplify expressions from Racah's algebra. Apart from the Wigner n−j symbols (j=3,6,9) and spherical harmonics, we now extended the code to allow for Wigner rotation matrices. This extension will support the study of those quantum processes especially where different axis of quantization occurs in course of the theoretical deviations.Summary of revisions: In a revised version of the Racah program [S. Fritzsche, Comput. Phys. Comm. 103 (1997) 51; S. Fritzsche, T. Inghoff, M. Tomaselli, Comput. Phys. Comm. 153 (2003) 424], we now also support the occurrence of the Wigner D-functions and reduced rotation matrices. By following our previous design, the (algebraic) properties of these rotation matrices as well as a number of summation and integration rules are implemented to facilitate the algebraic simplification of expressions from the theories of angular momentum and the spherical tensor operators.Restrictions onto the complexity of the problem: The definition as well as the properties of the rotation matrices, as used in our implementation, are based mainly on the book of Varshalovich et al. [D.A. Varshalovich, A.N. Moskalev, V.K. Khersonskii, Quantum Theory of Angular Momentum, World Scientific, Singapore, 1988], Chapter 4. From this monograph, most of the relations involving the Wigner D-functions and rotation matrices are taken into account although, in practice, only a rather selected set was needed to be implemented explicitly owing to the symmetries of these functions. In the integration over the rotation matrices, products of up to three Wigner D-functions or reduced matrices (with the same angular arguments) are recognized and simplified properly; for the integration over a solid angle, however, the domain of integration must be specified for the Euler angles α and γ. This restriction arose because Maple does not generate a constant of integration when the limits in the integral are omitted. For any integration over the angle β the range of the integration, if omitted, is always taken from 0 to π.Unusual features of the program: The Racah program is designed for interactive use that allows a quick and algebraic evaluation of (complex) expression from Racah's algebra. It is based on a number of well-defined data structures that are now extended to incorporate the Wigner rotation matrices. For these matrices, the transformation properties, sum rules, recursion relations, as well as a variety of special function expansions have been added to the previous functionality of the Racah program. Moreover, the knowledge about the orthogonality as well as the completeness of the Wigner D-functions is also implemented.Typical running time:All the examples presented in Section 4 take only a few seconds on a 1.5 GHz Pentium Pro computer.     

新颖的网格模型压缩算法——网格切片

针对三维(3D)网格模型的存储与网络传输问题,提出一种新颖的三维模型压缩算法。该算法基于对网格模型的切片处理,主要由以下三个步骤组成:切片顶点的计算、切片边界的均匀采样以及对切片所得图像的编码。对于一个给定的三维模型,首先,计算模型的包围盒;然后,沿包围盒长度最长的方向进行切片;同时计算切片与网格模型表面每条边的交点,构成一个多边形,这个多边形即为切片的边界;其次,对切片边界进行均匀的重采样,使每层切片具有相同的顶点数;最后,把每层的顶点坐标转化为极坐标形式,这样,所有层顶点的ρ-坐标以及θ-坐标能分别构成一张图像,原始的三维模型即能由这两张图像表示。这种表示方法具有以下两个明显的优势:第一,降低了数据的维度,有效减少了数据量;第二,具有极大的数据相关性,进一步减少了数据的熵。基于这两个优势,该算法对图像数据进行差值编码以及算术编码,最后得到压缩后的文件。与增量参数细化(IPR)方法相比,在解码模型同等质量的前提下,所提算法的编码效率提高了23%。实验结果表明,所提算法在模型存储和传输应用中能取得很好的压缩效率,有效减少了数据量。     

MDS codes in Doob graphs

The Doob graph D(m, n), where m > 0, is a Cartesian product of m copies of the Shrikhande graph and n copies of the complete graph K ₄ on four vertices. The Doob graph D(m, n) is a distance-regular graph with the same parameters as the Hamming graph H(2m + n, 4). We give a characterization of MDS codes in Doob graphs D(m, n) with code distance at least 3. Up to equivalence, there are m ³/36+7m ²/24+11m/12+1?(m mod 2)/8?(m mod 3)/9 MDS codes with code distance 2m + n in D(m, n), two codes with distance 3 in each of D(2, 0) and D(2, 1) and with distance 4 in D(2, 1), and one code with distance 3 in each of D(1, 2) and D(1, 3) and with distance 4 in each of D(1, 3) and D(2, 2).     

Structure-preserving numerical algorithm for solving discrete-time LMI and DARS

We present a numerical algorithm to solve a discrete-time linear matrix inequality (LMI) and discrete-time algebraic Riccati system (DARS). With a given system (A,B,C,D) we associate a para-hermitian matrix pencil. Then we transform it by an orthogonal transformation matrix into a block-triangular para-hermitian form. Under either of the two assumptions (1) matrix pair (A,B) is controllable or (2) matrix pair (A,B) is reachable and (A,B,C,D) is a left invertible system, we extract the solution of LMI and DARS by the entries of the orthogonal transformation matrix.     

Pattern decomposition for tessellation automata

In this paper we obtain the following two results about one-dimensional tessellation automata. (1) There is a class of one-dimensional monogenic cellular automata with an arbitrary neighborhood size and arbitrary number of states where any pattern is always decomposed into atomic pieces leaving no residue. (2) For any positive integer q there exists a one-dimensional strongly connected tessellation automation with q states even if we put the restriction that available parallel transformation must be bijective mappings on the set of finite configurations.     

Computer package for microscopic spin Hamiltonian analysis of the 3d4 and 3d6 (spin S = 2) ions at orthorhombic and tetragonal symmetry sites

A FORTRAN package based on the microscopic spin Hamiltonian expressions derived by computer algebra (ALTRAN) for the high spin (S = 2) 3d⁴ and 3d⁶ ions with an orbital singlet ground state at orthorhombic (point groups: C_2v, D₂, D_2h) and tetragonal (point groups: C_4v, D₄, D_4h, D₂d) symmetry sites is presented. The spin—orbit (λ) and the spin—spin (ρ) coupling contributions up to the fourth-order perturbation theory are taken into account within the ⁵D approximation. The package enables efficient numerical calculations of the Zeeman electronic (Ze) parameters and the zero-field splitting (ZFS) parameters for the S = 2 3d⁴ and 3d⁶ ions. The following terms are included: λ and λ² for the Ze parameters, λ², λ³ λ⁴, ρ, ρ² and λ²ρ for the second-rank ZFS ones, and λ⁴, ρ² and λ²ρ for the fourth-rank ZFS ones. The program is applicable to all possible energy level schemes with a ground orbital singlet arising from the ⁵D multiplet due to orthorhombic or tetragonal symmetry crystal fields. The input parameters are λ, ρ, the energy levels Δ_j (j = 1, 2, 3, 4) and the mixing coefficient s, which can be obtained from other spectroscopic data. The ZFS parameters output in the extended Stevens notation B_k^q and b_k^q, as well as the conventional notation (D, E; a, F, K), is provided.     

A simple approach to the problem of 3-D reconstruction

A relatively simple mathematical procedure for the reconstruction of the 3-dimensional (3D) image of the left ventricle (LV) of the heart is presented. The method is based on the assumption that every ray whoch emanates from the midpoint of the long axis of the 3D body crosses the surface boundary of the ventricle at one and only one point. The coordinates r_i, φ_i, θ_i of the data points on, say, the outer boundary, (i.e., the epicardium) are calculated in a spherical coordinate system having its origin in the midpoint of the long axis. The problem of defining the coordinates of a prescribed grid point on the boundary is treated as an interpolation problem for the function r = r(φ, θ), defined in the rectangle 0 ≤ φ ≤ 2π; 0 ≤ θ ≤ π with r_i given in the points (φ_i, θ_i).     

On a special transformation to a non-inertial, radially rigid reference frame

We discuss the conditions under which a body, moving non-inertially in Minkowski space, can preserve its size. Under these conditions, using a series expansion of the generalized Lorentz transformation, we find a coordinate transformation connecting the laboratory inertial reference frame S and the rigid non-inertial reference frame s which moves without its own rotation with respect to S. Direct consequences of this transformation are: (a) desynchronization, in system s, of the coordinate clocks of s which were previously synchronized in S, and (b) a kinematic contraction of a ruler of system s observed in S. We also consider the dependence of the transformation vector parameter on the proper coordinates of s.     

基于VTK的三维医学图像虚拟切片提取

目前大多数医学影像设备,如CT、MRI、超声等所产生的图像序列往往必须沿着某一固定方向,但这通常不能满足临床上多方位、任意角度诊查的需要。针对该问题,提出了一种三维医学图像的任意角度虚拟切片提取方法。在Visual C++平台下,结合可视化工具包VTK,对DICOM格式的CT图像序列进行三维重建,通过设置虚拟切面的法向量和内点来对重建后的三维物体进行切割并获得虚拟切片信息,在切割的同时可以同步显示出虚拟切片图像。通过简单的鼠标操作可以生成任意角度、任意部位的虚拟切片图像,并能对切割平面及虚拟切片图像进行移     

BOUT++: A framework for parallel plasma fluid simulations

A new modular code called BOUT++ is presented, which simulates 3D fluid equations in curvilinear coordinates. Although aimed at simulating Edge Localised Modes (ELMs) in tokamak x-point geometry, the code is able to simulate a wide range of fluid models (magnetised and unmagnetised) involving an arbitrary number of scalar and vector fields, in a wide range of geometries. Time evolution is fully implicit, and 3rd-order WENO schemes are implemented. Benchmarks are presented for linear and non-linear problems (the Orszag-Tang vortex) showing good agreement. Performance of the code is tested by scaling with problem size and processor number, showing efficient scaling to thousands of processors.Linear initial-value simulations of ELMs using reduced ideal MHD are presented, and the results compared to the ELITE linear MHD eigenvalue code. The resulting mode-structures and growth-rate are found to be in good agreement (γBOUT++=0.245ωA, γELITE=0.239ωA, with Alfvénic timescale 1/ωA=R/VA). To our knowledge, this is the first time dissipationless, initial-value simulations of ELMs have been successfully demonstrated.     

On Succinct Greedy Drawings of Plane Triangulations and 3-Connected Plane Graphs

Geometric routing by using virtual locations is an elegant way for solving network routing problems. In its simplest form, greedy routing, a message is simply forwarded to a neighbor that is closer to the destination. It has been an open conjecture whether every 3-connected plane graph has a greedy drawing in the Euclidean plane R ² (by Papadimitriou and Ratajczak in Theor. Comp. Sci. 344(1):3–14, 2005). Leighton and Moitra (Discrete Comput. Geom. 44(3):686–705, 2010) recently settled this conjecture positively. One main drawback of this approach is that the coordinates of the virtual locations require Ω(nlogn) bits to represent (the same space usage as traditional routing table approaches). This makes greedy routing infeasible in applications. In this paper, we show that the classical Schnyder drawing in R ² of plane triangulations is greedy with respect to a simple natural metric function H(u,v) over R ² that is equivalent to Euclidean metric D _E (u,v) (in the sense that $D_{E}(u,v) \leq H(u,v) \leq2\sqrt{2}D_{E}(u,v)$ ). The drawing uses two integer coordinates between 0 and 2n?5, which can be represented by logn bits. We also show that the classical Schnyder drawing in R ² of 3-connected plane graphs is weakly greedy with respect to the same metric function H(?,?). The drawing uses two integer coordinates between 0 and f (where f is the number of internal faces of G).     

Multiplicity of positive solutions of a class of nonlinear fractional differential equations

This paper is concerned with the nonlinear fractional differential equation L(D)u=f(x,u), u(0)=0, 0<x<1,where L(D) = Dsn − an−1Dsn−1 − … − a1Ds11 < s2 < … < sn < 1, and aj > 0, j = 1,2,…, n − 1. Some results are obtained for the existence, nonexistence, and multiplicity of positive solutions of the above equation by using Krasnoselskii's fixed-point theorem in a cone. In particular, it is proved that the above equation has N positive solutions under suitable conditions, where N is an arbitrary positive integer.