Clifford algebra and quantum logic gates

This paper discusses the properties of Quantum bit (Qubit) and Quantum logic gates (Quantum not-gate, Hadamard gate and Quantum controlled not-gate etc.) by the generating element of Pauli algebra (Clifford algebra Cl₃).     

Crystal planes and reciprocal space in Clifford geometric algebra

This paper discusses the geometry of kD crystal cells given by (k+ 1) points in a projective space ?^{n+ 1}. We show how the concepts of barycentric and fractional (crystallographic) coordinates, reciprocal vectors and dual representation are related (and geometrically interpreted) in the projective geometric algebra Cl(?^{n+ 1}) (see (Die Ausdehnungslehre von 1844 und die Geom. Anal. Teubner: Leipzig, 1894)) and in the conformal algebra Cl(?^{n+ 1, 1}). The crystallographic notions of d‐spacing, phase angle, structure factors, conditions for Bragg reflections, and the interfacial angles of crystal planes are obtained in the same context. Copyright © 2011 John Wiley & Sons, Ltd.     

Clifford Fourier Transformation and Uncertainty Principle for the Clifford Geometric Algebra Cl 3,0

First, the basic concept of the vector derivative in geometric algebra is introduced. Second, beginning with the Fourier transform on a scalar function we generalize to a real Fourier transform on Clifford multivector-valued functions Third, we show a set of important properties of the Clifford Fourier transform on Cl_3,0 such as differentiation properties, and the Plancherel theorem. Finally, we apply the Clifford Fourier transform properties for proving an uncertainty principle for Cl_3,0 multivector functions.     

Clifford 代数,几何计算和几何推理

Clifford代数是一种深深根植于几何学之中的代数系统,被它的创始人称为几何代数．历史上,E．Cartan,R．Brauer,H．Weyl,C．Chevalley等数学大师都曾研究和应用过Clifford代数,对它的发展起了重要作用．近年来,Clifford代数在微分几何、理论物理、经典分析等方面取得了辉煌的成就,是现代理论数学和物理的一个核心工具,并在现代科技的各个领域,如机器人学、信号处理、计算机视觉、计算生物学、量子计算等方面有广泛的应用．本文主要介绍Clifford代数在几何计算和几何推理中的应用．作为一种优秀的描述和计算几何问题的代数语言,Clifford代数对于几何体,几何关系和几何变换有不依赖于坐标的、易于计算的多种表示,因而应用它进行几何自动推理,不仅使困难定理的证明往往变得极为简单,而且能够解决一些著名的公开问题,目前在国际上,几何自动推理已经成为Clifford代数的一个重要应用领域。     

A method for constructing a recognition algorithm in algebra over an estimate calculation set

A method is proposed for constructing an algorithm in algebra over an estimate calculation set in an algebraic extension of the least degree.     

Gauss-Mean Value Formula for Triharmonic Functions and its Applications in Clifford Analysis

In this paper, the integral representation for some polyharmonic functions with values in a universal Clifford algebra Cl(V_n,n) is studied and Gauss-mean value formula for triharmonic functions with values in a Clifford algebra Cl(V_n,n) are proved by using Stokes formula and higher order Cauchy-Pompeiu formula. As application some results about growth condition at infinity are obtained.     

A Dijkstra-like shortest path algorithm for certain cases of negative arc lengths

If each negative length arc of a digraphG is acyclic, i.e., does not belong to any cycle, then we show that the shortest paths from a given node to all other nodes can be computed inO(V ²) time, whereV is the number of nodes inG.     

Mathematics of Clifford - a Maple package for Clifford and Graßmann algebras

CLIFFORD performs various computations in Gra?mann and Clifford algebras. It can compute with quaternions, octonions, and matrices with entries in C ℓ (B) - the Clifford algebra of a vector space V endowed with an arbitrary bilinear form B. Two user-selectable algorithms for the Clifford product are implemented: cmulNUM - based on Chevalley’s recursive formula, and cmulRS - based on a non-recursive Rota-Stein sausage. Gra?mann and Clifford bases can be used. Properties of reversion in undotted and dotted wedge bases are discussed.     

Clifford algebra and the Lorentz transformation with (p, q) form

In this paper, the concepts of Lorentz inner product with (p, q) form, the Lorentz space and the Lorentz transformation with (p, q) form are given by using Clifford algebra. It is shown that L_m^p,q is the Lorentz transformation with (p, q) form, and the matrix equality relation of Minkowski space with (n − 1, 1) form is given. The examples are given to illustrate the corresponding results.     

Clifford鞅的b+p-权不等式及其应用

如果Cl,S是一个Clifford代数,v是一个非负测度,ψ是一个Cl,S-值可测函数,那么dμ=ψdv是一个Cl,S-值测度.在这篇论文中将证明如果ψ满足某些b+p-权条件那么关于测度dμ的Clifford-值鞅的一些不等式成立,且应用这些不等式在权ψ∈b+∞∩a1时获得了一些关于Clifford-值鞅的权对偶空间.     

Martins' algorithm revisited for multi-objective shortest path problems with a MaxMin cost function

This paper presents a direct extension of the label setting algorithm proposed by Martins in 1984 for the shortest path problem with multiple objectives. This extended version computes all the efficient paths from a given source vertex, to all the other vertices of the network. The algorithm copes with problems in which the "cost values" associated with the network arcs are positive. The proposed extension can handle objective functions that are either of the "sum" type or of the "bottleneck" type. The main modifications to Martins' algorithm for multi-objective shortest path problems are linked to the dominance test and the procedure for identifying efficient paths. The algorithmic features are described and a didactic example is provided to illustrate the working principle. The results of numerical experiments concerning the number of efficient solutions produced and the CPU time consumed for several configurations of objectives, on a set of randomly generated networks, are also provided. Received: February 2005 / Revised version: June 2005 AMS classification: 90C29, 90C27, 05C38, 90B18, 68M12     

Multi-vector Spherical Monogenics, Spherical Means and Distributions in Clifford Analysis

New higher-dimensional distributions have been introduced in the framework of Clifford analysis in previous papers by Brackx, Delanghe and Sommen. Those distributions were defined using spherical co-ordinates, the ＂finite part＂ distribution Fp x＋^μ on the real line and the generalized spherical means involving vector-valued spherical monogenics. In this paper, we make a second generalization, leading to new families of distributions, based on the generalized spherical means involving a multivector-valued spherical monogenic. At the same time, as a result of our attempt at keeping the paper self-contained, it offers an overview of the results found so far.     

Speeding up the Floyd-Warshall algorithm for the cycled shortest path problem

On a network with a cycle, where at least one cycle exists, the Floyd-Warshall algorithm is one of the algorithms most used for determining the least cost path between every pair of nodes. In this work a new algorithm for this problem is developed that requires less computational effort than the Floyd-Warshall algorithm. Furthermore, we show that the basis of our algorithm is much easier to understand, which might be an advantage for educational purposes. A small example validates our algorithm and shows its implementation.     

结点有约束的交通网络最短路径模型

结点有约束的网络是一类特殊的网络，如具有禁止通行限制信息的交通路网等，由于最短路径的求解是有后效性的，经典的Dijkstra算法等不能直接用来求解该问题，本文提出了一种结点有约束的交通网络最短路径建模方法，该方法所建模型为一般网络模型，可用任一传统高效的算法求其最短路径，从根本上降低了问题的复杂性，为很好地解决交通、通信等领域中的此类问题提供了有益的方法。     

Implementation of a method for constructing a recognition algorithm in algebra over an estimate calculation set

A method is implemented for constructing an algorithm in algebra over an estimate calculation set in an algebraic extension of the least degree.     

一类网络最优路径的双向模拟扩散算法

基于模拟扩散算法的基本原理,文中提出了一种双向寻求网络最优路径的扩散算法,并介绍了该算法原理和具体计算过程,验证了该算法的正确性和合理性。该算法具有并行计算的能力,适合于分布式计算机,寻求大型复杂网络的最优路径。     

Clifford分析中双正则函数的Taylor展式及其性质

首先借助实Clifford分析中双正则函数的累次积分的换序公式,给出了双正则函数的Cauchy积分公式,然后由特征边界的Cauchy积分公式,得到了双正则函数的Taylor展式,并由此给出了双正则函数的唯一性定理,柯西不等式和Weierstrass定理.     

Computing inter-site distances for routing and scheduling problems

Routing and scheduling problems have received considerable attention in the literature in terms of model building and algorithm development. On these fronts, progress has been substantial. However, one often neglected (yet critical) aspect concerning the use of these models and algorithms is their data requirements. In particular, the distance matrix yielding the shortest distance between each pair of sites (nodes) represents a major portion of the data required by all such problems. Yet, such data are seldom available with the degree of accuracy desired and often are not available at all.This paper describes an efficient method for obtaining this distance matrix that is based on the underlying road structure for the geographic region in question. Thus, the distances obtained reflect ‘actual’ distances. Finally, the paper presents some brief computational experience and discusses an implementation concerning the routing of environmental inspectors in the state of Pennsylvania.     

Exact algorithms for multi-criteria multi-modal shortest path with transfer delaying and arriving time-window in urban transit network

This paper investigates the solution algorithms for the multi-criteria multi-modal shortest path problem (M-SPP), which belongs to the set of problems known as NP-hard, in urban transit network (UTN). The related M-SPP is one of the important and practical problems in several fields such as urban transportation system and freight transportation. The UTN is composed of multiple modes (e.g., automobile, bus, subway, light rail, pedestrian and so on). To get their destination, the passengers can alternate between different modes. As a special demand, the time-window is usually associated with the M-SPP. Because of the service time-limit of modes, the available modes at a stop are varied with the time. So the optimal M-SPP with arriving time-window cannot be simply obtained by finding the optimal M-SPP firstly and then reversely deducing the leaving time-window of the origin according to the arriving time-window of destination. In this paper, the M-SPP with arriving time-window is firstly proposed. To solve the multi-criteria M-SPPs (MM-SPP) with transfer delaying, an improved exact label correcting algorithm (LCA) is designed and, to solve the proposed MM-SPPs with both of transfer delaying and arriving time-window, an exact reverse LCA is designed. Finally, some computing examples are given to test the effectiveness of the designed algorithms.