Cosmological Simulations Using Special Purpose Computers: Implementing P3M on Grape

An adaptation of the Particle-Particle/Particle-Mesh (P3M) code to the special purpose hardware GRAPE is presented. The short range force is calculated by a four chip GRAPE-3A board, while the rest of the calculation is performed on a Sun Sparc 10/51 workstation. The limited precision of the GRAPE hardware and algorithm constraints introduce stochastic errors of the order of a few percent in the gravitational forces. Tests of this new P3MG3A code show that it is a robust tool for cosmological simulations. The code currently achieves a peak efficiency of one third the speed of the vectorized P3M code on a Cray C-90 and significant improvements are planned in the near future. Special purpose computers like GRAPE are therefore an attractive alternative to supercomputers for numerical cosmology.Comment: 9 pages (ApJS style); uuencoded compressed PostScript file (371 kb) Also available by anonymous 'ftp' to astro.Princeton.EDU [128.112.24.45] in: summers/grape/p3mg3a.ps (668 kb) and WWW at: http://astro.Princeton.EDU/~library/prep.html (as POPe-600) Send all comments, questions, requests, etc. to: [email protected]

Complex numbers from 1600 to 1840

This thesis uses primary and secondary sources to study advances in complex number theory during the 17th and 18th Centuries. Some space is also given to the early 19th Century. Six questions concerning their rules of operation, usage, symbolism, nature, representation and attitudes to them are posed in the Introduction. The main part of the thesis quotes from the works of Descartes, Newton, Wallis, Saunderson, Maclaurin, d'Alembert, Euler, Waring, Frend, Hutton, Arbogast, de Missery, Argand, Cauchy, Hamilton, de Morgan, Sylvester and others, mainly in chronological order, with comment and discussion. More attention has been given tp algebraists, the originators of most advances in complex numbers, than to writers in trigonometry, calculus and analysis, who tended to be users of them. The last chapter summarises the most important points and considers the extent to which the six questions have been resolved. The most important developments during the period are identified as follows: (i) the advance in status of complex numbers from 'useless' to 'useful'. (ii) their interpretation by Wallis, Argand and Gauss in arithmetic, geometric and algebraic ways. (iii) the discovery that they are essential for understanding polynomials and logarithmic, exponential and trigonometric functions. (iv) the extension of trigonometry, calculus and analysis into the complex number field. (v) the discovery that complex numbers are closed under exponentiation, and so under all algebraic operations. (vi) partial reform of nomenclature and symbolism. (vii) the eventual extension of complex number theory to n dimensions

Newton’s Law of Gravitational Force (NLGF) based Machine Learning Technique for Uneven Illuminated Face Detection

A photo gallery is crucial for organizing your photos, presenting them in beautiful categories, and doing sophisticated memory searches. The photo gallery is portrayed in a vocabulary of nonlinear similarities to the prototype face image collection. One of the difficult research ideas for machine learning technologies is the maintenance of a photo gallery using facial recognition. Based on changes in the faces' appearance, faces are identified. This research proposes novel machine learning algorithms to recognize faces by characterizing the majority of discriminating local characteristics, which maximizes the dissimilarity between face photos of different persons and reduces the dissimilarity between features between face images of the same person. This method relies on Newton's third law of gravitational force to determine the relationship between pixels to extract the features of noisy accurately and efficiently, unevenly illuminated, and rotationally invariant face images

Newtonian Potential in Quantum Regge Gravity

We show how the Newtonian potential between two heavy masses can be computed in simplicial quantum gravity. On the lattice we compute correlations between Wilson lines associated with the heavy particles and which are closed by the lattice periodicity. We check that the continuum analog of this quantity reproduces the Newtonian potential in the weak field expansion. In the smooth anti-de Sitter-like phase, which is the only phase where a sensible lattice continuum limit can be constructed in this model, we attempt to determine the shape and mass dependence of the attractive potential close to the critical point in $G$. It is found that non-linear graviton interactions give rise to a potential which is Yukawa-like, with a mass parameter that decreases towards the critical point where the average curvature vanishes. In the vicinity of the critical point we give an estimate for the effective Newton constant.Comment: (47 pages), CERN-TH.7314/9

A value estimation approach to Iri-Imai's method for constrained convex optimization.

Lam Sze Wan.Thesis (M.Phil.)--Chinese University of Hong Kong, 2002.Includes bibliographical references (leaves 93-95).Abstracts in English and Chinese.Chapter 1 --- Introduction --- p.1Chapter 2 --- Background --- p.4Chapter 3 --- Review of Iri-Imai Algorithm for Convex Programming Prob- lems --- p.10Chapter 3.1 --- Iri-Imai Algorithm for Convex Programming --- p.11Chapter 3.2 --- Numerical Results --- p.14Chapter 3.2.1 --- Linear Programming Problems --- p.15Chapter 3.2.2 --- Convex Quadratic Programming Problems with Linear Inequality Constraints --- p.17Chapter 3.2.3 --- Convex Quadratic Programming Problems with Con- vex Quadratic Inequality Constraints --- p.18Chapter 3.2.4 --- Summary of Numerical Results --- p.21Chapter 3.3 --- Chapter Summary --- p.22Chapter 4 --- Value Estimation Approach to Iri-Imai Method for Con- strained Optimization --- p.23Chapter 4.1 --- Value Estimation Function Method --- p.24Chapter 4.1.1 --- Formulation and Properties --- p.24Chapter 4.1.2 --- Value Estimation Approach to Iri-Imai Method --- p.33Chapter 4.2 --- "A New Smooth Multiplicative Barrier Function Φθ+,u" --- p.35Chapter 4.2.1 --- Formulation and Properties --- p.35Chapter 4.2.2 --- "Value Estimation Approach to Iri-Imai Method by Us- ing Φθ+,u" --- p.41Chapter 4.3 --- Convergence Analysis --- p.43Chapter 4.4 --- Numerical Results --- p.46Chapter 4.4.1 --- Numerical Results Based on Algorithm 4.1 --- p.46Chapter 4.4.2 --- Numerical Results Based on Algorithm 4.2 --- p.50Chapter 4.4.3 --- Summary of Numerical Results --- p.59Chapter 4.5 --- Chapter Summary --- p.60Chapter 5 --- Extension of Value Estimation Approach to Iri-Imai Method for More General Constrained Optimization --- p.61Chapter 5.1 --- Extension of Iri-Imai Algorithm 3.1 for More General Con- strained Optimization --- p.62Chapter 5.1.1 --- Formulation and Properties --- p.62Chapter 5.1.2 --- Extension of Iri-Imai Algorithm 3.1 --- p.63Chapter 5.2 --- Extension of Value Estimation Approach to Iri-Imai Algo- rithm 4.1 for More General Constrained Optimization --- p.64Chapter 5.2.1 --- Formulation and Properties --- p.64Chapter 5.2.2 --- Value Estimation Approach to Iri-Imai Method --- p.67Chapter 5.3 --- Extension of Value Estimation Approach to Iri-Imai Algo- rithm 4.2 for More General Constrained Optimization --- p.69Chapter 5.3.1 --- Formulation and Properties --- p.69Chapter 5.3.2 --- Value Estimation Approach to Iri-Imai Method --- p.71Chapter 5.4 --- Numerical Results --- p.72Chapter 5.4.1 --- Numerical Results Based on Algorithm 5.1 --- p.73Chapter 5.4.2 --- Numerical Results Based on Algorithm 5.2 --- p.76Chapter 5.4.3 --- Numerical Results Based on Algorithm 5.3 --- p.78Chapter 5.4.4 --- Summary of Numerical Results --- p.86Chapter 5.5 --- Chapter Summary --- p.87Chapter 6 --- Conclusion --- p.88Bibliography --- p.93Chapter A --- Search Directions --- p.96Chapter A.1 --- Newton's Method --- p.97Chapter A.1.1 --- Golden Section Method --- p.99Chapter A.2 --- Gradients and Hessian Matrices --- p.100Chapter A.2.1 --- Gradient of Φθ(x) --- p.100Chapter A.2.2 --- Hessian Matrix of Φθ(x) --- p.101Chapter A.2.3 --- Gradient of Φθ(x) --- p.101Chapter A.2.4 --- Hessian Matrix of φθ (x) --- p.102Chapter A.2.5 --- Gradient and Hessian Matrix of Φθ(x) in Terms of ∇xφθ (x) and∇2xxφθ (x) --- p.102Chapter A.2.6 --- "Gradient of φθ+,u(x)" --- p.102Chapter A.2.7 --- "Hessian Matrix of φθ+,u(x)" --- p.103Chapter A.2.8 --- "Gradient and Hessian Matrix of Φθ+,u(x) in Terms of ∇xφθ+,u(x)and ∇2xxφθ+,u(x)" --- p.103Chapter A.3 --- Newton's Directions --- p.103Chapter A.3.1 --- Newton Direction of Φθ (x) in Terms of ∇xφθ (x) and ∇2xxφθ(x) --- p.104Chapter A.3.2 --- "Newton Direction of Φθ+,u(x) in Terms of ∇xφθ+,u(x) and ∇2xxφθ,u(x)" --- p.104Chapter A.4 --- Feasible Descent Directions for the Minimization Problems (Pθ) and (Pθ+) --- p.105Chapter A.4.1 --- Feasible Descent Direction for the Minimization Prob- lems (Pθ) --- p.105Chapter A.4.2 --- Feasible Descent Direction for the Minimization Prob- lems (Pθ+) --- p.107Chapter B --- Randomly Generated Test Problems for Positive Definite Quadratic Programming --- p.109Chapter B.l --- Convex Quadratic Programming Problems with Linear Con- straints --- p.110Chapter B.l.1 --- General Description of Test Problems --- p.110Chapter B.l.2 --- The Objective Function --- p.112Chapter B.l.3 --- The Linear Constraints --- p.113Chapter B.2 --- Convex Quadratic Programming Problems with Quadratic In- equality Constraints --- p.116Chapter B.2.1 --- The Quadratic Constraints --- p.11

Laguerre-like methods for the simultaneous approximation of polynomial multiple zeros

Two new methods of the fourth order for the simultaneous determination of multiple zeros of a polynomial are proposed. The presented methods are based on the fixed point relation of Laguerre's type and realized in ordinary complex arithmetic as well as circular complex interval arithmetic. The derived iterative formulas are suitable for the construction of modified methods with improved convergence rate with negligible additional operations. Very fast convergence of the considered methods is illustrated by two numerical examples