4,478 research outputs found
A Randomized Sublinear Time Parallel GCD Algorithm for the EREW PRAM
We present a randomized parallel algorithm that computes the greatest common
divisor of two integers of n bits in length with probability 1-o(1) that takes
O(n loglog n / log n) expected time using n^{6+\epsilon} processors on the EREW
PRAM parallel model of computation. We believe this to be the first randomized
sublinear time algorithm on the EREW PRAM for this problem
Computing heights on weighted projective spaces
In this note we extend the concept height on projective spaces to that of
weighted height on weighted projective spaces and show how such a height can be
computed. We prove some of the basic properties of the weighted height and show
how it can be used to study hyperelliptic curves over Q. Some examples are
provided from the weighted moduli space of binary sextics and octavics
An Analytical Study on the Multi-critical Behaviour and Related Bifurcation Phenomena for Relativistic Black Hole Accretion
We apply the theory of algebraic polynomials to analytically study the
transonic properties of general relativistic hydrodynamic axisymmetric
accretion onto non-rotating astrophysical black holes. For such accretion
phenomena, the conserved specific energy of the flow, which turns out to be one
of the two first integrals of motion in the system studied, can be expressed as
a 8 degree polynomial of the critical point of the flow configuration.
We then construct the corresponding Sturm's chain algorithm to calculate the
number of real roots lying within the astrophysically relevant domain of
. This allows, for the first time in literature, to {\it
analytically} find out the maximum number of physically acceptable solution an
accretion flow with certain geometric configuration, space-time metric, and
equation of state can have, and thus to investigate its multi-critical
properties {\it completely analytically}, for accretion flow in which the
location of the critical points can not be computed without taking recourse to
the numerical scheme. This work can further be generalized to analytically
calculate the maximal number of equilibrium points certain autonomous dynamical
system can have in general. We also demonstrate how the transition from a
mono-critical to multi-critical (or vice versa) flow configuration can be
realized through the saddle-centre bifurcation phenomena using certain
techniques of the catastrophe theory.Comment: 19 pages, 2 eps figures, to appear in "General Relativity and
Gravitation
Development of symbolic algorithms for certain algebraic processes
This study investigates the problem of computing the exact greatest common divisor of two polynomials relative to an orthogonal basis, defined over the rational number field. The main objective of the study is to design and implement an effective and efficient symbolic algorithm for the general class of dense polynomials, given the rational number defining terms of their basis. From a general algorithm using the comrade matrix approach, the nonmodular and modular techniques are prescribed. If the coefficients of the generalized polynomials are multiprecision integers, multiprecision arithmetic will be required in the construction of the comrade matrix and the corresponding systems coefficient matrix. In addition, the application of the nonmodular elimination technique on this coefficient matrix extensively applies multiprecision rational number operations. The modular technique is employed to minimize the complexity involved in such computations. A divisor test algorithm that enables the detection of an unlucky reduction is a crucial device for an effective implementation of the modular technique. With the bound of the true solution not known a priori, the test is devised and carefully incorporated into the modular algorithm. The results illustrate that the modular algorithm illustrate its best performance for the class of relatively prime polynomials. The empirical computing time results show that the modular algorithm is markedly superior to the nonmodular algorithms in the case of sufficiently dense Legendre basis polynomials with a small GCD solution. In the case of dense Legendre basis polynomials with a big GCD solution, the modular algorithm is significantly superior to the nonmodular algorithms in higher degree polynomials. For more definitive conclusions, the computing time functions of the algorithms that are presented in this report have been worked out. Further investigations have also been suggested
Natural Density Distribution of Hermite Normal Forms of Integer Matrices
The Hermite Normal Form (HNF) is a canonical representation of matrices over
any principal ideal domain. Over the integers, the distribution of the HNFs of
randomly looking matrices is far from uniform. The aim of this article is to
present an explicit computation of this distribution together with some
applications. More precisely, for integer matrices whose entries are upper
bounded in absolute value by a large bound, we compute the asymptotic number of
such matrices whose HNF has a prescribed diagonal structure. We apply these
results to the analysis of some procedures and algorithms whose dynamics depend
on the HNF of randomly looking integer matrices
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