1,447 research outputs found
Solving the Least Squares Method problem in the AHP for 3 X 3 and 4 X 4 matrices
The Analytic Hierarchy Process (AHP) is one of the most popular methods used in Multi-Attribute Decision Making. The Eigenvector Method (EM) and some distance minimizing methods such as the Least Squares Method (LSM) are of the possible tools for computing the priorities of the alternatives. A method for generating all the solutions of the LSM problem for 3 × 3 and 4 × 4 matrices is discussed in the paper. Our algorithms are based on the theory of resultants
Multi-variable Polynomial Solutions to Pell's Equation and Fundamental Units in Real Quadratic Fields
For each positive integer it is shown how to construct a finite
collection of multivariable polynomials such that each positive integer whose squareroot has
a continued fraction expansion with period lies in the range of exactly
one of these polynomials. Moreover, each of these polynomials satisfy a
polynomial Pell's equation (where
and are polynomials in the variables ) and the fundamental solution can be written down.
Likewise, if all the 's and are non-negative then the continued
fraction expansion of can be written down. Furthermore, the
congruence class modulo 4 of depends in a simple way on the variables
so that the fundamental unit
can be written down for a large class of real quadratic fields. Along the way a
complete solution is given to the problem of determining for which symmetric
strings of positive integers do there exist positive
integers and such that .Comment: 13 page
epsilon: A tool to find a canonical basis of master integrals
In 2013, Henn proposed a special basis for a certain class of master
integrals, which are expressible in terms of iterated integrals. In this basis,
the master integrals obey a differential equation, where the right hand side is
proportional to in space-time dimensions. An
algorithmic approach to find such a basis was found by Lee. We present the tool
epsilon, an efficient implementation of Lee's algorithm based on the Fermat
computer algebra system as computational backend.Comment: 34 pages; changed reference to fuchsi
Semidefinite Representation of the -Ellipse
The -ellipse is the plane algebraic curve consisting of all points whose
sum of distances from given points is a fixed number. The polynomial
equation defining the -ellipse has degree if is odd and degree
if is even. We express this polynomial equation as
the determinant of a symmetric matrix of linear polynomials. Our representation
extends to weighted -ellipses and -ellipsoids in arbitrary dimensions,
and it leads to new geometric applications of semidefinite programming.Comment: 16 pages, 5 figure
Solving a "Hard" Problem to Approximate an "Easy" One: Heuristics for Maximum Matchings and Maximum Traveling Salesman Problems
We consider geometric instances of the Maximum Weighted Matching Problem
(MWMP) and the Maximum Traveling Salesman Problem (MTSP) with up to 3,000,000
vertices. Making use of a geometric duality relationship between MWMP, MTSP,
and the Fermat-Weber-Problem (FWP), we develop a heuristic approach that yields
in near-linear time solutions as well as upper bounds. Using various
computational tools, we get solutions within considerably less than 1% of the
optimum.
An interesting feature of our approach is that, even though an FWP is hard to
compute in theory and Edmonds' algorithm for maximum weighted matching yields a
polynomial solution for the MWMP, the practical behavior is just the opposite,
and we can solve the FWP with high accuracy in order to find a good heuristic
solution for the MWMP.Comment: 20 pages, 14 figures, Latex, to appear in Journal of Experimental
Algorithms, 200
Comments on the Links between su(3) Modular Invariants, Simple Factors in the Jacobian of Fermat Curves, and Rational Triangular Billiards
We examine the proposal made recently that the su(3) modular invariant
partition functions could be related to the geometry of the complex Fermat
curves. Although a number of coincidences and similarities emerge between them
and certain algebraic curves related to triangular billiards, their meaning
remains obscure. In an attempt to go beyond the su(3) case, we show that any
rational conformal field theory determines canonically a Riemann surface.Comment: 56 pages, 4 eps figures, LaTeX, uses eps
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