1,447 research outputs found

    Solving the Least Squares Method problem in the AHP for 3 X 3 and 4 X 4 matrices

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    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

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    For each positive integer nn it is shown how to construct a finite collection of multivariable polynomials {Fi:=Fi(t,X1,...,X⌊n+12⌋)}\{F_{i}:=F_{i}(t,X_{1},..., X_{\lfloor \frac{n+1}{2} \rfloor})\} such that each positive integer whose squareroot has a continued fraction expansion with period n+1n+1 lies in the range of exactly one of these polynomials. Moreover, each of these polynomials satisfy a polynomial Pell's equation Ci2−FiHi2=(−1)n−1C_{i}^{2} -F_{i}H_{i}^{2} = (-1)^{n-1} (where CiC_{i} and HiH_{i} are polynomials in the variables t,X1,...,X⌊n+12⌋t,X_{1},..., X_{\lfloor \frac{n+1}{2} \rfloor}) and the fundamental solution can be written down. Likewise, if all the XiX_{i}'s and tt are non-negative then the continued fraction expansion of Fi\sqrt{F_{i}} can be written down. Furthermore, the congruence class modulo 4 of FiF_{i} depends in a simple way on the variables t,X1,...,X⌊n+12⌋t,X_{1},..., X_{\lfloor \frac{n+1}{2} \rfloor} 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 a1,...,ana_{1},..., a_{n} do there exist positive integers DD and a0a_{0} such that D=[a0;a1,>...,an,2a0ˉ]\sqrt{D} = [ a_{0};\bar{a_{1}, >..., a_{n},2a_{0}}].Comment: 13 page

    epsilon: A tool to find a canonical basis of master integrals

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    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 ϵ\epsilon in d=4−2ϵd=4-2\epsilon 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 kk-Ellipse

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    The kk-ellipse is the plane algebraic curve consisting of all points whose sum of distances from kk given points is a fixed number. The polynomial equation defining the kk-ellipse has degree 2k2^k if kk is odd and degree 2k−(kk/2)2^k{-}\binom{k}{k/2} if kk is even. We express this polynomial equation as the determinant of a symmetric matrix of linear polynomials. Our representation extends to weighted kk-ellipses and kk-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

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    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

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    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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