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In this paper first we prove four fundamental theorems of the alternative, called scaling dualities, characterizing exact and approximate solvability of four significant conic problems in finite dimensional spaces, defined as: homogeneous programming (HP), scaling problem (SP), homogeneous scaling problem (HSP), and algebraic scaling problem (ASP). Let φ be(More)
Kalantari This book offers fascinating and modern perspectives into the theory and practice of the historical subject of polynomial root-finding, rejuvenating the field via polynomiography, a creative and novel computer visualization that renders spectacular images of a polynomial equation. Polynomiography will not only pave the way for new applications of(More)
Let p(x) be a polynomial of degree n 2 with coecients in a subeld K of the complex numbers. For each natural number m 2, let L m (x) be the m 2 m lower triangular matrix whose diagonal entries are p(x) and for each j = 1; : : : ; m 0 1, its j-th subdiagonal entries are p (j) (x)=j!. For i = 1; 2, let L (i) m (x) be the matrix obtained from L m (x) by(More)
Givenaset P with n points in R k ,its diameter, d P ,isthe maximum of the Euclidean distances between its points. We describe an algorithm that in m ≤ n iterations obtains 5 − 2 √  3 r m }. For k fixed, the cost of each iteration is O(n). Inparticular in the first iteration, the algorithm produces an approximation r 1 which is within √  3o f d P(More)
The general form of Taylor's theorem gives the formula, f = P n + R n , where P n is the New-ton's interpolating polynomial, computed with respect to a connuent vector of nodes, and R n is the remainder. When f 0 6 = 0, for each m = 2; : : :; n + 1, we describe a \determinantal interpolation formula", f = P m;n +R m;n , where P m;n is a rational function in(More)