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One of the author's favorite experiences in high school mathematics was learning how to solve quadratic equations. As is usual with life, things are not as simple as they were in high school. There are several problems with the qu...
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One of the author's favorite experiences in high school mathematics was learning how to solve quadratic equations. As is usual with life, things are not as simple as they were in high school. There are several problems with the quadratic formula. In this article, the author talks about these problems and various solutions to them.
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In this note, we improve the results proved by S.-M. Jung [8], and S.-M. Jung and P. K. Sahoo [9] on the stability of the Pexider-quadratic functional equation and Drygas' equation, respectively. We give sharper bounds on more gen...
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In this note, we improve the results proved by S.-M. Jung [8], and S.-M. Jung and P. K. Sahoo [9] on the stability of the Pexider-quadratic functional equation and Drygas' equation, respectively. We give sharper bounds on more general settings by making use of different and easier ways. An example is also given to illustrate that the quadratic equation (and so the Pexider-quadratic equation) is not stable on the free group generated by two elements.
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In this paper we study general solutions of the following Pexider functional equation on a vector space over a field of characteristic different from 2, for k ≥ 3.
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In this study we investigate the question of accurate determination of the root enclosures of quadratic equations whose coefficients constitute interval variables. We treat several special cases where either Sridhara's classical f...
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In this study we investigate the question of accurate determination of the root enclosures of quadratic equations whose coefficients constitute interval variables. We treat several special cases where either Sridhara's classical formula or Fagnano's alternative expression provide exact interval enclosures for the roots of the quadratic equation. In the case of a single coefficient serving as an interval, it is shown that the classical interval analysis by either Sridhara's, Fagnano's or specifically modified Fagnano's formulas provide lower and upper bounds of roots. Then we follow with the case of two coefficients being intervals whereas the third coefficient is a deterministic quantity. Numerous examples are provided in which the solutions are compared with the direct numerical evaluation of the roots' enclosures. In three arising cases either Sridhara's or Fagnano's expressions suffice to obtain exact enclosures. (C) 2015 Published by Elsevier Inc.
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We prove that the Diophantine equation x~2 - kxy + y~2 + lx = 0, l ∈ {1, 2, 4} has an infinite number of positive integer solutions x and y if and only if (k, l) = (3, 1), (3, 2), (4, 2), (3, 4), (4, 4), (6,4). Furthermore, we pr...
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We prove that the Diophantine equation x~2 - kxy + y~2 + lx = 0, l ∈ {1, 2, 4} has an infinite number of positive integer solutions x and y if and only if (k, l) = (3, 1), (3, 2), (4, 2), (3, 4), (4, 4), (6,4). Furthermore, we prove that the Diophantine equation x~2 -kxy+y~2 + x = 0 has infinitely many integer solutions x and y if and only if k ≠ 0, ±1, which answers a problem in Marlewski and Marzycki (2004) [ 1 ].
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The generalized Gaussian distribution with location parameter , scale parameter sigma, and shape parameter p contains the Laplace, normal, and uniform distributions as particular cases for p = 1, 2, +, respectively. Derivations of...
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The generalized Gaussian distribution with location parameter , scale parameter sigma, and shape parameter p contains the Laplace, normal, and uniform distributions as particular cases for p = 1, 2, +, respectively. Derivations of the true maximum-likelihood estimators of and sigma for these special cases are popular exercises in many university courses. Here, we show how the true maximum-likelihood estimators of and sigma can be derived for p = 3, 4, 5. The derivations involve solving of quadratic, cubic, and quartic equations.
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We derived an algorithm to find the real roots of the homogeneous quadratic equation, Ax/sup 2/+2Bxw+Cw/sup 2/=0. Because the equation is homogeneous, a root consists of an [x, w] pair where any nonzero multiple represents the sam...
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We derived an algorithm to find the real roots of the homogeneous quadratic equation, Ax/sup 2/+2Bxw+Cw/sup 2/=0. Because the equation is homogeneous, a root consists of an [x, w] pair where any nonzero multiple represents the same root. We strove to find an algorithm that didn't blow up no matter what values of A, B, and C we were given, including various combinations of zeroes. At the end of the article the author wrote the final algorithm in tabular form.
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In this paper, we prove the equation in the title has no positive integer solutions (x, y, n) with 2 X n and x not equal y apart from (x, y, n) = (5, 2, 5) (90, 2, 13). (c) 2005 Elsevier Inc. All rights reserved.
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We consider the dynamics of the three-dimensional N-body Schr?dinger equation in the presence of a quadratic trap. We assume the pair interaction potential is N ~(3β-1) V(N ~β _x). We justify the mean-field approximation and off...
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We consider the dynamics of the three-dimensional N-body Schr?dinger equation in the presence of a quadratic trap. We assume the pair interaction potential is N ~(3β-1) V(N ~β _x). We justify the mean-field approximation and offer a rigorous derivation of the three-dimensional cubic nonlinear Schr?dinger equation (NLS) with a quadratic trap. We establish the space-time bound conjectured by Klainerman and Machedon (Commun Math Phys 279:169-185, 2008) for β ∈ (0, 2/7] by adapting and simplifying an argument in Chen and Pavlovi? (Annales Henri Poincaré, 2013) which solves the problem for β ∈ (0,1/4) in the absence of a trap.
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The equation x~2 + x - 2 is a quadratic, because the highest power of x is 2, and the equation has two solutions: x = -2 and x = 1. These values are called the 'roots' of the equation and we teach a method and a formula that will ...
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The equation x~2 + x - 2 is a quadratic, because the highest power of x is 2, and the equation has two solutions: x = -2 and x = 1. These values are called the 'roots' of the equation and we teach a method and a formula that will find them if they exist. The quadratic is a fully solvable problem at high school level but I would not write an article on 'exploration into quadratics'. There is not really enough there to explore. When I was at school, I remember asking my teacher if there was a formula like the quadratic formula that would solve cubic equations, like x~3 + x~2 + x-2 = 0.
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