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Jim Blinn discusses a good practical closed-form solution process for cubic equations and makes some observations about the relation between iterative and closed-form solutions.
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In the previous four columns, the properties of the homogeneous cubic polynomial were studied. In this article, the author introduces two new algorithms that, at first, look quite different from what we''ve done so far. It will tu...
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In the previous four columns, the properties of the homogeneous cubic polynomial were studied. In this article, the author introduces two new algorithms that, at first, look quite different from what we''ve done so far. It will turn out, though, that they actually do fit into our solution scheme. In showing this, he has taken good ideas from a variety of authors and translated them into a common notation while also converting them to deal with homogeneous polynomials
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It should not be surprising that in solving cubic equations, sooner or later you are going to have to take the cube root of something. But it's not immediately obvious what cubic equations have to do with arc cosines. Well, in the...
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It should not be surprising that in solving cubic equations, sooner or later you are going to have to take the cube root of something. But it's not immediately obvious what cubic equations have to do with arc cosines. Well, in the general case of complex coefficients and roots we would need to take the cube root of a complex number
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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 consider the following Hamiltonian equation on the L2 Hardy space on the circle,ietu = II(|u|~2u)where II is the Szego projector. This equation can be seen as a toy model for totally non dispersive evo-lution equations. We disp...
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We consider the following Hamiltonian equation on the L2 Hardy space on the circle,ietu = II(|u|~2u)where II is the Szego projector. This equation can be seen as a toy model for totally non dispersive evo-lution equations. We display a Lax pair structure for this equation. We prove that it admits an infinite sequence of conservation laws in involution, and that it can be approximated by a sequence of finite dimensional completely integrable Hamiltonian systems. We establish several instability phenomena illustrating the degeneracy of this completely integrable structure. We also classify the traveling waves for this system.
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We consider the 3D cubic focusing nonlinear Schrodinger (NLS) equation iθ_tu+ Δu+ |u|~2u=0, which appears as a model in condensed matter theory and plasma physics. We construct a family of axially symmetric solutions, correspond...
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We consider the 3D cubic focusing nonlinear Schrodinger (NLS) equation iθ_tu+ Δu+ |u|~2u=0, which appears as a model in condensed matter theory and plasma physics. We construct a family of axially symmetric solutions, corresponding to an open set in H_(axial)~1(R~3) of initial data, that blow up in finite time with a singular circle in the xy-plane. Our construction is modeled on Raphael’s [33] construction of a family of solutions to the 2D quintic focusing NLS, iθ_tu+ Δu+ |u|~4u=0, that blows up on a circle.
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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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The theory of the cubic theta function ∑qm2+mn+n2z1mz2n is developed analogously to the theory of the classical Jacobian theta functions. Many inversion formulas, addition formulas, and modular equations are derived, and these re...
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The theory of the cubic theta function ∑qm2+mn+n2z1mz2n is developed analogously to the theory of the classical Jacobian theta functions. Many inversion formulas, addition formulas, and modular equations are derived, and these results extend previous work on cubic theta functions.
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