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A survey of modern algebra. Revised ed
... As we mentioned in Remark 5, f can be written as f (λ) = a 2 +λ a 4 c 2 g(λ), where g(λ) is a polynomial of degree three (see Lemma 13 for details); thus Cardano's formulas can be applied to g(λ) to detect contact between H and S by a simple calculation as follows. In general, for a monic polynomial x 3 +a 2 x 2 +a 1 x+a 0 , the quantities Q = (3a 1 −a 2 2 )/9 and R = (9a 2 a 1 − 27a 0 − 2a 3 1 )/54 are defined so that ∆ = Q 3 + R 2 detects complex and multiple roots (see, for instance, [1]). Thus, a direct analysis of the coefficients of the polynomial p(λ) gives the value for ∆ and detects contact between S and H as follows: ...
... For this to be possible it is necessary that c 2 < ar, as we saw in Remark 12. This shows assertion (1). Note that if the center of S is in the plane z = 0 then −a 2 and c 2 are roots of the characteristic polynomial, as follows from Lemmas 13 and 16. ...
... The interior of H for a fixed value of z is a disk, so it is convex. Hence we can build a path 1], so that α(0) is the center of S, α(1) is in the OZ axis, and we move the center of the sphere along α without touching H. As before, define f t (λ) to be the characteristic polynomial when the center of S is at α(t). ...
We characterize all possible relative positions between a circular hyperboloid of one sheet and a sphere through the roots of a characteristic polynomial associated to these quadrics.
As an application, this provides a method to detect contact between the 2 surfaces by a simple calculation in many real world applications.
... Where Σ is the 3-D stress tensor, Λ are the principal values, I is the Identity matrix and I 1, I 2 and I 3 are the stress invariants. An analytical expression for the principal values from Equation 7 would not be as simple as Equation 4. It is well documented in literature about mathematical procedures like Cardano's method (Birkhoff & MacLane 1997) involving transformation to get reduced cubic equations and subsequent reduction to a quadratic equation to find analytical solutions, but such an approach for solving Equation 7 would prove rather complex and cumbersome. Since the superposed stresses as in Equation 1, 2 and 3 would introduce a new variable λ, finding a closed form solution, to be able to implement in a finite element framework, becomes rather unrealistic. ...
This article presents a new non-proportional loading strategy for Sequentially Linear Analysis (SLA), which is a robust secant stiffness based procedure for nonlinear finite element analysis of quasi-brittle materials, like concrete and masonry. The strategy is based on finding the principal planes for a total strain based fixed cracking model, by searching for the critical plane where the normal stresses due to the scaled combination of two non-proportional loads is equal to the allowable strength. For a plane stress situation (2D), the scaling factor λ is expressed as a function of θ, the inclination of an arbitrary plane to the reference coordinate system, and a one dimensional (θ) optimization of λ is done to determine the principal plane and the resulting fixed crack coordinate system. This approach has been illustrated to match up to the closed form solution (Van de Graaf, 2017), obtained previously based on the principal stress theory, using single element tests and a quasi-static test pushover test on a masonry shear wall. Finally, the concept for the 3-D stress situation is presented, where the optimization problem becomes two-dimensional, with respect to l and m (two-directional cosines).
ARTÍCULO DE INVESTIGACIÓN, Mathesis III 2 (1), 2007: 73-97. ISSN 0185-6200.
After a general discussion of group actions, orbifolds, and "weak orbifolds" this note will provide elementary introductions to two basic moduli spaces over the real or complex numbers: First the moduli space of effective divisors with finite stabilizer on the projective space modulo the group of projective transformations of ; and then the moduli space of effective 1-cycles with finite stabilizer on modulo the group of projective transformations of .
An observable canonical form is formulated for the set of rational systems on a variety each of which is a single-input-single-output, affine in the input, and a minimal realization of its response map. The equivalence relation for the canonical form is defined by the condition that two equivalent systems have the same response map. A proof is provided that the defined form is well-defined canonical form. Special cases are discussed.
We present a collection of simple construction problems and determine conditions for which the construction can be carried out with compass and straightedge. Since the problems reduce to finding the roots of quartic polynomials, it is not sufficient to show that the polynomials are irreducible to conclude that the roots are not constructible. Some interesting results appear as we discover general conditions that determine when the solutions are or are not constructible.
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