International journal for innovation education and research, May 31, 2015
Cardano's Method is not easily understood by undergraduate students. In this research project, we... more Cardano's Method is not easily understood by undergraduate students. In this research project, we developed a method that students can understand without advanced mathematics skills. The method we developed only need to use the skills of factoring polynomials in complex field and finding cubic roots of a complex number. The procedures we developed are as following: 1) Write cubic equation in the form of A 3 + B 3 + C 3-3ABC = 0, where A is a function of x, B and C are complex numbers. 2) Solve the quadratic equation Z 2-(B 3 + C 3)Z + B 3 C 3 = 0, which gives B and C. 3) Factor equation of (1) into (A + B + C)(A +Bw + Cw 2)(A +Bw 2 +Cw) = 0, where w is a complex root of 1. 4) Solve equations A+B+C = 0, A+Bω+Cω2 = 0, and A+Bω2+Cω = 0
Let Vl be the class of all vector lattices, and let S and T be torsion classes of l-groups. T ∩ V... more Let Vl be the class of all vector lattices, and let S and T be torsion classes of l-groups. T ∩ Vl is a torsion class if and only if each divisible abelian l -group in T contains a largest l -ideal that is a vector lattice. Moreover, if T ∩ Vl is a torsion class, so is S ∩ T ∩ Vl. The following classes of vector lattices form torsion classes: the hyperarchimedean vector lattices; the finite-valued vector lattices; the class of all vector lattices of the form Σ(Δ,R). In particular, the principal torsion class \( \tilde \sum (\Delta, R) \) determined by Σ(Δ,R) consists of vector lattices; it consists of all cardinal sums of l-groups Σ(Λ, R) where Λ is a direct limit of connected, convex subsets of Λ. The following classes of vector lattices form pseudo torsion classes: the archimedean l -groups; the special-valued and conditionally laterally complete l -groups. Underlying this theory is the fact that if K is a finite-valued l -group or a conditionally laterally complete l -group, then K is a vector lattice if and only if each K(k) is a vector lattice, which is true if and only if each K(k), with k a special element, is a vector lattice.
A minimal clan is a lattice-ordered partial semigroup having the cancellation property and differ... more A minimal clan is a lattice-ordered partial semigroup having the cancellation property and difference property. Wyler (1966 [15]) proved every Abelian clan could be embedded into an Abelian lattice-ordered group. Conrad (1990) showed that every clan could be embedded into a lattice-ordered group. There are two main results in this paper. First theorem said a clan C is complete if and only if the lattice-ordered group generated by C is complete. The second theorem said every clan contains a greatest generalized Boolean algebra. That answered Schmit's question ([16]). The second part of this paper is to that if an l-group G has minimal generating clan such that {0} is the only subsemigroup then G is a Specker group. This theorem makes a connection between Specker groups and Clans.
International journal for innovation education and research, 2015
Cardano’s Method is not easily understood by undergraduate students. In this research project, ... more Cardano’s Method is not easily understood by undergraduate students. In this research project, we developed a method that students can understand without advanced mathematics skills. The method we developed only need to use the skills of factoring polynomials in complex field and finding cubic roots of a complex number. The procedures we developed are as following:1) Write cubic equation in the form of A3 + B3 + C3 – 3ABC = 0, where A is a function of x, B and C arecomplex numbers.2) Solve the quadratic equation Z2 – (B3 + C3)Z + B3 C3 = 0, which gives B and C.3) Factor equation of (1) into (A + B + C)(A +Bw + Cw2)(A +Bw2+Cw) = 0, where w is a complex rootof 1.4) Solve equations A+B+C = 0, A+BI‰+CI‰2 = 0, and A+BI‰2+CI‰ = 0
International journal for innovation education and research, May 31, 2015
Cardano's Method is not easily understood by undergraduate students. In this research project, we... more Cardano's Method is not easily understood by undergraduate students. In this research project, we developed a method that students can understand without advanced mathematics skills. The method we developed only need to use the skills of factoring polynomials in complex field and finding cubic roots of a complex number. The procedures we developed are as following: 1) Write cubic equation in the form of A 3 + B 3 + C 3-3ABC = 0, where A is a function of x, B and C are complex numbers. 2) Solve the quadratic equation Z 2-(B 3 + C 3)Z + B 3 C 3 = 0, which gives B and C. 3) Factor equation of (1) into (A + B + C)(A +Bw + Cw 2)(A +Bw 2 +Cw) = 0, where w is a complex root of 1. 4) Solve equations A+B+C = 0, A+Bω+Cω2 = 0, and A+Bω2+Cω = 0
Let Vl be the class of all vector lattices, and let S and T be torsion classes of l-groups. T ∩ V... more Let Vl be the class of all vector lattices, and let S and T be torsion classes of l-groups. T ∩ Vl is a torsion class if and only if each divisible abelian l -group in T contains a largest l -ideal that is a vector lattice. Moreover, if T ∩ Vl is a torsion class, so is S ∩ T ∩ Vl. The following classes of vector lattices form torsion classes: the hyperarchimedean vector lattices; the finite-valued vector lattices; the class of all vector lattices of the form Σ(Δ,R). In particular, the principal torsion class \( \tilde \sum (\Delta, R) \) determined by Σ(Δ,R) consists of vector lattices; it consists of all cardinal sums of l-groups Σ(Λ, R) where Λ is a direct limit of connected, convex subsets of Λ. The following classes of vector lattices form pseudo torsion classes: the archimedean l -groups; the special-valued and conditionally laterally complete l -groups. Underlying this theory is the fact that if K is a finite-valued l -group or a conditionally laterally complete l -group, then K is a vector lattice if and only if each K(k) is a vector lattice, which is true if and only if each K(k), with k a special element, is a vector lattice.
A minimal clan is a lattice-ordered partial semigroup having the cancellation property and differ... more A minimal clan is a lattice-ordered partial semigroup having the cancellation property and difference property. Wyler (1966 [15]) proved every Abelian clan could be embedded into an Abelian lattice-ordered group. Conrad (1990) showed that every clan could be embedded into a lattice-ordered group. There are two main results in this paper. First theorem said a clan C is complete if and only if the lattice-ordered group generated by C is complete. The second theorem said every clan contains a greatest generalized Boolean algebra. That answered Schmit's question ([16]). The second part of this paper is to that if an l-group G has minimal generating clan such that {0} is the only subsemigroup then G is a Specker group. This theorem makes a connection between Specker groups and Clans.
International journal for innovation education and research, 2015
Cardano’s Method is not easily understood by undergraduate students. In this research project, ... more Cardano’s Method is not easily understood by undergraduate students. In this research project, we developed a method that students can understand without advanced mathematics skills. The method we developed only need to use the skills of factoring polynomials in complex field and finding cubic roots of a complex number. The procedures we developed are as following:1) Write cubic equation in the form of A3 + B3 + C3 – 3ABC = 0, where A is a function of x, B and C arecomplex numbers.2) Solve the quadratic equation Z2 – (B3 + C3)Z + B3 C3 = 0, which gives B and C.3) Factor equation of (1) into (A + B + C)(A +Bw + Cw2)(A +Bw2+Cw) = 0, where w is a complex rootof 1.4) Solve equations A+B+C = 0, A+BI‰+CI‰2 = 0, and A+BI‰2+CI‰ = 0
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