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The spectral mapping theorem in a unital Banach algebra is examined for its constructive content. Mathematics Subject Classification: 03F60, 46S30.Let B be a (complex) Banach algebra that is unital -that is, has an identity e. Let a be an element of B, R a = {λ ∈ C : a − λe has a two-sided inverse}, the resolvent set of a, and σ a = {λ ∈ C : (∀λ ∈ R a ) (λ = λ )}, the spectrum of a. The Classical Spectral Mapping Theorem says that if p is a monic polynomial of degree ≥ 1, then σ p(a) = p(σ a ). The classical proof of this theorem hangs on the decomposition of p(z) − λ, where λ ∈ C, into linear factors, and the observation that p(a) − λe has no inverse if and only if at least one of its corresponding factors has no inverse. However, this proof is not valid in constructive mathematics -that is, mathematics based on intuitionistic logic (which leads to theorems valid in a wider range of models, including recursive and other computational ones, than those proved with classical logic). Our main aims in this paper are (i) to find a constructive proof of the inclusion p(σ a ) ⊂ σ p(a) , and (ii) to recover constructively as much as is possible of the reverse inclusion. In order to fulfil these aims, we need relatively little technical background in constructive mathematics. Much of what is needed will be stated in context below; for more detailed information on constructive mathematics, see [1,2,4,7]. The first half of the spectral mapping theoremOur proof of the inclusion p(σ a ) ⊂ σ p(a) will be based on some elementary constructive semigroup theory. First, we define an inequality on a set X to be a binary relation = on X with the following two properties:x = y ⇒ ¬(x = y), x = y ⇒ y = x.

This paper addresses the development of a set of novel autonomous control laws for the navigation of multiple mini or micro quadrotors in a workspace populated with cylindrical obstacles. To the authors' knowledge, this is the first time that such a set of control inputs for the autonomous control of multiple quadrotors and avoidance of cylindrical obstacles is being derived from a single Lyapunov function. The avoidance of the cylindrical obstacles is achieved via a Minimum Distance Technique that allows a quadrotor to avoid the closest point on the curved surface of a cylinder at every unit time. In addition, the novel controllers ensure a near-to-horizontal orientation of a quadrotor at every unit time and the hovering motion exhibited by the quadrotor in a neighborhood of its target. The hovering capability has been solved completely in this paper by solving the underactuatedness of the quadrotor which allows for the design of the maximum translational velocities needed for vertical take-off and landing, and hovering. This is important in applications where payloads are sensitive to acrobatic orientations. Computer simulations that mimic real-life scenarios using cylindrical towers as obstacles in a city-like environment illustrates the effectiveness of the controllers.

As with other Pacific Island nations, the scientific efforts of the Kingdom of Tonga have been hampered through a lack of local scientists and science graduates. As observed globally, the region appears to face a steady decline in student interest and achievement in science, resulting in reduced uptake of science subjects in schools and universities. This study aims to provide insight into the attitudes of Tongan senior secondary students toward science, using the validated Test of Scientific Related Attitudes (TOSRA) instrument. The sample population comprised 2636 students of approximately 15–18 years of age, from 26 schools across Tonga. Overall, the mean attitudes of Tongan senior secondary students toward science were lower than that previously observed for Australian secondary students (Grades 7–10) and Fijian senior secondary students. A significant reduction in attitude was found between forms 5 and 6 for female students from rural areas, but not those from urban areas. No significant changes across different form levels were found for male students. The greatest difference between students’ perspectives was found for the normality of scientists, which may be indicative of cultural views toward this topic. Combined with the similar results of previous research in Fijian students, this may point toward broader differences in the attitudes toward science between students in Pacific Island countries more generally.