An extension to the Hermite-Joubert problem

Let E/F be a field extension of degree n. A classical problem is to find a generating element in E whose characteristic polynomial over F is as simple is possible. An 1861 theorem of Ch. Hermite [5] asserts that for every separable field E/F of degree n there exists an element a ∈ E whose characteristic polynomial is of the form f(x) = x⁵ +b₂x³ +b₄x+b₅ or equivalently, tr{E/F}(a) = tr{E/F}(a³) = 0. A similar result for extensions of degree 6 was proven by P. Joubert in 1867; see [6]. In this thesis we ask if these results can be extended to field extensions of larger degree. Specifically, we give a necessary and sufficient condition for a field F, a prime p and an integer n ≥ 3 to have the following property: Every separable field extension E/F of degree n contains an element a ∈ E such that a generates E over F, and tr{E/F}(a) = tr{E/F}(a^p) = 0. As a corollary we show for infinitely many new values of n that the theorems of Hermite and Joubert do not extend to field extensions of degree n. We conjecture the same for more values of n and provide computational evidence for a large number of these.Science, Faculty ofMathematics, Department ofGraduat

A graph-theoretic approach to a conjecture of Dixon and Pressman

Given n×n matrices, A_1,...,A_k, define the linear operator L(A_1,...,A_k): Mat_n -> Mat_n by L(A_1,...,A_k)(A_(k+1)) = sum_sigma sgn(sigma) sgn(sigma)A_sigma(1)A_sigma(2)...A_sigma(k+1). The Amitsur-Levitzki theorem asserts that L(A_1,...,A_k) is identically 0 for every k > 2n − 1. Dixon and Pressman conjectured that if 2 <= k <= 2n − 2, then for A_1,...,A_k ∈ Mat_n(R) in general position, the kernel of L(A_1,...,A_k) has dimension k when k is even and either k +1 or k + 2 when k is odd (depending on whether n is even or odd). We prove this conjecture in the case where k is even. Our proof relies on graph-theoretic techniques.Science, Faculty ofMathematics, Department ofGraduat

Data from: Development and field validation of a regional, management-scale habitat model: a koala Phascolarctos cinereus case study

Species distribution models have great potential to efficiently guide management for threatened species, especially for those that are rare or cryptic. We used MaxEnt to develop a regional-scale model for the koala Phascolarctos cinereus at a resolution (250 m) that could be used to guide management. To ensure the model was fit for purpose, we placed emphasis on validating the model using independently-collected field data. We reduced substantial spatial clustering of records in coastal urban areas using a 2-km spatial filter and by modeling separately two subregions separated by the 500-m elevational contour. A bias file was prepared that accounted for variable survey effort. Frequency of wildfire, soil type, floristics and elevation had the highest relative contribution to the model, while a number of other variables made minor contributions. The model was effective in discriminating different habitat suitability classes when compared with koala records not used in modeling. We validated the MaxEnt model at 65 ground-truth sites using independent data on koala occupancy (acoustic sampling) and habitat quality (browse tree availability). Koala bellows (n = 276) were analyzed in an occupancy modeling framework, while site habitat quality was indexed based on browse trees. Field validation demonstrated a linear increase in koala occupancy with higher modeled habitat suitability at ground-truth sites. Similarly, a site habitat quality index at ground-truth sites was correlated positively with modeled habitat suitability. The MaxEnt model provided a better fit to estimated koala occupancy than the site-based habitat quality index, probably because many variables were considered simultaneously by the model rather than just browse species. The positive relationship of the model with both site occupancy and habitat quality indicates that the model is fit for application at relevant management scales. Field-validated models of similar resolution would assist in guiding management of conservation-dependent species

seL4: from general purpose to a proof of information flow enforcement

Abstract—In contrast to testing, mathematical reasoning and formal verification can show the absence of whole classes of security vulnerabilities. We present the, to our knowledge, first complete, formal, machine-checked verification of information flow security for the implementation of a general-purpose microkernel; namely seL4. Unlike previous proofs of information flow security for operating system kernels, ours applies to the actual 8,830 lines of C code that implement seL4, and so rules out the possibility of invalidation by implementation errors in this code. We assume correctness of compiler, assembly code, hardware, and boot code. We prove everything else. This proof is strong evidence of seL4’s utility as a separation kernel, and describes precisely how the general purpose kernel should be configured to enforce isolation and mandatory information flow control. We describe the information flow security statement we proved (a variant of intransitive noninterference), including the assumptions on which it rests, as well as the modifications that had to be made to seL4 to ensure it was enforced. We discuss the practical limitations and implications of this result, including covert channels not covered by the formal proof. I