UBC cIRcle
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The DSpace digital repository system captures, stores, indexes, preserves, and distributes digital research material.2014-04-24T20:02:28ZIntersective polynomials and Hensel's Lemma
http://hdl.handle.net/2429/46574
An intersective polynomial is a polynomial with integer coefficients that has no rational roots, but has a root modulo every integer greater than 1. These polynomials have been difficult to find using traditional methods. In this thesis, we employ elementary methods, namely Hensel’s Lemma and the Chinese remainder theorem, to allow us to create three new infinite families of intersective polynomials.
In order to create a candidate intersective polynomial, we employ methods from Galois theory. We multiply together carefully chosen polynomials that define subfields of a splitting field to create our candidate. We chose the subfields by first finding the n-cover, a collection of n proper subgroups, of the Galois group and identifying the corresponding subfields. We multiply the minimal polynomials of the subfields of the chosen splitting fields together to create the candidate intersective polynomial.
From our method of creating candidates we can see that intersective polynomials have at least two irreducible factors. In order to prove that these polynomials have a root modulo every integer greater than 1, we examine each factor modulo certain prime numbers and determine which sets of primes admit solutions for each factor. We then use Hensel’s Lemma to lift those solutions to infinitely high powers of the particular primes. Finally, we combine the solutions modulo prime powers by using the Chinese remainder theorem to show that the polynomial has solutions modulo every integer greater than 1.
Through the method outlined above, we have created three infinite families of intersective polynomials: (x³ − n)(x² + 3) when the prime factors of n are of the form 3k + 1 and n is congruent to 1 mod 9; (x² − a)(x² − b)(x² − a₁b₁) when at least one of a, b, a₁b₁ is congruent to 1 modulo 8 and one of the Legendre Symbols (a/p), (b/p), (a₁b₁/p) has the value +1; and (x^q −n)(x^(q-¹) +x^q-²+...+ x + 1) when the prime factors of n are kq + 1 and n is congruent to 1 modulo q² for an odd prime q.
In the last chapter, we treat some special cases of intersective polynomials which will be considered in detail in future work.
2014-04-24T00:00:00ZAre inversion, posture, motion and muscle effects important to spinal alignment?
http://hdl.handle.net/2429/46573
Rollover accidents are dynamic and complex events in which head contacts with the vehicle interior can cause catastrophic neck injuries through head-first impact. Ex vivo cadaver tests are valuable for studying these mechanisms of head-first axial loading neck injuries; however, they lack a biofidelic representation of neuromuscular control, postural stability, and overall spine posture. Computational modeling can be used to evaluate changes in the risk of neck injury under the influence of muscle forces, yet the exact muscles and levels of forces that are involved leading up to a head-first impact are unknown. Knowing the state of the neck prior to impact is critical to improving cadaveric and computational models of neck injury.
Four human volunteer experiments were conducted to determine whether inversion, head position, muscle tensing, and dynamic motion influence the cervical spine alignment. These four studies included: (1) static inversion, (2) muscle tensing, (3) moment generation, (4) dynamic flexion/extension. For each experiment, cervical alignment was captured using fluoroscopy and muscle activity was captured using electromyography.
The inverted posture and muscle activations were found to be different than the upright relaxed posture and the differences depend on the position of the head (study 1). Actively tensing the neck muscles in a free unconstrained task (study 2) and in generating flexion and extension forces with head constraint (study 3) resulted in different cervical alignment compared to the initial resting spine. Not only do these neck muscle contractions induce postural changes, they also provide a substantial stiffening effect to the neck. Finally, dynamically arriving at the neutral position did not result in the same cervical alignment as static neutral and the alignment depended on the direction that neutral is approached from (full flexion or full extension).
These findings suggest that it may not be sufficient to replicate the upright resting posture in cadaveric and computational models of neck injury. Adopting in vivo postures and muscle activations, relevant to head-first impact, in the laboratory may help in replicating the spectrum of injuries observed in real life rollovers, an important step toward injury prevention.
2014-04-24T00:00:00ZEssential dimension and linear codes
http://hdl.handle.net/2429/46572
The essential dimension of an algebraic group G is a measure of the complexity of G-torsors. One of the central open problems in the theory of essential dimension is to compute the essential dimension of PGL_n, whose torsors correspond to central simple algebras up to isomorphism.
In this thesis, we study the essential dimension of groups of the form G/μ, where G is a reductive algebraic group satisfying certain properties, and μ is a central subgroup of G. In particular, we consider the case G=GL_(n₁) × ⋯ × GL_(n_r ) where each n_i is a power of a single prime p, which is a generalization of the group PGL_(p^a )=GL_(p^a )/G_m. We will see that torsors for G/μ correspond to tuples of central simple algebras satisfying certain properties. Surprisingly, computing the essential dimension of G/μ becomes easier when r≥3.
Using techniques from Galois cohomology, representation theory and the essential dimension of stacks, we give upper and lower bounds for the essential dimension of G/μ. To do this, we first attach a linear ‘code’ to the central subgroup μ, and define a weight function on this code. Our upper and lower bounds are given in terms of a minimal weight generator matrix for the code. In some cases we can determine the exact value of the essential dimension of G/μ.
2014-04-24T00:00:00ZTourists' visual perceptions of forests and forest management in Vancouver Island and Tasmania
http://hdl.handle.net/2429/46571
In the past, forested areas have been used primarily for timber production. However, the recent growth of nature-based tourism has given monetary value to the recreation and scenic characteristics associated the forests in many places. This can lead to conflicts between forestry and tourism raising questions relating to the management of these two industries. What impact does forestry have on the perception of tourists in region’s that promote natural landscapes? Are certain tourist segments affected differently by the impacts of forestry in regions that market natural landscapes? How can forests be managed to ensure that tourism values are not compromised by other forest interests?
To gain a better perspective of tourism and forestry related conflicts this investigation utilized a comparative case study method. The case study locations selected include Vancouver Island, Canada and Tasmania, Australia. At each destination visitors were surveyed at three types of attractions to understand differences in forestry perceptions between user groups. Forestry and tourism professionals from both regions were also interviewed. Vancouver Island and Tasmania were chosen because of the important role that both forestry and nature-based tourism play in shaping the economies of both places. Despite these similarities, differences exist in the way these two industries are managed. These differences were important for providing insight into management strategies that could be used to address these conflicts.
Results suggest that forestry impacts have the potential to negatively impact upon visitor perceptions. However, this seems partly dependent upon the type of impact observed, as differences were noted between harvested areas, tree plantations, logging trucks and saw/pulp mills. Results from the different sample groups were compared to learn whether or not differences exist in the way that tourist user groups are affected by exposure to forestry impacts. Findings provided a limited amount of evidence to support this. Through the analysis of the semi-structured interviews a set of recommendations were developed to help assist forestry and tourism managers who may be dealing with similar land use conflicts. These recommendations would be useful for land managers in other jurisdictions where similar conflicts between forestry and tourism exist.
2014-04-24T00:00:00Z