Australian Mathematical Society (AustMS): E-Journals
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Proofs of Urysohn's lemma and the Tietze extension theorem via the Cantor function
No full textUrysohn's Lemma is a crucial property of normal spaces that deals with separation of closed sets by continuous functions. It is also a fundamental ingredient in proving the Tietze Extension Theorem, another property of normal spaces that deals with the existence of extensions of continuous functions. Using the Cantor function, we give alternative proofs for Urysohn's Lemma and the Tietze Extension Theorem
An optimal linear filter for estimation of random functions in Hilbert space
No full textLet be a square-integrable, zero-mean, random vector with observable realizations in a Hilbert space , and let be an associated square-integrable, zero-mean, random vector with realizations which are not observable in a Hilbert space . We seek an optimal filter in the form of a closed linear operator acting on the observable realizations of a proximate vector that provides the best estimate of the vector . We assume the required covariance operators are known. The results are illustrated with a typical example.
doi:10.1017/S144618112000018
The Rogers--Ramanujan continued fraction and related Eta-Quotient representations
No full textWe construct eta-quotient representations of two families of -series involving the Rogers--Ramanujan continued fraction by establishing related recurrence relations. We also display how these eta-quotient representations could be utilized to dissect certain -series identities
Do poor environmental conditions drive trachoma transmission in Burundi? A mathematical modelling study
No full textTrachoma is an infectious disease and it is the leading cause of preventable blindness worldwide. To achieve its elimination, the World Health Organization set a goal of reducing the prevalence in endemic areas to less than 55% by 2020, utilizing the SAFE (surgery, antibiotics, facial cleanliness, environmental improvement) strategy. However, in Burundi, trachoma prevalences of greater than 55% are still reported in 11 districts and it is hypothesized that this is due to the poor implementation of the environmental improvement factor of the SAFE strategy. In this paper, a model based on an ordinary differential equation, which includes an environmental transmission component, is developed and analysed. The model is calibrated to recent field data and is used to estimate the reductions in trachoma that would have occurred if adequate environmental improvements were implemented in Burundi. Given the assumptions in the model, it is clear that environmental improvement should be considered as a key component of the SAFE strategy and, hence, it is crucial for eliminating trachoma in Burundi.
doi:10.1017/S144618112100038
Further remarks on elementary radicals and associated filters of ideals: Radicals and filters
No full texthttp://dx.doi.org/10.1017/S000497271200033
Influence functions for dimension reduction methods
No full texthttp://dx.doi.org/10.1017/S000497271200033
Optimal location of an underground connector using discounted Steiner tree theory
No full textThe objective of this paper is to demonstrate that the gradient-constrained discounted Steiner point algorithm (GCDSPA) described in an earlier paper by the authors is applicable to a class of real mine planning problems, by using the algorithm to design a part of the underground access in the Rubicon gold mine near Kalgoorlie in Western Australia. The algorithm is used to design a decline connecting two ore bodies so as to maximize the net present value (NPV) associated with the connector. The connector is to break out from the access infrastructure of one ore body and extend to the other ore body. There is a junction on the connector where it splits in two near the second ore body. The GCDSPA is used to obtain the optimal location of the junction and the corresponding NPV. The result demonstrates that the GCDSPA can be used to solve certain problems in mine planning for which currently available methods cannot provide optimal solutions.
doi:10.1017/S144618112000023
Linearly implicit energy-preserving Fourier pseudospectral schemes for the complex modified Korteweg-de Vries equation
No full textWe propose two linearly implicit energy-preserving schemes for the complex modified Korteweg–de Vries equation, based on the invariant energy quadratization method. First, a new variable is introduced and a new Hamiltonian system is constructed for this equation. Then the Fourier pseudospectral method is used for the space discretization and the Crank–Nicolson leap-frog schemes for the time discretization. The proposed schemes are linearly implicit, which is only needed to solve a linear system at each time step. The fully discrete schemes can be shown to conserve both mass and energy in the discrete setting. Some numerical examples are also presented to validate the effectiveness of the proposed schemes.
doi:10.1017/S144618112000021
Note on minimum degree and proper connection number
No full texthttp://dx.doi.org/10.1017/S0004972712000330An edge-colored graph is called \textit{properly connected} if any two vertices are connected by a properly colored path. The \textit{proper connection number} of a graph , denoted by , is the smallest number of colors that are needed to color such that it is properly connected. Let denote the minimum value such that for any 2-connected incomplete graph of order with minimum degree at least . Brause et al. showed in [{\it Minimum degree conditions for the proper connection number of graphs, Graphs and Combinatorics, 33(2017), 833-843}] that \delta(n)>n/42. In this note, we show that \delta(n)>n/36