Rethinking the discovery function of proof within the context of proofs and refutations

Proof and proving are important components of school mathematics and have multiple functions in mathematical practice. Among these functions of proof, this paper focuses on the discovery function that refers to invention of a new statement or conjecture by reflecting on or utilizing a constructed proof. Based on two cases in which eighth and ninth graders engaged in proofs and refutations, we demonstrate that facing a counterexample of a primitive statement can become a starting point of students’ activity for discovery, and that a proof of the primitive statement can function as a useful tool for inventing a new conjecture that holds for the counterexample. An implication for developing tasks by which students can experience this discovery function is mentioned.ArticleInternational Journal of Mathematical Education in Science and Technology. 45(7):1053-1067 (2014)journal articl

Highly syntenic and yet divergent: a tale of two Theilerias

The published genomic sequences of the two major host-transforming Theileria species of cattle represent a rich resource of information that has allowed novel bioinformatic and experimental studies into these important apicomplexan parasites. Since their publication in 2005, the genomes of T. annulata and T. parva have been utilised for a diverse range of applications, ranging from candidate antigen discovery to the identification of genetic markers for population analysis. This has led to advancements in the quest for a sub-unit vaccine, while providing a greater understanding of variation among parasite populations in the field. The unique ability of these Theileria species to induce host cell transformation is the subject of considerable scientific interest and the availability of full genomic sequences has provided new insights into this area of research. This article reviews the data underlying published comparative analyses, focussing on the general features of gene expression, the major Tpr/Tar multi-copy gene family and a re-examination of the predicted macroschizont secretome. Codon usage between the Theileria species is reviewed in detail, as this underpins ongoing comparative studies investigating selection at the intra- and inter-species level. The TashAT/TpshAT family of genes, conserved between T. annulata and T. parva, encodes products targeted to the host nucleus and has been implicated in contributing to the transformed bovine phenotype. Species-specific expansion and diversification at this critical locus is discussed with reference to the availability, in the near future, of genomic datasets which are based on non-transforming Theileria species

Experiments on sedimentation in wide reservoirs and erosion following dam removal

River morphodynamics and sediment transportSedimentation in reservoir

Primary beam effects of radio astronomy antennas -- II. Modelling the MeerKAT L-band beam

After a decade of design and construction, South Africa's SKA-MID precursor MeerKAT has begun its science operations. To make full use of the widefield capability of the array, it is imperative that we have an accurate model of the primary beam of its antennas. We have taken available L-band full-polarization 'astro-holographic' observations of three antennas and a generic electromagnetic simulation and created sparse representations of the beams using principal components and Zernike polynomials. The spectral behaviour of the spatial coefficients has been modelled using discrete cosine transform. We have provided the Zernike-based model over a diameter of 10 deg averaged over the beams of three antennas in an associated software tool (EIDOS) that can be useful in direction-dependent calibration and imaging. The model is more accurate for the diagonal elements of the beam Jones matrix and at lower frequencies. As we get more accurate beam measurements and simulations in the future, especially for the cross-polarization patterns, our pipeline can be used to create more accurate sparse representations of MeerKAT beams.Comment: 16 pages, 18 figures. This is a pre-copyedited, author-produced PDF of an article accepted for publication in MNRAS following peer review. The version of record [K. M. B. Asad et al., 2021] is available online at: https://doi.org/10.1093/mnras/stab10

Scattering of Straight Cosmic Strings by Black Holes: Weak Field Approximation

The scattering of a straight, infinitely long string moving with velocity $v$ by a black hole is considered. We analyze the weak-field case, where the impact parameter ($b_{imp}$) is large, and obtain exact solutions to the equations of motion. As a result of scattering, the string is displaced in the direction perpendicular to the velocity by an amount $\Delta b\sim -2\pi GMv\gamma/c^3 -\pi (GM)^2/ (4c^3 v b_{imp})$, where $\gamma=(1-(v/c)^2)^{-1/2}$. The second term dominates at low velocities $v/c<(GM/b_{imp})^{1/2}$ . The late-time solution is represented by a kink and anti-kink, propagating in opposite directions at the speed of light, and leaving behind them the string in a new ``phase''. The solutions are applied to the problem of string capture, and are compared to numerical results.Comment: 19 pages, 5 figure

Frog eat frog: exploring variables influencing anurophagy

Background. Frogs are generalist predators of a wide range of typically small prey items. But descriptions of dietary items regularly include other anurans, such that frogs are considered to be among the most important of anuran predators. However, the only existing hypothesis for the inclusion of anurans in the diet of post-metamorphic frogs postulates that it happens more often in bigger frogs. Moreover, this hypothesis has yet to be tested