Graphs with the strong Havel-Hakimi property

The Havel-Hakimi algorithm iteratively reduces the degree sequence of a graph to a list of zeroes. As shown by Favaron, Mah\'eo, and Sacl\'e, the number of zeroes produced, known as the residue, is a lower bound on the independence number of the graph. We say that a graph has the strong Havel-Hakimi property if in each of its induced subgraphs, deleting any vertex of maximum degree reduces the degree sequence in the same way that the Havel-Hakimi algorithm does. We characterize graphs having this property (which include all threshold and matrogenic graphs) in terms of minimal forbidden induced subgraphs. We further show that for these graphs the residue equals the independence number, and a natural greedy algorithm always produces a maximum independent set.Comment: 7 pages, 3 figure

Counting elliptic curves with a cyclic mm-isogeny over Q\mathbb{Q}

Using methods from analytic number theory, for $m > 5$ and for $m = 4$, we obtain asymptotics with power-saving error terms for counts of elliptic curves with a cyclic $m$-isogeny up to quadratic twist over the rational numbers. For $m > 5$, we then apply a Tauberian theorem to achieve asymptotics with power saving error for counts of elliptic curves with a cyclic $m$-isogeny up to isomorphism over the rational numbers.Comment: PhD thesis. The official manuscript for this thesis can be found here: https://digitalcommons.dartmouth.edu/dissertations/214/ 285 pages, 14 graphic

Counting elliptic curves with a cyclic m-isogeny over Q

Using methods from analytic number theory, for m \u3e 5 and for m = 4, we obtain asymptotics with power-saving error terms for counts of elliptic curves with a cyclic m-isogeny up to quadratic twist over the rational numbers. For m \u3e 5, we then apply a Tauberian theorem to achieve asymptotics with power saving error for counts of elliptic curves with a cyclic m-isogeny up to isomorphism over the rational numbers

The arithmetic of modular grids

A modular grid is a pair of sequences $(f_m)_m$ and $(g_n)_n$ of weakly holomorphic modular forms such that for almost all $m$ and $n$, the coefficient of $q^n$ in $f_m$ is the negative of the coefficient of $q^m$ in $g_n$. Zagier proved this coefficient duality in weights $1/2$ and $3/2$ in the Kohnen plus space, and such grids have appeared for Poincar\'{e} series, for modular forms of integral weight, and in many other situations. We give a general proof of coefficient duality for canonical row-reduced bases of spaces of weakly holomorphic modular forms of integral or half-integral weight for every group $\Gamma \subseteq {\text{SL}}_2(\mathbb{R})$ commensurable with ${\text{SL}}_2(\mathbb{Z})$. We construct bivariate generate functions that encode these modular forms, and study linear operations on the resulting modular grids.Comment: Revised versio

A database of basic numerical invariants of Hilbert modular surfaces

We describe algorithms for computing geometric invariants for Hilbert modular surfaces, and we report on their implementation

Flood magnitude-frequency and lithologic control on bedrock river incision in post-orogenic terrain

Mixed bedrock-alluvial rivers - bedrock channels lined with a discontinuous alluvial cover - are key agents in the shaping of mountain belt topography by bedrock fluvial incision. Whereas much research focuses upon the erosional dynamics of such rivers in the context of rapidly uplifting orogenic landscapes, the present study investigates river incision processes in a post-orogenic (cratonic) landscape undergoing extremely low rates of incision (> 5 m/Ma). River incision processes are examined as a function of substrate lithology and the magnitude and frequency of formative flows along Sandy Creek gorge, a mixed bedrock-alluvial stream in arid SE-central Australia. Incision is focused along a bedrock channel with a partial alluvial cover arranged into riffle-pool macrobedforms that reflect interactions between rock structure and large-flood hydraulics. Variations in channel width and gradient determine longitudinal trends in mean shear stress (τb) and therefore also patterns of sediment transport and deposition. A steep and narrow, non-propagating knickzone (with 5% alluvial cover) coincides with a resistant quartzite unit that subdivides the gorge into three reaches according to different rock erodibility and channel morphology. The three reaches also separate distinct erosional styles: bedrock plucking (i.e. detachment-limited erosion) prevails along the knickzone, whereas along the upper and lower gorge rock incision is dependent upon large formative floods exceeding critical erosion thresholds (τc) for coarse boulder deposits that line 70% of the channel thalweg (i.e. transport-limited erosion). The mobility of coarse bed materials (up to 2 m diameter) during late Holocene palaeofloods of known magnitude and age is evaluated using step-backwater flow modelling in conjunction with two selective entrainment equations. A new approach for quantifying the formative flood magnitude in mixed bedrock-alluvial rivers is described here based on the mobility of a key coarse fraction of the bed materials; in this case the d84 size fraction. A 350 m3/s formative flood fully mobilises the coarse alluvial cover with τb200-300 N/m2 across the upper and lower gorge riffles, peaking over 500 N/m2 in the knickzone. Such floods have an annual exceedance probability much less than 10- 2 and possibly as low as 10- 3. The role of coarse alluvial cover in the gorge is discussed at two scales: (1) modulation of bedrock exposure at the reach-scale, coupled with adjustment to channel width and gradient, accommodates uniform incision across rocks of different erodibility in steady-state fashion; and (2) at the sub-reach scale where coarse boulder deposits (corresponding to <i>τ</i><sub>b</sub> minima) cap topographic convexities in the rock floor, thereby restricting bedrock incision to rare large floods. While recent studies postulate that decreasing uplift rates during post-orogenic topographic decay might drive a shift to transport-limited conditions in river networks, observations here and elsewhere in post-orogenic settings suggest, to the contrary, that extremely low erosion rates are maintained with substantial bedrock channel exposure. Although bed material mobility is known to be rate-limiting for bedrock river incision under low sediment flux conditions, exactly how a partial alluvial cover might be spatially distributed to either optimise or impede the rate of bedrock incision is open to speculation. Observations here suggest that the small volume of very stable bed materials lining Sandy Creek gorge is distributed so as to minimise the rate of bedrock fluvial incision over time

Observations of H-alpha, iron, and oxygen lines in B, Be, and shell stars

We have carried out a spectroscopic survey of several B, Be, and shell stars in optical and near-infrared regions. Line profiles of the H-alpha line and of selected Fe II and O I lines are presented.Comment: 23 pages, 45 figures; accepted in Astronomy & Astrophysic

The physiological linkage between molar inclination and dental macrowear pattern

Objectives: Exact symmetry and perfect balance between opposite jaw halves, as well as between antagonistic teeth, is not frequently observed in natural masticatory systems. Research results show that asymmetry in our body, skull, and jaws is often related to genetic, epigenetic, environmental and individual ontogenetic factors. Our study aims to provide evidence for a significant link between masticatory asymmetry and occlusal contact between antagonist teeth by testing the hypothesis that tooth inclination is one of the mechanisms driving distribution of wear in masticatory phases in addition to dietary and cultural habits. Materials and Methods: The present work investigates the relationship between dental macrowear patterns and tooth inclinations on a sample of complete maxillary and mandibular 3D models of dental arches from 19 young and adult Yuendumu Aboriginal individuals. The analysis was carried out on first molars (M1) from all quadrants. Occlusal Fingerprint Analysis was used for the quantification of macrowear patterns, and 2D cross-sectional geometric analysis was carried out to investigate asymmetry in dental arches. Results: The asymmetry is highly variable on both arches, and it is associated with differences in the inclination of upper M1 crowns. Each molar has variable inclination (buccal/lingual) which influence tooth to tooth contact, producing greater or lesser variation in wear pattern. Interindividual variability of morphological variation of the occlusal relationship has to be considered in macrowear analysis. Discussion: Our results suggest that overall asymmetry in the masticatory apparatus in modern humans affects occlusal contact areas between antagonist teeth influencing macrowear and chewing efficiency during ontogeny