Extension of Gutenberg-Richter Distribution to Mw -1.3, No Lower Limit in Sight

With twelve years of seismic data from TauTona Gold Mine, South Africa, we show that mining-induced earthquakes follow the Gutenberg-Richter relation with no scale break down to the completeness level of the catalog, at moment magnitude MW −1.3. Events recorded during relatively quiet hours in 2006 indicate that catalog detection limitations, not earthquake source physics, controlled the previously reported minimum magnitude in this mine. Within the Natural Earthquake Laboratory in South African Mines (NELSAM) experiment\u27s dense seismic array, earthquakes that exhibit shear failure at magnitudes as small as MW −3.9 are observed, but we find no evidence that MW −3.9 represents the minimum magnitude. In contrast to previous work, our results imply small nucleation zones and that earthquake processes in the mine can readily be scaled to those in either laboratory experiments or natural faults

Cellular spanning trees and Laplacians of cubical complexes

We prove a Matrix-Tree Theorem enumerating the spanning trees of a cell complex in terms of the eigenvalues of its cellular Laplacian operators, generalizing a previous result for simplicial complexes. As an application, we obtain explicit formulas for spanning tree enumerators and Laplacian eigenvalues of cubes; the latter are integers. We prove a weighted version of the eigenvalue formula, providing evidence for a conjecture on weighted enumeration of cubical spanning trees. We introduce a cubical analogue of shiftedness, and obtain a recursive formula for the Laplacian eigenvalues of shifted cubical complexes, in particular, these eigenvalues are also integers. Finally, we recover Adin's enumeration of spanning trees of a complete colorful simplicial complex from the cellular Matrix-Tree Theorem together with a result of Kook, Reiner and Stanton.Comment: 24 pages, revised version, to appear in Advances in Applied Mathematic

Simplicial matrix-tree theorems

We generalize the definition and enumeration of spanning trees from the setting of graphs to that of arbitrary-dimensional simplicial complexes $\Delta$, extending an idea due to G. Kalai. We prove a simplicial version of the Matrix-Tree Theorem that counts simplicial spanning trees, weighted by the squares of the orders of their top-dimensional integral homology groups, in terms of the Laplacian matrix of $\Delta$. As in the graphic case, one can obtain a more finely weighted generating function for simplicial spanning trees by assigning an indeterminate to each vertex of $\Delta$ and replacing the entries of the Laplacian with Laurent monomials. When $\Delta$ is a shifted complex, we give a combinatorial interpretation of the eigenvalues of its weighted Laplacian and prove that they determine its set of faces uniquely, generalizing known results about threshold graphs and unweighted Laplacian eigenvalues of shifted complexes.Comment: 36 pages, 2 figures. Final version, to appear in Trans. Amer. Math. So

Simplicial and Cellular Trees

Much information about a graph can be obtained by studying its spanning trees. On the other hand, a graph can be regarded as a 1-dimensional cell complex, raising the question of developing a theory of trees in higher dimension. As observed first by Bolker, Kalai and Adin, and more recently by numerous authors, the fundamental topological properties of a tree --- namely acyclicity and connectedness --- can be generalized to arbitrary dimension as the vanishing of certain cellular homology groups. This point of view is consistent with the matroid-theoretic approach to graphs, and yields higher-dimensional analogues of classical enumerative results including Cayley's formula and the matrix-tree theorem. A subtlety of the higher-dimensional case is that enumeration must account for the possibility of torsion homology in trees, which is always trivial for graphs. Cellular trees are the starting point for further high-dimensional extensions of concepts from algebraic graph theory including the critical group, cut and flow spaces, and discrete dynamical systems such as the abelian sandpile model.Comment: 39 pages (including 5-page bibliography); 5 figures. Chapter for forthcoming IMA volume "Recent Trends in Combinatorics

A non-partitionable Cohen-Macaulay simplicial complex

A long-standing conjecture of Stanley states that every Cohen-Macaulay simplicial complex is partitionable. We disprove the conjecture by constructing an explicit counterexample. Due to a result of Herzog, Jahan and Yassemi, our construction also disproves the conjecture that the Stanley depth of a monomial ideal is always at least its depth.Comment: Final version. 13 pages, 2 figure

Nuclear Mixing Meters for Classical Novae

Classical novae are caused by mass transfer episodes from a main sequence star onto a white dwarf via Roche lobe overflow. This material forms an accretion disk around the white dwarf. Ultimately, a fraction of this material spirals in and piles up on the white dwarf surface under electron-degenerate conditions. The subsequently occurring thermonuclear runaway reaches hundreds of megakelvin and explosively ejects matter into the interstellar medium. The exact peak temperature strongly depends on the underlying white dwarf mass, the accreted mass and metallicity, and the initial white dwarf luminosity. Observations of elemental abundance enrichments in these classical nova events imply that the ejected matter consists not only of processed solar material from the main sequence partner but also of material from the outer layers of the underlying white dwarf. This indicates that white dwarf and accreted matter mix prior to the thermonuclear runaway. The processes by which this mixing occurs require further investigation to be understood. In this work, we analyze elemental abundances ejected from hydrodynamic nova models in search of elemental abundance ratios that are useful indicators of the total amount of mixing. We identify the abundance ratios $\Sigma$CNO/H, Ne/H, Mg/H, Al/H, and Si/H as useful mixing meters in ONe novae. The impact of thermonuclear reaction rate uncertainties on the mixing meters is investigated using Monte Carlo post-processing network calculations with temperature-density evolutions of all mass zones computed by the hydrodynamic models. We find that the current uncertainties in the $^{30}$P($p$,$\gamma$)$^{31}$S rate influence the Si/H abundance ratio, but overall the mixing meters found here are robust against nuclear physics uncertainties. A comparison of our results with observations of ONe novae provides strong constraints for classical nova models

Broken-Symmetry Unrestricted Hybrid Density Functional Calculations on Nickel Dimer and Nickel Hydride

In the present work we investigate the adequacy of broken-symmetry unrestricted density functional theory (DFT) for constructing the potential energy curve of nickel dimer and nickel hydride, as a model for larger bare and hydrogenated nickel cluster calculations. We use three hybrid functionals: the popular B3LYP, Becke's newest optimized functional Becke98, and the simple FSLYP functional (50% Hartree-Fock and 50% Slater exchange and LYP gradient-corrected correlation functional) with two basis sets: all-electron (AE) Wachters+f basis set and Stuttgart RSC effective core potential (ECP) and basis set. We find that, overall, the best agreement with experiment, comparable to that of the high-level CASPT2, is obtained with B3LYP/AE, closely followed by Becke98/AE and Becke98/ECP. FSLYP/AE and B3LYP/ECP give slightly worse agreement with experiment, and FSLYP/ECP is the only method among the ones we studied that gives an unaceptably large error, underestimating the dissociation energy of nickel dimer by 28%, and being in the largest disagreement with the experiment and the other theoretical predictions.Comment: 17 pages, 7 tables, 7 figures; submitted to J. Chem. Phys.; Revtex4/LaTeX2e. v2 (8/5/04): New (and better) ECP results, without charge density fitting (which was found to give large errors). Subtracted the relativistic corrections from all experimental value

The Differential Warming Response of Britain's Rivers (1982-2011)

River water temperature is a hydrological feature primarily controlled by topographical, meteorological, climatological, and anthropogenic factors. For Britain, the study of freshwater temperatures has focussed mainly on observations made in England and Wales; similar comprehensive data sets for Scotland are currently unavailable. Here we present a model for the whole of mainland Britain over three recent decades (1982-2011) that incorporates geographical extrapolation to Scotland. The model estimates daily mean freshwater temperature for every river segment and for any day in the studied period, based upon physico-geographical features, daily mean air and sea temperatures, and available freshwater temperature measurements. We also extrapolate the model temporally to predict future warming of Britain's rivers given current observed trends. Our results highlight the spatial and temporal diversity of British freshwater temperatures and warming rates. Over the studied period, Britain's rivers had a mean temperature of 9.84°C and experienced a mean warming of +0.22°C per decade, with lower rates for segments near lakes and in coastal regions. Model results indicate April as the fastest-warming month (+0.63°C per decade on average), and show that most rivers spend on average ever more days of the year at temperatures exceeding 10°C, a critical threshold for several fish pathogens. Our results also identify exceptional warming in parts of the Scottish Highlands (in April and September) and pervasive cooling episodes, in December throughout Britain and in July in the southwest of England (in Wales, Cornwall, Devon, and Dorset). This regional heterogeneity in rates of change has ramifications for current and future water quality, aquatic ecosystems, as well as for the spread of waterborne diseases