Steiner t-designs for large t

One of the most central and long-standing open questions in combinatorial design theory concerns the existence of Steiner t-designs for large values of t. Although in his classical 1987 paper, L. Teirlinck has shown that non-trivial t-designs exist for all values of t, no non-trivial Steiner t-design with t > 5 has been constructed until now. Understandingly, the case t = 6 has received considerable attention. There has been recent progress concerning the existence of highly symmetric Steiner 6-designs: It is shown in [M. Huber, J. Algebr. Comb. 26 (2007), pp. 453-476] that no non-trivial flag-transitive Steiner 6-design can exist. In this paper, we announce that essentially also no block-transitive Steiner 6-design can exist.Comment: 9 pages; to appear in: Mathematical Methods in Computer Science 2008, ed. by J.Calmet, W.Geiselmann, J.Mueller-Quade, Springer Lecture Notes in Computer Scienc

Block-Transitive Designs in Affine Spaces

This paper deals with block-transitive $t$-$(v,k,\lambda)$ designs in affine spaces for large $t$, with a focus on the important index $\lambda=1$ case. We prove that there are no non-trivial 5-$(v,k,1)$ designs admitting a block-transitive group of automorphisms that is of affine type. Moreover, we show that the corresponding non-existence result holds for 4-$(v,k,1)$ designs, except possibly when the group is one-dimensional affine. Our approach involves a consideration of the finite 2-homogeneous affine permutation groups.Comment: 10 pages; to appear in: "Designs, Codes and Cryptography

Beyond Blobs in Percolation Cluster Structure: The Distribution of 3-Blocks at the Percolation Threshold

The incipient infinite cluster appearing at the bond percolation threshold can be decomposed into singly-connected ``links'' and multiply-connected ``blobs.'' Here we decompose blobs into objects known in graph theory as 3-blocks. A 3-block is a graph that cannot be separated into disconnected subgraphs by cutting the graph at 2 or fewer vertices. Clusters, blobs, and 3-blocks are special cases of $k$-blocks with $k=1$, 2, and 3, respectively. We study bond percolation clusters at the percolation threshold on 2-dimensional square lattices and 3-dimensional cubic lattices and, using Monte-Carlo simulations, determine the distribution of the sizes of the 3-blocks into which the blobs are decomposed. We find that the 3-blocks have fractal dimension $d_3=1.2\pm 0.1$ in 2D and $1.15\pm 0.1$ in 3D. These fractal dimensions are significantly smaller than the fractal dimensions of the blobs, making possible more efficient calculation of percolation properties. Additionally, the closeness of the estimated values for $d_3$ in 2D and 3D is consistent with the possibility that $d_3$ is dimension independent. Generalizing the concept of the backbone, we introduce the concept of a ``$k$-bone'', which is the set of all points in a percolation system connected to $k$ disjoint terminal points (or sets of disjoint terminal points) by $k$ disjoint paths. We argue that the fractal dimension of a $k$-bone is equal to the fractal dimension of $k$-blocks, allowing us to discuss the relation between the fractal dimension of $k$-blocks and recent work on path crossing probabilities.Comment: All but first 2 figs. are low resolution and are best viewed when printe

Size and shape analysis of error-prone shape data

We consider the problem of comparing sizes and shapes of objects when landmark data are prone to measurement error. We show that naive implementation of ordinary Procrustes analysis that ignores measurement error can compromise inference. To account for measurement error, we propose the conditional score method for matching configurations, which guarantees consistent inference under mild model assumptions. The effects of measurement error on inference from naive Procrustes analysis and the performance of the proposed method are illustrated via simulation and application in three real data examples. Supplementary materials for this article are available online

Correlation effects in ionic crystals: I. The cohesive energy of MgO

High-level quantum-chemical calculations, using the coupled-cluster approach and extended one-particle basis sets, have been performed for (Mg2+)n (O2-)m clusters embedded in a Madelung potential. The results of these calculations are used for setting up an incremental expansion for the correlation energy of bulk MgO. This way, 96% of the experimental cohesive energy of the MgO crystal is recovered. It is shown that only 60% of the correlation contribution to the cohesive energy is of intra-ionic origin, the remaining part being caused by van der Waals-like inter-ionic excitations.Comment: LaTeX, 20 pages, no figure

Uncertainties in the modelled CO2 threshold for Antarctic glaciation

A frequently cited atmospheric CO2 threshold for the onset of Antarctic glaciation of ∼780 ppmv is based on the study of DeConto and Pollard (2003) using an ice sheet model and the GENESIS climate model. Proxy records suggest that atmospheric CO2 concentrations passed through this threshold across the Eocene-Oligocene transition ∼34 Ma. However, atmospheric CO2 concentrations may have been close to this threshold earlier than this transition, which is used by some to suggest the possibility of Antarctic ice sheets during the Eocene. Here we investigate the climate model dependency of the threshold for Antarctic glaciation by performing offline ice sheet model simulations using the climate from 7 different climate models with Eocene boundary conditions (HadCM3L, CCSM3, CESM1.0, GENESIS, FAMOUS, ECHAM5 and GISS-ER). These climate simulations are sourced from a number of independent studies, and as such the boundary conditions, which are poorly constrained during the Eocene, are not identical between simulations. The results of this study suggest that the atmospheric CO2 threshold for Antarctic glaciation is highly dependent on the climate model used and the climate model configuration. A large discrepancy between the climate model and ice sheet model grids for some simulations leads to a strong sensitivity to the lapse rate parameter

Conformational Free-Energy Landscapes for a Peptide in Saline Environments

AbstractThe conformations that proteins adopt in solution are a function of both their primary structure and surrounding aqueous environment. Recent experimental and computational work on small peptides, e.g., polyK, polyE, and polyR, have highlighted an interesting and unusual behavior in the presence of aqueous ions such as ClO4−, Na+, and K+. Notwithstanding the aforementioned studies, as of this writing, the nature of the driving force induced by the presence of ions and its role on the conformational stability of peptides remains only partially understood. Molecular-dynamics simulations have been performed on the heptapeptide AEAAAEA in NaCl and KCl solutions at concentrations of 0.5, 1.0, and 2.0 M. Metadynamics in conjunction with a three-dimensional model reaction coordinate was used to sample the conformational space of the peptide. All simulations were run for 2 μs. Free-energy landscapes were computed over the model reaction coordinate for the peptide in each saline assay as well as in the absence of ions. Circular dichroism spectra were also calculated from each trajectory. In the presence of Na+ and K+ ions, no increase in helicity is observed with respect to the conformation in pure water

Logarithmic two-loop corrections to the Lamb shift in hydrogen

Higher order $(\alpha/\pi)^2 (Z \alpha)^6$ logarithmic corrections to the hydrogen Lamb shift are calculated. The results obtained show the two-loop contribution has a very peculiar behavior, and significantly alter the theoretical predictions for low lying S-states.Comment: 14 pages, including 2 figures, submitted to Phys. Rev. A, updated with minor change