Algorithmic construction of low-discrepancy point sets via dependent randomized rounding

We provide a deterministic algorithm that constructs small point sets exhibiting a low star discrepancy. The algorithm is based on recent results on randomized roundings respecting hard constraints and their derandomization. It is structurally much simpler than a previous algorithm presented for this problem in [B. Doerr, M. Gnewuch, A. Srivastav, Bounds and constructions for the star discrepancy via δ$\delta$-covers, J. Complexity, 21 (2005) 691–709]. Besides leading to better theoretical running time bounds, our approach also can be implemented with reasonable effort. We implemented this algorithm and performed numerical comparisons with other known low-discrepancy constructions. The experiments take place in dimensions ranging from 5 to 21 and indicate that our algorithm leads to superior results if the dimension is relatively high and the number of points that have to be constructed is rather small.     

Understanding How Students Develop Mathematical Models

in this article, we discuss findings from a research study designed to characterize students' development of significant mathematical models by examining the shifts in their thinking that occur during problem investigations. These problem investigations were designed to elicit the development of mathematical models that can be used to describe and explain the relations, patterns, and structure found in data from experienced situations. We were particularly interested in a close examination of the student interactions that appear to foster the development of such mathematical models. This classroom-based qualitative case study was conducted with precalculus students enrolled in a moderate-sized private research university. We observed several groups of 3 students each as they worked together on 5 different modeling tasks. In each task, the students were asked to create a quantitative system that could describe and explain the patterns and structures in an experienced situation and that could be used to make predictions about the situation. Our analysis of the data revealed 4 sources of mismatches that were significant in bringing about the occurrence of shifts in student thinking: conjecturing, questioning, impasses to progress, and the use of technology-based representations. The shifts in thinking in turn led to the development of mathematical models. These results suggest that students would benefit from learning environments that provide them with ample opportunity to express their ideas, ask questions, make reasoned guesses, and work with technology while engaging in problem situations that elicit the development of significant mathematical models.     

An η3-Bound Allyl Ligand on Magnesium in a Mechanochemically Generated Mg/K Allyl Complex

Milling two equivalents of K[1,3-(SiMe₃)₂C₃H₃] (=K[A′]) with MgX₂ (X=Cl, Br) produces the allyl complex [K₂MgA′₄] ( 1 ). Crystals grown from toluene are of the solvated species [((η⁶-tol)K)₂MgA′₄] ([ 1 ⋅2(tol)]), a trimetallic monomer with both bridging and terminal (η¹) allyl ligands. When recrystallized from hexanes, the unsolvated 1 forms a 2D coordination polymer, in which the Mg is surrounded by three allyl ligands. The C−C bond lengths differ by only 0.028 Å, indicating virtually complete electron delocalization. This is an unprecedented coordination mode for an allyl ligand bound to Mg. DFT calculations indicate that in isolation, an η³-allyl configuration on Mg is energetically preferred over the η¹- (σ-bonded) arrangement, but the Mg must be in a low coordination environment for it to be experimentally realized. Methyl methacrylate is effectively polymerized by 1 , with activities that are comparable to K[A′] and greater than the homometallic magnesium complex [{MgA′₂}₂].     

Lattice distortions in layered type arsenides LnTAs2 (Ln=La-Nd, Sm, Gd, Tb; T=Ag, Au): Crystal structures, electronic and magnetic properties

The lanthanide coinage-metal diarsenides LnTAs₂ (Ln=La, Ce-Nd, Sm; T=Ag, Au) have been reinvestigated and their structures have been refined from single crystal X-ray data. Two different distortion variants of the HfCuSi₂ type are found: PrAgAs₂, NdAgAs₂, SmAgAs₂, GdAgAs₂, TbAgAs₂, NdAuAs₂ and SmAuAs₂ crystallize as twofold superstructures in space group Pmcn with the As atoms of their planar layers forming zigzag chains, whereas LaAgAs₂, CeAgAs₂ and PrAuAs₂ adopt a fourfold superstructure (space group Pmca) with cis-trans chains of As atoms. The respective atomic positions can be derived from the HfCuSi₂ type by group-subgroup relations. The compounds with zigzag chains of As atoms exhibit metallic behaviour while those with cis-trans chains are semiconducting as measured on powder pellets. The majority of the compounds including 4f elements show antiferromagnetic ordering at T_N<20 K.     

QM/MM study of the absorption spectra of DsRed.M1 chromophores

We report geometries and vertical excitation energies for the red and green chromophores of the DsRed.M1 protein in the gas phase and in the solvated protein environment. Geometries are optimized using density functional theory (DFT, B3LYP functional) for the isolated chromophores and combined quantum mechanical/molecular mechanical (QM/MM) methods for the protein (B3LYP/MM). Vertical excitation energies are computed using DFT/MRCI, OM2/MRCI, and TDDFT as QM methods. In the case of the red chromophore, there is a general blue shift in the excitation energies when going from the isolated chromophore to the protein, which is caused both by structural changes and by electrostatic interactions with the environment. For the lowest ππ* transition, these two factors contribute to a similar extent to the overall DFT/MRCI shift of 0.4 eV. An enlargement of the QM region to include active‐site residues does not change the DFT/MRCI excitation energies much. The DFT/MRCI results are closest to experiment for both chromophores. OM2/MRCI and TDDFT overestimate the first vertical excitation energy by 0.3–0.5 and 0.2–0.4 eV, respectively, relative to the experimental or DFT/MRCI values. The experimental gap of 0.35 eV between the lowest ππ* excitation energies of the red (cis‐acylimine) and green (trans‐peptide) forms is well reproduced by DFT/MRCI and TDDFT (0.32 and 0.37 eV, respectively). A histogram spectrum for an equal mixture of the two forms, generated by OM2/MRCI calculations on 450 snapshots along molecular dynamics trajectories, matches the experimental spectrum quite well, with a gap of 0.23 eV and an overall blue shift of about 0.3 eV. DFT/MRCI appears as an attractive choice for calculating excitation energies in fluorescent proteins, without the shortcomings of TDDFT and computationally more affordable than CASSCF‐based approaches. © 2009 Wiley Periodicals, Inc. J Comput Chem, 2010     

Giant magnetostrain based on strong single ion anisotropy of rare earth materials

The volume, shape and microstructure of solids can be influenced by magnetic fields. Much effort is focused on magnetic shape memory (MSM) materials. Recently, the MSM effect has been discovered to occur also in the paramagnetic state, e.g. in RCu₂ compounds (R = rare earth). RMSM materials distinguish themselves from conventional MSM materials by the new origin of the magnetoic anisotropy: the strong rare-earth single ion anisotropy. Due to the pseudo-hexagonal symmetry of RCu₂, three orientational variants exists, each of them rotated by about 60 deg with respect to the others. Switching these variants by an external field results in a change of the macroscopic shape. The strain is in the order of one percent (= Giant MagnetoStrain). The variant's fraction remains unchanged when ramping down the field. The virgin state can be recovered by heating or by a perpendicularly directed field. We present temperature and field dependent measurements of magnetostrain and magentization at the model substance Tb_0.5Dy_0.5Cu₂. The macroscopic characterization of the sample is complemented by a detailed microscopic analysis done by elastic neutron scattering. Although the GMS effect of RCu₂ was worked out at single crystals, the principle of this magneto-mechanical coupling phenomenon is also useful for polycrystalline or microscaled applications. The existence of this structural irreversibility shows the potential to construct field controlled actuators or switches.