Neither even nor odd: Sixth grade students’ dilemmas regarding the parity of zero

This study investigates two sixth grade students’ dilemmas regarding the parity of zero. Both students originally claimed that zero was neither even nor odd. Interviews revealed a conflict between students’ formal definitions of even numbers and their concept images of even numbers, zero, and division. These images were supported by practically based explanations relying on everyday contexts. By using mathematically based explanations that rely solely on mathematical notions, students were able to correctly conclude that zero is an even number. Extending the natural number system in elementary school to include zero can be used as springboard to encourage the use of mathematically based explanations.     

Über die Konstruktion von Parallelen mit Hilfe des Lineals und ortsfester rechter Winkel

Ohne ZusammenfassungHerrnRobert Sauer zum 60. Geburtstag gewidmet     

Changes in vibrational modes near the magnetic phase transition in magnetostrictive materials

Mössbauer studies of ⁵⁷Fe in RFe₂ (R = Y, Tb, Dy, Ho) at temperatures 300 to 800 K and in R(Co_0.98Fe_0.02)₂ (R = Tb, Dy) at temperatures 85 to 300 K have been performed. The spectra yield the temperature dependence of the easy axis of magnetization, the magnetic hyperfine fields, electric field gradients and isomer shift. From the total Mössbauer absorption area the recoil free fraction was determined. The relative intensities of the various Mössbauer absorption lines yield the anisotropy in the recoil free fraction. In all compounds, except in YFe₂, all measured quantities show large variations near the magnetic phase transition temperature (T_c). The variation in the recoil free fraction and in its anisotropy and the variation in the electric field gradient are all consistent, quantitatively, with anisotropic softening in the vibration modes of the Fe nucleus near T_c In RFe₂ the modes in which the iron vibrates along the local C_3h symmetry axes are softened less than the perpendicular modes. In RCo₂ it is the other way around. It seems, that these critical softening phenomena are due to the strong magnon-phonon coupling present in these materials and absent in YFe₂. This conclusion is confirmed by a theoretical analysis given in an adjacent paper. In it we use the Green function method to calculate the average of the square of vibrational amplitudes in a Debye solid with a strong magnon-phonon coupling at temperatures close to the magnetic phase transition. We show that these amplitudes may increase or decrease near T_c depending on the details of the coupled system.     

Stabilization of the tautomers HP(OH)2 and P(OH)3 of hypophosphorous and phosphorous acids as ligands

Treatment of [CpRu(PPh(3))(2)Cl] 1 with the stoichiometric amount of H(3)PO(2) or H(3)PO(3) in the presence of chloride scavengers (AgCF(3)SO(3) or TlPF(6)) yields compounds of formula [CpRu(PPh(3))(2)(HP(OH)(2))]Y (Y = CF(3)SO(3) 2a or PF(6) 2b) and [CpRu(PPh(3))(2)(P(OH)(3))]Y (Y = CF(3)SO(3) 3aor PF(6) 3b) which contain, respectively, the HP(OH)(2) and P(OH)(3) tautomers of hypophosphorous and phosphorous acids bound to ruthenium through the phosphorus atom. The triflate derivatives 2a and 3a react further with hypophosphorous or phosphorous acids to yield, respectively, the complexes [CpRu(PPh(3))(HP(OH)(2))(2)]CF(3)SO(3) 4 and [CpRu(PPh(3))(P(OH)(3))(2)]CF(3)SO(3) 5 which are formed by substitution of one molecule of the acid for a coordinated triphenylphosphine molecule. The compounds 2 and 3 are quite stable in the solid state and in solutions of common organic solvents, but the hexafluorophosphate derivatives undergo easy transformations in CH(2)Cl(2): the hypophosphorous acid complex 2b yields the compound [CpRu(PPh(3))(2)(HP(OH)(2))]PF(2)O(2) 6, whose difluorophosphate anion originates from hydrolysis of PF(6)(-); the phosphorous acid complex 3b yields the compound [CpRu(PPh(3))(2)(PF(OH)(2))]PF(2)O(2) 7, which is produced by hydrolysis of hexafluorophosphate and substitution of a fluorine for an OH group of the coordinated acid molecule. All the compounds have been characterized by elemental analyses and NMR measurements. The crystal structures of 2a, 3a and 7 have been determined by X-ray diffraction methods.     

Mean values with cubic characters

We investigate various mean value problems involving order 3 primitive Dirichlet characters. In particular, we obtain an asymptotic formula for the first moment of central values of the Dirichlet L-functions associated to this family, with a power saving in the error term. We also obtain a large sieve-type result for order 3 (and 6) Dirichlet characters.     

A task that elicits reasoning: A dual analysis

This paper reports on the forms of reasoning elicited as fourth grade students in a suburban district and sixth grade students in an urban district worked on similar tasks involving reasoning with the use of Cuisenaire rods. Analysis of the two data sets shows similarities in the reasoning used by both groups of students on specific tasks, and the tendency of a particular task to elicit numerous forms of reasoning in both groups of students. Attributes of that task and ways that those attributes can be replicated in other domains may have implications in the teaching of early reasoning.     

Kinetic analysis of the reaction between (V2O5)n=1,2+ and ethylene

A systematic experimental and theoretical investigation of the influence of reactant energy on the reactivity of (V(2)O(5))(n)=1,2(+) clusters with ethylene (Justes, D. R.; Mitri?, R.; Moore, N. A.; Bonaci?-Koutecky, V.; Castleman, A. W., Jr. J. Am. Chem. Soc., 2003, 125, 6289) provided evidence of the rate controlling steps in the reaction. Herein, we present further experimental and theoretical evidence for our recently proposed radical cation mechanism for oxygen atom transfer from (V(2)O(5))(n)=1,2(+) clusters to ethylene. In particular the results of ab initio molecular dynamics simulations are found to further support the radical cation mechanism. Experimental reaction cross sections at the zero pressure limit and rate coefficients show that the energy dependence of the reaction cross section is in accord with the Langevin formula. Evidence is presented that ion-molecule association is the rate determining step, whereas subsequent hydrogen transfer and formation of acetaldehyde proceed without significant barriers. We propose a kinetic model for the reaction cross section that fully accounts for the experimental findings. The model offers the prospect of elucidating the details of the general reaction mechanisms through a study of the energy dependence of the reaction cross sections.     

Lepton-pair production from deep inelastic scatteringin peripheral relativistic heavy ion collisions

We calculate the inelastic electron- and muon-pair production in peripheral relativistic heavy ion collisions in the region of large Q² of one of the photons. This offers a possibility to study the quark distribution functions in ions in "ultraperipheral heavy ion collisions". The calculations are compared with those making use of the equivalent photon and the equivalent lepton approximation. We compare the results for Pb-Pb and Pb-p collisions at RHIC (γ ≈ 100) and LHC (γ ≈ 3000) energies. Furthermore we include nuclear modifications to the parton distribution functions in our calculations to study their effect on the cross sections.     

Divalent Sialylated Precision Glycooligomers Binding to Polyomaviruses and the Effect of Different Linkers

Divalent precision glycooligomers terminating in N‐acetylneuraminic acid (Neu5Ac) or 3′‐sialyllactose (3′‐SL) with varying linkers between scaffold and the glycan portions are synthesized via solid phase synthesis for co‐crystallization studies with the sialic acid‐binding major capsid protein VP1 of human Trichodysplasia spinulosa‐associated Polyomavirus. High‐resolution crystal structures of complexes demonstrate that the compounds bind to VP1 depending on the favorable combination of carbohydrate ligand and linker. It is found that artificial linkers can replace portions of natural carbohydrate linkers as long as they meet certain requirements such as size or flexibility to optimize contact area between ligand and receptor binding sites. The obtained results will influence the design of future high affinity ligands based on the structures presented here, and they can serve as a blueprint to develop multivalent glycooligomers as inhibitors of viral adhesion.