Iconicity and contraction: a semiotic investigation of forms of algebraic generalizations of patterns in different contexts

The aim of this paper is to investigate the progressive manner in which students gain fluency with cultural algebraic modes of reflection and action in pattern generalizing tasks. The first section contains a short discussion of some epistemological aspects of generalization. Drawing on this section, a definition of algebraic generalization of patterns is suggested. This definition is used in the subsequent sections to distinguish between algebraic and arithmetic generalizations and some elementary naïve forms of induction to which students often resort to solve pattern problems. The rest of the paper discusses the implementation of a teaching sequence in a Grade 7 class and focuses on the social, sign-mediated processes of objectification through which the students reached stable forms of algebraic reflection. The semiotic analysis puts into evidence two central processes of objectification—iconicity and contraction.     

Dipole and quadrupole moments of molecules in crystals: a novel approach based on integration over Hirshfeld surfaces

Elegant expressions are derived for the computation of dipole and quadrupole moments of molecules using the electrostatic potential and electric field evaluated on an oriented molecular surface. These expressions are implemented for Hirshfeld surfaces, applied to various molecular crystals, and compared with the results from the quantum theory of atoms in molecules. The effect of intermolecular interactions is also explored by examining the differences between electrostatic moments derived from a periodic Hartree-Fock electron density and an electron density resulting from a superposition of noninteracting molecules. The enhancement of the dipole moment for hydrogen bonded molecular crystals is typically 30%-40% and shown to be largely independent of the partitioning scheme. Dipole moments calculated from Hirshfeld surfaces systematically underestimate those from zero-flux surfaces, a result attributed to the translation of the Hirshfeld surface relative to the zero-flux surfaces for these molecules. For acetylene and benzene, the differences between a crystal calculation and the sum of noninteracting molecules are small, and both partitioning schemes yield quadrupole and second moment results in close agreement.     

Pointed Hopf Algebras—from Enveloping Algebras to Quantum Groups and Beyond

A fundamental example of a pointed Hopf algebra is the universal enveloping algebra of a Lie algebra. This example finds its generalization in quantum groups. The class of pointed Hopf algebras is rather extensive and has been the subject of intense study. We recount some of the basic ideas in the development of the theory. A fundamental structure in the general theory is an analog of the enveloping algebra in a certain category, the Nichols algebra.     

Physics with heavy neutron-rich RIBs at the HRIBF

The Holifield Radioactive Ion Beam Facility at Oak Ridge National Laboratory has recently produced the world's first post-accelerated beams of heavy neutron-rich nuclei. The first experiments with these beam are described, and the results discussed. B(E2;0⁺ → 2⁺) values for neutron-rich ^126,128Sn and ^132,134,136Te isotopes have been measured by Coulomb excitation in inverse kinematics. The results for ¹³²Te and ¹³⁴Te (N = 80, 82) show excellent agreement with systematics of lighter Te isotopes, but the B(E2) value for ¹³⁶Te (N = 84) is unexpectedly small. Single-neutron transfer reactions with a ¹³⁴Te beam on ^natBe and ¹³C targets at energies just above the Coulomb barrier have also been studied. Received: 21 March 2002 / Accepted: 16 May 2002 / Published online: 31 October 2002 RID="a" ID="a"e-mail: radford@mail.phy.ornl.gov     

B(E2) upward arrow measurements for radioactive neutron-rich ge isotopes: reaching the N=50 closed shell

The B(E2;0(+)(1)-->2(+)(1)) values for the radioactive neutron-rich germanium isotopes (78,80)Ge and the closed neutron shell nucleus 82Ge were measured at the HRIBF using Coulomb excitation in inverse kinematics. These data allow a study of the systematic trend between the subshell closures at N=40 and 50. The B(E2) behavior approaching N=50 is similar to the trend observed for heavier isotopic chains. A comparison of the experimental results with a shell model calculation demonstrates persistence of the N=50 shell gap and a strong sensitivity of the B(E2) values to the effective interaction.     

Donor-acceptor interactions in crystal engineering

The molecular complex formed between 4-methyltolane and bis(4-N-methylpyridinium)ethyne ditriflate is reported. The X-ray crystal structure indicates that the crystalline superstructure consists of infinite zigzag ribbons of interlocked donor-acceptor complexes separated by triflate counterions.     

Analysis of the laser magnetic resonance spectrum of HO2

An analysis of the previously detected laser magnetic resonance spectrum of HO₂ is carried out by (i) assigning M_J quantum numbers to each observed Zeeman line, (ii) determining the quantum numbers (N′_Ka′Kc′-N″_Ka″Kc″) and energies of the zero-field asymmetric rotor transitions involved, and (iii) determining the values of the zero-field spin-rotation doublet splittings in the upper and lower states of each asymmetric rotor transition. The rotational transitions obtained lie in the region 50–150 cm^?1, with quantum numbers 4 ≤ N ≤ 19 and 1 ≤ K_a ≤ 4. They are fit to an asymmetric rotor program to obtain the three rotational constants A, B, C and the three symmetric-top centrifugal distortion constants D_K, D_NK, D_N. The spin splittings are fit to an approximate theoretical expression involving two adjustable linear combinations of components of the spin-rotation interaction tensor ?. Because of the lack of spectra from other isotopic species, a unique molecular geometry cannot be derived.     

Rotational laser magnetic resonance spectroscopy of PH2(X 2B1)

Laser magnetic resonance spectra arising from eight rotational transitions of PH₂ in its vibronic ground state have been measured in the far infrared region. Analysis of the spectra yields more accurate rotational, centrifugal distortion and fine structure parameters than previous spectroscopic results. Determination of the isotropic an anisotropic hyperfine constant from the resolved ³¹P and proton hyperfine structure shows that the unpaired electron is essentially located in a 3p atomic orbital on the phosphorus atom.