Weak convergence of Galerkin approximations for fractional elliptic stochastic PDEs with spatial white noise

The numerical approximation of the solution to a stochastic partial differential equation with additive spatial white noise on a bounded domain is considered. The differential operator is assumed to be a fractional power of an integer order elliptic differential operator. The solution is approximated by means of a finite element discretization in space and a quadrature approximation of an integral representation of the fractional inverse from the Dunford–Taylor calculus. For the resulting approximation, a concise analysis of the weak error is performed. Specifically, for the class of twice continuously Fréchet differentiable functionals with second derivatives of polynomial growth, an explicit rate of weak convergence is derived, and it is shown that the component of the convergence rate stemming from the stochasticity is doubled compared to the corresponding strong rate. Numerical experiments for different functionals validate the theoretical results.     

Modification and validation of the mixed-format Engineering Concept Assessment for middle school students using many-facet Rasch measurement

Integrating engineering into the K-12 science curriculum continues to be a focus in national reform efforts in science education. Although there is an increasing interest in research in and practice of integrating engineering in K-12 science education, to date only a few studies have focused on the development of an assessment tool to measure students’ understanding of engineering design. Most of the existing measures focus only on knowledge and understanding of engineering design concepts using multiple-choice items with the exception of the mixed-format Engineering Concept Assessment (ECA). Also, advanced measurement models are lacking application in the testing of such mixed-format assessments in science education. This study applied many-faceted Rasch measurement to the modified ECA for eighth-grade (ECA/M8) and a newly constructed rubric applied by five judges across 497 eighth-grade students’ responses after experiencing an integrated learning unit on the engineering design process. The results supported the fit of the items and rubric rating scales to the Rasch specifications. Recommendations are made for item wording, and further reliability and validity testing of the ECA/M8, and use of the ECA/M8 in science education and research.     

Experimental charge density study of estrogens: 17beta-estradiol.urea

To relate the molecular electrostatic potential to the biological activities of estrogens, a comparative charge density study of different derivatives has been initiated. The second completed charge density analysis of this series for 17beta-estradiol*urea is presented here. This is a large organic system with 52 atoms in a noncentrosymmetric space group, therefore special tools such as an optimal coordinate system and slow, initially constrained refinement have been used to accomplish this study. Our results for the urea molecule reasonably agree with previous experimental and theoretical results. In the 17beta-estradiol molecule, the oxygen atoms appear to be close to sp3 in shape, exhibiting two consistent, distinct lone pairs despite different chemical environments. No significant interaction of the hydroxyl group oxygen with the orbitals of the aromatic ring is observed. Analysis of the electrostatic potential revealed that the negative potential in the lone pair region of the two oxygen atoms is quite different. The topological analysis of the electron density has been performed, and the atomic charges have been estimated.     

Ba6(MN4)N2–x (M = MoVI/TaV), a Subvalent Nitridometalate with Perovskite‐like Crystal Structure

The subvalent nitridometalate Ba₆[(Mo_1–xTa_x)N₄]N_0.86 was prepared from mixtures of Mo powder with Ba, Na, and Ba₂N at 600 °C in Ta ampoules. It crystallizes in space group Cmcm with a = 11.672(3), b = 10.177(2) and c = 10.8729(19) Å. Its crystal structure exhibits an orthorhombically distorted Perovskite topology with [Ba₆N] building units forming the ReO₃‐type lattice via common vertices, and the nitridometalate anions occupying half of the available distorted cuboctahedral interstices. [MN₄] anions show statistically mixed occupancy of M by Mo^VI and Ta^V. They show no notable deviation from nitridometalate anions in known ionic nitridomolybdates and ‐tantalates, and the metrics of the [Ba₆N] octahedra correspond to those found in similar subvalent compounds. The nitrogen atom position centering the [Ba₆N] octahedra is underoccupied. Band structure calculations corroborate the subvalent character of the compound and the two individual anionic structural building units.     

Structures of divalent transition metal salts of 4-aminonaphthalene-1-sulfonate: unexpected variations in layering pattern

Salts of 4-aminonaphthalene-1-sulfonate with divalent Mg, Mn, Co, and Ni cations have been crystallized and their structures determined by single crystal X-ray methods. The Mg and Mn salts are isostructural. Crystal data for hexa-aquamagnesium(II) 4-aminonaphthalene-1-sulfonate dihydrate, [Mg(H₂O)₆](H₂NC₁₀H₆SO₃)₂ 2H ₂O: monoclinic, P2₁/c, Z = 2, a = 8.622(3), b = 7.043(3), c = 23.178(3) Å, =93.78(2)°, V = 1404.3(7) ų; hexa-aquamanganese(II) 4-aminonaphthalene-1-sulfonate dihydrate, [Mn(H₂O)₆](H₂NC₁₀H₆SO₃)₂ 2H ₂O: monoclinic, P2₁/c, Z = 2, a = 8.652(3), b = 7.031(4), c = 23.402(2) Å, =93.09(2)°, V = 1421.5(9) ų. The structures are composed of alternating layers of octahedral metal–aqua complexes and sulfonate anions linked by an extensive network of hydrogen bonds. The extra water molecules of crystallization are located in the hexa-aquametal cation layers. The repeat unit along the c axis is a double layer. The Co and Ni compounds are isostructural with each other, but compared to the Mg and Mn compounds, have a strikingly different structure. Crystal data for hexa-aquacobalt(II) 4-aminonaphthalene-1-sulfonate trihydrate, [Co(H₂O)₆](H₂NC₁₀H₆SO₃)₂ 3H ₂O: orthorhombic, Pbca, Z = 8, a = 8.518(1), b = 14.327(2), c = 45.367(6) Å, V = 5536(1) ų; hexa-aquanickel(II) 4-aminonaphthalene-1-sulfonate trihydrate, [Ni(H₂O)₆](H₂NC₁₀H₆SO₃)₂ 3H ₂O: orthorhombic, Pbca, Z = 8, a = 8.4976(6), b = 14.288(1), c = 45.076(3) Å, V = 5472.9(7) ų. These structures also contain layers of octahedral hexa-aquametal complexes and additional water molecules of crystallization sandwiched by layers of sulfonate anions, however the stacking pattern is more complex with a quadruple layer repeat unit and two different types of anion layers.     

Energetic materials: The preparation and structural characterization of melaminium dinitramide and melaminium nitrate

Two energetic salts of the melaminium cation have been prepared and structurally characterized from room temperature X-ray single crystal diffraction data. Melaminium dinitramide (I), triclinic, P1¯, a = 6.6861(11), b = 6.9638(16), c = 10.447(2) Å , = 99.07(3), = 98.30(3), = 108.50(3)°, V = 445.6(2) ų, and Z = 2. Melaminium nitrate (II), monoclinic, P2₁/c, a = 3.5789(7), b = 20.466(4), c = 10.060(2) Å, = 94.01(2)°, V = 735.0(3) ų, and Z = 4. The crystal structures of both salts show distinct monoprotonated melaminium cations and dinitramide- or nitrate anions, respectively. Efficient packing in the solid state is achieved by extensive hydrogen bonding between two-dimensional zigzag ribbons of the melaminium cations and the respective anions resulting in high densities of the solid state structures of 1.74 (I) and 1.71 g/cm³ (II).     

Di- and triindolylmethanes: molecular structures and spectroscopic characterization of potentially bidentate and tridentate ligands

4-Bromophenyldi(3-methylindol-2-yl)methane (2) and 2-methoxyphenyldi(3-methylindol-2-yl)methane (3) were prepared by sulfuric-acid-catalyzed reactions of 3-methylindole with 4-bromobenzaldehyde and o-anisaldehyde, respectively. Di(3-methylindol-2-yl)phenylmethane (1) and tri(3-methylindol-2-yl)methane (4) were similarly prepared as described previously. Spectroscopic data (¹H, ¹³C NMR) and the X-ray crystal structures for 1 C₂H₅OH and 2–4 are reported. The molecular structure of 1 C₂H₅OH shows hydrogen bonding of both indolyl NH protons to the oxygen of an ethanol molecule. Crystal data for 1 C₂H₅OH: Orthorhombic, Pca2₁, a = 23.9782(17) Å, b = 8.4437(7) Å, c = 11.3029(9) Å, V = 2288.4(3) ų, R ₁ = 0.0597. Crystal data for 2: Orthorhombic, P2₁2₁2₁, a = 8.911(3) Å, b = 9.584(4) Å, c = 24.040(11) Å, V = 2053.0(14) ų, R ₁ = 0.0454. Crystal data for 3: Monoclinic, P2₁/c, a = 9.737(2) Å, b = 25.035(6) Å, c = 9.359(2) Å, = 114.853(4), V = 2070.2(8) ų, R ₁ = 0.0511. Crystal data for 4: Trigonal, R3, a = 14.2214(10) Å, c = 9.6190(10) Å, V = 1684.8(2) ų, R ₁ = 0.0425.