Crystallochemistry of the novel two-layer RECuMg4 (RE=La, Tb) ternary compounds

The crystal structures of the new ternary compounds LaCuMg₄ and TbCuMg₄ were studied by X-ray powder diffraction and single-crystal methods, respectively. Scanning electron microscopy (SEM) coupled with energy dispersive X-ray spectroscopy (EDXS) was used for examining microstructure and phase composition. LaCuMg₄ crystallizes in the UCoAl₄ structure type (space group P6¯2m, Pearson code hP18, a=1.03911(1), c=0.45126(1) nm, Z=3, R_F=0.0654), while TbCuMg₄ exhibits a new structure (space group Cmmm, Pearson code oS48, a=1.35797(6), b=2.03333(9), c=0.39149(2) nm, Z=8, wR₂=0.0426). Both structures represent a family of two-layer compounds. All interatomic distances indicate metallic type bonding. The structural peculiarities of these compounds and their relations are discussed.     

Phase-fixed double-group 3-Γ symbols. I. A novel exposition of the general theory of 3-Γ symbols and coupling coefficients

The present paper is the first in a series aiming at the establishment of a transparent and readily applicable Wigner-Racah algebra for all the noncommutative double groups.Starting from the Wigner-Eckart theorem in a very general setting, the theory of the fundamental quantities called here triple coefficients — and the closely related coupling coefficients — is developed and leads through a careful discussion of permutational symmetries to the concept of 3- symbols. By basing the exposition on triple coefficients and by consistently using matrix representations, we obtain a notation and a terminology which enable a clear separation of permutational properties and problems concerning complex conjugation, and a more transparent discussion of tensor (Kronecker) product multiplicities.A particularly elegant formalism is obtained for a situation which generalizes that of the classical rotation-group Wigner-Racah algebra, viz., in which there is a fixed group element effecting (through the inner automorphism it defines) complex conjugation of all the standard irreducible matrix representations.     

Flexible thinking and met-befores: Impact on learning mathematics

In this paper we study the difficulties resulting from changes in meaning of the minus sign, from an operation on numbers, to a sign designating a negative number, to the additive inverse of an algebraic symbol on students in two-year colleges and universities. Analysis of the development of algebra reveals that these successive meanings that the student has met before often become problematic, leading to a fragile knowledge structure that lacks flexibility and leads to confusion and long-term disaffection. The problematic aspects that arise from changes in meaning of the minus sign are identified and the iconic function machine is utilized as a supportive strategy, along with formative assessment to encourage teachers and learners to seek more flexible and effective ways of making sense of increasingly sophisticated mathematics.     

Classification and Representations of Low-Dimensional Nanomaterials: Terms and Symbols

A simple scheme, with special terms and symbols useful in categorizing various nanostructures, is introduced. Using “n-D in/on m-D” composite nanomaterials where n,m ≤ 2 as examples, we illustrate how these terms and symbols can be used to represent various hetero nanostructures. This simple nomenclature system also allows a systematization of a wide variety of multi-dimensional nanocomposite heterostructures.     

A -analogue of the 9- symbols and their orthogonality

We introduce a q-analogue of Wigner’s 9-j symbols following the notational scheme used by Wilson in identifying the 6-j symbols with Racah polynomials, which eventually led Askey and Wilson to obtain a q-analogue of them, namely, the q-Racah polynomials. Most importantly, we prove the orthogonality of our analogues in complete generality, as well as derive an explicit polynomial expression for these new functions.     

多圆盘上的交换对偶Toeplitz算子

We completely characterize commutativity of Sρ and Sψ on L2α (Dn)⊥ for bounded pluriharmonic symbols ρ and ψ on Dn, and prove that SρSψ=Sψρ if and only if ρ is analytic or ψ is analytic.     

Module supersingulier et homologie des courbes modulaires

We study here the free group generated by isomorphism classes of supersingular elliptic curves in positive characteristic. We compare this -module to the homology of the modular curve X₀(p). We give an interpretation of Gross formula for special values of L-functions of modular forms in this context. As an application, we obtain a formula for the sum of the squares of the optimal embeddings numbers of quadratic orders in a definite quaternion algebra.     

Power Residue Symbols and the Central Sections of SL(2, A)

In TheE(2,A) sections of SL(2,A) (Ann. of Math. 134 (1991), 159–188), we locate the E(A) normalized subgroups of SL(2,A) in central sections of SL(2,A) for all subrings of Q and all commutative rings satisfying SR ₂ In solving this problem we introduced the notion of radix (see (1.1)) and the group C(P_x) = [E(2,A),E(2,A;P_x)] = [SL(2,A), SL(2,A;P_x)] for the rings considered here.The purpose of this paper is to determine SL(2,A;P_xC(P_x) for SR ₂ rings and number rings with infinitely many units.In Section 2, Mennicke symbols for Jordan ideals are defined. They are determined for number rings and shown to be connected to power residue symbols in a delicate way. This extends the work of Bass, Milnor and Serre.In Section 3, an explicit homomorphism from E(2,Al;P_x) into an additive section of A is given for all commutative rings A. If A satisfiesSR ₂ the kernel of this map is C(P_x.The main problem for number rings is solved by giving an explicit homomorphism on SL(2,A;P_x) whose kernel is C(P_x).     

Phase-fixed double-group 3-Γ symbols. V. 3-Γ symbols and coupling coefficients for all the octahedral double group-subgroup hierarchies

For all the subgroup hierarchies descending from the octahedral double group O ^*, we have obtained sets of 3- symbols and discuss here their properties. We have entirely real sets of 3- symbols for the tetrahedral and tetragonal hierarchies as well as for O ^* C ₃ ^* . For the latter hierarchy and the tetragonal ones, formalisms almost as powerful as the classical one for the rotation group may be established. We also discuss results obtained for cases with strict adaption to D ₃ ^* where it is now known that non-real 3- symbols are unavoidable.The 3- symbols are phase-fixed by the specification of basis functions (or, equivalently, subduction coefficients) generating them.The significance of the concept of associated representations of O* is discussed. The problems raised by the two multiplicity triples UT₁U and UT₂U in O* are given particular attention.