Large tilting modules and representation type

We study finiteness conditions on large tilting modules over arbitrary rings. We then turn to a hereditary artin algebra R and apply our results to the (infinite dimensional) tilting module L that generates all modules without preprojective direct summands. We show that the behaviour of L over its endomorphism ring determines the representation type of R. A similar result holds true for the (infinite dimensional) tilting module W that generates the divisible modules. Finally, we extend to the wild case some results on Baer modules and torsion-free modules proven in Angeleri Hügel, L., Herbera, D., Trlifaj, J.: Baer and Mittag-Leffler modules over tame hereditary algebras. Math. Z. 265, 1–19 (2010) for tame hereditary algebras.     

Mathematics identity and achievement among Black students

Employing data from the National Center of Educational Statistics' High School Longitudinal Study and utilizing critical race theory and intersectionality as theoretical frameworks, this article interrogates the relationship between mathematics identity and math success for a nationwide sample of Black secondary school students. More specifically, hierarchical regression modeling is employed to examine the relative impact of math identity, demographic variables, and school/parent social capital variables on the math grade point averages of this sample. The article ends with a discussion of specific steps for teaching mathematics that put the identity of those from traditionally marginalized communities at the center of mathematics instruction. Thus making experiences, histories, culture, and abilities essential elements of students' learning, that are to be supported and built upon.     

TTF Triples in Functor Categories

We characterize the hereditary torsion pairs of finite type in the functor category of a ring R that are associated to tilting torsion pairs in the category of R-modules. Moreover, we determine a condition under which they give rise to TTF triples.     

Orthogonal polynomials in several variables for measures with mass points

Let dν be a measure in ? ^d obtained from adding a set of mass points to another measure dμ. Orthogonal polynomials in several variables associated with dν can be explicitly expressed in terms of orthogonal polynomials associated with dμ, so are the reproducing kernels associated with these polynomials. The explicit formulas that are obtained are further specialized in the case of Jacobi measure on the simplex, with mass points added on the vertices, which are then used to study the asymptotics kernel functions for dν.     

Morphology of the f ? f intensity parameters

The structure of the intensity parameters understood within the framework of ab initio     

Cellulose beads: a new versatile solid support for microwave- assisted synthesis. Preparation of pyrazole and isoxazole libraries

The synthesis of libraries of substituted pyrazoles and isoxazoles has been developed via in situ generation of polymer-bound enaminones. The synthetic protocol makes use of commercially available aniline cellulose, a low-cost and versatile biopolymer, under very mild conditions. This new support allowed us to carry out reactions in polar solvents under both conventional heating and MW irradiation without degradation of the polymer. The reaction between cellulose-bound enaminone and hydroxylamine or hydrazines to afford the target heterocycles in high yields directly in solution is the key step. The support can be conveniently recycled.     

Spiroimidazolidinone library derivatives on SynPhase lanterns

Optimization and solid-phase synthesis of new spiroimidazolidinone derivatives as highly functionalized templates is reported. The synthesis of 1,4,8-triazaspiro[4.5]decan-2-one derivatives was performed on SynPhase lanterns from dipeptides anchored on the solid support and N-protected piperidone. A library of 180 discrete compounds was prepared.     

Perturbative corrections to basis incompleteness in molecular SCF calculations

A unified summary is presented of the mathematical approach developed by McDowell for employing perturbation theory to correct for basis-set incompleteness in ab initio SCF calculations. Revised expressions for the corrections to the wavefunction both in terms of orbitals and spin-orbitals are presented with explicit incorporation of the spin variables. Employing H₂O as an example, we show that this approach is considerably more powerful for computing molecular energies with standard basis sets than was indicated by previous work. In particular at the higher levels of approximation it accurately reproduces the effect of polarization functions in sets such as 6-31G^* and 6-31G^**. The equilibrium molecular structure of H₂O was also computed by this approach and found to give good accuracy. In each case perturbing functions coupled to both occupied and virtual orbitals are required for acceptable results.