The robertson-taussky inequality revisited

Let a positive definite Hermitian matrix H?Mⁿ (C) be decomposed as H=A + iB, with A, B ? M_nm (R). We give two new proofs of the inequality det H ≤ det A (with equality iff B = 0. each of which vields something futher. One exhibits majorization between the eigenvalues of A and H the other allows proof of the permanental analog per H≥per A.     

Hadamard product submultiplicativity of certain induced norms

In this paper we ask which norms on M_d induced by an absolute vector norm are sub-multiplicative with respect to the Hadamard product. We provide a simple necessary condition for submultiplicativity. We demonstrate that each norm on M_d induced by an l_p norm Hadamard submultiplicative and that the norms induced by certain polyhedral norms are Hadamard submultiplicative. We also consider some related inequalities.     

Designing and scaling up an innovation in abstract algebra

In this paper, we describe the process of designing and scaling up the TAAFU group theory curriculum. This work unfolded in three overlapping stages of research and design. The initial designs emerged along with local instructional theories as the result of small-scale design experiments conducted with pairs of students. A second stage of the research and design process focused on generalizing from the initial laboratory design context to an authentic classroom setting. The third (ongoing) stage involves generalizing to instructors (mathematicians) who were not involved in the design process. We describe each of these stages, and our efforts to investigate the efficacy of the resulting curriculum, in order to provide an illustrative example of the process of scaling up an innovation.     

Implementing inquiry-oriented curriculum: From the mathematicians’ perspective

As part of an effort to scale up an instructional innovation in abstract algebra, several mathematicians have implemented an inquiry-oriented, group theory curriculum. Three of those mathematicians (co-authors here) also participated in iterative rounds of interviews designed to document and investigate their experiences as they worked to implement this curriculum. Analyses of these interviews uncovered three themes that were important to these mathematicians: coverage, goals for student learning, and the role of the teacher. Here, by drawing on interview data, classroom data, and first person commentaries, we will present and discuss each teacher's perspective on these three themes.     

Unprecedented Five‐Coordinate Iron(IV) Imides Generate Divergent Spin States Based on the Imide R‐Groups

Three five‐coordinate iron(IV) imide complexes have been synthesized and characterized. These novel structures have disparate spin states on the iron as a function of the R‐group attached to the imide, with alkyl groups leading to low‐spin diamagnetic (S=0) complexes and an aryl group leading to an intermediate‐spin (S=1) complex. The different spin states lead to significant differences in the bonding about the iron center as well as the spectroscopic properties of these complexes. Mössbauer spectroscopy confirmed that all three imide complexes are in the iron(IV) oxidation state. The combination of diamagnetism and ¹⁵N labeling allowed for the first ¹⁵N NMR resonance recorded on an iron imide. Multi‐reference calculations corroborate the experimental structural findings and suggest how the bonding is distinctly different on the imide ligand between the two spin states.     

A Short and Efficient Synthesis of the [3]Triangulene Ring System

Triangulenes are of current interest for potential applications in molecular electronics. We describe here a three step synthesis of the 4,8,12‐trihydro[3]triangulenium cation by cascade cyclization of a tetra‐benzyl alcohol precursor in triflic acid solution. This stable carbocation is easily observed by NMR and optical spectroscopy and is highly fluorescent. Quenching of the cation into basic solutions or by hydride transfer from triethylsilane provides access to stable dihydro and tetrahydro[3]triangulenes. These neutral species interconvert with cations in a complex series of proton and hydride transfers. This route provides several important [3]triangulene precursors. Preliminary experiments designed to generate [3]triangulene in the solution phase provide evidence for its formation and rapid oligomerization.     

Effect of monovalent ion binding on molecular dynamics of the S100-family calcium-binding protein calbindin D9k

Calbindin D_9k is a member of the S100 subfamily of EF-hand calcium binding proteins, and has served as an important model system for biophysical studies. The fast timescale dynamics of the calcium-free (apo) state is characterized using molecular dynamics simulations. Order parameters for the backbone NH bond vectors are determined from the simulations and compared with experimentally derived values, with a focus on the dynamics of calcium-binding site I. There is a significant discrepancy between simulated and experimental order parameters for site I residues in the case of no ion bound in site I. However, it was found in the simulations that a Na⁺ ion can bind in site I, and the resulting order parameters determined from the simulations are in excellent agreement with experiment. Comparisons are made to X-ray structures of other S100 family members in which Na⁺ ions were observed or suggested to be bound in site I. © 2019 Wiley Periodicals, Inc.     

Dynamics of topological defects in the Lβ′ phase of 1,2-dipalmitoylphosphatidycholine bilayers

We use the optical birefringence of 1,2-dipalmitoylphosphatidycholine bilayers (DPPC) in the gel (L_β′) phase to study recombination dynamics of topological defects. The birefringence of anisotropic thin films, such as the L_β′ phase of DPPC bilayers, is related to their molecular polarizability, different on the heads and the acyl chains. When the sample is cooled down into the L_β′ phase, a period of rapid recombination (taking place over a few seconds) is followed by slow dynamics with metastable states existing in excess of several minutes. After this, either another metastable state or a truly stable state remains where no further change is observed, although a spatially non-uniform distribution of the orientation of the birefringence remains. We compare our results with a model for the free energy and the dynamics of the lipid bilayer in the gel state, finding good qualitative agreement.