Widder’s Representation Theorem for Symmetric Local Dirichlet Spaces

In classical PDE theory, Widder’s theorem gives a representation for non-negative solutions of the heat equation on \(\mathbb{R }^n\) . We show that an analogous theorem holds for local weak solutions of the canonical “heat equation” on a symmetric local Dirichlet space satisfying a local parabolic Harnack inequality.     

Relaxed fabrication tolerance for self-imaging photonic crystal waveguide splitters using a tapered multimode interference region

The power imbalance between different waveguide outputs is compensated by manipulating the dispersion of the guided propagation in the multimode interference (MMI) region. This is attainable using a tapered region at the beginning of the MMI region that has been verified through simulation and experiment. From this, the fabrication tolerance for the diameters of holes in a tapered 1×3 photonic crystal waveguide (PhCW) splitter is relaxed up to a range of at least 27 nm. The output power is well-balanced to within 1 dB. The effective bandwidth of the splitters shifts only around 13 nm, for a reduction of 10 nm in the diameter of the PhCW holes. The optimized component is an outstanding ultracompact 1×3 splitter for the photonic integrated circuit (PIC).     

Elimination of standing wave effects in ultrasound radiation force excitation in air using random carrier frequency packets

The ultrasound radiation force has been used for noncontact excitation of devices ranging from microcantilevers to acoustic guitars. For ultrasound radiation force excitation, one challenge is formation of standing waves between the ultrasound transducer and the device under test. Standing waves result in constructive/destructive interference causing significant variations in the intensity of the ultrasound field. The standing-wave induced intensity variations in the radiation force can result from minor changes in the transducer position, carrier frequency, or changes in the speed of sound due to changes in ambient temperature. The current study demonstrates that by randomly varying the ultrasound carrier frequency in packets, it is possible to eliminate the negative consequences resulting from the formation of standing waves. A converging ultrasound transducer with a central frequency of 550 kHz was focused onto a brass cantilever. The 267 Hz resonance was excited with the ultrasound radiation force with a carrier frequency that randomly varied between 525 kHz to 575 kHz in packets of 10 cycles. Because each packet had a different carrier frequency, the amplitude of standing wave artifacts was reduced by a factor of 20 compared to a constant frequency excitation of 550 kHz.     

Minimal and maximal operator spaces and operator systems in entanglement theory

We examine k-minimal and k-maximal operator spaces and operator systems, and investigate their relationships with the separability problem in quantum information theory. We show that the matrix norms that define the k-minimal operator spaces are equal to a family of norms that have been studied independently as a tool for detecting k-positive linear maps and bound entanglement. Similarly, we investigate the k-super minimal and k-super maximal operator systems that were recently introduced and show that their cones of positive elements are exactly the cones of k-block positive operators and (unnormalized) states with Schmidt number no greater than k, respectively. We characterize a class of norms on the k-super minimal operator systems and show that the completely bounded versions of these norms provide a criterion for testing the Schmidt number of a quantum state that generalizes the recently-developed separability criterion based on trace-contractive maps.     

The Influence of Science Summer Camp on African‐American High School Students' Career Choices

This study explored if a weeklong science camp changed Louisiana African‐American high school students' perception of science. A semi‐structured survey was used before and after the camp to determine the changes in science attitudes and career choices. Among the perceived benefits were parental involvement, increased science academic ability, and deepened scientific knowledge. These perceived benefits influenced the identities that students constructed for themselves in relation to science in their lives. Students who reported doing well in school science courses believed that science was more relevant to their lives. Female students who cited doing well in science reported less parental involvement in their schoolwork than males. This study draws attention to gender differences in science and to designing informal science learning experiences for African‐American high school students that can change attitudes toward career choices in science‐related fields.     

A modular phase transfer and ligand exchange protocol for quantum dots

In this paper, we describe a quantum dot (qdot) phase transfer protocol using ligand exchange and the amino acid histidine. The phase transfer from nonpolar solvents to aqueous buffers is homogeneous, and no appreciable precipitation occurs. The molecule histidine was chosen in order to first displace the organic encapsulation and second to provide a weakly chemisorbing intermediate at the qdot ionic interface. This allows the histidine to act as an intermediate shell upon which further direct ligand exchange can occur. Since this intermediate encapsulation is easily displaced by an assortment of different molecules while in aqueous buffers, we refer to this approach as modular. Characterization via FTIR and NMR revealed the extent of ligand exchange, and provides insights into the interfacial binding mechanism. The colloidal stability and photostability of the qdots was probed via UV-vis and steady state fluorescence, which revealed promising quantum yield stability of greater than 1 year. The qdots have hydrodynamic diameters of <12 nm and surface charges dependent upon ligand type and coverage. The modularity of this approach is shown by tailoring the qdot surface charge via sequential ligand exchange using mixed monolayers of carboxylic acid and poly(ethylene glycol)-terminated thiols.     

Enantioselective decarboxylative alkylation reactions: catalyst development, substrate scope, and mechanistic studies

α-Quaternary ketones are accessed through novel enantioselective alkylations of allyl and propargyl electrophiles by unstabilized prochiral enolate nucleophiles in the presence of palladium complexes with various phosphinooxazoline (PHOX) ligands. Excellent yields and high enantiomeric excesses are obtained from three classes of enolate precursor: enol carbonates, enol silanes, and racemic β-ketoesters. Each of these substrate classes functions with nearly identical efficiency in terms of yield and enantioselectivity. Catalyst discovery and development, the optimization of reaction conditions, the exploration of reaction scope, and applications in target-directed synthesis are reported. Experimental observations suggest that these alkylation reactions occur through an unusual inner-sphere mechanism involving binding of the prochiral enolate nucleophile directly to the palladium center.     

Octadentate cages of Tb(III) 2-hydroxyisophthalamides: a new standard for luminescent lanthanide labels

The synthesis, structure, and photophysical properties of several Tb(III) complexes with octadentate, macrotricyclic ligands that feature a bicapped topology and 2-hydroxyisophthalamide (IAM) chelating units are reported. These Tb(III) complexes exhibit highly efficient emission (Φ(total) ≥ 50%), large extinction coefficients (ε(max) ≥ 20,000 M(-1) cm(-1)), and long luminescence lifetimes (τ(H(2)O) ≥ 2.45 ms) at dilute concentrations in standard biological buffers. The structure of the methyl-protected ligand was determined by single-crystal X-ray diffraction and confirms the macrotricyclic structure of the parent ligand; the amide groups of the methyl-protected cage compound generate an anion binding cavity that complexes a chloride anion. Once the ligand is deprotected, a conformational change generates a similar cavity, formed by the phenolate and ortho amide oxygen groups that strongly bind lanthanide ions. The Tb(III) complexes thus formed display long-term stability, with little if any change in their spectral properties (including lifetime, quantum yield, and emission spectrum) over time or in different chemical environments. Procedures to prepare functionalized derivatives with terminal amine, carboxylate, and N-hydroxysuccinimide groups suitable for derivatization and protein bioconjugation have also been developed. These bifunctional ligands have been covalently attached to a number of different proteins, and the terbium complexes' exceptional photophysical properties are retained. These compounds establish a new aqueous stability and quantum yield standard for long-lifetime lanthanide reporters.     

Polymerization Induced Microphase Separation for the Fabrication of Nanostructured Materials

Polymerization induced microphase separation (PIMS) is a strategy used to develop unique nanostructures with highly useful morphologies through the microphase separation of emergent block copolymers during polymerization. In this process, nanostructures are formed with at least two chemically independent domains, where at least one domain is composed of a robust crosslinked polymer. Crucially, this synthetically simple method is readily used to develop nanostructured materials with the highly coveted co-continuous morphology, which can also be converted into mesoporous materials by selective etching of one domain. As PIMS exploits a block copolymer microphase separation mechanism, the size of each domain can be tightly controlled by modifying the size of block copolymer precursors, thus providing unparalleled control over nanostructure and resultant mesopore sizes. Since its inception 11 years ago, PIMS has been used to develop a vast inventory of advanced materials for an extensive range of applications including biomedical devices, ion exchange membranes, lithium-ion batteries, catalysis, 3D printing, and fluorescence-based sensors, among many others. In this review, we provide a comprehensive overview of the PIMS process, summarize latest developments in PIMS chemistry, and discuss its utility in a wide variety of relevant applications.