Infinite Dimensional Bicomplex Spectral Decomposition Theorem

This paper presents a bicomplex version of the Spectral Decomposition Theorem on infinite dimensional bicomplex Hilbert spaces. In the process, the ideas of bounded linear operators, orthogonal complements and compact operators on bicomplex Hilbert spaces are introduced and treated in relation with the classical Hilbert space M′ imbedded in any bicomplex Hilbert space M.     

Re-weighted random series path integral simulations of molecular clusters: Applications to lithium solvated by a mixed Stockmayer cluster

Nuclear quantum effects in finite temperature simulations of molecular clusters are determined by taking advantage of a recently developed method based on the Feynman Path Integral. The structural and thermodynamic properties, including the nuclear quantum effects are determined for three Stockmayer clusters. The ionic system contain a lithium ion solvated by six strong dipoles and 12 weaker ones. The presence of the ion in the mixed Stockmayer cluster drastically enhances the fluxional nature of the less polar components which occupy the second solvation layer, whereas the neutral counterpart has the effect of reducing it. The nuclear quantum effects are significant at room temperature and above for the solvated ionic system. These are attributable to two factors: (a) the lightness of the lithium ion and (b) the stiffness of the ion-dipole interactions. At 300 K, the difference between the fully converged quantum and the classical heat capacities is about 1.3 K_B for the ionic cluster. This difference is about 10 SDs obtained from 95% confidence estimates of the statistical fluctuations. Cubic convergence is confirmed for temperatures as low as 50 K by regression analysis. The nuclear quantum effects do not change the peak melting temperature of the cluster.     

Rapid Assembly of Saturated Nitrogen Heterocycles in One‐Pot: Diazo‐Heterocycle “Stitching” by N–H Insertion and Cyclization

Methods that provide rapid access to new heterocyclic structures in biologically relevant chemical space provide important opportunities in drug discovery. Here, a strategy is described for the preparation of 2,2‐disubstituted azetidines, pyrrolidines, piperidines, and azepanes bearing ester and diverse aryl substituents. A one‐pot rhodium catalyzed N–H insertion and cyclization sequence uses diazo compounds to stitch together linear 1,m‐haloamines (m=2–5) to rapidly assemble 4 ‐, 5 ‐, 6 ‐, and 7 ‐membered saturated nitrogen heterocycles in excellent yields. Over fifty examples are demonstrated, including examples with diazo compounds derived from biologically active compounds. The products can be functionalized to afford α,α‐disubstituted amino acids and applied to fragment synthesis.     

Spontaneous trans‐Selective Transfer Hydrogenation of Apolar Boron–Boron Double Bonds

The transfer hydrogenation of N‐heterocyclic carbene (NHC)‐supported diborenes with dimethylamine borane proceeds with high selectivity for the trans‐1,2‐dihydrodiboranes. DFT calculations, supported by kinetic studies and deuteration experiments, suggest a stepwise proton‐first‐hydride‐second reaction mechanism via an intermediate μ‐hydrodiboronium dimethylaminoborate ion pair.     

Reconciling Electrostatic and n→π* Orbital Contributions in Carbonyl Interactions

Interactions between carbonyl groups are prevalent in protein structures. Earlier investigations identified dominant electrostatic dipolar interactions, while others implicated lone pair n→π* orbital delocalisation. Here these observations are reconciled. A combined experimental and computational approach confirmed the dominance of electrostatic interactions in a new series of synthetic molecular balances, while also highlighting the distance-dependent observation of inductive polarisation manifested by n→π* orbital delocalisation. Computational fiSAPT energy decomposition and natural bonding orbital analyses correlated with experimental data to reveal the contexts in which short-range inductive polarisation augment electrostatic dipolar interactions. Thus, we provide a framework for reconciling the context dependency of the dominance of electrostatic interactions and the occurrence of n→π* orbital delocalisation in C=O⋅⋅⋅C=O interactions.     

Preparation and Characterization of a π-Conjugated Donor–Acceptor System Containing the Strongly Electron-Accepting Tetraphenylborolyl Unit

A novel thiophene-bridged donor–acceptor system was synthesized with a carbazole as donor and a borole as acceptor unit. The borole group was successfully installed via the tin–boron exchange reaction of 1,1-dimethyl-2,3,4,5-tetraphenylstannole with 9-(5-(dibromoboryl)thiophen-2-yl)carbazole. The effect of the borole on the optoelectronic properties of the donor–acceptor system was explored by spectroscopic (UV/Vis and fluorescence spectroscopy), electrochemical (cyclic voltammetry) and theoretical (TD-DFT) methods as well as by modifying its structure. The corresponding donor–acceptor compound bearing the widely employed dimesitylboryl acceptor group was also synthesized for comparison.     

Targeted Photodynamic Therapy of Human Head and Neck Squamous Cell Carcinoma with Anti-epidermal Growth Factor Receptor Antibody Cetuximab and Photosensitizer IR700DX in the Mouse Skin-fold Window Chamber Model

Targeted photodynamic therapy (PDT) in head/neck cancer patients with a conjugate of the anti-epidermal growth factor receptor (EGFR) antibody, Cetuximab and a phthalocyanine photosensitizer IR700DX is under way, but the exact mechanisms of action are still not fully understood. In this study, the EGFR-overexpressing human head/neck OSC-19-luc2-cGFP tumor with transfected GFP gene was used in a skin-fold window chamber model in BALB/c nude mice. The uptake and localization of the conjugate in the tumor and its surrounding normal tissues were studied by an intravital confocal laser scanning microscopy with image analyses. The tumor was also irradiated with 690 nm laser light 24 h after conjugate administration. The vascular and tumor responses were examined by morphological evaluation and immunohistochemistry (IHC). The amount of conjugate in the tumor peaked at 24–48 h after injection. Image analyses of colocalization correlation parameters demonstrated a high fraction of the conjugate IR700DX colocalized in the GFP-expressing tumor cells. PDT-treated tumors showed extensive necrotic/apoptotic destruction with little vascular damage, while IHC showed no HIF-1α expression and decreased EGFR and Ki67 expression with activated caspase-3 overexpression, indicating a direct killing of tumor cells through both necrotic and apoptotic cell death.     

Investigating secondary students beliefs about mathematical problem-solving

Many studies over the past 30 years have highlighted the important role of students’ beliefs for successful problem-solving in mathematics. Given the recent emphasis afforded to problem-solving on the reformed Irish secondary school mathematics curriculum, the main aim of this study was to identify Irish students’ (n = 975) beliefs about the field. A quantitative measure of these beliefs was attained through the use of the Indiana Mathematical Belief Scale, an existing 30-item (five-scale) self-report questionnaire. A statistical analysis of the data revealed that students who were further through their secondary education had a stronger belief that not all problems could be solved by applying routine procedures. In contrast, the same students held less positive beliefs than their younger counterparts that they could solve time-consuming problems and that conceptual understanding was important. The analysis also indicated that gender had a significant impact on three of the five belief scales.     

Bottleable Neutral Analogues of [B2H5]− as Versatile and Strongly Binding η2 Donor Ligands

Herein we report the discovery that two bottleable, neutral, base‐stabilized diborane(5) compounds are able to bind strongly to a number of copper(I) complexes exclusively through their B?B bond. The resulting complexes represent the first known complexes containing unsupported, neutral σ_B?B diborane ligands. Single‐crystal X‐ray analyses of these complexes show that the X?Cu moiety (X=Cl, OTf, C₆F₅) lies opposite the bridging hydrogen atom of the diborane and is near perpendicular to the B?B bond, interacting almost equally with both boron atoms and causing a B?B bond elongation. DFT studies show that σ donation from and π backdonation to the pseudo‐π‐like B?B bond account for their formation. Astoundingly, these copper σ_B?B complexes are inert to ligand exchange with pyridine under either heating or photoirradiation.