The finite element-Galerkin method for singular self-adjoint differential equations

We investigate the finite element-Galerkin method for singular self-adjoint second-order differential expressions. The weak formulation of the problem involves integration by parts, which allows the use of the usual piecewise linear functions. Our analysis shows that the method produces the solution corresponding to a particular self-adjoint realization of the differential expression. We also propose two algorithms to approximate the solution of any self-adjoint realization. Numerical examples are given to illustrate the analysis as well as the proposed algorithms.     

Barrier strips and positive solutions of first order singular and regular initial value problems

We use the barrier strip method to prove sufficient conditions for the global solvability of the initial value problem f(t, x, x′) = 0, x(0) = A, including the case in which the function (t, x, y) → f(t, x, y) has a singularity at x = A.     

Structure of positive solution sets of sub-linear semi-positone operator equations

In this paper, first we obtain some results on the structure of the positive solution set of a nonlinear operator equation. Then using these results, we obtain some existence results for positive solutions of nonlinear operator equations. We use global bifurcation theories to show our main results.     

Surgery obstructions from Khovanov homology

For a 3-manifold with torus boundary admitting an appropriate involution, we show that Khovanov homology provides obstructions to certain exceptional Dehn fillings. For example, given a strongly invertible knot in S ³, we give obstructions to lens space surgeries, as well as obstructions to surgeries with finite fundamental group. These obstructions are based on homological width in Khovanov homology, and in the case of finite fundamental group depend on a calculation of the homological width for a family of Montesinos links.     

Mode coherence measurements across a 1.5 THz spectral bandwidth of a passively mode-locked quantum dash laser

The mode coherence of adjacent and non-adjacent spectral modes of a passively mode locked quantum dash (QDash) semiconductor laser are deduced through radio frequency beat-tone linewidth measurements. A wavelength conversion scheme that uses degenerate four wave mixing in a semiconductor optical amplifier is proposed which considerably extends the mode spacing beyond the limit imposed by conventional fast-photodetection and electrical spectrum analysis of around 100 GHz. Using this scheme, the mode coherence of the QDash laser was measured out to the thirty-first harmonic, or a mode separation of 1.5 THz.     

Photochemically Mediated Ring Expansion of Indoles and Pyrroles with Chlorodiazirines: Synthetic Methodology and Thermal Hazard Assessment

We demonstrate that arylchlorodiazirines serve as photo-activated halocarbene precursors for the selective one-carbon ring expansion of N-substituted pyrroles and indoles to the corresponding pyridinium and quinolinium salts. Preliminary investigations indicate that the same strategy also enables the conversion of N-substituted pyrazoles to pyrimidinium salts. The N-substituent of the substrate plays an essential role in: (1) increasing substrate scope by preventing product degradation, (2) enhancing yields by suppressing co-product inhibition, and (3) activating the azinium products towards subsequent synthetic manipulations. This latter point is illustrated by subjecting the quinolinium salts to four complementary partial reductions, which provide concise access to ring-expanded products with different degrees of increased C(sp³) character. Thermal analysis of the diazirines by differential scanning calorimetry (DSC) provides detailed insight into their energetic properties, and highlights the safety benefits of photolyzing—rather than thermolyzing—these reagents.     

Skeletal Editing: Interconversion of Arenes and Heteroarenes

Skeletal editing involves making specific point-changes to the core of a molecule through the selective insertion, deletion or exchange of atoms. It thus represents a potentially powerful strategy for the step-economic modification of complex substrates and is a perfect complement to methods such as C−H functionalization that target the molecular periphery. Given their ubiquity in biologically active compounds, the ability to perform skeletal editing on – and therefore interconvert between – aromatic heterocycles is especially valuable. This review summarizes both recent and key historical examples of skeletal editing as applied to interconversion of aromatic rings; we anticipate that it will serve to highlight not only the innovative and enabling nature of current skeletal editing methods, but also the tremendous opportunities that still exist in the field.     

Continuous selecting families and collectively fixed point theory

Aequationes mathematicae - A variety of existence theorems for a collective family of set-valued maps are presented. Our results are deduced from fixed point theory for single valued maps...     

Vinyl Halide-Modified Unsaturated Cyclitols are Mechanism-Based Glycosidase Inhibitors

Suitably configured allyl ethers of unsaturated cyclitols act as substrates of β-glycosidases, reacting via allylic cation transition states. Incorporation of halogens at the vinylic position of these carbasugars, along with an activated leaving group, generates potent inactivators of β-glycosidases. Enzymatic turnover of these halogenated cyclitols (F, Cl, Br) displayed a counter-intuitive trend wherein the most electronegative substituents yielded the most labile pseudo-glycosidic linkages. Structures of complexes with the Sulfolobus β-glucosidase revealed similar enzyme-ligand interactions to those seen in complexes with a 2-fluorosugar inhibitor, the lone exception being displacement of tyrosine 322 from the active site by the halogen. Mutation of Y322 to Y322F largely abolished glycosidase activity, consistent with lost interactions at O5, but minimally affected (7-fold) rates of carbasugar hydrolysis, yielding a more selective enzyme for unsaturated cyclitol ether hydrolysis.     

Continuous selections and collectively fixed point theory with applications to generalized games

Aequationes mathematicae - Using a combination of new and old ideas we present some new collectively fixed point theorems which will then generate some new equilibrium results for generalized games.