Dictums for problem solving and approximation in mathematical acoustics: examples involving low-frequency vibration and radiation

A sequence of dictums for mathematical acoustics is given representing opinions intended to be regarded as authoritative, but not necessarily universally agreed upon. The dictums are presented in the context of the detailed solution for a class of problems involving the forced vibration of a long cylinder protruding half-way into a half-space bounded by a compliant surface (impedance boundary) characterized by a spring constant. One limiting case corresponds to a cylinder vibrating within an infinite rigid baffle, and another limiting case corresponds to a vibrating cylinder on the compliant surface of an incompressible fluid. The second limiting case is identified as analogous to that of a floating half-submerged cylinder whose vibrations cause water waves to propagate over the surface. Attention is focused on vibrations at very low frequencies. Difficulties with insuring a causal solution are pointed out and dictums are given as to how one overcomes such difficulties. Various approximation techniques are described. The derivations involve application of the theory of complex variables and the method of matched asymptotic expansions, and the results include the apparent entrained mass in the near field of the cylinder and the radiation resistance per unit length experienced by the vibrating cylinder.     

Counting rational points on smooth cyclic covers

A conjecture of Serre concerns the number of rational points of bounded height on a finite cover of projective space ${\mathbb{P}}^{n?1}$. In this paper, we achieve Serre?s conjecture in the special case of smooth cyclic covers of any degree when $n?10$, and surpass it for covers of degree $r?3$ when $n>10$. This is achieved by a new bound for the number of perfect r-th power values of a polynomial with nonsingular leading form, obtained via a combination of an r-th power sieve and the q-analogue of van der Corput?s method.     

Alpha-lithiosilanes: III. The Reaction of Vinylalkoxysilanes with Organolithium Reagents

The reaction of two equivalents of vinyldimethylethoxysilane or vinyldimethylmethoxysilane with hydrocarbon soluble alkyllithium reagents; $\text{t}$-BuLi, $\text{s}$-BuLi, or $\text{n}$-BuLi; in hexane at low temperature gives high yields of 1,1-dimethyl-2-alkyl-4-(dimethylalkoxysilyl)silacyclobutanes. With methyl- or phenyllithium substituted vinylsilanes are obtained. The stereochemistry of the silacyclobutanes is assigned on the basis of Si-29 and H-1 NMR. For vinyldimethylethoxysilane the ratio of cis to trans silacyclobutane is about $\text{57}\text{43}$, and is independent of the alkyllithium reagent used. In the reaction of vinyldimethylmethoxysilane with $\text{t}$-BuLi a $\text{sol70}\text{30}$ ratio of the cis and trans silacyclobutane is obtained. A pathway is proposed which is consistent with the stereochemical results and with the products isolated in the reaction run in THF.     

A Riemann–Hilbert problem for skew-orthogonal polynomials

We find a local (d+1)×(d+1)$(d+1)\times (d+1)$ Riemann–Hilbert problem characterizing the skew-orthogonal polynomials associated to the partition function of orthogonal ensembles of random matrices with a potential function of degree d  . Our Riemann–Hilbert problem is similar to a local d×d$d\times d$ Riemann–Hilbert problem found by Kuijlaars and McLaughlin characterizing the bi-orthogonal polynomials. This gives more motivation for finding methods to compute asymptotics of high order Riemann–Hilbert problems, and brings us closer to finding full asymptotic expansions of the skew-orthogonal polynomials.     

Nuclear Spin Hyperpolarization of NH2- and CH3-Substituted Pyridine and Pyrimidine Moieties by SABRE

Hyperpolarization of N-heterocycles with signal amplification by reversible exchange (SABRE) induces NMR sensitivity gains for biological molecules. Substitutions with functional groups, in particular in the ortho-position of the heterocycle, however, result in low polarization using a typical Ir catalyst with a bis-mesityl N-heterocyclic carbene ligand for SABRE, presumably due to steric hindrance. With the addition of allylamine or acetonitrile as coligands to the precatalyst chloro(1,5-cyclooctadiene)[4,5-dimethyl-1,3-bis(2,4,6-trimethylphenyl)imidazol-2-ylidene] iridium, the ¹H signal enhancement increased in several substrates with ortho NH₂ substitutions. For example, for a proton in 2,4-diaminopyrimidine, the enhancement factors increased from −7±1 to −210±20 with allylamine or to −160±10 with acetonitrile. CH₃ substituted molecules yielded maximum signal enhancements of −25±7 with acetonitrile addition, which is considerably less than the corresponding NH₂ substituted molecules, despite exhibiting similar steric size. With the more electron-donating NH₂ substitution resulting in greater enhancement, it is concluded that steric hindrance is not the only dominant factor in determining the polarizability of the CH₃ substituted compounds. The addition of allylamine increased the signal enhancement for the 290 Da trimethoprim, a molecule with a 2,4-diaminopyrimidine moiety serving as an antibacterial agent, to −70.     

Coupling of thioamides with 4-bromocrotonate esters and subsequent conjugate addition for the rapid one-pot synthesis of functionalized thiazolines

An efficient one-pot synthesis of functionalized thiazolines via the coupling of thioamides with 4-bromocrotonate esters is described. A range of aryl- and alkyl-thioamides with various substitutions are well-tolerated. Additionally, the presence of the exocyclic ester functionality provides a convenient handle for the synthesis of more complex thiazoline-containing natural products and biologically-relevant compounds.     

A Practical Photochemical Synthesis of 6-AZA-1, 10-Phenanthroic Anhydride

A high yield synthesis of 6-aza-1, 10-phenanthroic anhydride has been developed. The key step is a volume-efficient photochemical cyclization of azastilbene 5 to the photochemically stable dihydroazaphenanthrene 9.