On the approximation of spectra of linear operators on Hilbert spaces

We present several new techniques for approximating spectra of linear operators (not necessarily bounded) on an infinite-dimensional, separable Hilbert space. Our approach is to take well-known techniques from finite-dimensional matrix analysis and show how they can be generalized to an infinite-dimensional setting to provide approximations of spectra of elements in a large class of operators. We conclude by proposing a solution to the general problem of approximating the spectrum of an arbitrary bounded operator by introducing the n-pseudospectrum and argue how that can be used as an approximation to the spectrum.     

A note on the Jacobian conjecture

In this paper we consider the Jacobian conjecture for a map of complex affine spaces of dimension . It is well known that if is proper, then the conjecture will hold. Using topological arguments, specifically Smith theory, we show that the conjecture holds if and only if is proper onto its image.

     

Detecting completeness from Ext-vanishing

Motivated by work of C. U. Jensen, R.-O. Buchweitz, and H. Flenner, we prove the following result. Let be a commutative noetherian ring and an ideal in the Jacobson radical of . Let be the -adic completion of . If is a finitely generated -module such that for all , then is -adically complete.

     

Unsaturated aldehydes as alkene equivalents in the Diels-Alder reaction

A one-pot procedure is described for using alpha,beta-unsaturated aldehydes as olefin equivalents in the Diels-Alder reaction. The method combines the normal electron demand cycloaddition with aldehyde dienophiles and the rhodium-catalyzed decarbonylation of aldehydes to afford cyclohexenes with no electron-withdrawing substituents. In this way, the aldehyde group serves as a traceless control element to direct the cycloaddition reaction. The Diels-Alder reactions are performed in a diglyme solution in the presence of a catalytic amount of boron trifluoride etherate. Subsequent quenching of the Lewis acid, addition of 0.3% of [Rh(dppp)2Cl] and heating to reflux achieves the ensuing decarbonylation to afford the product cyclohexenes. Under these conditions, acrolein, crotonaldehyde and cinnamaldehyde have been reacted with a variety of 1,3-dienes to afford cyclohexenes in overall yields between 53 and 88%. In these transformations, the three aldehydes serve as equivalents of ethylene, propylene and styrene, respectively.     

Accuracy of typical approximations in classical models of intermolecular polarization

One of the largest limitations of standard molecular-mechanics force fields is the neglect of intermolecular polarization. Several attempts to cure this problem have been made, but the results have not always been fully satisfactory. In this paper, we present a quantitative study of the fundamental approximations that underlie polarization models for classical force fields. The induced charge density of a large set of molecular dimers is compared to supermolecular calculations for a hierarchy of simplified models. We study the effect of the Pauli principle, the local inhomogeneity of the electric field, the intramolecular coupling of the polarization response, and the fact that the induced density is a continuous function. We show that standard point-polarizability models work rather well, despite their lack of all these effects, because (1) there is a systematic error cancellation between the neglect of effects of the Pauli principle and the locally inhomogeneous electric field, and (2) the lack of intramolecular coupling and the use of a dipole expansion of the induced density have only minor effects on the polarization. However, the cancellation in (1) is not perfect, and therefore polarizable force-fields could be improved if both effects are explicitly treated.     

Direct vinylation and difluorovinylation of arylboronic acids using vinyl- and 2,2-difluorovinyl tosylates via the Suzuki-Miyaura cross coupling

General reaction conditions were developed for the Pd(0)-catalyzed Suzuki-Miyaura coupling reaction of aryl boronic acids with a simple electrophilic vinylation reagent, vinyl tosylate, providing access to styrene derivatives in good yields. The easily accessible vinyl tosylate represents a stable and less toxic alternative to the vinyl halides and the triflate/nonaflate derivatives. Furthermore, this methodology was expanded to provide a facile and straightforward approach for the introduction of a gem-difluorovinyl substituent onto an aromatic ring using the similar and also readily available 2,2-difluorovinyl tosylate as the electrophilic complement.     

Nucleic acid structural engineering using pyrene-functionalized 2'-amino-alpha-L-LNA monomers and abasic sites

Oligonucleotides (ONs) modified with a 2'-N-(pyren-1-yl)acetyl-2'-amino-alpha-L-LNA thymine monomer Y flanked on the 3'-side by an abasic site Phi (i.e., YPhi-unit) exhibit unprecedented increases in thermal affinity (DeltaT(m) values) toward target strands containing abasic sites (DeltaT(m) per YPhi unit >+33.0 degrees C in 9-mer duplexes relative to unmodified ONs). Biophysical studies along with force field calculations suggest that the conformationally locked 2-oxo-5-azabicyclo[2.2.1]heptane skeleton of monomer Y, in concert with the short rigid acetyl linker, efficiently forces the thymine and pyrene moieties to adopt an interplanar distance of approximately 3.4 A. This precisely positions the pyrene moiety in the duplex core void formed by abasic sites (Phi:Phi pair) for optimal pi-pi overlap. Duplexes with multiple YPhi: APhi units separated by one base pair are tolerated extraordinarily well, as exemplified by a 13-mer duplex containing four separated YPhi: APhi units (8 abasic sites distributed over 13 "base pairs"), which exhibit a thermal denaturation temperature of 60.5 degrees C. The YPhi probes display up to 16-fold increases in fluorescence intensity at 380 nm upon hybridization with abasic target strands, whereby self-assembly of these complex architectures can be easily monitored. This study underlines the potential of N2'-functionalized 2'-amino-alpha-L-LNA as building blocks in nucleic acid based diagnostics and nanomaterial engineering.     

Combined experimental and theoretical mechanistic investigation of the Barbier allylation in aqueous media

The Barbier allylation of a series of para-substituted benzaldehydes with allylbromide in the presence of Zn, In, Sn, Sb, Bi, and Mg was investigated using competition experiments. In all cases, the slope of the Hammett plots indicated a build-up of negative charge in the selectivity-determining step. For Zn, In, Sn, Sb, and Bi, an inverse secondary kinetic isotope effect was found (kH/kD = 0.75-0.95), which was compatible with the formation of a discrete organometallic species prior to allylation via a closed six-membered transition state. With Mg, a significantly larger build-up of negative charge along with a small positive secondary kinetic isotope effect (kH/kD = 1.06) indicated that the selectivity-determining step was the generation of the radical anion of benzaldehyde. The reaction through a six-membered transition state was modeled using density functional theory with the effect of solvent described by a polarized continuum model. The calculated secondary deuterium isotope effects based on this mechanism were found to be in good agreement with experimental values, thus adding further support to this mechanistic scenario.     

Formation of an ion-pair molecule with a single NH(+)...Cl(-) hydrogen bond: Raman spectra of 1,1,3,3-tetramethylguanidinium chloride in the solid state, in solution, and in the vapor phase

Some ionic compounds (salts) form liquids when heated to temperatures in the range of 200-300 degrees C. They may be referred to as moderate temperature ionic liquids. An example of such a compound is the 1,1,3,3-tetramethylguanidinium chloride, [TMGH]Cl, melting at approximately 212 degrees C. The chemistry of this compoundcontaining a dimeric ion-pair "molecule"was investigated in the solid state, in solutions in water and ethanol, and in the vapor phase, based on ab initio molecular orbital density functional theory (DFT)-type calculations with 6-311+G(d,p) basis sets. Calculations on the monomeric [TMGH] (+) ion and the dimeric chloride ion-pair salt converged to give geometries near the established crystal structure of [TMGH]Cl. The structures and their binding energies are given as well as calculated vibrational harmonic normal modes (IR and Raman band wavenumbers and intensities). Experimentally obtained Raman scattering spectra are presented and assigned, by comparing to the quantum mechanical calculations. It is concluded that dimeric molecular ion pairs with four N-H (+)...Cl (-) hydrogen bonds probably exist in the solutions and are responsible for the relatively high solubility of the "salt" in ethanol. It was discovered that the compound can be easily sublimed by heating to about 200-230 degrees C. In the Raman spectrum of the vapor at 225 degrees C, a characteristic strong band at 2229 cm (-1) was found and interpreted to show that the gas phase consists of monomeric ion-pair "molecules" held together by a single N-H (+)...Cl (-) hydrogen bond, the stretching band of which is causing the band.     

A monolithic PNGase F enzyme microreactor enabling glycan mass mapping of glycoproteins by mass spectrometry

We demonstrate a simple and rapid single-step method to fabricate an enzyme microreactor incorporating the N-glycosidase PNGase F (peptide-N-glycosidase F) into a porous polymer-based monolith. The monolith is contained in a capillary format, while the enzyme reactor is ready to use within 1 h of preparation. The monomers making up the monolith, including N-acryloxysuccinimide for covalent immobilization of the enzyme, are mixed with PNGase F and introduced into the column by capillary force for polymerization/immobilization. Glycoproteins (ribonuclease B, asialofetuin, alpha1-acid glycoprotein, and ovalbumin) perfused through the PNGase F reactor were shown to be effectively deglycosylated on a time-scale of seconds/low minutes using low nanogram to microgram per microliter concentration (corresponding to a total sample consumption of 0.1-20 microg of a glycoprotein). The reactor enzyme activity was shown to be reproducible for at least 8 weeks when used and stored at room temperature. Evaluation was performed by matrix-assisted laser desorption/ionization time-of-flight mass spectrometry.