Letters to the Editor

The Mathematical Intelligencer encourages comments about the material in this issue. Letters to the editor should be sent to either of the editors-in-chief, Chandler Davis or Marjorie Senechal.     

Scaling of the energy gap withT c in Bi2Sr2CaCu2O8+δ

The results of a high-resolution photoemission study using synchrotron radiation of two single crystals of Bi₂Sr₂CaCu₂O₈₊, with different critical temperaturesT _c due to a variation in oxygen stoichiometry are reported. Within experimental accuracy, the energy gap 2 is found to scale withT _c , amounting to a reduced gap parameter of 2/k _BT_c7.4. Employing resonant photoemission at the O–2s and Cu–3p thresholds, two spectral peaks at binding energies of 180 meV and 320 meV were identified as predominantly O–2p-and Cu–3d 4s-derived states, respectively.     

Another look at complete positivity in generalized quantum dynamics: Reply to Raggio and Primas

In this rejoinder to a critique by Raggio and Primas of our paper, On Completely Positive Maps in Generalized Quantum Dynamics, we acknowledge that, contrary to our original assertion, the Bloch equations are indeed completely positive. We then explain briefly why this modification of our analysis does not alter its main conclusions.     

A comparative study on methods of optimal sample preparation for the analysis of oligonucleotides by matrix-assisted laser desorption/ionization mass spectrometry

Metal adducts (e.g., Na+ and K+) significantly hinder the analysis of oligonucleotides by matrix-assisted laser desorption/ionization time-of-flight mass spectrometry (MALDI-TOF-MS). Although a number of sample purification methods exist, to date no comparative study exists to determine the most efficient method for purifying oligonucleotides. The objective of this work was to perform such a study. Several different oligonucleotide samples were synthesized. Aliquots of these samples were then purposely contaminated with sodium acetate to generate representative contaminated (salted) oligonucleotide samples. A number of popular oligonucleotide purification techniques were then tested as to their effectiveness at removing Na+ from the salted samples. The effectiveness of Na+ removal was qualitatively assessed by comparing the MALDI mass spectra of the original sample, the salted sample, and the salted sample after purification. Micropipet tips packed with C18 reversed-phase packing material (e.g., Zip Tips) appear to be the most effective means of purifying the oligonucleotides investigated. Minidialysis was found to be an effective alternative for purifying higher molecular weight oligonucleotides (> 10,000 u).     

Nanostructured Materials Synthesis in a Mixed Surfactant Mesophase

The mixed surfactant system of bis (2-ethylhexyl) sodium sulfosuccinate (AOT) and α-phosphatidylcholine (lecithin) forms a rigid gel-like mesophase in the presence of equal volumes of water and a hydrocarbon (isooctane). Small angle neutron scattering (SANS) results indicate that these structures undergo transitions from columnar hexagonal geometries to lamellar geometries depending on the water content and/or the temperature. The system is used to synthesize nanostructured ceramics (silica) in the aqueous microphase. Interpenetrating networks of poly-(hydroxyethylmethacrylate) and poly(styrene) are also synthesized using the aqueous microphase to support the water soluble monomer (hydroxyethyl methacrylate) and the organic microphase (styrene). SANS results indicate that the template structure is maintained during materials synthesis.     

Influence of connectivity and porosity on ligand-based luminescence in zinc metal-organic frameworks

Applications of metal-organic frameworks (MOFs) require close correlation between their structure and function. We describe the preparation and characterization of two zinc MOFs based on a flexible and emissive linker molecule, stilbene, which retains its luminescence within these solid materials. Reaction of trans-4,4'-stilbene dicarboxylic acid and zinc nitrate in N,N-dimethylformamide (DMF) yielded a dense 2-D network, 1, featuring zinc in both octahedral and tetrahedral coordination environments connected by trans-stilbene links. Similar reaction in N,N-diethylformamide (DEF) at higher temperatures resulted in a porous, 3-D framework structure, 2. This framework consists of two interpenetrating cubic lattices, each featuring basic zinc carboxylate vertices joined by trans-stilbene, analogous to the isoreticular MOF (IRMOF) series. We demonstrate that the optical properties of both 1 and 2 correlate with the local ligand environments observed in the crystal structures. Steady-state and time-resolved spectroscopic measurements reveal that the stilbene linkers in the dense structure 1 exhibit a small degree of interchromophore coupling. In contrast, the stilbenoid units in 2 display very little interaction in this low-density 3-D framework, with excitation and emission spectra characteristic of monomeric stilbenes, similar to the dicarboxylic acid in dilute solution. In both cases, the rigidity of the stilbene linker increases upon coordination to the inorganic units through inhibition of torsion about the central ethylene bond, resulting in luminescent crystals with increased emission lifetimes compared to solutions of trans-stilbene. The emission spectrum of 2 is found to depend on the nature of the incorporated solvent molecules, suggesting use of this or related materials in sensor applications.     

Novel perturbation expansion for the Langevin equation

We discuss the randomly driven systemdx/dt= -W(x) +f(t), wheref(t) is a Gaussian random function or stirring force withf(t)f(t)=2(t–t), andW(x) is of the formgx ¹⁺². The parameter is a measure of the nonlinearity of the equation. We show how to obtain the correlation functionsx(t)f(t)···x(t( n)) _f as a power series in. We obtain three terms in the expansion and show how to use Padé approximants to analytically continue the answer in the variable. By using scaling relations, we show how to get a uniform approximation to the equal-time correlation functions valid for allg and.     

Clutter interference and the integration time of echoes in the echolocating bat, Eptesicus fuscus

The ability of the echolocating bat, Eptesicus fuscus, to detect a sonar target is affected by the presence of other targets along the same axis at slightly different ranges. If echoes from one target arrive at about the same delay as echoes from another target, clutter interference occurs and one set of echoes masks the other. Although the bat's sonar emissions and the echoes themselves are 2 to 5 ms long, echoes (of approximately equal sensation levels--around 15 dB SL) only interfere with each other if they arrive within 200 to 400 microseconds of the same arrival time. This figure is an estimate of the integration time of the bat's sonar receiver for echoes. The fine structure of the clutter-interference data reflects the reinforcement and cancellation of echoes according to their time separation. When clutter interference first occurs, the waveforms of test and cluttering echoes already overlap for much of their duration. The masking effect underlying clutter interference appears specifically due to overlap, not between raw echo waveforms, but between the patterns of mechanical excitation created when echoes pass through bandpass filters equivalent to auditory-nerve tuning curves. While the time scale of clutter interference is substantially shorter than the duration of echo waveforms, it still is much longer than the eventual width of a target's range-axis image expressed in terms of echo delay.