The dual Brunn-Minkowski theory for bounded Borel sets: Dual affine quermassintegrals and inequalities

This paper develops a significant extension of E. Lutwak's dual Brunn-Minkowski theory, originally applicable only to star-shaped sets, to the class of bounded Borel sets. The focus is on expressions and inequalities involving chord-power integrals, random simplex integrals, and dual affine quermassintegrals. New inequalities obtained include those of isoperimetric and Brunn-Minkowski type. A new generalization of the well-known Busemann intersection inequality is also proved. Particular attention is given to precise equality conditions, which require results stating that a bounded Borel set, almost all of whose sections of a fixed dimension are essentially convex, is itself essentially convex.     

Instantaneous frequency decomposition: an application to spectrally sparse sounds with fast frequency modulations

Classical time-frequency analysis is based on the amplitude responses of bandpass filters, discarding phase information. Instantaneous frequency analysis, in contrast, is based on the derivatives of these phases. This method of frequency calculation is of interest for its high precision and for reasons of similarity to cochlear encoding of sound. This article describes a methodology for high resolution analysis of sparse sounds, based on instantaneous frequencies. In this method, a comparison between tonotopic and instantaneous frequency information is introduced to select filter positions that are well matched to the signal. Second, a cross-check that compares frequency estimates from neighboring channels is used to optimize filter bandwidth, and to signal the quality of the analysis. These cross-checks lead to an optimal time-frequency representation without requiring any prior information about the signal. When applied to a signal that is sufficiently sparse, the method decomposes the signal into separate time-frequency contours that are tracked with high precision. Alternatively, if the signal is spectrally too dense, neighboring channels generate inconsistent estimates-a feature that allows the method to assess its own validity in particular contexts. Similar optimization principles may be present in cochlear encoding.     

Unravelling overlaps and torsion-facilitated coupling using two-dimensional laser-induced fluorescence

Two-dimensional laser-induced fluorescence (2D-LIF) spectroscopy is employed to identify contributions to fluorescence excitation spectra that arise from both overlapping bands and coupling between zero-order states (ZOSs). Evidence is found for the role of torsional motion in facilitating the coupling between vibrations that particularly involves the lowest-wavenumber out-of-plane vibrational modes. The experiments are carried out on jet-cooled p-fluorotoluene, where the molecules are initially in the lowest two torsional levels. Here we concentrate on the 390–420?cm^?1 features in the S₁?←?S₀ excitation spectrum, assigning the features seen in the 2D-LIF spectrum, aided by separate dispersed fluorescence spectra. The 2D-LIF spectra allow the overlapping contributions to be cleanly separated, including some that arise from vibrational-torsional coupling. Various coupling routes open up because of the different symmetries of the lowest two torsional modes; these combine with the vibrational symmetry to provide new symmetry-allowed vibration-torsion (‘vibtor’) interactions, and the role of the excited m?=?1 torsional level is found to be significant.     

Solvent-dependent stabilization of the E configuration of propargylic secondary amides

Secondary amides typically exist 98-99% in the Z rotamer to avoid steric repulsion between the substituent on the carbonyl carbon and the nitrogen. In contrast, secondary amide 3a displays 24% E rotamer at room temperature in aqueous solution. The analogous ester displays 6% E rotamer in chloroform, which suggests that the relatively high E conformer population observed for 3a in water results in part from the low steric bulk of the sp-hybridized carbons and in part from the hydrophobic effect.     

The dynamics of ring rotation in ferrocene,nickelocene and ruthenocene by incoherent quadi-elastic neutron scattering

Incoherent quasi-elastic neutron scattering has been used to study the reorientational motions of the cyclopentadienyl rings in ferrocene, nickelocene and ruthenocene. The results for ferrocene show that the activation energy for ring rotation drops above the 164 K phase transition to 4.4 ± 0.5 kJ mol^?1 (which is approximately half its low temperature value) but the rings still appear to jump between only five orientations on the observable time scale. At room temperature, the rings in nickelocene appear to behave the same as in ferrocene but in ruthenocene they reorientate much less frequently and resemble those in ferrocene below 164 K.     

Synthesis of des-n-tetramethyltriostin a from C-terminal Z-d-serine tetra-and octadepsipeptide intermediates

Des-N-tetramethyltriostin A (1), a known DNA-intercalation agent, has been synthesized from tetra- and octapeptide intermediates that have Z-d-serine at the C-terminal position. The procedure thus allows the fragment coupling and cyclization reactions leading to the synthesis of the title compound to occur without racemization at the C-terminal amino acid. Esterification of Boc-Val-OH with the p-bromophenacyl ester of Z-d-serine provided didepsipeptide Z-d-Ser(Boc-Val)-OBpa (4). Stepwise addition of the requisite amino acids provided tetradepsipeptide Z-d-Ser[Boc-Ala-Cys(Acm)-Val]-OBpa (6). Fragment coupling of the respective C- and N-deprotected tetradepsipeptides 7 and 8, derived from 6, furnished linear octadepsipeptide 9, which upon cyclization and disulfide formation gave the bicyclic octadepsipeptide 11, a known synthetic precursor to 1. The degree of racemization incurred in the alanine and valine residues of selected depsipeptides was measured and the results compared with those obtained in previous studies. It was concluded that alanine, perhaps because of sequence effects, undergoes a degree of racemization (4–10%) during hydrolysis of tetradepsipeptide 6 and octadepsipeptide 9.     

Sound cancellation by the use of secondary multipoles: experiments

Theory related to global, free-field cancellation of a primary monopole field by the use of a displaced, secondary multipole was presented previously: a corresponding experimental investigation is presented here. The construction of multipole source components to octopole order is described, as are procedures for determining their source strengths. Dipoles, longitudinal quadrupoles, and longitudinal octopoles that conformed closely to their theoretical models were constructed using arrays of unbaffled loudspeakers. Two methods of calculating the multipole strengths required to cancel a primary monopole field were implemented in an open-loop manner: a "direct" approach based on a multipole expansion of the primary field, and a least-squares procedure based on fitting the secondary field to the primary field either along a circle enclosing a secondary source, or along a segment of that circle. Cancellation measurements were made on a 1-m-radius circle centered on the secondary source: the primary-to-secondary source separation was approximately 0.2 wavelengths. It was found both that a secondary multipole could provide far greater cancellation than a monopole placed at the same distance from the primary source and that the least-squares approach resulted in greater far-field cancellation than did the direct approach.     

An acoustic study of soils that model seabed sediments containing gas bubbles

The acoustic response of gassy seabed sediment is unique. It is a dispersive and extraordinarily attenuative natural material at frequencies which cause gas bubble resonance. It conceals the structure of the seabed from seismic profiling and it dampens acoustic signals that, for example, trigger acoustic mines. In the past, theoretical studies have formulated the probable cause of this response and crude experimental work has partially corroborated theory. This study measures compressional wave velocity and attenuation in a laboratory soil simulating natural gassy soil, and it investigates the structural properties that cause the unique acoustic response. It was confirmed that below the frequencies which cause resonance the soil behaves as a compressible material (containing gas), and above as a relatively incompressible material (containing no gas). Over the frequency range producing bubble resonance it is suggested that the soil should be modeled as a biphasic material of gas and a relatively incompressible saturated soil matrix (particles and fluid). Velocities for gassy soil were found to be as low as 220 m/s at frequencies below resonance and 1500 m/s above resonance; attenuations were found to be as high as 60 dB/cm for moderately gassy soil and as low as 1 dB/cm for soil with almost no gas.     

Selection of mandrel wrap mode filters for optical fiber loss measurements

A selection criterion for mode filters based on far-field patterns is developed. The criterion is used to select a mandrel wrap filter for 50 μm core diameter, 125 μm outside diameter fibers with δ=1.1%. When cut-back loss measurements were made using the mandrel wrap chosen in this way, loss scaled with length to within ± 0.05 dB/km in three 2-km fibers.