Exact and Approximate Values of the Period for a "Truly Nonlinear" Oscillator: $\ddot{x} + x + x^{1/3} = 0$

We investigate the mathematical properties of a "truly nonlinear" oscillator differential equation. In particular, using phase-space methods, it is shown that all solutions are periodic and the fixed-point is a nonlinear center. We calculate both exact and approximate analytical expressions for the period, where the exact solution is given in terms of elliptic functions and the method of harmonic balance is used to calculate the approximate formula.     

Selected stable states in large quantum systems

The utility of an infinite number of degrees of freedom is examined in the context of quantum systems. Such systems are known to allow inequivalent representations of the commutation relations, qualitative differentiation (spontaneously broken symmetries) arising from cooperative effects, and superselected stable states. This freedom is discussed and then used in an exactly soluble model of the measurement process.     

On the stability of some classical operators from approximation theory

We obtain some results on Hyers–Ulam stability for some classical operators from approximation theory. For Bernstein operators we determine the Hyers–Ulam constant using a result concerning coefficient bounds of Lorentz representation for a polynomial.     

Precise Arrhenius Law for p-forms: The Witten Laplacian and Morse–Barannikov Complex

Accurate asymptotic expressions are given for the exponentially small eigenvalues of Witten Laplacians acting on p-forms. The key ingredient, which replaces explicit formulas for global quasimodes in the case p = 0, is Barannikov’s presentation of Morse theory in Barannikov (Adv Soviet Math 21:93–115, 1994).     

Simultaneously measured behavioral and electrophysiological hearing thresholds in a bottlenose dolphin (Tursiops truncatus)

Dolphin auditory thresholds obtained via evoked potential audiometry may deviate from behavioral estimates by 20 dB or more. Differences in the sound source, stimulus presentation method, wave form, and duration may partially explain these discrepancies. To determine the agreement between behavioral and auditory evoked potential (AEP) threshold estimates when these parameters are held constant, behavioral and AEP hearing tests were simultaneously conducted in a bottlenose dolphin. Measurements were made in-air, using sinusoidal amplitude-modulated tones continuously projected via a transducer coupled to the pan region of the dolphin's lower jaw. Tone trials were presented using the method of constant stimuli. Behavioral thresholds were estimated using a 50% correct detection. AEP thresholds were based on the envelope following response and 50% correct detection. Differences between AEP and behavioral thresholds were within +/-5 dB, except at 10 kHz (12 dB), 20 kHz (8 dB), 30 kHz (7 dB), and 150 kHz (24 dB). In general, behavioral thresholds were slightly lower, though this trend was not significant. The results demonstrate that when the test environment, sound source, stimulus wave form, duration, presentation method, and analysis are consistent, the magnitude of the differences between AEP and behavioral thresholds is substantially reduced.     

A simple system for staining protein and nucleic acid electrophoresis gels

Researchers in molecular biology spend a significant amount of time tending to the staining and destaining of electrophoresis gels. Here we describe a simple system, costing approximately $100 and taking approximately 1 h to assemble, that automates standard nucleic acid and protein gel staining protocols. Staining is done in a tray or, with DNA gels, in the electrophoresis chamber itself following automatic detection of the voltage drop. Miniature pumps controlled by a microcontroller chip exchange the necessary solutions at programmed time intervals. We demonstrate efficient and highly reproducible ethidium bromide and methylene blue staining of DNA in agarose gels and Coomassie blue and silver staining of proteins in polyacrylamide gels.     

California sea lion (Zalophus californianus) aerial hearing sensitivity measured using auditory steady-state response and psychophysical methods

Although electrophysiological methods of measuring the hearing sensitivity of pinnipeds are not yet as refined as those for dolphins and porpoises, they appear to be a promising supplement to traditional psychophysical procedures. In order to further standardize electrophysiological methods with pinnipeds, a within-subject comparison of psychophysical and auditory steady-state response (ASSR) measures of aerial hearing sensitivity was conducted with a 1.5-yr-old California sea lion. The psychophysical audiogram was similar to those previously reported for otariids, with a U-shape, and thresholds near 10 dB re 20 μPa at 8 and 16 kHz. ASSR thresholds measured using both single and multiple simultaneous amplitude-modulated tones closely reproduced the psychophysical audiogram, although the mean ASSR thresholds were elevated relative to psychophysical thresholds. Differences between psychophysical and ASSR thresholds were greatest at the low- and high-frequency ends of the audiogram. Thresholds measured using the multiple ASSR method were not different from those measured using the single ASSR method. The multiple ASSR method was more rapid than the single ASSR method, and allowed for threshold measurements at seven frequencies in less than 20 min. The multiple ASSR method may be especially advantageous for hearing sensitivity measurements with otariid subjects that are untrained for psychophysical procedures.     

Well‐Balanced Designs for Data Placement

The problem we consider in this article is motivated by data placement, in particular data replication in distributed storage and retrieval systems. We are given a set V of v servers along with b files (data, documents). Each file is replicated on exactly k servers. A placement consists in finding a family of b subsets of V (representing the files) called blocks, each of size k. Each server has some probability to fail and we want to find a placement that minimizes the variance of the number of available files. It was conjectured that there always exists an optimal placement (with variance better than any other placement for any value of the probability of failure). We show that the conjecture is true, if there exists a well‐balanced design—that is, a family of blocks—each of size k, such that each j‐element subset of V, , belongs to the same or almost the same number of blocks (difference at most one). The existence of well‐balanced designs is a difficult problem as it contains, as a subproblem, the existence of Steiner systems. We completely solve the case and give bounds and constructions for and some values of v and b.     

Synthesis of the silaisocyanoacetylene molecule

The hitherto elusive silaisocyanoacetylene molecule (HCCNSi)-a member of the silaisocyanide family-has been synthesized for the first time through the reaction of the silicon nitride radical (SiN) with acetylene (C(2)H(2)) in the gas phase under single collision conditions. Compared to the isoelectronic reaction of the cyano radical (CN) with acetylene, the replacement of the carbon atom in the cyano group by an isovalent silicon atom has a pronounced effect on the reactivity. Whereas the silicon nitride radical was found to pass an entrance barrier and adds with the nitrogen atom to the acetylene molecule, the cyano radical adds barrierlessly with the carbon atom forming the HCCH(NSi) and HCCH(CN) intermediates, respectively. These structures undergo hydrogen loss to form the linear silaisocyanoacetylene (HCCNSi) and cyanoacetylene molecules (HCCCN), respectively. Therefore, the isovalency of the silicon atom was found to bear little resemblance with the carbon atom having a dramatic effect not only on the reactivity, but also on the reaction mechanism, thermochemistry, and chemical bonding of the isoelectronic silaisocyanoacetylene and cyanoacetylene products, effectively reversing the thermodynamical stability of the nitrile versus isonitrile and silanitrile versus isosilanitrile isomer pairs.