The effect of two distinct fast time scales in the rotating,stratified Boussinesq equations: variations from quasi-geostrophy

Inspired by the use of fast singular limits in time-parallel numerical methods for a single fast frequency, we consider the limiting, nonlinear dynamics for a system of partial differential equations when two fast, distinct time scales are present. First-order slow equations are derived via the method of multiple time scales when the two small parameters are related by a rational power. We find that the resultant system depends only on the relationship of the two fast time scales, i.e. which fast time is fastest? Using the theory of cancellation of fast oscillations, we show that with the appropriate assumptions on the nonlinear operator of the full system, this reduced slow system is exactly that which the solution will converge to if each asymptotic limit is considered sequentially. The same result is also obtained via the method of renormalization. The specific example of the rotating, stratified Boussinesq equations is explored in detail, indicating that the most common distinguished limit of this system—quasi-geostrophy, is not the only limiting asymptotic system.     

Efficient p-adic Cell Decompositions for Univariate Polynomials

Cell decompositions are constructed for polynomials f(x)Z_p[x] of degree n, such that n<p, using O(n²) cells. When f is square-free this yields a polynomial-time algorithm for counting and approximating roots in Z_p. These results extend to give a polynomial-time algorithm in the bit model for fZ[x].     

Invariants of matrix pairs over discrete valuation rings and Littlewood-Richardson fillings

Let M and N be two r×r matrices of full rank over a discrete valuation ring R with residue field of characteristic zero. Let P,Q and T be invertible r×r matrices over R. It is shown that the orbit of the pair (M,N) under the action (M,N)?(PMQ^-1,QNT^-1) possesses a discrete invariant in the form of Littlewood-Richardson fillings of the skew shape λ/μ with content ν, where μ is the partition of orders of invariant factors of M, ν is the partition associated to N, and λ the partition of the product MN. That is, we may interpret Littlewood-Richardson fillings as a natural invariant of matrix pairs. This result generalizes invariant factors of a single matrix under equivalence, and is a converse of the construction in Appleby (1999) [1], where Littlewood-Richardson fillings were used to construct matrices with prescribed invariants. We also construct an example, however, of two matrix pairs that are not equivalent but still have the same Littlewood-Richardson filling. The filling associated to an orbit is determined by special quotients of determinants of a matrix in the orbit of the pair.     

Single-Molecule Force Spectroscopy of DNA-Based Reversible Polymer Bridges: Surface Robustness and Homogeneity

Single-molecule force spectroscopy, as implemented in an atomic force microscope, provides a rarely used method by which to monitor dynamic processes that occur near surfaces. Here, a methodology is presented and characterized that facilitates the study of polymer bridging across nanometer-sized gaps. The model system employed is that of DNA-based reversible polymers, and an automated procedure is introduced that allows the AFM tip–surface contact point to be automatically determined, and the distance d between opposing surfaces to be actively controlled. Using this methodology, the importance of several experimental parameters was systematically studied, e.g. the frequency of repeated tip/surface contacts, the area of the substrate surface sampled by the AFM, and the use of multiple AFM tips and substrates. Experiments revealed the surfaces to be robust throughout pulling experiments, so that multiple touches and pulls could be carried out on a single spot with no measurable affect on the results. Differences in observed bridging probabilities were observed, both on different spots on the same surface and, more dramatically, from one day to another. Data normalization via a reference measurement allows data from multiple days to be directly compared.     

BEEM-π calculations for a series of pyridines and their 14N N.Q.R.

The ⁵⁷Fe Mössbauer spectrum of a powdered sample of phthalocyanine-iron(II) in an applied magnetic field of 3·0 teslas has been measured as a function of temperature in the range 4·2 K to 100 K. Measurements have also been made at 4·2 K with 6·0 teslas applied, and on a single crystal specimen at 4·2 K with 3·0 teslas applied. Independent computer fits to the three measurements taken at 4·2 K were found to be consistent with one another, and showed that detailed information concerning magnetic anisotropy can be obtained even from powdered samples of paramagnets by Mössbauer spectroscopy. Although the asymmetry parameter in the electric field gradient tensor was found to be small, there was a significant departure from tetragonal symmetry in the magnetic properties of the molecule. The magnetic hyperfine field at the ⁵⁷Fe nucleus was found to be positive in all directions, indicating that all three electronic g values are significantly greater than 2·0.