The concept of wavelength-dependent absorption Ångström coefficients (AACs) is discussed and clarified for both single and two-wavelengths AACs and guidance for their implementation with noisy absorption spectra is provided. This discussion is followed by application of the concept to models for brown carbon bulk absorption spectra including the damped simple harmonic oscillator model, its Lorentzian approximation, and the band-gap model with and without Urbach tail. We show that the band-gap model with Urbach tail always has an unphysical discontinuity in the first derivative of the AAC at the band-gap – Urbach-tail matching wavelength. Complex refractive indices obtained from the bulk damped simple harmonic oscillator model are used to calculate absorption spectra for spherical particles, followed by a discussion of their features. For bulk material and small particles, this model predicts a monotonic decrease of the AAC with wavelength well above the resonance wavelength; the model predicts a monotonic increase for large particles

Abstract.A nonholonomic system, for short "NH,'' consists of a configuration space Q n , a Lagrangian L(q,q, t), a nonintegrable constraint distribution H ⊂ T Q, with dynamics governed by Lagrange-d'Alembert's principle. We present here two studies, both using adapted moving frames. In the first we explore the affine connection viewpoint. For natural Lagrangians L = T −V , where we take V = 0 for simplicity, NH-trajectories are geodesics of a (nonmetric) connection ∇ NH which mimics Levi-Civita's. Local geometric invariants are obtained by Cartan's method of equivalence.As an example, we analyze Engel's (2-4) distribution. This is the first such study for a distribution that is not strongly nonholonomic. In the second part we study * The authors thank the Brazilian funding agencies CNPq and

Bacteria that swim without the benefit of flagella might do so by generating longitudinal or transverse surface waves. For example, swimming speeds of order 25 ,um/s are expected for a spherical cell propagating longitudinal waves of 0.2 ,um length, 0.02 ,um amplitude, and 160 ,im/s speed. This problem was solved earlier by mathematicians who were interested in the locomotion of ciliates and who considered the undulations of the envelope swept out by ciliary tips. A new solution is given for spheres propagating sinusoidal waveforms rather than Legendre polynomials. The earlier work is reviewed and possible experimental tests are suggested.Strains of the cyanobacterium Synechococcus swim in seawater at speeds of up to 25 ,Lm/s (1). They are rod-shaped organisms measuring about 1 ,tm in diameter by 2 ,um long. Synechococcus swim in the direction of their long axis, following an irregular helical track. Their means of locomotion is not known, and they have no flagella, either external or internal. As far as one can see by light microscopy, they do not change shape. Under certain growth conditions, long asymmetric cells appear, but these just roll rigidly about an axis parallel to their long axis, the direction of locomotion (T. P. Pitta, personal communication). An electrophoretic, or "ion drive," mechanism has been proposed for other bacteria (2) but has been ruled out for Synechococcus (3). The only propulsive mechanisms that remain possible appear to be surface flow or undulation. Here, we note that the requisite thrust might be generated by small-amplitude, high-frequency waves that travel along the outer cell membrane.The traveling waves that we envisage are surface oscillations. They can be either normal or tangential to the surface, or a combination of the two. Even tangential waves can yield the requisite thrust. In retrospect, this propulsive mechanism for cyanobacteria could have been suggested by a number of researchers much earlier. Lighthill (4), Blake (5), Brennen (6), and Shapere and Wilczek (7) developed quantitative theories of swimming in low Reynolds number fluids by means of small surface waves. These theories were developed for ciliated organisms, the surface wave being the "envelope", or smooth approximation to, the tips of the many cilia. However, the cilia themselves were not essential for the theories. This paper is a review of these past results with an eye toward application to cyanobacterial swimming. We have tried to put the results in a framework that will be useful to microbiologists. We have also included a technical extension of the previous theories: we can expand our surface waveforms in a Fourier basis to obtain swimming velocities, as opposed to the traditional expansions in terms of a Legendre basis. RESULTSSwimming Speeds. Imagine the organism as a sphere or ellipsoid with tangential waves traveling from one pole to the other, with the wave amplitudes constant along lines of latitude, as shown in Fig. 1. c denotes the speed, A denotes the wavelength, and a denotes the ...

The concept of wavelength dependent absorption Ångström coefficients (AACs) is discussed and clarified for both single and two-wavelengths AACs and guidance for their implementation with noisy absorption spectra is provided. This discussion is followed by application of the concept to models for brown carbon bulk absorption spectra including the damped simple harmonic oscillator model, its Lorentzian approximation, and the band-gap model with and without Urbach tail. We show that the band-gap model with Urbach tail always has an unphysical discontinuity in the first derivative of the AAC at the band-gap – Urbach-tail matching wavelength. Complex refractive indices obtained from the bulk damped simple harmonic oscillator model are used to calculate absorption spectra for spherical particles, followed by a discussion of their features. For bulk material and small particles, this model predicts a monotonic decrease of the AAC with wavelength well above the resonance wavelength; the model predicts a monotonic increase for large particles

``Rubber'' coated rolling bodies satisfy a no-twist in addition to the no slip satisfied by ``marble'' coated bodies. Rubber rolling has an interesting differential geometric appeal because the geodesic curvatures of the curves on the surfaces at the corresponding points are equal. The associated distribution in the 5 dimensional configuration space has 2-3-5 growth (these distributions were first studied by Cartan; he showed that the maximal symmetries occurs for rubber rolling of spheres with 3:1 diameters ratio and materialize the exceptionalgroup G_2. The 2-3-5 nonholonomic geometries are classified in a companion paper via Cartan's equivalence method. Rubber rolling of a convex body over a sphere defines a generalized Chaplygin system with SO(3) symmetry group, total space Q = SO(3) X S^2 that can be reduced to an almost Hamiltonian system in T^*S^2 with a non-closed 2-form \omega_{NH}. In this paper we present some basic results on this reduction and as an example we discuss the sphere-sphere problem. In this example the 2-form is conformally symplectic so the reduced system becomes Hamiltonian after a coordinate dependent change of time. In particular there is an invariant measure. Using sphero-conical coordinates we verify the results by Borisov and Mamaev that the system is integrable for a ball over a plane and a rubber ball with twice the radius of a fixed internal ball.Comment: 22 pages; submitted to Regular and Chaotic Dynamic

We propose a model for the self-propulsion of the marine bacterium Synechococcus utilizing a continuous looped helical track analogous to that found in Myxobacteria [1]. In our model cargo-carrying protein motors, driven by proton-motive force, move along a continuous looped helical track. The movement of the cargo creates surface distortions in the form of small amplitude traveling ridges along the S-layer above the helical track. The resulting fluid motion adjacent to the helical ribbon provides the propulsive thrust. A variation on the helical rotor model of [1] allows the motors to be anchored to the peptidoglycan layer, where they drive rotation of the track creating traveling helical waves along the S-layer. We derive expressions relating the swimming speed to the amplitude, wavelength, and velocity of the surface waves induced by the helical rotor, and show that they fall in reasonable ranges to explain the velocity and rotation rate of swimming Synechococcus.

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Cartan's moving frames method is a standard tool in riemannian geometry. We set up the machinery for applying moving frames to cotangent bundles and its sub-bundles defined by non-holonomic constraints.Comment: 13 pages, to appear in Rep. Math. Phy