A mathematical theory of stochastic microlensing. II. Random images, shear, and the Kac-Rice formula

Continuing our development of a mathematical theory of stochastic microlensing, we study the random shear and expected number of random lensed images of different types. In particular, we characterize the first three leading terms in the asymptotic expression of the joint probability density function (pdf) of the random shear tensor due to point masses in the limit of an infinite number of stars. Up to this order, the pdf depends on the magnitude of the shear tensor, the optical depth, and the mean number of stars through a combination of radial position and the star's mass. As a consequence, the pdf's of the shear components are seen to converge, in the limit of an infinite number of stars, to shifted Cauchy distributions, which shows that the shear components have heavy tails in that limit. The asymptotic pdf of the shear magnitude in the limit of an infinite number of stars is also presented. All the results on the random microlensing shear are given for a general point in the lens plane. Extending to the general random distributions (not necessarily uniform) of the lenses, we employ the Kac-Rice formula and Morse theory to deduce general formulas for the expected total number of images and the expected number of saddle images. We further generalize these results by considering random sources defined on a countable compact covering of the light source plane. This is done to introduce the notion of global expected number of positive parity images due to a general lensing map. Applying the result to microlensing, we calculate the asymptotic global expected number of minimum images in the limit of an infinite number of stars, where the stars are uniformly distributed. This global expectation is bounded, while the global expected number of images and the global expected number of saddle images diverge as the order of the number of stars. © 2009 American Institute of Physics.

Multiple Dynamic Processes Contribute to the Complex Steady Shear Behavior of Cross-Linked Supramolecular Networks of Semidilute Entangled Polymer Solutions.

Molecular theories of shear thickening and shear thinning in associative polymer networks are typically united in that they involve a single kinetic parameter that describes the network -- a relaxation time that is related to the lifetime of the associative bonds. Here we report the steady-shear behavior of two structurally identical metallo-supramolecular polymer networks, for which single-relaxation parameter models break down in dramatic fashion. The networks are formed by the addition of reversible cross-linkers to semidilute entangled solutions of PVP in DMSO, and they differ only in the lifetime of the reversible cross-links. Shear thickening is observed for cross-linkers that have a slower dissociation rate (17 s(-1)), while shear thinning is observed for samples that have a faster dissociation rate (ca. 1400 s(-1)). The difference in the steady shear behavior of the unentangled vs. entangled regime reveals an unexpected, additional competing relaxation, ascribed to topological disentanglement in the semidilute entangled regime that contributes to the rheological properties.

Mechanism of Shear Thickening in Reversibly Cross-linked Supramolecular Polymer Networks.

We report here the nonlinear rheological properties of metallo-supramolecular networks formed by the reversible cross-linking of semi-dilute unentangled solutions of poly(4-vinylpyridine) (PVP) in dimethyl sulfoxide (DMSO). The reversible cross-linkers are bis-Pd(II) or bis-Pt(II) complexes that coordinate to the pyridine functional groups on the PVP. Under steady shear, shear thickening is observed above a critical shear rate, and that critical shear rate is experimentally correlated with the lifetime of the metal-ligand bond. The onset and magnitude of the shear thickening depend on the amount of cross-linkers added. In contrast to the behavior observed in most transient networks, the time scale of network relaxation is found to increase during shear thickening. The primary mechanism of shear thickening is ascribed to the shear-induced transformation of intrachain cross-linking to interchain cross-linking, rather than nonlinear high tension along polymer chains that are stretched beyond the Gaussian range.

Contrasting the efficiency of radiation belt losses caused by ducted and nonducted whistler-mode waves from ground-based transmitters

It has long been recognized that whistler-mode waves can be trapped in plasmaspheric whistler ducts which guide the waves. For nonguided cases these waves are said to be "nonducted", which is dominant for L < 1.6. Wave-particle interactions are affected by the wave being ducted or nonducted. In the field-aligned ducted case, first-order cyclotron resonance is dominant, whereas nonducted interactions open up a much wider range of energies through equatorial and off-equatorial resonance. There is conflicting information as to whether the most significant particle loss processes are driven by ducted or nonducted waves. In this study we use loss cone observations from the DEMETER and POES low-altitude satellites to focus on electron losses driven by powerful VLF communications transmitters. Both satellites confirm that there are well-defined enhancements in the flux of electrons in the drift loss cone due to ducted transmissions from the powerful transmitter with call sign NWC. Typically, ∼80% of DEMETER nighttime orbits to the east of NWC show electron flux enhancements in the drift loss cone, spanning a L range consistent with first-order cyclotron theory, and inconsistent with nonducted resonances. In contrast, ∼1% or less of nonducted transmissions originate from NPM-generated electron flux enhancements. While the waves originating from these two transmitters have been predicted to lead to similar levels of pitch angle scattering, we find that the enhancements from NPM are at least 50 times smaller than those from NWC. This suggests that lower-latitude, nonducted VLF waves are much less effective in driving radiation belt pitch angle scattering. Copyright 2010 by the American Geophysical Union.

Intracardiac acoustic radiation force impulse (ARFI) and shear wave imaging in pigs with focal infarctions.

Four pigs, three with focal infarctions in the apical intraventricular septum (IVS) and/or left ventricular free wall (LVFW), were imaged with an intracardiac echocardiography (ICE) transducer. Custom beam sequences were used to excite the myocardium with focused acoustic radiation force (ARF) impulses and image the subsequent tissue response. Tissue displacement in response to the ARF excitation was calculated with a phase-based estimator, and transverse wave magnitude and velocity were each estimated at every depth. The excitation sequence was repeated rapidly, either in the same location to generate 40 Hz M-modes at a single steering angle, or with a modulated steering angle to synthesize 2-D displacement magnitude and shear wave velocity images at 17 points in the cardiac cycle. Both types of images were acquired from various views in the right and left ventricles, in and out of infarcted regions. In all animals, acoustic radiation force impulse (ARFI) and shear wave elasticity imaging (SWEI) estimates indicated diastolic relaxation and systolic contraction in noninfarcted tissues. The M-mode sequences showed high beat-to-beat spatio-temporal repeatability of the measurements for each imaging plane. In views of noninfarcted tissue in the diseased animals, no significant elastic remodeling was indicated when compared with the control. Where available, views of infarcted tissue were compared with similar views from the control animal. In views of the LVFW, the infarcted tissue presented as stiff and non-contractile compared with the control. In a view of the IVS, no significant difference was seen between infarcted and healthy tissue, whereas in another view, a heterogeneous infarction was seen to be presenting itself as non-contractile in systole.

The island-scale internal wave climate of Moorea, French Polynesia

Analysis of five-year records of temperatures and currents collected at Moorea reveal strong internal wave activity at predominantly semi-diurnal frequencies impacting reef slopes at depths 30m around the entire island. Temperature changes of 1.5C to 3C are accompanied by surges of upward and onshore flow and vertical shear in onshore currents. Superimposed on annual temperature changes of approximately 3C, internal wave activity is high from Oct-May and markedly lower from Jun-Sep. The offshore pycnocline is broadly distributed with continuous stratification to at least 500m depth, and a subsurface fluorescence maximum above the strong nutricline at approximately 200m. Minimum buoyancy periods range from 4.8 to 6min, with the maximum density gradient occurring at 50 to 60m depth in summer and deepening to approximately 150 to 200m in winter. The bottom slope angle around all of Moorea is super-critical relative to the vertical stratification angle suggesting that energy propagating into shallow water is only a portion of total incident internal wave energy. Vertical gradient Richardson numbers indicate dominance by density stability relative to current shear with relatively limited diapycnal mixing. Coherence and lagged cross-correlation of semi-diurnal temperature variation indicate complex patterns of inter-site arrival of internal waves and no clear coherence or lagged correlation relationships among island sides. Semi-diurnal and high frequency internal wave packets likely arrive on Moorea from a combination of local and distant sources and may have important impacts for nutrient and particle fluxes in deep reef environments. © 2012 American Geophysical Union. All Rights Reserved.