BDDC and FETI-DP under Minimalist Assumptions

The FETI-DP, BDDC and P-FETI-DP preconditioners are derived in a particulary simple abstract form. It is shown that their properties can be obtained from only on a very small set of algebraic assumptions. The presentation is purely algebraic and it does not use any particular definition of method components, such as substructures and coarse degrees of freedom. It is then shown that P-FETI-DP and BDDC are in fact the same. The FETI-DP and the BDDC preconditioned operators are of the same algebraic form, and the standard condition number bound carries over to arbitrary abstract operators of this form. The equality of eigenvalues of BDDC and FETI-DP also holds in the minimalist abstract setting. The abstract framework is explained on a standard substructuring example.Comment: 11 pages, 1 figure, also available at http://www-math.cudenver.edu/ccm/reports

Interference fringes with maximal contrast at finite coherence time

Interference fringes can result from the measurement of four-time fourth-order correlation functions of a wave field. These fringes have a statistical origin and, as a consequence, they show the greatest contrast when the coherence time of the field is finite. A simple acoustic experiment is presented in which these fringes are observed, and it is demonstrated that the contrast is maximal for partial coherence. Random telegraph phase noise is used to vary the field coherence in order to highlight the problem of interpreting this interference; for this noise, the Gaussian moment theorem may not be invoked to reduce the description of the interference to one in terms of first-order interference.M.W. Hamilto

Polar Varieties and Efficient Real Equation Solving: The Hypersurface Case

The objective of this paper is to show how the recently proposed method by Giusti, Heintz, Morais, Morgenstern, Pardo \cite{gihemorpar} can be applied to a case of real polynomial equation solving. Our main result concerns the problem of finding one representative point for each connected component of a real bounded smooth hypersurface. The algorithm in \cite{gihemorpar} yields a method for symbolically solving a zero-dimensional polynomial equation system in the affine (and toric) case. Its main feature is the use of adapted data structure: Arithmetical networks and straight-line programs. The algorithm solves any affine zero-dimensional equation system in non-uniform sequential time that is polynomial in the length of the input description and an adequately defined {\em affine degree} of the equation system. Replacing the affine degree of the equation system by a suitably defined {\em real degree} of certain polar varieties associated to the input equation, which describes the hypersurface under consideration, and using straight-line program codification of the input and intermediate results, we obtain a method for the problem introduced above that is polynomial in the input length and the real degree.Comment: Late

Calcium Pyrophosphate Crystal Deposition: The Effect of Soluble Iron in a Kinetic Study Using a Gelatin Matrix Model

The kinetics of calcium pyrophosphate dihydrate (CPPD) crystal growth was studied by allowing calcium and pyrophosphate (PPi-4) ions to diffuse through a denatured collagen matrix (biological grade gelatin) in the presence of either ferric or ferrous ions. Ferric and, to some extent, ferrous ions blocked the migration of the PPi-4 diffusion gradient. This retardation in the [PPi-4] gradient led to numerous changes in the patterns of CPPD crystal formation. At the initial stages of crystal growth, the iron ions induced more crystal growth compared to control. At later incubation times, ferrous and ferric ions enhanced crystal growth at the expense of crystal nucleation. The presence of both ferrous and ferric ions resulted in the more rapid formation of the two crystals observed in vivo, triclinic CPPD and monoclinic CPPD. Further, both ferrous and ferric ions also reduced the solubility of the crystalline material in the broad diffuse band which formed when the Ca+2 and PPi-4 gradients first met. In this system, the presence of either ferrous or ferric ions increased the amount of hydroxyproline included in the crystalline precipitates. Iron was also incorporated into the crystals, particularly into the triclinic CPPD and monoclinic CPPD crystals

Calcium Pyrophosphate Crystal Deposition: The Effect of Monosodium Urate and Apatite Crystals in a Kinetic Study Using a Gelatin Matrix Model

The kinetics of calcium pyrophosphate dihydrate (CPPD) crystal growth was studied by allowing calcium and pyrophosphate (PPi-4) ions to diffuse through a denatured collagen matrix (biological grade gelatin) in the presence of either monosodium urate monohydrate (MSU) or hydroxyapatite (HA) crystals. In this in vitro model system, MSU crystals significantly altered the kinetics of PPi-4 ionic diffusion through the gelatin matrix by allowing the [PPi-4] gradient to fall off much more rapidly, suggesting an increased level of scavenging of PPi-4 ions into crystalline materials. Even more significantly, the presence of MSU crystals markedly influenced the crystal growth morphology of triclinic CPPD, producing that observed in vivo. A large number of epitaxially dimensional matches between MSU and triclinic (t) and monoclinic (m) CPPD were identified, suggesting that MSU crystals can epitaxially induce CPPD crystal growth. This finding supports the hypothesis that the association of urate gout and CPPD crystal deposition disease is based on the nucleating potential of MSU crystals for CPPD crystal growth. In contrast, the HA crystal structure did not appear to serve as a nucleating agent for CPPD crystals. However, HA crystals did serve as effective traps for PPi-4 ions and their presence led to more stable CPPD crystal growth

Calcium Pyrophosphate Crystal Deposition: A Kinetic Study Using a Type I Collagen Gel Model

Calcium pyrophosphate dihydrate (CPPD) crystal deposition disease is characterized by deposits of triclinic (t) and monoclinic (m) CPPD crystals in articular and fibrocartilage. Many investigators have attempted to model CPPD crystal growth using both solution and a variety of gel systems. We have investigated the effect of type I collagen fibrils on CPPD crystal nucleation and growth using an ionic diffusion model. Collagen was isolated from porcine menisci using a pepsin solubilization procedure and gelled in three layers, with one containing 10 mM pyrophosphate (PPi) plus physiologic ions, the middle containing only the ions, while the third contained 25 mM Ca plus physiologic ions. Initially, amorphorous calcium pyrophosphate formed at the Ca-PPi interface. Monoclinic CPPD crystallized in 6 weeks when the [Ca] was between 2 and 3 mM and the [PPi] was between 50 and 75 μM. At 13 weeks, t-CPPD formed when the [Ca] was also between 2 and 3 mM, but the PPi was less than 25 μM. One of the most striking differences between this system and all previous solution and gel model systems is the total absence of orthorhombic calcium pyrophosphate tetrahydrate (o-CPPT) from the gels made of collagen fibrils in near native conformation. Further, crystals of t-CPPD appear as large single crystals with the classic prismatic growth habit observed in vivo, and crystals of m-CPPD also evidence the in vivo rod habit. In contrast, the crystal growth habits of t-CPPD, m-CPPD, and o-CPPT grown in all of the other model systems never matched that observed in vivo. When compared to the previous studies, these results, particularly the crystal growth habit data, suggest that the native collagen fibrils themselves can nucleate CPPD crystal formation

Signal-to-noise ratio of Gaussian-state ghost imaging

The signal-to-noise ratios (SNRs) of three Gaussian-state ghost imaging configurations--distinguished by the nature of their light sources--are derived. Two use classical-state light, specifically a joint signal-reference field state that has either the maximum phase-insensitive or the maximum phase-sensitive cross correlation consistent with having a proper $P$ representation. The third uses nonclassical light, in particular an entangled signal-reference field state with the maximum phase-sensitive cross correlation permitted by quantum mechanics. Analytic SNR expressions are developed for the near-field and far-field regimes, within which simple asymptotic approximations are presented for low-brightness and high-brightness sources. A high-brightness thermal-state (classical phase-insensitive state) source will typically achieve a higher SNR than a biphoton-state (low-brightness, low-flux limit of the entangled-state) source, when all other system parameters are equal for the two systems. With high efficiency photon-number resolving detectors, a low-brightness, high-flux entangled-state source may achieve a higher SNR than that obtained with a high-brightness thermal-state source.Comment: 12 pages, 4 figures. This version incorporates additional references and a new analysis of the nonclassical case that, for the first time, includes the complete transition to the classical signal-to-noise ratio asymptote at high source brightnes

Comparison of Gravitational Wave Detector Network Sky Localization Approximations

Gravitational waves emitted during compact binary coalescences are a promising source for gravitational-wave detector networks. The accuracy with which the location of the source on the sky can be inferred from gravitational wave data is a limiting factor for several potential scientific goals of gravitational-wave astronomy, including multi-messenger observations. Various methods have been used to estimate the ability of a proposed network to localize sources. Here we compare two techniques for predicting the uncertainty of sky localization -- timing triangulation and the Fisher information matrix approximations -- with Bayesian inference on the full, coherent data set. We find that timing triangulation alone tends to over-estimate the uncertainty in sky localization by a median factor of $4$ for a set of signals from non-spinning compact object binaries ranging up to a total mass of $20 M_\odot$, and the over-estimation increases with the mass of the system. We find that average predictions can be brought to better agreement by the inclusion of phase consistency information in timing-triangulation techniques. However, even after corrections, these techniques can yield significantly different results to the full analysis on specific mock signals. Thus, while the approximate techniques may be useful in providing rapid, large scale estimates of network localization capability, the fully coherent Bayesian analysis gives more robust results for individual signals, particularly in the presence of detector noise.Comment: 11 pages, 7 Figure