A grid redistribution method for singular problems

Many physical phenomena develop singular, or nearly singular behavior in localized regions, e.g. boundary layers or blowup solutions. Using uniform grids for such problems becomes computationally prohibitive as the solution approaches singularity. Ren and Wang developed a semi-static adaptive grid method [W. Ren, X.P. Wang, An iterative grid redistribution method for singular problems in multiple dimensions, J. Comput. Phys. 159 (2000) 246–273] for the solution of these problems, known as the iterative grid redistribution (IGR) method. In this study we develop a theoretical basis for semi-static adaptive grid method for singular problems. Based on this theory, we obtain the key result of this study – a methodology for designing robust weight functionals which ensures grid resolution in the singular region, as well as control of the maximal grid spacing in the outer region. Using this methodology, we introduce a semi-static adaptive grid method, which does not involve an iterative procedure for grid redistribution, as in the IGR method. We demonstrate the efficacy of this method with numerical examples of solutions which localize by more than nine orders of magnitude.     

Fast Dimension Reduction Using Rademacher Series on Dual BCH Codes

The Fast Johnson–Lindenstrauss Transform (FJLT) was recently discovered by Ailon and Chazelle as a novel technique for performing fast dimension reduction with small distortion from ℓ ₂ ^d to ℓ ₂ ^k in time O(max {dlog d,k ³}). For k in [Ω(log d),O(d ^1/2)], this beats time O(dk) achieved by naive multiplication by random dense matrices, an approach followed by several authors as a variant of the seminal result by Johnson and Lindenstrauss (JL) from the mid 1980s. In this work we show how to significantly improve the running time to O(dlog k) for k=O(d ^1/2−δ ), for any arbitrary small fixed δ. This beats the better of FJLT and JL. Our analysis uses a powerful measure concentration bound due to Talagrand applied to Rademacher series in Banach spaces (sums of vectors in Banach spaces with random signs). The set of vectors used is a real embedding of dual BCH code vectors over GF(2). We also discuss the number of random bits used and reduction to ℓ ₁ space. The connection between geometry and discrete coding theory discussed here is interesting in its own right and may be useful in other algorithmic applications as well.     

Forces between mica surfaces, prepared in different ways, across aqueous and nonaqueous liquids confined to molecularly thin films

We have measured normal and lateral interactions across a range of different liquids between mica surfaces using a surface force balance (SFB). The mica surfaces were prepared either by melt cutting using Pt wire and standard procedures in our laboratories or by tearing sheets (that had not been exposed to Pt) off from a freshly cleaved sheet of mica. AFM micrographs revealed the substantial absence of Pt nanoparticles on the melt cut and torn-off mica surfaces. Normal-force versus surface-separation (D) profiles and shear force versus D measurements for purified water (no added salt), for concentrated aqueous NaCl solutions, and for cyclohexane revealed that in all cases the behavior of the highly confined liquids between melt-cut and between torn-off mica sheets was identical within experimental scatter. These results demonstrate directly that interactions measured between melt-cut mica surfaces as routinely prepared using established procedures in our laboratories and in other laboratories are free of the effect of any Pt contamination.     

Nonequilibrium membrane fluctuations driven by active proteins

We extend a model for nonthermal membrane undulations driven by active (adenosine triphosphate-dependent or light-harvesting) membrane proteins [N. Gov, Phys. Rev. Lett. 93, 268104 (2004)]. The present model accounts for the fact that proteins can diffuse laterally across the membrane surface and that individual proteins are expected to exert forces preferentially in one normal direction over the other (due to their orientation within the bilayer). The addition of these effects alters the scaling of fluctuation amplitudes with system size. Additionally, theoretical arguments and dynamic simulations both suggest that, in certain regimes, the probability distribution of fluctuation amplitudes is expected to be non-Gaussian (in contrast to thermal systems).     

Generation of a hollow dark spherical spot by 4pi focusing of a radially polarized Laguerre-Gaussian beam

The properties of the focal spot for 4pi focusing with radially polarized first-order Laguerre-Gaussian beams are calculated. It is shown that a focal spot that has an extremely sharp dark region at the center and an almost-perfect spherical symmetry can be achieved. When such a hollow dark spherical spot is used in 4pi fluorescence depletion microscopy, an axial FWHM spot size of approximately 39 nm and a transverse FWHM spot size of approximately 64 nm can be achieved simultaneously in a practical system.     

Nonsymmetrized Hyperspherical Harmonics with Realistic Potentials

The Schrödinger equation is solved for an A-nucleon system using an expansion of the wave function in nonsymmetrized hyperspherical harmonics. The present approach is based on a formalism developed by Gattobigio et al. (Phys. Rev. A 79:032513, 2009; Few-Body Syst. 45:127–131, 2009; Phys. Rev. C 83:024001, 2011). Spin and isospin degrees of freedom are included; this makes possible calculations with realistic NN potential models. The fermionic ground state is determined by introducing an additional potential term involving the Casimir operator such that the antisymmetric ground state becomes the lowest eigenstate of the A-body system. Results are discussed for ⁴He with the realistic AV18 NN potential and for ⁶Li with the semirealistic MTI/III NN potential.     

Estimating the distance to a monotone function

In standard property testing, the task is to distinguish between objects that have a property 𝒫 and those that are ε‐far from 𝒫, for some ε > 0. In this setting, it is perfectly acceptable for the tester to provide a negative answer for every input object that does not satisfy 𝒫. This implies that property testing in and of itself cannot be expected to yield any information whatsoever about the distance from the object to the property. We address this problem in this paper, restricting our attention to monotonicity testing. A function f : {1,…,n} ↦ R is at distance ε_f from being monotone if it can (and must) be modified at ε_fn places to become monotone. For any fixed δ > 0, we compute, with probability at least 2/3, an interval [(1/2 − δ)ε,ε] that encloses ε_f. The running time of our algorithm is O(ε_f⁻¹ log log ε_f^{− 1} log n), which is optimal within a factor of loglog ε_f⁻¹ and represents a substantial improvement over previous work. We give a second algorithm with an expected running time of O(ε_f⁻¹ log nlog log log n). Finally, we extend our results to multivariate functions. © 2007 Wiley Periodicals, Inc. Random Struct. Alg., 2007     

In vivo effects of porphyrins on bacterial DNA.

The DNA damage in intact Staphylococcus aureus and E. coli cells induced by photosensitized deuteroporphyrin or hemin is described. Treatment of S. aureus cultures with hemin or photosensitized deuteroporphyrin (Dp) caused time-dependent changes in the plasmidial DNA profiles. The major observation was the disappearance of the plasmid supercoiled fraction. The chromosomal DNA was also affected by hemin and by photosensitized Dp, since its degradation products were detected after exposing the bacterial cells to the porphyrin drugs. Photosensitization of E. coli cells, pretreated with Dp and polymyxin B nonapeptide (PMBNP), also resulted in plasmidial damage. No such damage occurred when E. coli cultures were treated with hemin and PMBNP. The above results can be tightly correlated with the antimicrobial action of porphyrins. Their damage to the bacterial DNA seems to reflect one of the in vivo effects of these porphyrins.     

Dense Fast Random Projections and Lean Walsh Transforms

Random projection methods give distributions over k×d matrices such that if a matrix Ψ (chosen according to the distribution) is applied to a finite set of vectors x _i ∈ℝ ^d the resulting vectors Ψx _i ∈ℝ ^k approximately preserve the original metric with constant probability. First, we show that any matrix (composed with a random ±1 diagonal matrix) is a good random projector for a subset of vectors in ℝ ^d . Second, we describe a family of tensor product matrices which we term Lean Walsh. We show that using Lean Walsh matrices as random projections outperforms, in terms of running time, the best known current result (due to Matousek) under comparable assumptions.