Adaptive Mollifiers for High Resolution Recovery of Piecewise Smooth Data from its Spectral Information

We discuss the reconstruction of piecewise smooth data from its (pseudo-) spectral information. Spectral projections enjoy superior resolution provided the data is globally smooth, while the presence of jump discontinuities is responsible for spurious O (1) Gibbs oscillations in the neighborhood of edges and an overall deterioration of the unacceptable first-order convergence in rate. The purpose is to regain the superior accuracy in the piecewise smooth case, and this is achieved by mollification. Here we utilize a modified version of the two-parameter family of spectral mollifiers introduced by Gottlieb and Tadmor [GoTa85]. The ubiquitous one-parameter, finite-order mollifiers are based on dilation . In contrast, our mollifiers achieve their high resolution by an intricate process of high-order cancellation . To this end, we first implement a localization step using an edge detection procedure [GeTa00a, b]. The accurate recovery of piecewise smooth data is then carried out in the direction of smoothness away from the edges, and adaptivity is responsible for the high resolution. The resulting adaptive mollifier greatly accelerates the convergence rate, recovering piecewise analytic data within exponential accuracy while removing the spurious oscillations that remained in [GoTa85]. Thus, these adaptive mollifiers offer a robust, general-purpose ``black box' procedure for accurate post-processing of piecewise smooth data. March 29, 2001. Final version received: August 31, 2001.     

On the stability of the unsmoothed Fourier method for hyperbolic equations

Summary. It has been a long open question whether the pseudospectral Fourier method without smoothing is stable for hyperbolic equations with variable coefficients that change signs. In this work we answer this question with a detailed stability analysis of prototype cases of the Fourier method. We show that due to weighted -stability, the -degree Fourier solution is algebraically stable in the sense that its amplification does not exceed . Yet, the Fourier method is weakly -unstable in the sense that it does experience such amplification. The exact mechanism of this weak instability is due the aliasing phenomenon, which is responsible for an amplification of the Fourier modes at the boundaries of the computed spectrum. Two practical conclusions emerge from our discussion. First, the Fourier method is required to have sufficiently many modes in order to resolve the underlying phenomenon. Otherwise, the lack of resolution will excite the weak instability which will propagate from the slowly decaying high modes to the lower ones. Second -- independent of whether smoothing was used or not, the small scale information contained in the highest modes of the Fourier solution will be destroyed by their amplification. Happily, with enough resolution nothing worse can happen. Received December 14, 1992/Revised version received March 1, 1993     

Global regularity of the 4D restricted Euler equations

We are concerned with the critical threshold phenomena in the restricted Euler (RE) equations. Using the spectral and trace dynamics we identify the critical thresholds for the 3D and 4D restricted Euler equations. It is well known that the 3D RE solutions blow up. Projected on the 3-sphere, the set of initial eigenvalues which give rise to bounded stable solutions is reduced to a single point, which confirms that the 3D RE blowup is generic. In contrast, we identify a surprisingly rich set of the initial spectrum on the 4-sphere which yields global smooth solutions; thus, 4D regularity is generic.     

Efficient algorithms for discrete lattice calculations

We discuss algorithms for lattice-based computations, in particular lattice reduction, the detection of nearest neighbors, and the computation of clusters of nearest neighbors. We focus on algorithms that are most efficient for low spatial dimensions (typically d=2,3$d=2\text{,}3$) and input data within a reasonably limited range. This makes them most useful for physically oriented numerical simulations, for example of crystalline solids. Different solution strategies are discussed, formulated as algorithms, and numerically evaluated.     

Sliding friction with polymer brushes

Using high-resolution shear force measurements, we examine in detail the frictional drag between rubbing surfaces bearing end-tethered polymeric surfactants (brushes). The drag attains a maximum on initial motion, attributed to elastic stretching of the chains, which falls by a cascade of relaxations to a value characteristic of kinetic friction. This has a very weak velocity dependence, attributed to chain moieties dragging within a self-regulating, mutual interpenetration zone. When sliding stops, the shear stress across the polymer layers decays logarithmically with time, consistent with the relaxation of a network of dangling ends.     

Eka-elements as chemical pure possibilities

From Mendeleev’s time on, the Periodic Table has been an attempt to exhaust all the chemical possibilities of the elements and their interactions, whether these elements are known as actual or are not known yet as such. These latter elements are called “eka-elements” and there are still some of them in the current state of the Table. There is no guarantee that they will be eventually discovered, synthesized, or isolated as actual. As long as the actual existence of eka-elements is predicted, they cannot be considered as actual but only as purely possible. Given that eka-elements are chemical pure possibilities, a possibilist approach, entitled “panenmentalism,” can gain support as well as an important implication.     

Line energy and the relation between advancing, receding, and young contact angles

The line energy associated with the triple phase contact line is a function of local surface defects (chemical and topographical); however, it can still be calculated from the advancing and receding contact angles to which those defects give rise. In this study an expression for the line energy associated with the triple phase contact line is developed. The expression relates the line energy to the drop volume, the interfacial energies, and the actual contact angle (be it advancing, receding, or in between). From the expression we can back calculate the equilibrium Young contact angle, theta0, as a function of the maximal advancing, thetaA, and minimal receding, thetaR, contact angles. To keep a certain maximal hysteresis between advancing and receding angles, different line energies are required depending on the three interfacial energies and the drop's volume V. We learn from the obtained expressions that the hysteresis is determined by some dimensionless parameter, K, which is some normalized line energy. The value of K required to keep a constant hysteresis (thetaA-thetaR) rises to infinity as we get closer to theta0 = 90 degrees.     

Measurement of the long- and short-range hydrophobic attraction between surfactant-coated surfaces

We have measured the attractive long-range 'hydrophobic' forces in water between double-chained surfactant monolayers physisorbed on mica. We used both normal and high-speed video cameras to follow the dynamics and possible rate-dependence of force-distance profiles in the distance regime from 1000 A to adhesive contact, including the short-distance regime below 100 A-the regime of greatest biological interest. We find that the hydrophobic interaction follows a double-exponential function down to separations of approximately 50 A, after which point the attractive force appears to become considerably stronger.