Nonlinear stability and convergence of finite-difference methods for the “good” Boussinesq equation

Summary The good Boussinesq equationu _tt =–u _xxxx +u _xx +(u ²) _xx has recently been found to possess an interesting soliton-interaction mechanism. In this paper we study the nonlinear stability and the convergence of some simple finite-difference schemes for the numerical solution of problems involving the good Boussinesq equation. Numerical experimentas are also reported.     

Best constants for approximations of periodic functions by Fejér operators

We obtain the best estimate for approximations of continuous 2-periodic functions by Fejér summation operators. The estimate is expressed in terms of the moduli of continuity of functions to be approximated. In the class of functions satisfying Lipschitz condition, we obtain the best constant for the approximation by Fejer integral and summation operators.Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 42, No. 1, pp. 75–82, 1990.     

Impulse systems with fixed moments of shocks of general position: The structure of the set of moments of the shocks

Ordinary differential equations with impulse action are considered at fixed moments of time when finite limits points are admitted to the set of moments of the shocks. The topological structure of this set is clarified: it must be resolvable, in particular nowhere dense in R.Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 42, No. 3, pp. 378–383, March, 1990.     

Tracking poles and representing Hankel operators directly from data

Summary We propose and analyse a method of estimating the poles near the unit circleT of a functionG whose values are given at a grid of points onT: we give an algorithm for performing this estimation and prove a convergence theorem. The method is to identify the phase for an estimate by considering the peaks of the absolute value ofG onT, and then to estimate the modulus by seeking a bestL ² fit toG over a small arc by a first order rational function. These pole estimates lead to the construction of a basis ofL ² which is well suited to the numerical representation of the Hankel operator with symbolG and thereby to the numerical solution of the Nehari problem (computing the bestH , analytic, approximation toG relative to theL norm), as analysed in [HY]. We present the results of numerical tests of these algorithms.Partially supported by grants from the AFOSR and NSF     

Delaunay graphs are almost as good as complete graphs

LetS be any set ofN points in the plane and let DT(S) be the graph of the Delaunay triangulation ofS. For all pointsa andb ofS, letd(a, b) be the Euclidean distance froma tob and let DT(a, b) be the length of the shortest path in DT(S) froma tob. We show that there is a constantc (≤((1+√5)/2) π≈5.08) independent ofS andN such that $$\frac{{DT(a,b)}}{{d(a,b)}}< c.$$      

On the long term behavior of some finite particle systems

Summary We consider the problem of comparing large finite and infinite systems with locally interacting components, and present a general comparison scheme for the case when the infinite system is nonergodic. We show that this scheme holds for some specific models. One of these is critical branching random walk onZ ^d . Let _t denote this system, and let _t ^N denote a finite version of _t defined on the torus [–N,N] ^d Z ^d . Ford3 we prove that for stationary, shift ergodic initial measures with density , that ifT(N) andT(N)/(2N+1)^d s[0,] asN, then {v }, 0 is the set of extremal invariant measures for the infinite system _t andQ _s is the transition function of Feller's branching diffusion. We prove several extensions and refinements of this result. The other systems we consider are the voter model and the contact process.Work supported in part by the National Science Foundation under Grant DMS-8802055, by the U.S. Army Research Office through the Mathematical Sciences Institute at Cornell University and by the Deutsche Forschungsgemeinschaft through the SFB 123 at the Universität Heidelberg     

Variations on the theme of repeated distances

We give an asymptotically sharp estimate for the error term of the maximum number of unit distances determined byn points in ^{d, d}4. We also give asymptotically tight upper bounds on the total number of occurrences of the favourite distances fromn points in ^{d, d}4. Related results are proved for distances determined byn disjoint compact convex sets in ².At the time this paper was written, both authors were visiting the Technion — Israel Institute of Technology.     

Electron Emission from Clean Solid Surfaces by Fast Ions

Total backward electron yields from 27 elemental, non-crystalline, clean solids were measured during bombardment by H⁺-, H-, H-, He⁺- and Ar⁺-ions in the energy range from 100 keV to 800 keV. The yields were found to exhibit an oscillatory dependence on the atomic number of the target material correlated with the periods of the periodic system. These Z₂-oscillations are relatively insensitive to the type of projectile and the impact energy at the high projectile energies of this experiment. Present theories of electron emission cannot explain the main experimental results. The reasons for this failure are discussed.     

Effects of flanking band proximity, number, and modulation pattern on comodulation masking release

Comodulation masking release for a 700-Hz pure-tone signal was investigated as a function of the number and spectral positions of 20-Hz-wide comodulated flanking bands. In the first experiment, all stimuli were presented diotically. CMR was examined as a function of the number of flanking bands present, in conditions where the bands were arranged symmetrically around the signal frequency, were below the signal frequency, or were above the signal frequency. The number of flanking bands ranged from one to eight, and the magnitude of the diotic CMR ranged from approximately 5-16 dB. The results indicated: (1) bands closer to the signal resulted in larger masking release, and (2) more bands gave rise to larger CMR (but with diminishing returns above two flanking bands). Two additional sets of diotic conditions were examined and compared to the condition where all eight comodulated flanking bands were present: In one set of conditions, two of the eight flanking bands were removed; in the other set of conditions, two of the eight flanking bands were replaced with bands (termed "deviant" bands) that were not comodulated with respect to the other bands. There was very little effect of reducing eight bands to six, even when the removed bands were relatively near the signal frequency; however, CMR was substantially reduced when deviant bands were introduced, particularly when the deviant bands were placed relatively near the signal frequency. These reductions in CMR were slightly greater when each of the deviant bands had a unique modulation pattern (bideviant bands) than when the two deviant bands themselves shared the same modulation pattern (codeviant bands). In the second experiment, dichotic conditions were examined where the number and spectral positions of the flanking bands in the nonsignal ear were varied (the signal ear received only a 20-Hz-wide noise band centered on the signal frequency). The magnitude of the dichotic CMR ranged from approximately 2-10 dB, depending on condition. Effects of proximity and the number of flanking bands were similar to the effects obtained in diotic conditions. For both the diotic and the dichotic data, the effects of proximity were more consistent with an interpretation based upon across-channel processing than upon a within-channel interaction. The results obtained using deviant bands indicate that it is difficult for the auditory system to disregard the modulation pattern of flanking bands that differ from the modulation pattern of the on-signal band, particularly if such bands are proximal to the signal frequency.     

Vibrotactile temporal gap detection as a function of age

The ability of subjects to detect temporal gaps between bursts of sinusoids or bursts of bandlimited noise was measured to evaluate the phenomenon of tactile "sensory persistence" in older persons. Vibratory stimuli were delivered to the right thenar eminence of 27 subjects ranging in age from 8-75 years. The subjects' task was to detect the presence of a silent interval or "gap" between flanking 350-ms vibrotactile stimuli. The gap-detection threshold, expressed as the amplitude of vibration relative to the absolute detection threshold, decreased as the gap duration increased and was higher for gaps in noise than for gaps in sinusoids. The threshold for detecting short gaps increased with age for noise stimuli, but not for sinusoidal stimuli. Furthermore, the gap-detection threshold recovered more rapidly in older subjects for noise stimuli, but less rapidly in older subjects for sinusoidal stimuli. Because of these differences, it appears that the effects of age on gap detection cannot be due to a simple increase in sensory persistence, but may be due to multiple processes.