Interpolation of scattered data: Distance matrices and conditionally positive definite functions

Among other things, we prove that multiquadric surface interpolation is always solvable, thereby settling a conjecture of R. Franke.     

Chaos and Shadowing Around a Heteroclinically Tubular Cycle With an Application to Sine-Gordon Equation

The theory of chaos and shadowing developed recently by the author is amplified to the case of a heteroclinically tubular cycle. Specifically, let F be a   C ³  diffeomorphism on a Banach space. F has a heteroclinically tubular cycle that connects two normally hyperbolic invariant manifolds. Around the heteroclinically tubular cycle, a Bernoulli shift dynamics of submanifolds is established through a shadowing lemma. As an example, a sine-Gordon equation with a chaotic forcing is studied. Existence of a heteroclinically tubular cycle is proved. Also proved are chaos associated with the heteroclinically tubular cycle, and chaos cascade referring to the embeddings of smaller-scale chaos in larger-scale chaos.     

Determinantal formulae and nonsymmetric gaussian perfect elimination

Based upon certain key principal minors, a formula is given for the determinant of a nonsingular matrix in the case its inverse has a perfect elimination directed graph. The main result of [8] may be deduced from this formula. Our method allows us to construct a counterexample to a conjecture made in [8].     

Comparison of a subrank to a full-rank time-reversal operator in a dynamic ocean

This paper investigates the application of time-reversal techniques to the detection and ensonification of a target of interest. The focusing method is based on a generalization of time-reversal operator techniques. A subrank time-reversal operator is derived and implemented using a discrete set of transmission beams to ensonify a region of interest. In a dynamic ocean simulation, target focusing using a subrank matrix is shown to be superior to using a full-rank matrix, specifically when the subrank matrix is captured in a period shorter than the coherence time of the modeled environment. Backscatter from the point target was propagated to a vertical 64-element source-receiver array and processed to form the sub-rank time-reversal operator matrix. The eigenvector corresponding to the strongest eigenvalue of the time-reversal operator was shown to focus energy on the target in simulation. Modeled results will be augmented by a limited at-sea experiment conducted on the New Jersey shelf in April-May 2004 measured low-frequency backscattered signal from an artificial target (echo repeater).     

Effects of acoustic parameters on bubble cloud dynamics in ultrasound tissue erosion (histotripsy)

High intensity pulsed ultrasound can produce significant mechanical tissue fractionation with sharp boundaries ("histotripsy"). At a tissue-fluid interface, histotripsy produces clearly demarcated tissue erosion and the erosion efficiency depends on pulse parameters. Acoustic cavitation is believed to be the primary mechanism for the histotripsy process. To investigate the physical basis of the dependence of tissue erosion on pulse parameters, an optical method was used to monitor the effects of pulse parameters on the cavitating bubble cloud generated by histotripsy pulses at a tissue-water interface. The pulse parameters studied include pulse duration, peak rarefactional pressure, and pulse repetition frequency (PRF). Results show that the duration of growth and collapse (collapse cycle) of the bubble cloud increased with increasing pulse duration, peak rarefactional pressure, and PRF when the next pulse arrived after the collapse of the previous bubble cloud. When the PRF was too high such that the next pulse arrived before the collapse of the previous bubble cloud, only a portion of histotripsy pulses could effectively create and collapse the bubble cloud. The collapse cycle of the bubble cloud also increased with increasing gas concentration. These results may explain previous in vitro results on effects of pulse parameters on tissue erosion.     

Wavelength-selective core-to-jacket coupling in optical fiber

Mode coupling between the fundamental core mode of an as-drawn optical fiber operating far from cutoff and higher-order modes in the plastic jacketing has been observed. In one fiber, the transmission spectrum of the core exhibited five distinct minima between 1200 and 1900 nm. By approximating the fiber jacket as an asymmetric slab waveguide of infinite lateral extent, the wavelengths of these transmission features were successfully predicted using coupled-mode theory. Since conditions at the jacket surface influence the transmission spectrum, this coupling effect potentially enables new applications for environmental sensing.     

Turbulent flow of non-Newtonian substances

Summary A unique shear stress-shear rate relationship exists for laminar flow of any time independent substance in a tube, whereas this is not the case for turbulent flow. In order to obtain a unique relationship for turbulent flow, a new approach based on the elementary theoretical interpretation of experimental data is adopted in the present paper. In particular, wall shear stress is found to be a unique function of a new turbulent pseudo shear rate term. In this relationship therè are two parameters which characterize a given substance — the limiting viscosity at high shear rateµ _m and a factor _m which takes into account modification of turbulent structure by the non-Newtonian properties. Both of these parameters must be determined experimentally. Methods of predicting pressure gradients and of scaling up are outlined. In applying the approach to suspensions in which the solid phase has a density greater than that of the liquid medium, it may be important to determine the increment in shear stress equivalent to the energy required to maintain the solid particles in suspension.The validity of this approach is confirmed by data for the flow of a variety of substances including kaolin suspensions and Carbopol solutions in tubes ranging in diameter from 1.5 to 20 mm. Nomenclature C volume fraction solid in suspension - D tube diameter - f Darcy-Weisbach friction factor - g gravitational acceleration - K _s proportionality constant defined by eq. [10] - L length of tube - P pressure - Re Reynolds number - t exponent defined by eq. [1] - V mean velocity - V ^* volume of particles in pipe lengthL - W settling velocity of particles - _m factor defined by eq. [1] - shear rate - turbulent pseudo shear rate defined by eqs. [8] and [9] - _w wall shear stress - ( _w) _s increment in wall shear stress due to presence of settling particles - µ _m limiting viscosity at high rate of shear - ₁ density of carrier liquid - _m density of mixture - _s density of solid Professor of Chemical Engineering, University of Toronto and scientific advisor to Worthington (Canada) Ltd.With 8 figures