Abstract. We present a preliminary study of a new phenomena associated with the Euler-Poisson equations -the so called critical threshold phenomena, where the answer to questions of global smoothness vs. finite time breakdown depends on whether the initial configuration crosses an intrinsic, O(1) critical threshold.We investigate a class of Euler-Poisson equations, ranging from one-dimensional problem with or without various forcing mechanisms to multi-dimensional isotropic models with geometrical symmetry. These models are shown to admit a critical threshold which is reminiscent of the conditional breakdown of waves on the beach; only waves above certain initial critical threshold experience finite-time breakdown, but otherwise they propagate smoothly. At the same time, the asymptotic long time behavior of the solutions remains the same, independent of crossing these initial thresholds.A case in point is the simple one-dimensional problem where the unforced inviscid Burgers' solution always forms a shock discontinuity except for the non-generic case of increasing initial profile, u ′ 0 ≥ 0. In contrast, we show that the corresponding one dimensional Euler-Poisson equation with zero background has global smooth solutions as long as its initial (ρ0, u0)-configuration satisfies u ′ 0 ≥ − √ 2kρ0 -see (2.11) below, allowing a finite, critical negative velocity gradient. As is typical for such nonlinear convection problems one is led to a Ricatti equation which is balanced here by a forcing acting as a 'nonlinear resonance', and which in turn is responsible for this critical threshold phenomena.

Abstract. We consider the problem of detecting edges-jump discontinuities in piecewise smooth functions from their N -degree spectral content, which is assumed to be corrupted by noise. There are three scales involved: the "smoothness" scale of order 1/N , the noise scale of order √ η, and the O(1) scale of the jump discontinuities. We use concentration factors which are adjusted to the standard deviation of the noise √ η 1/N in order to detect the underlying O(1)-edges, which are separated from the noise scale √ η 1.

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The assessment of oxygen saturation in arterial blood by pulse oximetry (SpO2) is based on the different light absorption spectra for oxygenated and deoxygenated hemoglobin and the analysis of photoplethysmographic (PPG) signals acquired at two wavelengths. Commercial pulse oximeters use two wavelengths in the red and infrared regions which have different pathlengths and the relationship between the PPG-derived parameters and oxygen saturation in arterial blood is determined by means of an empirical calibration. This calibration results in an inherent error, and pulse oximetry thus has an error of about 4%, which is too high for some clinical problems. We present calibration-free pulse oximetry for measurement of SpO2, based on PPG pulses of two nearby wavelengths in the infrared. By neglecting the difference between the path-lengths of the two nearby wavelengths, SpO2 can be derived from the PPG parameters with no need for calibration. In the current study we used three laser diodes of wavelengths 780, 785 and 808 nm, with narrow spectral line-width. SaO2 was calculated by using each pair of PPG signals selected from the three wavelengths. In measurements on healthy subjects, SpO2 values, obtained by the 780–808 nm wavelength pair were found to be in the normal range. The measurement of SpO2 by two nearby wavelengths in the infrared with narrow line-width enables the assessment of SpO2 without calibration.

Pulse oximetry is an optical technique for the assessment of oxygen saturation in arterial blood and is based on the different light absorption spectra for oxygenated and deoxygenated hemoglobin and on two-wavelength photoplethysmographic (PPG) measurement of arterial blood volume increase during systole. The technique requires experimental calibration for the determination of the relationship between PPG-derived parameters and arterial oxygen saturation, and this calibration is a source of error in the method. We suggest a three-wavelength PPG technique for the measurement of arterial oxygen saturation that has no need for calibration if the three wavelengths are properly selected in the near-infrared region. The suggested technique can also be implemented for the assessment of venous oxygen saturation by measuring the decrease in transmission of light through a tissue after increasing its blood volume by venous occlusion. The oxygen saturation in venous blood is a parameter that is related to oxygen consumption in tissue and to tissue blood flow. The three-wavelength method has the potential to provide accurate oxygen saturation measurements in arterial and venous blood, but experimental validation of the theory is still required to confirm this claim.