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[1] We extend theories of thermoremanent magnetization (TRM) and partial TRM (pTRM) in multidomain (MD) grains to model thermal demagnetization and pTRM acquisition steps in Thellier paleointensity determination. Because of the interleaving of zero-field and in-field heating-cooling steps to increasing temperatures the initial state for any step is complex, and theoretical modeling is more intricate than for pTRM production or thermal demagnetization separately. At low to moderate temperature T, TRM lost exceeds pTRM regained, causing convex down Arai plots sagging below the ideal single-domain (SD) line. At moderate to high T, pTRM acquisition outweighs TRM loss. As T approaches T Curie , pTRM gain exactly equals TRM loss, and the Arai plot becomes ideal. When pTRMs are produced perpendicular to the original TRM and measured directly rather than by differencing field-on and field-off results, there is less deviation from ideality at low to moderate T. Our theory agrees semiquantitatively with results for parallel and perpendicular pTRMs for large MD magnetites (135 mm). Smaller MD magnetites (6 and 20 mm) have less curved Arai plots, and the smallest magnetites (0.6 and 1 mm) have almost linear plots. Positive pTRM checks demonstrate that curved Arai plots of MD grains are reproducible, unlike curved plots for rocks that alter physicochemically in the Thellier experiment, while negative pTRM tail checks indicate undemagnetized pTRM residuals. Low-temperature demagnetization improves linearity only slightly. Practical applications of this work include using the predicted threshold T below which no net pTRM is produced in a parallel Thellier experiment to screen data used for paleointensity fits. Straight line fits through low-and medium-T points in Arai plots of MD (135 mm) grains overestimated the paleofield by as much as 100% and for small pseudo-single-domain (PSD) (0.6 and 1 mm) grains overestimated by about 25%. However, by using linear segments of medium-to high-T data with f values !0.5 it may be possible to obtain reasonable paleointensity estimates even for larger PSD (6 and 20 mm) and MD grains. Middle-to high-T fits for 0.6 mm grains gave paleointensities within 4% of the correct value, utilizing essentially the entire data set ( f > 0.9). Perpendicular data always gave superior linear fits. Orienting samples with their natural remanent magnetizations perpendicular to the laboratory field is therefore recommended for rocks containing PSD and MD grains. However, double heatings are preferable to single heatings because they allow pTRM tail checks to be carried out. INDEX TERMS: 1521

Although microstress is well‐recognized as playing a very important, and sometimes dominant, role in “pinning” domain walls in minerals of interest to palaeomagnetists, it has often been treated in rock magnetism in a grossly oversimplified and incorrect manner. This paper attempts to review and extend the knowledge of how microstress in general and dislocations in particular affect the motion of domain walls. It is found that only such unrealistic cases as a large grain containing a single dislocation is the microcoercivity, hc, independent of domain wall thickness, w. In a first order approximation, hc is linearly proportional to (λwm)/Ms, where Ms is the saturation magnetization, λ is some appropriate magnetostriction constant, and m is a constant determined by the distribution of microstress. A reasonable upper limit for m for magnetite based on experimental studies is found to be 0.5. Both positive and negative values for m are allowed. Microstress acts like a narrow‐band‐pass filter in the “pinning” of domain walls. Domain walls that have thicknesses roughly 1/5 of the microstress wavelength are far more effectively impeded by the stress than are larger or smaller walls. This implies that scientists can reduce their focus to a narrow range of stress wavelengths that effectively “pin” domain walls and that it is inappropriate to use sinusoidal ranging stress fields to model microstress. Moreover, because domain wall thicknesses usually vary with temperature, the most effective pinning sites are also temperature dependent, even when defects are completely immobile.

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A noninterior path-following algorithm is proposed for the linear complementarity problem. The method employs smoothing techniques introduced by Kanzow. If the LCP is P 0 / R 0 and satisfies a nondegeneracy condition due to Fukushima, Luo, and Pang, then the algorithm is globally linearly convergent. As with interior point path-following methods, the convergence theory relies on the notion of a neighborhood for the central path. However, the choice of neighborhood differs significantly from that which appears in the interior point literature. Numerical experiments are presented that illustrate the significance of the neighborhood concept for this class of methods.