• Bulletin of the Korean Mathematical Society
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• v.56 no.4
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• pp.1077-1097
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• 2019

Let M be a finitely generated module over a regular local ring (R, n). We will fix an n-stable filtration for M and show that the minimal free resolution of M can be obtained from any filtered free resolution of M by zero and negative consecutive cancellations. This result is analogous to [10, Theorem 3.1] in the more general context of filtered free resolutions. Taking advantage of this generality, we will study resolutions obtained by the mapping cone technique and find a sufficient condition for the minimality of such resolutions. Next, we give another application in the graded setting. We show that for a monomial order ${\sigma}$, Betti numbers of I are obtained from those of $LT_{\sigma}(I)$ by so-called zero ${\sigma}$-consecutive cancellations. This provides a stronger version of the well-known cancellation "cancellation principle" between the resolution of a graded ideal and that of its leading term ideal, in terms of filtrations defined by monomial orders.

• Journal of the Korean Society of Surveying, Geodesy, Photogrammetry and Cartography
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• v.14 no.2
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• pp.127-139
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• 1996

This paper presents a procedure to recover sea surface heights (SSH) and free-air (FA) gravity anomalies from dense satellite altimeter SSH data with enhanced accuracies over the full spectrum of the gravity field. A wavenumber correlation filtering (WCF) of co-linear SSH tracks is developed for the coherent signals of sub-surface geological masses. Orbital cross-over adjustments with bias parameters are applied to the filtered SSH data, which are then separated into two groups of ascending and descending tracks and gridded with tensioned splines. A directional sensitive filter (DSF) is developed to reduce residual errors in the orbital adjustments that appear as track patterned SSH. Finally, FA gravity anomalies can be obtained by the application of a gradient filter on a high resolution estimate of geoid undulations after subtracting dynamic sea surface topography (DSST) from the SSH. These procedures are applied to the Geosat Geodetic Mission (GM) data of the southern oceans in a test area of ca. $900km\;\times{1,200}\;km$ to resolve geoid undulations and FA gravity anomalies to wavelengths of-10 km and larger. Comparisons with gravity data from ship surveys, predictions by least squares collocation (LSC), and 2 versions of NOAA's predictions using vertical deflections illustrate the performance of this procedure for recovering all elements of the gravity spectrum. Statistics on differences between precise ship data and predicted FA gravity anomalies show a mean of 0.1 mgal, an RMS of 3.5 mgal, maximum differences of 10. 2 mgal and -18.6 mgal, and a correlation coefficient of 0.993 over four straight ship tracks of ca. 1,600 km where gravity changes over 150 mgals.