Asymptotic behavior of the least common multiple of consecutive arithmetic progression terms

Let $l$ and $m$ be two integers with $l>m\ge 0$, and let $a$ and $b$ be integers with $a\ge 1$ and $a+b\ge 1$. In this paper, we prove that $\log {\rm lcm}_{mn<i\le ln}\{ai+b\} =An+o(n)$, where $A$ is a constant depending on $l, m$ and $a$.Comment: 8 pages. To appear in Archiv der Mathemati

The least common multiple of a sequence of products of linear polynomials

Let $f(x)$ be the product of several linear polynomials with integer coefficients. In this paper, we obtain the estimate: $\log {\rm lcm}(f(1), ..., f(n))\sim An$ as $n\rightarrow\infty$, where $A$ is a constant depending on $f$.Comment: To appear in Acta Mathematica Hungaric

Radar systems for the water resources mission, volume 2

The application of synthetic aperture radar (SAR) in monitoring and managing earth resources was examined. The function of spaceborne radar is to provide maps and map imagery to be used for earth resource and oceanographic applications. Spaceborne radar has the capability of mapping the entire United States regardless of inclement weather; however, the imagery must have a high degree of resolution to be meaningful. Attaining this resolution is possible with the SAR system. Imagery of the required quality must first meet mission parameters in the following areas: antenna patterns, azimuth and range ambiguities, coverage, and angle of incidence

Radar systems for the water resources mission. Volume 4: Appendices E-I

The use of a scanning antenna beam for a synthetic aperture system was examined. When the resolution required was modest, the radar did not use all the time the beam was passing a given point on the ground to build a synthetic aperture, so time was available to scan the beam to other positions and build several images at different ranges. The scanning synthetic-aperture radar (SCANSAR) could achieve swathwidths of well over 100 km with modest antenna size. Design considerations for a SCANSAR for hydrologic parameter observation are presented. Because of the high sensitivity to soil moisture at angles of incidence near vertical, a 7 to 22 deg swath was considered for that application. For snow and ice monitoring, a 22 to 37 deg scan was used. Frequencies from X-band to L-band were used in the design studies, but the proposed system operated in C-band at 4.75 GHz. It achieved an azimuth resolution of about 50 meters at all angles, with a range resolution varying from 150 meters at 7 deg to 31 meters at 37 deg. The antenna required an aperture of 3 x 4.16 meters, and the average transmitter power was under 2 watts

Radar systems for the water resources mission, volume 1

The state of the art determination was made for radar measurement of: soil moisture, snow, standing and flowing water, lake and river ice, determination of required spacecraft radar parameters, study of synthetic-aperture radar systems to meet these parametric requirements, and study of techniques for on-board processing of the radar data. Significant new concepts developed include the following: scanning synthetic-aperture radar to achieve wide-swath coverage; single-sideband radar; and comb-filter range-sequential, range-offset SAR processing. The state of the art in radar measurement of water resources parameters is outlined. The feasibility for immediate development of a spacecraft water resources SAR was established. Numerous candidates for the on-board processor were examined

Radar systems for a polar mission, volume 3, appendices A-D, S, T

Success is reported in the radar monitoring of such features of sea ice as concentration, floe size, leads and other water openings, drift, topographic features such as pressure ridges and hummocks, fractures, and a qualitative indication of age and thickness. Scatterometer measurements made north of Alaska show a good correlation with a scattering coefficient with apparent thickness as deduced from ice type analysis of stereo aerial photography. Indications are that frequencies from 9 GHz upward seem to be better for sea ice radar purposes than the information gathered at 0.4 GHz by a scatterometer. Some information indicates that 1 GHz is useful, but not as useful as higher frequencies. Either form of like-polarization can be used and it appears that cross-polarization may be more useful for thickness measurement. Resolution requirements have not been fully established, but most of the systems in use have had poorer resolution than 20 meters. The radar return from sea ice is found to be much different than that from lake ice. Methods to decrease side lobe levels of the Fresnel zone-plate processor and to decrease the memory requirements of a synthetic radar processor are discussed

Asymptotic Statistics of Poincar\'e Recurrences in Hamiltonian Systems with Divided Phase Space

By different methods we show that for dynamical chaos in the standard map with critical golden curve the Poincar\'e recurrences P(\tau) and correlations C(\tau) asymptotically decay in time as P ~ C/\tau ~ 1/\tau^3. It is also explained why this asymptotic behavior starts only at very large times. We argue that the same exponent p=3 should be also valid for a general chaos border.Comment: revtex, 4 pages, 3 ps-figure