Narrow Results

The aim of this work is to compare symbolic and ordinary powers of monomial ideals using commutative algebra and combinatorics. Monomial ideals whose symbolic and ordinary powers coincide are called Simis ideals. Weighted monomial ideals are defined by assigning linear weights to monomials. We examine Simis and normally torsion-free ideals, relate some of the properties of monomial ideals and weighted monomial ideals, and present a structure theorem for edge ideals of $d$-uniform clutters whose ideal of covers is Simis in degree $d$. One of our main results is a combinatorial classification of when the dual of the edge ideal of a weighted oriented graph is Simis in degree $2$.

We define the chordal simplicial complex by using the definition of chordal clutter introduced by Woodroofe. We show that the facet ideal of the chordal simplicial complex is Cohen–Macaulay if and only if it is unmixed. Moreover, we prove that the facet ideal of a chordal simplicial complex has infinitely many nontrivial Cohen–Macaulay modifications.

Recently, a 206.5MHz Stratosphere Troposphere (ST) Radar system was successfully installed and made operationalized at Aryabhatta Research Institute of Observational Sciences (ARIES) (29.4N, 79.2E, 1793m amsl), Nainital, India. It is the first such unique observational facility located in the central Himalayan region and will play an important role in understanding the meteorological conditions of the region that has a vital role in atmospheric studies in South Asia. The entire ST radar system is indigenously built and installed in a compact 30$\mathrm{\text{m}}\times 30$m two-storey building, making maximum use of the available space in the hilly terrain. A metal fence of 3.5–4m height was designed and installed along the perimeter of the array to attenuate the clutter returns from the nearby mountains with the shielding efficiency 22–25dB. Since its operation, the radar has obtained useful data of neutral atmosphere, precipitation, convection, and hailstorm events for scientific research. The technical details of different sub-systems, radar integration and calibration methodology are presented here. A dedicated off-line GUI based data processing tool has been developed and is being used for the data analysis. A comparison of wind components derived from ARIES ST Radar with collocated GPS-radiosonde observations indicates a good agreement with correlation coefficients for zonal (0.92), meridional (0.76), wind speed (0.86), and wind direction (0.7). The change in wind patterns is demonstrated up to a height of about 31km amsl and the tropopause was marked to be at 16–17km on 2020 June 20. A dramatic reversal of winds from westerly (below the tropopause) to easterly (above the tropopause) was also observed. ARIES ST Radar could capture the signature of the precipitation in addition to neutral air in the same Doppler spectrum and the height of the starting point of precipitation is identified to be $\sim 6$km. This ability to detect atmospheric scattering from both neutral wind (Bragg) and precipitation (Rayleigh) in the same spectrum makes the 200MHz band radar a unique instrument in the wind profiler application for atmospheric research.