On twists of modules over non-commutative Iwasawa algebras

It is well known that, for any finitely generated torsion module M over the Iwasawa algebra Z_p [[{\Gamma} ]], where {\Gamma} is isomorphic to Z_p, there exists a continuous p-adic character {\rho} of {\Gamma} such that, for every open subgroup U of {\Gamma}, the group of U-coinvariants M({\rho})_U is finite; here M( {\rho}) denotes the twist of M by {\rho}. This twisting lemma was already applied to study various arithmetic properties of Selmer groups and Galois cohomologies over a cyclotomic tower by Greenberg and Perrin-Riou. We prove a non commutative generalization of this twisting lemma replacing torsion modules over Z_p [[ {\Gamma} ]] by certain torsion modules over Z_p [[G]] with more general p-adic Lie group G.Comment: submitte

Spatio-temporal Trends of Standardized Precipitation Index for Meteorological Drought Analysis across Agroclimatic Zones of India

1. Introduction:
Drought is a normal part of climate of India and every year it affects one or the other State. Droughts, like other meteorological phenomena, have spatial and temporal characteristics that vary significantly from one region to another. The understanding of the spatio-temporal trends of meteorological drought helps in undertaking informed decisions on their preparedness and mitigation measures. Though no significant trends have been reported in the Indian Summer Monsoon Rainfall (IMSR) over long periods, the spatio-temporal trends in drought indices reveals the anomaly in rainfall across regions over different time scales which may be related to climate change induced extreme rainfall events. So, a study was carried out to compute spatio-temporal trends in Standardized Precipitation Index (SPI), an index of rainfall anomaly, using gridded monthly rainfall datasets of CRU TS 3.0 for the 1951 – 2006 period for Indian landmass and the results are reported here.

2. Methodology:
SPI is simply the difference in standardized precipitation from its mean for a specified time period divided by the standard deviation. As precipitation is typically not normally distributed for accumulation periods of 12 months or less, SPI overcomes this disadvantage by fitting an incomplete gamma distribution and then transforming it to normal distribution. Negative values of SPI due to less than normal rainfall indicate dryness while SPI less than -1 indicate drought. Delineation of homogeneous regions for climate change / trend analysis has been a debatable matter due to unwise delineation of the homogeneous regions based on a single climatic variable, mostly the isohyets. Therefore, in this study 14 Agroclimatic Zones (ACZs) of India were selected for SPI trend analysis as homogeneous regions (Fig 1) due to commonality of climatic parameters and their extremes, soil types and water resources. The results of drought trends at ACZ level also can be directly translated into plans for agricultural sectors. SPI were computed for individual months (June, July, August and September) and for the whole Indian summer monsoon duration (June-July-August-September i.e. JJAS). Mean SPIs of various ACZs for the individual months and JJAS over a 56 year period were analyzed for temporal trends using the Mann Kendall test and regional temporal trends across all ACZ together using the Regional Kendall test.

3. Results:
Significant temporal trends in monthly & JJAS SPI at 10% or lower level of significance were observed for ACZ4 (Middle Gangetic plains), ACZ5 (Upper Gangetic plains), ACZ6 (Trans Gangetic plains), ACZ7 (Eastern plateau and hills), ACZ8 (Central plateau and hills), ACZ12 (West coast) and ACZ14 (Western dry) regions. The rest of the ACZs did not show any significant trend in SPI for the 56 year study period. In ACZ4, SPI showed a consistently negative trend for JUN, JUL, AUG and JJAS, whereas in ACZ5, SPI showed a significant negative trend for JUL, AUG & JJAS. SPI showed a significant negative trend in SEP & JJAS for ACZ7, AUG & JJAS in ACZ8, JUL in ACZ12 and AUG in ACZ14. On the other hand, a significantly positive trend in SPI was observed in JUN in ACZ6, ACZ8, and ACZ12. 
The analysis of temporal trends in SPI for all ACZs taken together using the Regional Kendall test showed a significant positive trend in JUN SPI, while significant negative trends in SPI were observed for JUL, AUG and JJAS across India. No significant trend was observed for SEP SPI. The rate of increase in JUN SPI was 0.75E-02 per year, while the rate of change of SPI for JUL, AUG, & JJAS was -0.76E-02, -0.54E-02, and -0.65E-02 per year, respectively.

4. Conclusions:
Among the different ACZ of India, there was an increase in probability of meteorological drought hazard in ACZ4, ACZ5, ACZ7 and ACZ8 covering the States of Bihar, Uttar Pradesh, Madhya Pradesh, Orissa and Eastern Rajasthan. Results also point to a significant decreasing trend in rainfall in these regions for different months as well as for the JJAS monsoon period. In the dry western parts of India (Western Rajasthan and Gujarat), which are traditionally water scare regions, there is no change in probability of occurrence of meteorological drought. 
For India as a whole, there is an increase in probability of meteorological drought in the future due to decreasing trends in rainfall for the JJAS period. The months of July and August will become drier, while June will become wetter. This change in rainfall distribution towards the early period and overall drier months of July and August have an important implication for the productivity of the main crop season of India thus impacting its food security negatively

Binary Cubic Forms and Rational Cube Sum Problem

The classical Diophantine problem of determining which integers can be written as a sum of two rational cubes has a long history; from the earlier works of Sylvester, Satg{\'e}, Selmer etc. and up to the recent work of Alp{\"o}ge-Bhargava-Shnidman. In this note, we use integral binary cubic forms to study the rational cube sum problem. We prove (unconditionally) that for any positive integer $d$, infinitely many primes in each of the residue classes $1 \pmod {9d}$ as well as $-1 \pmod {9d}$, are sums of two rational cubes. Among other results, we prove that every non-zero residue class $a \pmod {q}$, for any prime $q$, contains infinitely many primes which are sums of two rational cubes. Further, for an arbitrary integer $N$, we show there are infinitely many primes $p$ in each of the residue classes $8 \pmod 9$ and $1 \pmod 9$, such that $Np$ is a sum of two rational cubes

Impact of Climate Change on Northeast Monsoon System of India - Role of Siberian Teleconnection

The Northeast monsoon (NEM) precipitation has undergone a significant change in pattern over the last half century. The changing pattern of climatic variables over the moisture source (i.e. west Pacific High) and the heat sink (Siberian High) of the winter monsoon dynamics over time has caused a shift in the pattern of NEM precipitation over the southern peninsular region of India. There is no significant trend in NEM precipitation in the Indian region whereas the surface temperature has a significantly increasing trend over India during the NEM season. There is no significant trend in outgoing long wave radiation (OLR) of Siberian High (SH) while the west Pacific High (WPH) has a significantly increasing trend in OLR. Surface pressure of both the SH & WPH has no significant trend for the last 59 years (1948-2006). Surface temperature over SH & WPH has a significantly increasing trend for the last 59 years (1948-2006).There is a high correlation of NEM precipitation with El Nino & Southern Oscillation (ENSO), Indian Ocean Dipole(IOD) and OLR of WPH during the period 1970-2000. It signifies that convectional activity in the moisture source region of the NEM, warm SST in the western Indian Ocean and the ENSO have a deep bearing on the NEM precipitation during the three decades 1970-2000. The correlation of NEM precipitation with ENSO, IOD during the last period 2000-2006 has undergone changes where the NEM precipitation has shown a shift that is negatively correlated with ENSO & IOD. The change is much more in IOD than ENSO which signifies that the conventional trend of bearing of warm or cold SST of West Indian ocean on NEM precipitation has decreased during this period of 2000-2006. The correlation of NEM precipitation with the convectional activity of the moisture sink region of the NEM has been gradually decreasing since the 1970s and the moisture source of NEM has a significantly decreasing convectional activity trend. The correlation of NEM precipitation with all three variables (OLR, surface pressure & surface temperature) has shown a comparatively higher value for the heat sink regions (Siberian High) than for the moisture source region (West Pacific High) during the period 2000-2006. Thus the NEM precipitation over India has faced a deep bearing by the role of Siberian High interference