26 FebA closer look into a well-known quotient ring

The ring \$\$Z[i]={a+ib: a,b in Z, i^{2}=-1}\$\$  is quite a well- known ring in Algebra. Further, being a principal ideal domain, every ideal of \$\$Z[i]\$\$  is generated by a single element say \$\$a+ib in Z[i]\$\$ and so can be taken in the form \$\$<a+ib>={(x+iy)(a+ib): x,y...

11 FebThe Millennium Problems

Recently Prof. Malay Dutta (Department of Computer Science and Engineering, Tezpur University) gave a talk on the Millennium Problems in mathematics. We present to you a series of 8 articles written by Prof. Dutta where he speaks about the problems in detail. This is the...

11 FebIndian National Mathematical Olympiad 2012

The Indian National Mathematical Olympiad 2012 was held on 5th February, 2012 at various centres all over the country. The North East had three centres at Guwahati, Shillong and Agartala. The questions  are given below: 1. Let \$\$ABCD\$\$ be a quadrilateral inscribed in a circle. Suppose...

10 JanApplications of Sylow Theorems

Sylow Theorem's is Group Theory are an important mathematical tool. Below we give a simple application of the theorems. Theorem: If \$\$H\$\$ and \$\$K\$\$ are subgroups of \$\$G\$\$ and \$\$Hleq N_G(K)\$\$, then \$\$HK\$\$ is a subgroup of \$\$G\$\$. In particular, if \$\$K\$\$ is normal is \$\$G\$\$...

National Entrance Screening Test (NEST) is a compulsory test for students seeking admission to National Institute of Science Education and Research (NISER), Bhubaneswar and University of Mumbai - Department of Atomic Energy Centre for Excellence in Basic Sciences (UM-DAE CBS), Mumbai. Both NISER and UM-DAE...

16 DecTIFR PhD Entrance 2012 Question paper

The Tata Institute of Fundamental Research (TIFR) conducts  a graduate programme leading to the award of PhD degree in certain subjects.  The 2012 Mathematics question paper can be found here. We had also earlier posted the 2011 question paper here. [ad#ad-2]...

09 DecMorley's Theorem

INTRODUCTION: One of the most elementary theorems in elementary geometry was discovered about 1899 by F. Morley (whose son Cristopher wrote novels such as Thunder on the Left). He mentioned it to his friends, who spread it over the world in the form of mathematical gossip....

30 NovWeirstrass' M-Test

We shall state and prove a very important result in Real Analysis called the Weirstrass' M-Test. The statement of the theorem is give below. Let \$\$\$\$ be a sequence of positive real numbers such that \$\$mid f_n(x) mid leq M_n\$\$ for \$\$x in D\$\$, \$\$n in...