The following is the extended version of my notes from my ATC talk given on June 4, 2014 at UCLA. I begin with a basic introduction to sums-of-squares formulas, and move on to giving motivation for studying these formulas and discussing some results about them over the reals. More recent techniques have made it possible to obtain similar results over arbitrary fields, and some of these are disc...

n = 1: 1 = 0 + 1; n = 2 (prime): 2 = 1 + 1; n = 3 (prime) is not a sum of two squares. n = 4: 4 = 2 + 0. n = 5 (prime): 5 = 2 + 1. n = 6 is not a sum of two squares. n = 7 (prime) is not a sum of two squares. n = 8: 8 = 2 + 2. n = 9: 9 = 3 + 0. n = 10: 10 = 3 + 1. n = 11 (prime) is not a sum of two squares. n = 12 is not a sum of two squares. n = 13 (prime): 13 = 3 + 2. n = 14 is not a sum of t...

Since landscape attributes show different patterns at different spatial extents, it is fundamental to identify how the relation between landscape structure and plant species diversity at local scale varies with scale. Then, it is fundamental to assess the appropriate extent at which landscape factors affect plant species richness at the local scale. To investigate this relation, data on plant s...

In this paper we have studied the role of varying sampling fractions on relative bias of conventional ratio estimator and also for the linear combination of ratio and PPS estimators to the first and second degree of approximations for a wide variety of populations. It will give the survey practitioners an idea whether it is worthwhile to ignore the expressions of mean sum of squares to the orde...

The article is devoted to the problem of finding the triples of integer numbers generating equal sums of the squares in the linear combination. At first the theorem giving the conditions of existing two equal sums of the squares is presented. Based on it the algorithm for finding the triples of the integer numbers generating equal sums of the squares is built. The time complexity is calculated ...