Investigation of the Prince Rupert’s drop properties

Prince Rupert’s drops (PRDs) are very interesting from both scientific and commercial points of view. The main achievements of my theoretical and experimental research are studying the way of PRDs obtainment, defining how much shape and optic-mechanical properties of a drop can vary and achieving very accurate data of how much pressure can PRDs stand. The most important commercial aspects are creation of PRD-based drilling bits, which are muchcheaperthan nowadays’ diamond onesbut still as good as them, composite body armor prototypes and experiments with PRD + concrete composites.

On the number of points on an algebraic curve in a ring of residues

In our paper we obtained the following results:

1. We introduce a new function R_(f,I) (n), which is the number of all possible combinations of giving coordinates of the solutions of the congruence f(x_1,…,x_k )≡0 mod n, and proved that this function is multiplicative.

2. We obtain the exact number of points on a curve x^m-y^k≡0 modn.

3. We introduce a new definition m/k-power residue modulo n and found an exact formula for their number modulo n. From this formula as a corollary we obtained the full results about m-power residues.

4. We calculated the number of points on a curve ax^2+bxy+cy^2≡0modulo a prime number.

5. We found an exact formula for the number of all possible values of quadratic polynomial mod n.

These results can be useful algebraic geometry and asymmetric cryptography.