Summer research project, Middlebury College.
Advisor Dr. Noah Graham.

The abstract (written by Dr. Graham).

After his first year at Middlebury, Pavlo worked as a summer research
student developing a Mathematica software package for the computation of
spheroidal wave functions. These functions arise when one considers waves
confined within or reflecting from oblate (pancake-shaped) or prolate
(cigar-shaped) spheroidal bodies. They generalize the more common
spherical harmonics and spherical Bessel functions. Pavlo quickly mastered
both the advanced mathematics and the complex computer code involved in
this work. He rewrote sections of the code to compute Taylor expansions of
spheroidal wave functions, fixing several bugs and significantly improving
the code’s efficiency — remarkable contributions, particularly for a
first-year student. Pavlo’s careful testing and benchmarking ensured the
reliability of these changes in the wide variety of situations in which the
code could be used. This code is currently used in Dr. Graham’s
calculations of Casimir energies and forces for spheroidal geometries.

The second year of internship research project for the Terrestrial Planet Finder (TPF) lab of Mechanical and Aerospace Engineering (MAE) department, at Princeton University. Advisor Dr. Belikov.

The abstract.

Terrestrial Planet Finder (TPF) is a project with the objective of finding earthlike planets in the vicinity of other stars. Such planets cannot be detected using simple telescopes, because for the planet to be earthlike, it has to orbit the star close enough. Planets at such a close radius cannot be seen through a telescope because once the starlight gets to the telescope lens, it diffracts, and this diffracted light is much brighter than the light reaching the telescope from the planet. To see the planet, the physicists at Princeton designed, through various optimization techniques, a telescope mask which creates a dark region very close to the star image, in which the weak light from the planet can be detected.

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Project for the Newark Academy. Advisor Dr. DiBianca. The Newark Academy is an independent school which has students from all over northern NJ Since many students drive long distances, we have a large per capita greenhouse gas emission. Pavlo’s idea was to calculate a carpool plan for the school that would result in the least total mileage driven by the school community in the morning commute. This was an optimization problem, which he solved using genetic algorithm programmed in Matlab. The project was tested on the limited set of data but was not finished for the complete school because of the technical problem of calculating the distance between two addresses. The attempt to use various web mapping engines failed because of server shutting multiple automated requests.

Internship research project for the Terrestrial Planet Finder (TPF) lab of Mechanical and Aerospace Engineering (MAE) department, at Princeton University. Advisor Dr. R. Belikov.

The abstract.

Terrestrial Planet Finder (TPF) is a project with the objective of finding earthlike planets in the vicinity of other stars. Such planets cannot be detected using simple telescopes, because for the planet to be earthlike, it has to orbit the star close enough. Planets at such a close radius cannot be seen through a telescope because once the starlight gets to the telescope lens, it diffracts, and this diffracted light is much brighter than the light reaching the telescope from the planet. To see the planet, the physicists at Princeton designed, through various optimization techniques, a telescope mask which creates a dark region very close to the star image, in which the weak light from the planet can be detected.

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The project for the “Biotechnology” class at Newark Academy about artificial nano-scale biochemical devices that mimic the process of photosynthesis. Advisor Ms. Tavarez.

To see the project click on the link: Biotech Project

This is his own project, which he did secretly and presented as New Year gift to his dad. There he tried to understand how salt crystals can melt ice, by numerically solving the time dependent Schrödinger equation. In order to do this he constructed the numerical approximation for first and second order derivatives, and used it to numerically solve the differential Schrödinger equation. He learned how to deal with discontinuities in solutions, and how to solve math problems using MS Excel spreadsheet.

To see the complete project, click the link: Schrodinger equation

This is a project Pavlo did in 8th grade at Copland Middle school for the “Research Project” class. In it he compared Newtonian and modern physics and described how the Newtonian physics failed to describe extremes.
To see the complete project, click the link: Newtonian and modern physics