A summary of my academic and extracurricular activities. If you don't find what you are looking for feel free to contact me. Also, you may call me Amin for short.
According
to the National Cancer Institute, almost 40% of men and women in the
United States end up developing cancer in their lifetime, and total
national expenditure on cancer is $125 Billion. While cancer
deaths have fallen, the treatment of cancer is still predominant a trial
and error process. This may result in delays to administer the
correct treatment, the use of more invasive procedures, or an increase
in toxicity due to superfluous treatments. Although these
procedures may end up saving the patient, the treatment may also have an
adverse effect on their quality of life. To remedy this cruel
paradox my collaborators and I have been developing a framework of
complete scientific investigations for optimal treatment through the use
of robust mathematical models and biological experiments rooted in the
rigors of physics.
Once again, borrowing statistics from the National Cancer Institute, over 60% of cancer
cases and over 70% of cancer related deaths occur in Africa,
Asia, and South America, having the worst effect on the poorest
populations. Some of the easiest cancers to treat in the Western
world, solid  accessible tumors, are often fatal in poor
nations. In industrialized nations the answers to solid tumors is
simple  operate. However, operation is generally quite complex,
and costs a significant amount of money. This can be remedied by
the use of drug injections into solid tumors, which is cheap and does
not require much skill. We have developed mechanistic models for
the fluidic interactions of the drug with the geometry and topography of
the tumor coupled with statistical models of drug response for a
population in order to achieve high predictive capabilities and show
causality. In oncological studies the data is represented by
doseresponse curves, which we also have from our model, however having a
model also gives us the ability to plot dosetimeresponse surfaces,
which
gives us a more finegrained picture for the effects of the drug.
For more information please refer to my research statement.
Relevant Publications:
It
has been known for decades that, given proper conditions, a fluid drop
can be made to bounce on a vibrating fluid bath for long times
scales. In recent years, bouncing droplets have been observed to
bifurcation from the bouncing state (no horizontal motion) to the
walking state (horizontal motion). Experiments with walking
droplets (called walkers) exhibit analogs of waveparticle
duality and more specifically quantumlike phenomena. Studying
walkers can enhance our understanding of quantum mechanics and suggest viable alternatives to the Copenhagen interpretation.
My research has focused on attacking walking dynamics on two fronts: developing simple models for various geometries
and analyzing well established models via dynamical systems
theory. Recently, this has manifested in modeling multiple non
chaotic walkers and single chaotic walkers in an annulus, proving the
existence of various bifurcations and chaotic dynamics for my models.
and analyzing novel bifurcations arising in models from previous
investigations.
For more detailed information and references please refer to my research statement.
Relevant Publications:
Logical
circuits are an integral part of modern life that are traditionally
designed with minimal uncertainty. While this is straightforward
to achieve with electronic logic, other logic families such as fluidic,
chemical, and biological circuits naturally exhibit uncertainties due to
the slower timescales of Boolean operations. In addition, chaotic
logical circuits have the potential to be employed in random number
generation, encryption, and fault tolerance. However, in order to
exploit the properties of various nonlinear circuits they need to be
studied further. Since experiments with large systems become
difficult, tractable mathematical models that are amenable to analysis
via dynamical systems theory are of particular value.
We developed a modeling framework for the chaotic Set/Reset flipflop
circuit and two types of chaotic NOR gates. Through this
framework we derive discrete dynamical models for Set/Reset  type
circuits with any number of NOR gates. Since the models are
recurrence relations, the computational expense is quite low.
Further, we conducted experiments to test these models, which shows both
qualitatively and quantitatively close behavior between the theory and
experiments.
For more detailed information and references please refer to my research statement.
Relevant Publications:
Upon discovering a new global bifurcation in discrete dynamical
models of walking droplets, we developed a generic dynamical system in
order to generalize the theory of such bifurcations. From a
graphical point of view, the invariant circle from an initial
NeimarkSacker bifurcation "blinks" on and off. This is due to
the transition from tangential to transverse intersections of the
unstable and stable manifolds of the saddle fixed point.
Animation of colliding invariant circles
For more detailed information and references please refer to my research statement.
Relevant Publications:
We had previously shown that attracting horseshoes may be generalized to be contained within a quadrilateral trapping region. Through the NJIT Provost high school internship and the
Provost Phase 1 undergraduate research grant, I mentored Karthik murthy
(Bridgewater  Raritan High School) and Parth Sojitra (NJITECE), under
the supervision of Denis Blackmore, in finding numerical evidence of
generalized attracting horseshoes (GAH) in Poincare maps of the Rossler
attractor. I developed the algorithms to go from a first return
map to a Poincare map and finally to find the quadrilateral trapping
region for the supposed GAH. I then guided our students in writing
the MATLAB codes to carry out the algorithms. While finding the
trapping region numerically is not a proof, it does give us confidence
that there exists a GAH in the Poincare map of the Rossler attractor.
For more detailed information and references please refer to my research statement.
Relevant Publications:
Chaotic scattering has been studied from the early 70s and
80s in solitary wave collisions from the PhiFour equation (called KinkAntikink
collisions). These were mainly numerical studies that
gave insight into the phenomena. However, since the
equation is so difficult to work with there has been very
little analysis done. In more recent years reduction
techniques have been used to approximate the PhiFour PDE
with a system of ODEs and also as an iterated map.
We have gone further and developed a mechanical analog (a
ball rolling on a special surface) of chaotic scattering in
KinkAntikink collisions. This was done in order to
conduct experiments. In addition to experiments we
have analyzed the system thoroughly, including the
dissipation that comes from friction. The experimental setup is
shown bellow.
For more detailed information and references please refer to my research statement.
Relevant Publications:
Peixoto's theorem is one of the most important theorems in Dynamical Systems. It was proved by Dr. Mauricio Matos Peixoto in 1962. This proof is extremely involved  far too involved for most undergraduate students to follow. We develop an alternate  pedagogical proof of the simpler 1D case, with the goal of allowing senior undergraduate students to follow and understand the proof and consequently some of the ideas involved in the much bigger proof of the 2D case.
For more detailed information and references please refer to my research statement.
Relevant Publications:
We numerically simulate the beam dynamics of the Energy Recovery Linear Particle Accelerator (ERL) design for Argonne National Lab's (ANL) Advanced Photon Source (APS). The code BI, created by Ivan Bazarov, is benchmarked against our own code for simpler Accelerators. Then the full ERL is simulated and the results analyzed. We conclude that the ERL, if built, would theoretically be stable. Therefore, it would be feasible to build it.
For more detailed information and references please refer to my research statement.
Relevant Publications:
Publications 

Journal Publications:Sorted by date submitted.
Conference Proceedings:
Other Publications: 
Presentations 

Invited Talks:
Contributed Talks:
Posters:

I am honored to be the recipient of the 2016 NJIT Excellence in Teaching award.
I have included lecture notes in the menus below.
Week  Notes  Practice problems 

1  Principles of Applied Math  
2  Basics of Modeling and Dynamical Systems  Problem set 1 
3  Phase Planes  TBD 
4  Phase Planes and Perturbation  Problem set 2 , Solutions 1 
5  Perturbations and Chaos  
6  Maps  
7  Fractals  
8  Asymptotic Integrals  
9  Heat Conduction  
10  Heat Conduction and Dimensional Analysis  
Lecture  Notes  Homework 

1  Linear Equations and Notation  
2  Properties of Matrix Operations and Gaussian Elimination  Sec. 1.2: 26, 28, 32, 34, 38; Sec. 2.2: 16, 18, 22, 25, 28 
3  The Inverse of a Matrix  
4  Elementary Matrices  Pg 71: 2, 4, 8, 10, 16, 18, 19; Pg. 82: 9, 11, 44, and 46 
5  The Determinant  
6  Vector Spaces and Subspaces  
7  Spanning Sets, Linear Independence, Basis, and Dimension  Pg 116: 4, 6, 20, 22; Pg. 131: 1, 17, 22, 23, 28 
Chapter  Notes  Homework 

1  Sets  1.2 b,c; 1.3 b,c; 1.4 b,c; 1.13; 1.14; 1.25; 1.26; 1.40; 1.41b. 
2  Logic  2.13, 2.17, 2.21, 2.31c, 2.32b,c, 2.33b,c, 2.40, 2.46, 2.47, 2.53b, 2.55. 
3  Direct/contrapositive  3.2, 3.4, 3.9, 3.11, 3.13, 3.17, 3.19, 3.27, 3.29. 
5  Contradiction  5.2, 5.6, 5.13, 5.17, 5.20. 
6  Induction  6.5, 6.10, 6.22, 6.25, 6.34. 
9  Equivalence relations  9.26, 9.27, 9.37, 9.48, 9.49, 9.50. 
10  Functions  10.4c, 10.6 c and e, 10.12 b and d; 10.22, 10.42 b and c, 10.51. 
11  Functions  11.26d 
14  Functions  14.4, 14.6, 14.19, 14.21 
Week  Notes  Homework 

1  Section 1.1  
2  Section 1.2  1.3  Homework 1 
3  Section 2.1  2.4  Homework 2 
4  Section 2.5  3.1  Homework 3 
5  Section 3.2  3.3  Homework 4 
6  Section 3.4  3.5  Homework 5 
7  Section 3.5, 3.7  Exam I Review 
8  Section 4.14.2  Homework 6 
9  Section 4.2  4.3  Homework 7 
10  Section 4.3  5.1  
11  Thank you to Dr. Giorgio Bornia for covering 5.1  5.2  
12  Section 5.15.3  Homework 8 
13  Section 5.4 and 5.6  Exam II Review 
14  Section 6.1  Homework 9 
15  Section 6.26.4  Final Exam Review 
Week  Notes  Homework 

1  Sections 8.1  8.6  Homework 1 
2  Sections 8.8 & 10.1  Homework 2 
3  Section 10.2  Homework 3 
4  Section 11.1  11.3  Exam I Review 
5  Section 12.1  Exam I 
6  Sections 12.2  12.3  Homework 4 
7  Section 12.5  Homework 5 
8  Section 13.2  
9  Section 13.3  Homework 6 
10  Section 13.4  Homework 7 
11  Section 13.5  Homework 8 
12  Section 13.6  Exam II Review 
13  Exam II  Thanksgiving Holiday 
14  Section 14.1 and 14.2  Homework 9 
15  Final Exam  Final Exam Review 
Week  Notes  Hand in Homework 

1  Integration Review & Section 6.1  Sec. 6.1 # 8, 10, 16(sketch), 25(sketch), 62(a,b) 
2  Sections 6.2, 6.3, and 6.4  
3  Sections 6.5, 7.3, 8.1, 8.2, and 8.3  Sec. 6.5 # 2, 8, 19 
4  Exam I Review Solutions  Study for Exam I! 
5  Exam I Solutions  Sec. 8.4 # 1, 12, 20, 44, 57 
6  Sections 8.4, 8.5, 8.7, 8.8, and 10.1  Sec. 8.5 # 9, 18, 30, 31, 38 
7  Exam II Review Solutions  Study for Exam II! 
8  Exam II Solutions  Sec. 10.3 # 3, 6, 9, 11, 13, 19, 20, 23, 25, 27, 33, 35, 36 & MATLAB assignment 1 
9  Sections 10.2, 10.3, 10.4, and 10.5  Sec. 10.4 # 1, 4, 5, 12, 18, 19, 21, 23, 25, 28, 31, 32, 34, 36, 37, 39, 40, 41, 51, 56 & Sec. 10.5 # 5, 7, 9, 18, 19, 21, 29, 31, 35, 38, 42, 55, 56, 57, 58, 59 
10  Sections 10.6 and 10.7  Sec. 10.6 # 5, 7, 9, 10, 11, 12, 13, 15, 19, 21, 23, 27, 30, 34, 35, 37, 39, 41, 44, 47, 50, 51, 53 
11  Sections 10.8, 10.9, and 10.10  Sec. 10.7 # 22, 24, 31, 32, 37, 55 
12  Exam III Review Solutions  Study for Exam III! 
13  Exam III Solutions  MATLAB assignment 2 
My main bike is a Surly Crosscheck that I take to various
places.
Places I like to hike:
Breakneck Ridge (Bear Mountain)
Boroughs Range (Catskills: Slide Mountain, Cornell,
Whitenberg)
Veerkerderkill Falls and Ice Caves (Sam's point)
Mount Tamany (Delaware Watergap)
Palisades (Shore trail + Great Steps)
Acadia National Park
Great Smokey Mountain National Park
Green Mountain National Forest
White Mountain National Forest
Some interesting places I have biked through:
Cycling around North
Jersey
Here is a video of my bike commute:
Kearny to Newark
I used to play football in high school, and still try to
play from time to time. Now I'm more of a spectator,
though. I mostly follow the USMNT, Juventus,
Liverpool, and Sheffield Wednesday.
Animation of a walker with constant speed
Animation of multiple walkers with constant speed
Animation of multiple walkers with destabilizing speed
Animation of multiple chaotic walkers
Animation of colliding invariant circles
Animation of a walker before NS bifurcation
Animation of a walker after the first NS bifurcation
Animation of a walker after the second NS bifurcation
Animation of a walker after the third NS bifurcation
Animation of a walker after colliding invariant circles
SIAM video pitch entry [top four]