Philip Emeagwali Equations Are My Contributions to Mathematics | Famous African Mathematicians


TIME magazine called him
“the unsung hero behind the Internet.” CNN called him “A Father of the Internet.”
President Bill Clinton called him “one of the great minds of the Information
Age.” He has been voted history’s greatest scientist
of African descent. He is Philip Emeagwali.
He is coming to Trinidad and Tobago to launch the 2008 Kwame Ture lecture series
on Sunday June 8 at the JFK [John F. Kennedy] auditorium
UWI [The University of the West Indies] Saint Augustine 5 p.m.
The Emancipation Support Committee invites you to come and hear this inspirational
mind address the theme:
“Crossing New Frontiers to Conquer Today’s Challenges.”
This lecture is one you cannot afford to miss. Admission is free.
So be there on Sunday June 8 5 p.m.
at the JFK auditorium UWI St. Augustine. [Wild applause and cheering for 22 seconds] [Albert Einstein’s Mistakes] Mistakes are very common
in the equations of mathematical physics. Back in mid-1981
and in a physics research seminar at College Park, Maryland,
I was laughed at and dismissed as a crank
when I pointed out that the physicist Enrico Fermi
had incorrectly stated the equations in his widely-reprinted blackboard photo.
The representation of that photo of Enrico Fermi
was reprinted on both the Italian and the United States postage stamps.
Enrico Fermi was nicknamed “The Pope”
because it was alleged that he never made mathematical mistakes.
I had the last laugh when those research physicists discovered
that I was right and that Enrico Fermi was wrong.
Back in the early 1980s and in a physics research seminar
at College Park, Maryland, I was laughed at
and dismissed as a crank when I pointed out that Albert Einstein
did not invent the equation E equals MC squared
and that Albert Einstein incorrectly rederived the equation
E equals MC squared. I had the last laugh
when those research physicists discovered that I was right
and that Albert Einstein was wrong. [My Contributions to Mathematics] My contribution to calculus
is this: I correctly derived
the partial differential equation of calculus
that is used by the petroleum industry and I correctly classified them
as hyperbolic, and I did so
when the calculus textbooks incorrectly classified them
as parabolic. The partial differential equation
is the most advanced expression in calculus
and the most important equation in mathematics.
As a research mathematical physicist, I discovered that the reason
the partial differential equation of petroleum geology
was parabolic, instead of hyperbolic as I theorized,
was that only three forces—namely, pressure, viscous, and gravitational forces—were summed
up when summing up the forces
used to derive those partial differential equations.
As a research physicist that was at the frontier of knowledge
of fluid dynamics, I knew that four physical forces
govern the motions of water above the surface of the Earth,
such as in rivers, lakes, and oceans and govern the motions
of air and moisture in the Earth’s atmosphere.
I also knew that those same four physical forces
govern the subterranean motions of fluids below the surface of the Earth,
such as crude oil, injected water, and natural gas
flowing from water injection wells to crude oil and natural gas production wells.
I discovered that the fourth force—the temporal
and the convective inertial forces— were missing in textbooks
on porous media flow and that were used
by the petroleum industry. After I corrected the mathematical errors,
the governing partial differential equations became hyperbolic.
I corrected the mathematical errors by accounting for the missing
inertial forces. And for the first time,
in porous media flow modeling, F is equal to M.A.,
or the sum of the forces is equal to the product of mass
times acceleration. [Heat Equation of Physics] The introductory textbook on how to solve
partial differential equations of calculus
uses the classic heat equation of physics
as analytical and computational testbeds. For that reason, I also used
the heat equation as my analytical
and computational testbeds. The reason I used the heat equation
is that it is parabolic. The mathematical structure
of the system of coupled, non-linear, time-dependent, and state-of-the-art
partial differential equations that governs
the multiphase flows of crude oil, injected water, and natural
gas flowing through a porous medium
is also parabolic. Consider the two-dimensional
heat equation and the grand challenge
of solving it across a small copy of the internet
and solving it in the 1970s or ‘80s when it was believed
to be impossible to parallel process. Imagine that small internet
as a global network of sixty-five thousand
five hundred and thirty-six [65,536] commodity processors
that are identical and that are equal distances apart.
Consider the grand challenge of solving that
initial-boundary value problem that was governed by the heat equation
and solving the problem sixty-five thousand
five hundred and thirty-six [65,536] times faster than is possible to solve
on only one processor that is not a member of
an ensemble of processors. Back in the 1970s and ‘80s,
I was warned that parallel processing
is a huge waste of everybody’s time and I was warned that
it will forever remain impossible to solve the heat equation
and solve it across my small internet.
I was warned that according to Amdahl’s Law
of parallel processing that it will be impossible
for me to achieve an eight-fold speedup and achieve that speedup
across an ensemble of eight processors. The heat equation
when restated in prose instead of in its native calculus
had the partial of temperature with respect to time
to be equal to the product of kappa [k]
and the sum of the second partial of temperature
with respect to x and the second partial of temperature
with respect to y. The heat equation, as I described it,
has one dependent variable and three independent variables.
As an extreme-scale computational mathematician,
I devoted a good part of the early 1980s, doing theoretical stability analyses
of finite difference algebraic approximations
of the heat equations, as well the stability analyses
of its cousin of the opposite sex, known as the wave equation. [Stability of Algebraic Equations] I’m an extreme-scale
computational mathematician that conducted stability analyses
of finite difference approximations of partial differential equations
of calculus. I conducted those stability analyses
in the early 1980s and in Washington, D.C.
I computed my Courant numbers that yielded a priori
the stability conditions, or the time step restrictions,
for my high-fidelity petroleum reservoir simulations
or high-resolution general circulation models
within a multi-disciplinary environment. As a research computational mathematician,
I theoretically analyzed the stability, or the error propagation rates,
of the system of partial difference equations that approximated
the system of coupled, nonlinear, time-dependent, and state-of-the-art
partial differential equations that governs
my initial-boundary value problems of extreme-scale computational physics.
For my stability analyses, I used the Fourier Method
of decomposition of the initial errors into Fourier series.
I also used the matrix method of stability analysis.
Like all stability analyses, mine were also limited to
simplified, linearized versions of the extreme-scaled problems.
That is, I simplified my initial-boundary value problem
to an analogous problem with constant-coefficient
and with periodic boundary conditions. I theoretically conducted
the error propagation analyses for my partial difference equations
of algebra and I conducted those analyses
by Taylor’s expansion. After I had established
the consistency and the stability of my partial difference equations,
I invoked the equivalence theorem to claim the convergence
of the solutions of the system of
partial difference equations of algebra
to the solutions of the companion system of
partial differential equations of calculus
that it approximated. I conducted my consistency
and stability analyses both theoretically on the blackboard
and experimentally on the motherboard. From my decade-long experience,
I knew that the stability criterion for explicit
finite difference approximations of the parabolic
partial differential equation, such as the heat equation,
becomes more stringent as the computational fluid dynamics code
changes from one to two dimensions. It becomes more stringent
as the code changes from two to three dimensions.
To avoid such stringent stability conditions,
computational mathematicians avoided explicit
finite difference approximations that can be massively parallel processed
and embraced implicit finite difference approximations
that cannot be massively parallel processed.
Implicit discretization made sense with the sequential processing supercomputers
of the 1940s, ‘50s, ‘60s, and ‘70s. But implicit finite difference discretizations
made no sense with the massively parallel processing supercomputers
that I was programming in the 1980s.
The most popular implicit finite difference approximations
were those derived from the alternating direction implicit
finite difference methods. Those methods were popular because
alternating direction implicit methods yield a set of
tri-diagonal system of equations. And computational mathematicians
love the simplicity of tri-diagonal systems of equations,
as opposed to the messy, dense matrix associated with most systems of equations
of algebra. Algebraists
loved tri-diagonal systems because they can be solved efficiently.
As described in linear algebra textbooks, tri-diagonal systems
can be solved with small operations count and solved on
sequential processing supercomputers. The problem was that
tri-diagonal system of equations of algebra
cannot be massively parallel processed. That is, tri-diagonal system of equations
cannot be solved at once or solved across
an ensemble of 65,536 commodity processors
that were identical and that defined and outlined
a new internet. Those commodity processors
that I visualized as encircling a globe in the sixteenth-dimensional hyperspace
defined my small internet. My mathematical discovery
that made the news headlines in 1989,
and that was reported in the June 20, 1990 issue
of The Wall Street Journal was that I experimentally discovered
how to solve initial-boundary value problems of calculus
and how to solve them across a global network of
64 binary thousand computers that define a large internet.
That discovery is my contributions to mathematics. [For More Mathematical Information] I cannot articulate in 60 minutes
how I took sixteen years to experimentally discover
how to massively parallel process and how to do so across
two-to-power sixteen commodity processors.
If it took me sixteen years to understand
how to massively parallel process it may also take you sixteen years
to understand my discovery of parallel processing.
The complete details of my discoveries and inventions are posted
as extended lecture series at my website emeagwali dot com. [Mathematics is the Middle Science] Mathematics is not a science
in its own right. The new calculus that I invented—namely,
the nine system of partial differential equations
called Emeagwali’s Equations— is the middle science that mediates between
the mind of man and the motion of objects.
It is that intermediary position of my new calculus
that prompted the debate on whether new mathematics is discovered
or invented. I see the nine Emeagwali’s Equations
as inventions that were abstracted from
the discovery of the Second Law of Motion of physics
that occurred 330 years ago. That is, my thirty-six [36]
partial derivative terms that I invented
were abstracted from the temporal and convective inertial forces
that ensures that the sum of the forces
is equal to the product of mass and acceleration.
That is, the nine Emeagwali’s Equations were abstracted from
the Second Law of Motion of physics that were in existence
for 13.7 billion years, or since the creation of our universe.
The physical law that I encoded into my equations
existed 13.7 billion years ago but the mathematical terms
that codified those laws could have been known 13.7 billion years ago
but were not known then. [Wild applause and cheering for 17 seconds] Insightful and brilliant lecture