# 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

Mistakes are very common

in the equations of mathematical physics.

I'm Philip Emeagwali. 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.

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.