| Eusebio “Seb” Legarda Koh, Ph.D. Regina, Saskatchewan, Canada Dedicated To Highlight Positive Filipino Image Columnist • Mathematician • Dedicated Leader |
Dr. Eusebio (Seb) L. Koh has been a professor emeritus of
mathematics at the University of Regina since 1999. He has
taught mechanical engineering and mathematics in the
Philippines, the United State, Canada, Germany and Saudi
Arabia. He holds masters degrees from Purdue University and
the University of Birmingham in England and a Ph. D. from SUNY,
Stony Brook. He has published over fifty refereed research papers
in mathematics and sundry articles in engineering. He has given
lectures and conference seminars in the Philippines, other Asian
countries, North America and Europe.
In May, 1998, he was honored in Toronto as one of the
Outstanding Filipino-Canadians. Later that year, he was honored
in Washington D. C. as one of Twenty Outstanding
Filipino-Americans (TOFA) by the Filipino Image Magazine.
Dr. Koh is an organizer. In 1971, he was co-founder and the first
President of the Philippine Association of Saskatchewan. This
organization is still very active doing cultural activities and helping
Filipinos in need. In 1999, he co-founded the Knights of
Columbus Santo Nino Council #12415 of which he was elected
as the Charter Grand Knight. This council has won the Star
mathematics at the University of Regina since 1999. He has
taught mechanical engineering and mathematics in the
Philippines, the United State, Canada, Germany and Saudi
Arabia. He holds masters degrees from Purdue University and
the University of Birmingham in England and a Ph. D. from SUNY,
Stony Brook. He has published over fifty refereed research papers
in mathematics and sundry articles in engineering. He has given
lectures and conference seminars in the Philippines, other Asian
countries, North America and Europe.
In May, 1998, he was honored in Toronto as one of the
Outstanding Filipino-Canadians. Later that year, he was honored
in Washington D. C. as one of Twenty Outstanding
Filipino-Americans (TOFA) by the Filipino Image Magazine.
Dr. Koh is an organizer. In 1971, he was co-founder and the first
President of the Philippine Association of Saskatchewan. This
organization is still very active doing cultural activities and helping
Filipinos in need. In 1999, he co-founded the Knights of
Columbus Santo Nino Council #12415 of which he was elected
as the Charter Grand Knight. This council has won the Star
Council Award five times in the last seven years. The award is the highest award given by the Knights of
Columbus for achievements in service and charitable works as well as membership and insurance
promotions. He was also a founding member of the Philippines-American Academy of Science and
Engineering (PAASE), chartered in Indiana. In 2004 he was selected to present the most prestigious Founder’
s Lecture of PAASE.
Since his retirement from university teaching, he has published short stories and poems in Our Own Voice,
The Best Philippine Short Stories and the Prairie Messenger, the Catholic weekly in Saskatchewan. Since
1993, he writes a column for Filipino Journal, a Philippine semimonthly in Winnipeg.
When not writing, he is into chess, tennis, crossword and Sudoku puzzles, volunteer church work, Knights of
Columbus or learning to cook.
Columbus for achievements in service and charitable works as well as membership and insurance
promotions. He was also a founding member of the Philippines-American Academy of Science and
Engineering (PAASE), chartered in Indiana. In 2004 he was selected to present the most prestigious Founder’
s Lecture of PAASE.
Since his retirement from university teaching, he has published short stories and poems in Our Own Voice,
The Best Philippine Short Stories and the Prairie Messenger, the Catholic weekly in Saskatchewan. Since
1993, he writes a column for Filipino Journal, a Philippine semimonthly in Winnipeg.
When not writing, he is into chess, tennis, crossword and Sudoku puzzles, volunteer church work, Knights of
Columbus or learning to cook.
Washington D.C. Since 1987
 |
 |
 |
 |
 |
 |
 |
August 24, 2006 issue of Philippine Star which published Founder's Lecture:
"Math and Aftermath"
Many years ago while taking an electrical engineering class at the University of the Philippines in Diliman, I
was thoroughly impressed by my professor, a Prof. Bartolome Blanco, who solved a differential equation with
initial conditions by some sort of a “trick.” He called his trick the Laplace transform.
With thirty some bright-eyed classmates agog at learning new ideas, I was wondering if he was pulling our
legs. Ever quick on the uptake, Prof. Blanco said, “If any of you wish to impugn what I taught here, you can
check out my solution by plugging it into the equation or you can run to the Math Department to see what was
going on.”
Wow! Some of us had brushed elbows with the simplest differential equations, never heard of transforms
except in reference to converting high voltage to low voltage in ac circuits. And “impugn”? I had read that word
somewhere in a long forgotten book but that was the first time I heard it used. Now you understand why I was
impressed with my prof and had visions of following in his footsteps.
So much so that I jumped on the opportunity to teach engineering at UP upon my graduation when Dean C.
Ortigas offered me an instructorship. At the same time I took more math classes on the side: a class on
differential equations here, a class on advanced engineering math there. It certainly gave me a step up when
UP later sent me abroad for graduate studies.
Now fast forward to what I ended up doing. My doctoral dissertation at the State University of New York at Stony
Brook was on the Hankel transformation of generalized functions with applications to sorts of differential
equations. This is really a mouthful and I am going too far ahead of myself. With your indulgence, let me start
from scratch.
1. Mathematics is the study of relations and their properties. A relation is a correspondence between two sets
of objects. These objects can be anything: numbers, figures, sets, or even relations too. When the
correspondence assigns a unique object from the second set to each element of the first set, we call that
relation a function. A function is usually restricted to a relation between sets of numbers. If the relation is
between sets of functions, we call it an operator. If the relation is from a set of functions to a set of numbers,
we call it a functional. Here are some examples.
Functions: linear, quadratic, rational, exponential, trigonometric, special.
Operators: differential operator - d/dx, integral operator- ∫…dx.
Functionals: definite integrals, Dirac delta.
2. Calculus is a branch of math dealing with change. The rate of change of a function f is its derivative f’ and
the process of finding the derivative is called differentiation. Because our world is always in a state of flux,
processes, natural or otherwise, can be described by equations involving derivatives. These are called
differential equations, which are creatures more intractable than algebraic equations. Their solutions are
functions while the latter has numbers for solutions. For example:
Algebraic equation: x square +4x+3=0 is solved by x = -1 or x = -3.
Differential equation: y”+4y’+3y=0 is solved by y=a[exp(-x)]+b[exp(-3x)], a & b constants.
The last equation describes a simple damped spring mass system or a simple electric circuit with a battery, a
capacitor and a resistance and the solution is a combination of two decaying exponential functions. The
solution is found by the observation that an exponential function replicates itself on differentiation. Thus the
task is reduced to solving an algebraic equation. More complicated equations are solved by such techniques
as infinite series, integral transformation, Fourier series, special functions and numerical methods and
computer codes.
3. An integral transformation is an operator that takes a set of functions and maps it into a set of transforms
via integration. The most famous is the Laplace transform given by F(s) = L[f(t)] = ∫exp(-st)f(t)dt with the
important property L[f’(t)] = sL[f(t)] – f(0). Thus when f(0) = 0, differentiation corresponds to multiplication by s in
the transform domain. The Laplace transform has been used to justify Heaviside’s calculus used by
engineers. The term exp(-st) is called the kernel of the transform. By changing the kernel as well as the range
of integration, other integral transforms were developed and used for solving other equations. The Hankel
transform replaces exp(-st) with √st Jμ(st) where Jμ(st) is Bessel function of order μ. This transform is
suitable for differential equations with variable coefficients that arise in physical problems with axial symmetry.
Other transforms go by such names as Fourier, Weierstrass, Stieltjes, Meijer, Hilbert, Jacobi, convolution, etc.
each with its suitability for certain differential operators. The definitions, properties and applications are
enough to fill several books.
4. In 1926, Dirac introduced his δ-function as δ(x)=0 when x≠0, ∫f(x) δ(a-x)dx=f(a) and used it in quantum
mechanics. This is not a function in the classical sense. To be sure there were other improper functions such
as Hadamard’s pseudofunction pf x, Cauchy’s pv of a divergent integral, derivatives of Dirac- δ that have
appeared in some physicists’ works. In the 1950’s, Laurent Schwartz came up with his Theory of Distributions
to explain all these as continuous linear functionals on test functions with compact supports. In the 60’s the
Russians Gelfand and Shilov came up with their four-volume treatise on Generalized Functions not drastically
different from Schwartz.
5. In the 60’s and 70’s, Armen Zemanian and others developed various integral transformations of
generalized functions (gfs). Zemanian and I developed the generalized Hankel transform under some
restrictions. Later, I went farther than our work going into representations of Hankel transforms, transforms of
negative orders, transforms in higher dimensions. I later developed the generalized Meijer transform (with my
student M. Ali), Hankel Transforms of Banach-space-valued gfs (with my student C. K. Li).
6. There are two methods of extending the integral transformation to generalized functions. The Adjoint
Method uses a Parseval –type relation to define the transform of a gf F as the application of F on the transform
of a suitable test function. Schwartz used this method for the Fourier transform, Gelfand and Shilov for the
Hilbert transform, Zemanian for the Hankel and other transforms. What’s involved is the construction of two
test function spaces such that one is the transform of the other. Details can be found in their books.
The other method defines the transform as the application of the generalized function F on the kernel as an
element of a test function space. This necessitates the construction of a suitable complete seminormed
space that contains the kernel. Zemanian used this method for the Laplace transform and I did it for the
Hankel transform. Details may be found in Zemanian’s book and Brychkov and Prudnikov (1998).
The University of the Philippines is a great source of inspiration for our youth despite its poverty of material
resources. It is however rich in a faculty that is a fountain of wisdom and knowledge. I credit Prof. Blanco in
igniting a spark to my career. But there were many excellent profs I had. I would be remiss not to acknowledge
Professor Josefina Constantino who instilled in me a love of the English language. From the time I became
Professor Emeritus in 1999, I have published short stories, poems and essays online and in print. In these
writings my math background seems to creep in.
American poet Adelaide Crapsey invented a verse form in the 1920s called cinquain, which is a poem of five
iambic lines of two, four, six, eight and two syllables. There are no restrictions on rhyme but there is a central
theme in the cinquain. In 1927, Angela M. Gloria published three cinquains in the Philippine Herald. Here are
some unpublished cinquains of mine on scientists.
"Math and Aftermath"
Many years ago while taking an electrical engineering class at the University of the Philippines in Diliman, I
was thoroughly impressed by my professor, a Prof. Bartolome Blanco, who solved a differential equation with
initial conditions by some sort of a “trick.” He called his trick the Laplace transform.
With thirty some bright-eyed classmates agog at learning new ideas, I was wondering if he was pulling our
legs. Ever quick on the uptake, Prof. Blanco said, “If any of you wish to impugn what I taught here, you can
check out my solution by plugging it into the equation or you can run to the Math Department to see what was
going on.”
Wow! Some of us had brushed elbows with the simplest differential equations, never heard of transforms
except in reference to converting high voltage to low voltage in ac circuits. And “impugn”? I had read that word
somewhere in a long forgotten book but that was the first time I heard it used. Now you understand why I was
impressed with my prof and had visions of following in his footsteps.
So much so that I jumped on the opportunity to teach engineering at UP upon my graduation when Dean C.
Ortigas offered me an instructorship. At the same time I took more math classes on the side: a class on
differential equations here, a class on advanced engineering math there. It certainly gave me a step up when
UP later sent me abroad for graduate studies.
Now fast forward to what I ended up doing. My doctoral dissertation at the State University of New York at Stony
Brook was on the Hankel transformation of generalized functions with applications to sorts of differential
equations. This is really a mouthful and I am going too far ahead of myself. With your indulgence, let me start
from scratch.
1. Mathematics is the study of relations and their properties. A relation is a correspondence between two sets
of objects. These objects can be anything: numbers, figures, sets, or even relations too. When the
correspondence assigns a unique object from the second set to each element of the first set, we call that
relation a function. A function is usually restricted to a relation between sets of numbers. If the relation is
between sets of functions, we call it an operator. If the relation is from a set of functions to a set of numbers,
we call it a functional. Here are some examples.
Functions: linear, quadratic, rational, exponential, trigonometric, special.
Operators: differential operator - d/dx, integral operator- ∫…dx.
Functionals: definite integrals, Dirac delta.
2. Calculus is a branch of math dealing with change. The rate of change of a function f is its derivative f’ and
the process of finding the derivative is called differentiation. Because our world is always in a state of flux,
processes, natural or otherwise, can be described by equations involving derivatives. These are called
differential equations, which are creatures more intractable than algebraic equations. Their solutions are
functions while the latter has numbers for solutions. For example:
Algebraic equation: x square +4x+3=0 is solved by x = -1 or x = -3.
Differential equation: y”+4y’+3y=0 is solved by y=a[exp(-x)]+b[exp(-3x)], a & b constants.
The last equation describes a simple damped spring mass system or a simple electric circuit with a battery, a
capacitor and a resistance and the solution is a combination of two decaying exponential functions. The
solution is found by the observation that an exponential function replicates itself on differentiation. Thus the
task is reduced to solving an algebraic equation. More complicated equations are solved by such techniques
as infinite series, integral transformation, Fourier series, special functions and numerical methods and
computer codes.
3. An integral transformation is an operator that takes a set of functions and maps it into a set of transforms
via integration. The most famous is the Laplace transform given by F(s) = L[f(t)] = ∫exp(-st)f(t)dt with the
important property L[f’(t)] = sL[f(t)] – f(0). Thus when f(0) = 0, differentiation corresponds to multiplication by s in
the transform domain. The Laplace transform has been used to justify Heaviside’s calculus used by
engineers. The term exp(-st) is called the kernel of the transform. By changing the kernel as well as the range
of integration, other integral transforms were developed and used for solving other equations. The Hankel
transform replaces exp(-st) with √st Jμ(st) where Jμ(st) is Bessel function of order μ. This transform is
suitable for differential equations with variable coefficients that arise in physical problems with axial symmetry.
Other transforms go by such names as Fourier, Weierstrass, Stieltjes, Meijer, Hilbert, Jacobi, convolution, etc.
each with its suitability for certain differential operators. The definitions, properties and applications are
enough to fill several books.
4. In 1926, Dirac introduced his δ-function as δ(x)=0 when x≠0, ∫f(x) δ(a-x)dx=f(a) and used it in quantum
mechanics. This is not a function in the classical sense. To be sure there were other improper functions such
as Hadamard’s pseudofunction pf x, Cauchy’s pv of a divergent integral, derivatives of Dirac- δ that have
appeared in some physicists’ works. In the 1950’s, Laurent Schwartz came up with his Theory of Distributions
to explain all these as continuous linear functionals on test functions with compact supports. In the 60’s the
Russians Gelfand and Shilov came up with their four-volume treatise on Generalized Functions not drastically
different from Schwartz.
5. In the 60’s and 70’s, Armen Zemanian and others developed various integral transformations of
generalized functions (gfs). Zemanian and I developed the generalized Hankel transform under some
restrictions. Later, I went farther than our work going into representations of Hankel transforms, transforms of
negative orders, transforms in higher dimensions. I later developed the generalized Meijer transform (with my
student M. Ali), Hankel Transforms of Banach-space-valued gfs (with my student C. K. Li).
6. There are two methods of extending the integral transformation to generalized functions. The Adjoint
Method uses a Parseval –type relation to define the transform of a gf F as the application of F on the transform
of a suitable test function. Schwartz used this method for the Fourier transform, Gelfand and Shilov for the
Hilbert transform, Zemanian for the Hankel and other transforms. What’s involved is the construction of two
test function spaces such that one is the transform of the other. Details can be found in their books.
The other method defines the transform as the application of the generalized function F on the kernel as an
element of a test function space. This necessitates the construction of a suitable complete seminormed
space that contains the kernel. Zemanian used this method for the Laplace transform and I did it for the
Hankel transform. Details may be found in Zemanian’s book and Brychkov and Prudnikov (1998).
The University of the Philippines is a great source of inspiration for our youth despite its poverty of material
resources. It is however rich in a faculty that is a fountain of wisdom and knowledge. I credit Prof. Blanco in
igniting a spark to my career. But there were many excellent profs I had. I would be remiss not to acknowledge
Professor Josefina Constantino who instilled in me a love of the English language. From the time I became
Professor Emeritus in 1999, I have published short stories, poems and essays online and in print. In these
writings my math background seems to creep in.
American poet Adelaide Crapsey invented a verse form in the 1920s called cinquain, which is a poem of five
iambic lines of two, four, six, eight and two syllables. There are no restrictions on rhyme but there is a central
theme in the cinquain. In 1927, Angela M. Gloria published three cinquains in the Philippine Herald. Here are
some unpublished cinquains of mine on scientists.
| Newton The man Felt the apple Dropped on his head with such Gravity, he couldn’t help yell, “Aha!” Einstein He did Tinker with his Relatives; speed boggles The mind as energy and mass Equate. Archimedes He ran Out in the buff Hollering “Eureka!” For he had found that buoyant force Pushed up. Leibniz Symbols He used for thoughts, Logic was the richer, And calculus was brought to the Limit. |
| Copyright© 2006 Filipino Image. All rights Reserved. Advertise | Contract | Subscribe | Publisher | Contact Us |