Transcript Part V

Dr. Richard Tapia Receives
National Medal of Science
The Isoperimetric Problem Revisited:
Extracting a Short Proof of Sufficiency
from Euler’s 1744 Proof of Necessity
Richard Tapia
Rice University
University Professor
Maxfield-Oshman Professor in Engineering
SIAM Annual Meeting
Minneapolis, Minnesota
July 11, 2012
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Dedicated to Peter Lax in Recognition of
his Numerous Mathematical Contributions
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This talk and a paper with the same title,
written in support of this talk, can be
found on my website
http://www.caam.rice.edu/~rat/
Just Google Richard Tapia
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Optimization: The Cradle of
Contemporary Mathematics
Optimization problems are relatively easy to
understand when compared with problems in many
other branches of mathematics. Controversy
invariably leads to interest. Hence, important
optimization problems embedded in some
controversy have played major roles in motivating
and promoting mathematical activity. Mathematical,
indeed scientific, activity can be motivated by many
factors, and not all are removed from human
emotion, as some might have us believe.
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Talk Objectives
 Promote the belief that the isoperimetric problem
has been the most impactful mathematics problem
in history
 Give a brief historical development of the
isoperimetric problem and its solution.
 Demonstrate that Euler’s 1744 necessity proof is
but an observation away from establishing a
sufficiency proof that we believe to be the shortest,
the most elementary, and the most teachable proof
in history of the isoperimetric problem.
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Talk Objectives
 Contrast our proof with the currently accepted
most elementary proof, that given by Peter
Lax in 1995.
 Argue that the process of solving the
isoperimetric problem was greatly
compromised by the fact that mathematicians
of the golden era 1630 – 1890 did not pursue
functional convexity, it was a 20th century
construct.
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Outline
I.
The isoperimetric problem, its origins and
promotion.
II. Proofs: Zenodorus, Euler, Steiner,
Weierstrass, Carathéodory, and Hurwitz.
III. Lax’s short proof.
IV. Our short proofs.
V. Mysteries from the Golden Era.
VI. Meet the players: Fermat, Euler, Lagrange,
Steiner, and Weierstrass
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The Isoperimetric Problem

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Queen Dido and the Isoperimetric Problem
 The Aeneid: written by Virgil in period 29-19 BC
 Dido – Life in danger flees her homeland with
wealth and entourage
 Finds new land and bargains with local king for a
piece of land that she can mark out with the hide of
a bull
 The Dido trick: cut hide into as many thin strips as
possible to form a long cord, using the seashore as
one edge, lay out the cord in the form of a semicircle.
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Dido Purchases Land for the Foundation of Carthage.
Engraving by Matth¨aus Merian the Elder, in Historiche
Chronica, Frankfurt a.M., 1630. Dido’s people cut the
hide of an ox into thin strips and try to enclose a
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maximal domain.
Application of the Dido Maximum Priciple
Medieval map of Cologne
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Yet Another Application of
the Dido Maximum Principle
Medieval map of Paris
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II
The Early Greeks
Zenodorus (Greek mathematician who lived 200-120 BC)
• Wrote work entitled On Isometric Figures
• Work lost, but referenced by Pappus and Theon 300 years later
• Studied figures with equal perimeters and different shapes
• Proved
•
•
The circle has a greater area than the regular polygon on the same perimeter
Of two regular polygons of the same perimeter, the one with the greater
number of angles has the greatest area
• Stated as scolia
In 2-D the circle solves the Isoperimetric Problem
• In 3- D the sphere solves the Isoperimetric Problem
• Can not demonstrate that he gave an incorrect or incomplete proof as many
historians believe
•
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II
Euler (1744)

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II
Euler (1744)
Goldstine states: It is interesting that Euler did
not completely understand the fact that his
condition[satisfaction of the Euler-Lagrange
equation] is a necessary but not a sufficient
one. In his discussion it is clear that he felt
his condition was sufficient to ensure an
extremum, and that by evaluating the integral
along an extremal [and one other curve] he
could decide whether it was s a maximum or a
minimum.
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Euler
II
General Isoperimetric Problem

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Queen Dido form of the Isoperimetric Problem
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Euler’s 1744 Proof of Necessity as Sufficiency
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II
Steiner (1838)
Jakob Steiner (1796 -1867 AD) was one of the most brilliant
and creative geometers in history. His mathematical inquiries
were confined to geometry to the total exclusion of analysis. In
fact he hated analysis and doubted if anything that could not be
proved with geometry could be proved with analysis. In 1838
Steiner gave the first of his five equivalent proofs that the
circle solved the isoperimetric problem. His proofs used
synthetic geometry and were mathematically quite elegant.
Mathematical historians embrace and promote his proofs and
call them models of mathematical ingenuity. They became
very visible in the mathematical community. Steiner boasted
that he had done with geometry what had not been done,
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II
Steiner (1838)
and could not be done, with analysis, i.e., solve the
isoperimetric problem. What Steiner proved was that any curve
which was not the circle could be modified using a geometric
procedure now called Steiner Symmetrization to obtain a curve
with the same perimeter but a larger area. He then concluded
that as a consequence the circle must be the solution to the
isoperimetric problem. As such he fell into the use of necessity
as sufficiency trap and made the trap rather infamous. The
analysts of the time, led by Peter Dirichlet, pointed out to
Steiner that his proof is not valid unless he assumes that the
isoperimetric problem has a solution, i.e. existence. Steiner did
not accept the criticism well, and rebutted with a very
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II
Steiner (1838)
superficial argument that he claimed demonstrated that the
isoperimetric problem must have a solution. Actually, at best
he demonstrated that existence of an upper bond for the area
functional. Now analysts observed that if the Steiner
symmetrization process could be applied repeatedly creating a
sequence of curves with the same perimeter but increasing area
that converges to the area of the circle, the so-called process of
completing Steiner’s proof, then the flaw in Steiner’s proof
would be removed.
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The Steiner 3 Step Proof
Step1 The curve must be convex:
Step2 Perimeter bisector divides the curve into equal
areas and we can therefore symmetrize across the
bisector
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Preliminary Results from Geometry
needed to complete Steiner’s Proof
Lemma 1 (Thales Theorem 600 BC) If AC is a diameter, than
the angle at B is a right angle.
Lemma 2 (Converse to Thales Theorem) A right triangle’s
hypotenuse is a diameter of its circumcircle.
Lemma 3 Of all possible triangles with two sides of specified
lengths the triangle of maximum area is the right triangle.
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The Steiner 3 Step Proof continued
“STEINER SYMMETRIZATION”
Step 3 All inscribed angles determined by the
perimeter bisector must be right angles.
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II
Weierstrass (1879)
Concerning the solution of the isoperimetric
problem, Weierstrass was quite aware of the
shortcomings of Steiner’s proof and somewhat
bothered by Steiner’s arrogantly promoted negative
view of analysis and analysts. Hence he boldly and
proudly placed himself in the noble role of defender
of analysis and vowed to solve the isoperimetric
problem using analysis. He introduces his work
with the following statements concerning the
solution of the isoperimetric problem:
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II
Weierstrass (1879)
“A detailed discussion of this problem is desirable,
since Steiner was of the opinion that the methods of
the calculus of variations were not sufficient to give a
complete proof, but the calculus of variations is in a
position to prove all this, as we will show later;
furthermore it can show what Steiner could not – that
such a maximum really exists.”
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II
Weierstrass (1879)
Weierstrass first builds an elegant and sophisticated
sufficiency theory for the simplest problem from the
calculus of variations employing such subtle notions as
Jacobi’s notion of conjugate points and his own notion of
fields of extremals. He then extends this sufficiency theory
to the isoperimetric problem by turning to Euler’s
multiplier rule and applying his sufficiency theory to the
Euler’s auxiliary problem. Using this theory he
demonstrates that Euler’s auxiliary problem for the
isoperimetric problem has the circle as solution; hence the
isoperimetric problem has the circle as solution. So,
according to the literature, some 135 years after
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II
Weierstrass (1879)
Euler’s proof of necessity we have the first sufficiency
proof. While this notable work gave the world its first
sufficiency proof for the isoperimetric problem, we expect
to convince the audience that Weierstrass really used a
sludge hammer to pound a nail. His sophisticated
sufficiency theory is not needed to merely demonstrate that
the circle solves Euler’s auxiliary problem.
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II Interesting Post – Weierstrass Proof
Carathéodory’s (1910) completion of
Steiner’s Proof.
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II Interesting Post – Weierstrass Proof
Hurwitz (student of Weierstrass) 1902
proof using Fourier series.
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“A Competition for the World’s Most
Elementary Solution of the
Isoperimetric Problem”
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Pedro Lax y Ricardo Tapia
“Mano a Mano”
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The Winner!
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Now on to more serious things like
Mathematical Proofs
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III
A Short Proof of Sufficiency Given by
Peter Lax (1995)
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IV Our Short Proof of Sufficiency Motivated by Euler
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Question:
Could Euler have made our observation at the time of his
1744 writing? The foundation of our observation is
Taylor’s theorem with remainder. The literature tells us
that Taylor published his theorem in 1715. So Euler most
likely was aware of Taylor’s theorem in 1744. However,
the rub is that Taylor’s theorem with remainder was not
known at that time. It is somewhat ironic that the form of
the remainder that we used in our proof is credited to
Lagrange in 1797, and is actually referred to today as the
Lagrange form of the remainder. So Euler would not have
been in good position to make our observation.
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IV Our Short Proof of Sufficiency Motivated by Lagrange
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Remark
Lagrange could have made this proof because he was
familiar with the form of Taylor’s theorem that we
used, indeed it is due to him. While this hypothesized
proof would have been made 50 years after Euler, it
would still have been some 80 years before
Weierstrass.
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Mysteries from the Golden Era (1630-1890)
• Euler thinking that all extremals were
either minimizers or maximizers.
• The use of necessity as sufficiency
• Lagrange not attempting to solve the
isoperimetric problem
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The Greatest Mystery from the
Golden Era
• The failure to develop functional convexity and consequently the
powerful optimization sufficiency theory that follows
Historical Development of Convexity
Archimedes (287 BC – 212 BC) On the Sphere and the Cylinder
A convex arc is a plane curve which lies on side of the line joining its
end points and all cords of which lie on the same side.
Fenchel’s Explanation for the failure to develop functional convexity
When in the seventeenth century Archimedes methods were taken up
again, convexity played still a role, for instance in the work Fermat.
But the development of the calculus let them recede to the
background and even be forgotten.
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Golden Era Uses of Convex Arc
(Not Convex Function)
Fermat 1630
Steiner 1838
Cauchy 1850
Minkowski (1910) in his book initiates the
field of convex geometry and promotes the
role of convexity in mathematics.
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Introduction of Convexity
First Definition of
• A convex function (on interval of reals) – Jensen
(1905)
• A general convex set – Steinitz (1913)
Use of convexity in optimization followed linear
programing contributions from 1940’s
Why so late?
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The Use of Convexity in Optimization Theory Today
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The Use of Convexity in Optimization Theory Today
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Convexity Observations
• The isoperimetric problem is not a convex
program
• But its Euler Multiplier Rule formulation is a
convex program.
• The iso-area program is a convex program.
For purposes of illustration lets follow this path.
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
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Conclusions
• Isoperimetric problem is history’s most impactful
mathematical problem
• Our short proof is most elementary and most
teachable.
• Effective solution of the isoperimetric problem
was seriously delayed because the great minds of
the Golden Era (1960- 1890) did not develop
functional convexity.
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Lets meet the players
(in order of performance)
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A Snapshot of Fermat
“The Prince of Amatuers”
• Lived 1601–1665
• Born and educated and functioned as a lawyer and legal official in
Toulouse, France
• Did mathematics for recreation and considered it a hobby. He dabbled
and rarely produced proofs. As such he was sloppy and chose not to
include detail or polish his work. Much of his work required “a fill in the
blanks” activity as is characterized by his last theorem. He did not
publish and communicated his work in the form of letters to important
people.
• Directly and naively engaged in controversy with prominent
mathematicians of the times and in particular with the powerful
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Descartes.
• Conceived and applied the differential calculus in a 1628
unpublished work entitled Minima and Maxima and the tangent to
a curve. Note that this was 15 years before Newton was born and
18 years before Leibniz was born.
• Ten years later in 1638 he made his work semi-public in a letter to
Descartes. The work stepped strongly on the toes of work that
Descartes was doing on tangents to curves.
• When asked for an official evaluation of Descartes work he wrote
“he is groping around, in the shadows.”
• Descartes responded with the public statement “Fermat is
inadequate as a mathematician and as a thinker.” This damaged
Fermat’s reputation.
• In prominent competition between Descartes and Fermat on the
notion of a tangent to a curve, Fermat won and Descartes lost.
• After the dust settles Descartes writes to Fermat “your work on
tangents is very good, if you had explained it well from the onset,
I would not have had to criticize it or you.”
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• Fermat had little to no interest in physical applications of
mathematics, he just loved the math. This is in strong
contrast to Newton who did the mathematics for his love of
physical applications. Even his notation and terminology
showed this.
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A snapshot of Euler
• Born in Switzerland in 1707. Died in Russia in 1783 at the age of 76
• Clearly the most prolific mathematician the world has ever known.
Laplace -“Read Euler, read Euler, he is the master of us all.”
• Very fond of children. He had 13, but 8 died at early ages.
• He could work anywhere, under any conditions. It is said that he
once wrote a math paper in the 30 minutes before dinner with one
child on his lap and others running around his chair.
• Phenomenal memory and mental calculation power. He memorized
the entire Aeneid.
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• Had sight problems his entire life. Was blind the last 17 years of
his life. These years were very productive. During this time
period he developed the lunar theory, something that escaped
Newton. All the necessary analysis and calculation was done in
his head.
• Very supportive of other mathematicians, especially the young
Lagrange. He dropped his multiplier theory in favor of that of
Lagrange and he called it Lagrange Multiplier Theory
• He was a devout Christian and argued with the prominent
atheists of the time. Since Euler knew no philosophy, Voltaire
tied him into metaphysical knots and made great fun of him in
Fredericks Court. Euler himself laughed when others laughed at
him
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A snapshot of Lagrange
• Born in Italy to wealthy parents in 1736. Died in France in 1813
at the age of 77.
• Quite precocious and at the age of 23 he was considered one of
the greatest mathematicians of the time including Euler and the
Bernoullis.
• During his early years, his parents lost their wealth, In later
years Lagrange said that this was good for him, “If I had
inherited a fortune, I would not have cast my lot with
mathematics.”
• Generously appreciative of the work of others, and dissatisfied
with his own. Considered his early work on the calculus of
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variations performed at the age of 19 his best.
• Befriended Napolean
• Married for social convenience, but grew very fond of his wife.
• Around the age of 50 he had become quite disillusioned with
mathematics. At 51 his wife died. He became depressed and did
nothing for 5 years.
• At age 56 he married the 17 year old daughter of a colleague. His
excitement for math returned and very productive years followed.
She rescued him from his twilight between life and death.
• Honors were showered on him by the French. Concerning these
awards he said my greatest trophy is having found such a tender
and devoted companion as my young wife. Here he probably
coined the phrase “trophy wife.”
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A snapshot of Steiner
• (1796 – 1863) Born and raised on a farm in Bern Switzerland.
• He was extremely poor, did not go to school and did not learn to read or write
until the age of 14.
• At the age of 22 Steiner took classes in mathematics at the University of
Heidelberg. He supported himself by tutoring in mathematics.
• At the age of 26 he decided to move to the more prestigious University, The
University of Berlin. There he befriended Abel, Jacob, and Crelle of Crelle’s
Journal. Leaving the University two years later, Steiner found a post at a Berlin
technical school and spent the next 10 years teaching mathematics there. He
taught basic concepts to the young students and the working atmosphere was
not good he wrote and published many significant paper, mostly in Crelle’s
Journal. He became the world leader in projective geometry.
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• At the age of 37 he was awarded an honorary doctorate by the
University of Königsberg on the recommendation of Jacobi
• At age of 38 he was elected to the Prussian Academy of
Sciences, again on the recommendation of Jacobi and other
leading mathematicians.
• At the the age of 38 he was appointed to a new extraordinary
professorship of geometry at the university of Berlin, again on
the recommendation of Jacobi and other.
• Steiner remained an extraordinary professor at the university
until his death 29 years later. He never married, but dedicated
much of his adult life to his students. A blunt and somewhat
coarse manner combined with surprisingly liberal social
attitudes made Steiner a unique individual, particularly to
students. He influenced many of his students, including the
noted mathematician Georg Friedrich Bernhard Riemann.
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A Snapshot of Weierstrass
• Born in Germany in 1815
• Died in Germany in 1897 at the age of 82
• Excelled in High school despite having to work part time as a
bookkeeper
• At the age of 19 his father sent him to university of Bonn to
study commerce and law. He did not attempt either, he
devoted his time to fencing, drinking beer, and studying math
on his own. After 4 years he returned home without a degree
• At age 26 started teaching high school
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• At the age of 38, as a high school teacher, he wrote a paper on
Abelian functions that created a sensation.
• At the age of 41 he was awarded an assistant professorship at
the University of Berlin where he became an outstanding
lecturer, teacher, and mentor
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Weierstrass
Weierstrass, in addition to his numerous mathematical contributions,
is known for introducing rigor of proof and cleanliness of definition
into the calculus of variations at a time that it was sorely needed. He
did not publish his work in this area but developed a complete and
polished set of lecture notes that he used in his university courses at
the University of Berlin. Today we know about his many
contributions in the calculus of variations from his collected works
which was constructed primarily from the lecture notes of his
numerous students during the time period 1865 – 1890. It is alleged
that Weierstrass had 40 or so students during this time period and
may of these students became quite distinguished in their own right;
for example Cantor, Frobenius, (Sofia) Kowalewski ( as a women
she was not officially accepted as a student at the University of
Berlin and was given an honorary degree), Mittag-Leffler, Runge,
Schur, and Schwarz.
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Weierstrass
Weierstrass’ critical sense and need to base his analysis on such a
firm foundation led him to continually revise and perfect his writings
to the point that publication was precluded. In spite of this his work
he became so well known that today he is called the father of
analysis.
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A Story of Mystery and Intrigue
“Karl Weierstrass and Sofia Kowalewski”
In 1870, at the age of 20, Sofia Kowalewski came to Berlin
to study with Weierstrass who was 55 years old at the time.
He taught her privately since she was not allowed admission
to the University. She was an unusually gifted young
woman in so many ways,
Weierstrass: Sofia is my favorite pupil and my “weakness”.
She died at the age of 41, he burnt all of her correspondence.
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A Story of Mystery and Intrigue
I would love to see a book written
focusing exclusively on the
Weierstrass-Kowalewski Relationship
In fact I have a title for this book.
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The Integral and the Integrand:
A Story of Love and Respect
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Thank you
for your attention.
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