romanian mathematicians - ayz.pl mathematicians ... andrei stefanescu 7f grade, ichb, romania ......
TRANSCRIPT
Romanian Mathematicians
Let’s Learn Maths Together
Multilateral Comenius Project
2011 - 2013
This project has been funded with support from the European Commission.
This publication [communication] reflects the views only of the author, and the Commission cannot be held
responsible for any use which may be made of the information contained therein.
Famous Romanian
Mathematicians
By Filip Chirica 7G Grade, ICHB, Romania
Alexandru Dobromir 7G Grade, ICHB, Romania
Adam Popescu 7F Grade, ICHB, Romania
Stefania Spandonide 7F Grade, ICHB, Romania
Andrei Stefanescu 7F Grade, ICHB, Romania
Maria Yaelle Toader 7G Grade, ICHB, Romania
Ana-Maria Tudorache 7F Grade, ICHB, Romania
.
Motto:
"All what is correct thinking is either
mathematics or feasible to be
transposed in a mathematical model.“
Grigore Moisil
“Gazeta Matematica” The most prestigious Romanian math publications The world longest running mathematical magazine of its kind Published monthly since 1895 by Romanian Mathematical Society.
Romanian Academy Founded on 1/13 April 1866 The quality of an academician is synonymous with absolute intellectual preeminence in modern Romanian society. There are 181 acting members, elected for life.
Three Symbols of Romanian
Mathematic World
“Learning maths, you learn to think. ”
Grigore Moisil
And …. Olympiads
Romania was the International Mathematics Olympiad promoter •Romania was host for IMO five times with a continuous increased number of contestants
•Over time 50 Romanian team attended IMO and won:
68 gold medals / 113 silver medals / 90 bronze medals / two honorable mentions.
•Olympics with the most gold medals (3 gold medals each) are: Ciprian Manolescu, Mihai Manea, Stephen Lawrence Hornet, Teodor Banica and the person who participated in the Olympics often (4 times - 2004-2007 ) is Adrian-Ioan Zahariuc. Moreover, Romanian mathematicians, internationally recognized professors at prestigious American universities such as Dan Voiculescu, George Lusztig and Daniel Tataru obtained, as students, the International Olympic gold medal.
Romanians love solving problems
• Any area of life is:
– to "solve a problem“
– to find a way out of a difficulty
– to find a way to achieve a goal that is not
directly accessible.
• To find the solution of such problems is a specific
feature of intelligence, and intelligence of the
human species is a distinctive peak.
• Mathematics, of all sciences, is that man learns
best how to solve a problem. And Romanians love
this.
What is a mathematician?
• A researcher
• A person who create new mathematical facts (concepts, methods, theories)
• A teacher with high degrees
The most Romanian mathematicians are all of these. They are not included in the 100 most famous mathematicians of the world, but they are international renowned for their contributions. We are proud of them.
Seven generations of Romanian
Mathematicians
First half of XIXth century – Janos Bolyai, Transilvania
(then was part of Austro – Hungarian Empire)
Middle of XIXth century - Spiru Haret and David Emmanuel (Ph. at Sorbonne)
Second half of XIXth century – Gh. Titeica, D. Pompeiu (Ph. at Sorbonne),
Al. Myller, Vera Myller (Ph. at Gottingen)
First half of XXth century – V. Valcovici, Traian Lalescu, Simion Stoilow
Middle of XXth century - O. Onicescu, P. Sergescu, Dan Barbilian,
A. Froda, Gh. Vranceanu
Second half of XXth century – Grigore Moisil, G. Calugareanu, Dramba
Contemporary – C. Calude, I. Cuculescu, Titu Andreescu, L. Tataru
János Bolyai
(1802 – 1860)
• Romanian (ethnic Hungarian)
• Born and lived in Transilvania,
Romania
• One of the founders of non-
Euclidean geometry (in parallel
and independent of Lobacevski).
The discovery of a consistent alternative geometry that might correspond
to the structure of the universe helped mathematicians to study abstract
concepts that later were applied in physics (Einstein theory).
János Bolyai and the Euclid’s Postulate
• Non-Euclidean geometry— a geometry that differs from Euclidean geometry in
its definition of parallel lines.
• By the age of 13, he had mastered calculus and other forms of analytical
mechanics. He became so obsessed with Euclid's parallel postulate that his
father wrote to him: "For God's sake, I beseech you, give it up. "
• János, however, persisted in his quest and eventually came to the conclusion
that the postulate is independent of the other axioms of geometry and that
different consistent geometries can be constructed on its negation. He wrote to
his father: "Out of nothing I have created a strange new universe".
• Between 1820 and 1823 he prepared a treatise on a complete system of non-
Euclidean geometry. Bolyai's work was published in 1832 as an appendix to a
mathematics textbook by his father.
• Gauss, on reading the Appendix, wrote to a friend saying "I regard this young
geometer Bolyai as a genius of the first order". In 1848 Bolyai discovered that
Lobachevsky had published a similar piece of work in 1829. Though
Lobachevsky published his work a few years earlier than Bolyai, it contained
only hyperbolic geometry. Bolyai and Lobachevsky did not know each other or
each other's works.
Others Works
• In addition to his work in geometry, Bolyai
developed a rigorous geometric concept of
complex numbers as ordered pairs of real
numbers.
• Although he never published more than the 24
pages of the Appendix, he left more than 20,000
pages of mathematical manuscripts when he
died. These can now be found in the Bolyai-
Teleki library in Târgu Mureş, where Bolyai died.
Mathematical discoveries, like springtime violets in the woods, have their season, which no man can hasten or retard. J. Bolyai
Beginning of Maths in Romania
Only since 2 centuries in the world of maths.
Begun in 1830, mathematical education was regarded as a model of thinking, an innovative idea for that time. Gheorghe Lazar and George Asachi have implemented this by introducing the first math books in Romanian.
Their work was continued in the late nineteenth century by Spiru Haret, math teacher, considered the father of modern Romanian education reform and David Emmanuel, both with Ph. degree at Sorbonne.
Spiru Haret
DIMITRE POMPEIU
(4 October 1873 – 8 October 1954)
From a little Moldavian town to Sorbonne and
back to Bucharest
Dimitrie Pompeiu
University of Iaşi
University of Bucharest
He grew up and followed primary school in in a little Moldavian town, Dorohoi and then the secondary school in Bucharest
1893 – 1898 – primary school teacher
1898 - University of Paris (the Sorbonne) –graduated in Mathematics
1905 - Ph.D. degree in Mathematics with thesis “On the continuity of complex variable functions”, written under the direction of Henri Poincaré.
1934 – is elected member of Romanian Academy
- Chair at Universities from Iasi and Bucharest
Pompeiu’s Contributions
• His contributions were mainly in the field of mathematical analysis, complex functions theory, and rational mechanics.
• In an article published in 1929, he posed a challenging conjecture in integral geometry, now widely known as the Pompeiu problem.
• Among his contributions to real analysis there is the construction, dated 1906, of non-constant, everywhere differentiable functions, with derivative vanishing on a dense set. Such derivatives are now called Pompeiu derivatives.
Hausdorff - Pompeiu Distance
• In mathematics, the Hausdorff distance, also called Pompeiu–Hausdorff distance or metric was introduced in 1914.
• Hausdorff – Pompeiu distance measures how far two subsets of a metric space are from each other. It turns the set of non-empty compact subsets of a metric space into a metric space in its own right.
• Informally, two sets are close in the Hausdorff distance if every point of either set is close to some point of the other set. The Hausdorff distance is the longest distance you can be forced to travel by an adversary who chooses a point in one of the two sets, from where you then must travel to the other set. In other words, it is the greatest of all the distances from a point in one set to the closest point in the other set.
Pompeiu’s theorem
Given an equilateral triangle ABC in the plane, and a point P in the plane
of the triangle ABC, the lengths PA, PB, and PC form the sides of a
(maybe, degenerate) triangle.
Proof 1. (Simple, but not classical)
Consider a rotation of 60° about the point C. Assume A maps to B, and B maps
to B '. Then we have PA=P’B and PC=P’C . Hence triangle PCP ' is
equilateral and PP’=PC . It is obvious that . Thus, triangle PBP ' has sides
equal to PA, PB, and PC and the proof by construction is complete.
Further investigations reveal that if P is not in the interior of the triangle,
but rather on the circumcircle, then PA, PB, PC form a degenerate triangle,
with the largest being equal to the sum of the others.
Pompeiu’s Theorem
Proof 2. (with complex numbers)
Let P(z), A(a), B(b) ,C(c) the four points with their afixes. Whereby ABC is equilateral => |a-c|=|b-c|=|a-b| By calculating we have (z-b)(a-c)=(z-a)(b-c)+(z-c)(a-b) Passing to absolute values and using 𝑥 + 𝑦 ≤ 𝑥 + 𝑦 we have:
|(z-b)(a-c)|=|(z-a)(b-c)+(z-c)(a-b)| <=> |z-b|=|(z-a)+(z-c)|<=|z-a|+|z-c) Now we obtain 𝑃𝐴 + 𝑃𝐶 ≥ 𝑃𝐵 , 𝑃𝐴 + 𝑃𝐵 ≥ 𝑃𝐶, 𝑎𝑛𝑑 𝑃𝐵 + 𝑃𝐶 ≥ 𝑃𝐴
OBS. We have equality in these relations then P is on the circumcircle. Then we have the Ptolemeu relation. This the explanation for the identity written at the beginning of the proof.
Gheorghe Titeica
1873-1939
• Founder of the Romanian
school of affine differential
geometry.
• He introduced a new class of
curves and surfaces that today
bears his name.
• Mate in Paris with H. Lebesgue
and Montel, he sustained his
master's degree thesis on the
framework of oblique curvature
with G. Darboux.
Gheorghe Titeica
Problem of the 5 lei coin (Titeica discovered it while pulling hung with a pencil outline of such coins during a meeting)
Statement Three congruent circles have a common
point H and twos longer intersect at points A, B, C.
Prove that the circle ABC is congruent with the
three circles and H is orthocenter of the triangle
ABC. (Long list, OBMJ, Izmir, Turkey, 2003)
The problem was the logo for
40th IMO, 1999, Romania
Titeica - Problem of the 5 lei coin
R- common length rays circles.
O1,O2 and O3 centers of the three circles.
Quadrilaterals O1AO2H, O2CO3H and O3BO1H are rhombus, with all sides congruent.
So O1B ∥O3H ∥ O2C and O1B ≡O3H ≡O2C. Then the quadrilateral O1BCO2 is parallelogram.
Therefore triangles ΔO1O2O3 ≡ Δ CBA ( S.S.S.). As O1H=O2H=O3H=O1A=R, observe that H circumcenter of triangle ΔO1O2O3. The circle (O1O2O3) has radius R.
Gheorghe Vranceanu
30 June 1900- 27 April 1979
• Famous Romania mathematician
• Best known for his work in differential geometry
• Romanian Academy full member from 1955 to 1979
• From 1964 to 1979 he was President of the Mathematics Section of the Romanian Academy.
• over 300 articles published in journals throughout the world
• “Gheorghe Vranceanu” school in Bacau, Romania.
Dan Barbilian- Ion Barbu 18 March 1895 –11 August 1961
• Distinguished Romanian mathematician ,
poet and professor at the University of Bucharest
• As a mathematician he is known for his publications in “Gazeta Matematica”
• As a poet, he is best known for his volume ”Second Game”
• The most important contributions - two papers that appeared
in Časopis Mathematiky a Fysiky.
• ”Barbilian” spaces in geometry were named after him.
• “Dan Barbilian-Ion Barbu mathematic and literature contest”
takes place yearly.
Romanian mathematician,
considered the father of
Romanian computer science.
His research was mainly in
the fields of mathematical
logic, (Łukasiewicz–Moisil
algebra), algebraic logic, and
differential equations.
Grigore Moisil
1906 - 1973
• Brilliant Math teacher, he is known not only for his
studies, but also for his jokes and for his great
sense of humor.
Professor Grigore Moisil, during a Math class, after
reading crap stuff written by a student, called the
student, kissed his forehead and said: "You can tell
everyone that Professor Grigore Moisil kissed your
ass, because that's not a head. "
Grigore Moisil - anecdotes
Grigore Moisil - anecdotes
During an interview, a reporter says:
- You know the truth hurts.
Moisil: I was never hurt by a math theorem
• An anecdote which is not understood by the listener, nor the narrator is
called psychological book.
• The marriage is the only escape for an unsuccessful man and a too
successful woman.
• - Hello, professor, do you believe in dreams?
- Of course! A while ago I dreamed that I became academician, I was in a
big conference room and I was the chair-man.
And when I woke up, I was an academician, I was in a big conference room
and I was the chair-man.
• During Moisil’s exams to become a professor, the commission evaluated
him. Professor Procopiu voted against Moisil because he considered Moisil
too young to get the “professor” title.
Moisil: It is a defect that I am correcting daily, said Moisil.
• A friend is telling to a math teacher:
- I am full of the math you are teaching; I am full up to my neck.
Moisil: But the math starts from the neck up!
.
• In 1960’s, Grigore Moisil participated at a National Meteorology Institute
meeting. Somebody stated:
- Based on the latest discoveries of Russian researchers, we are able to
forecast weather with a 40% probability..
- Moisil: Then, why don’t you forecast it the other way around? That way
you would have a 60% probability.
Grigore Moisil - anecdotes
Alive Famous Romanian
Mathematicians
• Mathematicians with international reputations in the
scientific world work in Romania and many are currently
professors in the great universities of the world.
• In Romania, world-renowned mathematical schools were
created and are still active; some of their fields of study are
the theory of operators, complex analysis, the theory of
potential, the theory of manifolds, differential equations and
the theory of the optimal central, fluid mechanics and the
theory of elasticity, the theory of probabilities and
mathematical statistics.
Cristian S. Calude Born 21 April 1952-
Romania-New Zealand mathematician and
computer scientist
• Mathematical Student Prize, Faculty of Mathematics,
Bucharest University, Romania, 1975.
• Honorificum Membrum, Black Sea University, Bucharest, Romania, December 2002.
• Excellence in Research Award, University of Bucharest, Romania, 2007
• Dean's Award for Excellence in Teaching, Faculty of Science, University of Auckland, 2007.
• The “C. S. Calude Mathematics Regional Contest” takes place every year in Galati, Romania.
Titu Andreescu -
Coach for the Romanian IMO team in 1983
Although he coached Romanian IMO team and consultant of the Education Minister, in 1985 Titu Andreescu was not allowed to go at Helsinki, for IMO.
The reason?! His grandparents emigrated early last century, but his mother and grandmother returned to Romania after the World War I.
He is born in 1956
As a high school student, he excelled in
mathematics: in 1973, 1974 and 1975 he
won the Romanian national problem solving
contests organized by the journal Mathematical Review .
Titu Andreescu
In 1990, as the Eastern Bloc began to collapse, Andreescu emigrated with his mother to the United States.
National award of "Distinguished Professor”
Associate professor of mathematics at the University of Texas at Dallas.
Director of AMC (as appointed by the Mathematical Association of America), Director of MOP, Head Coach of the USA IMO Team and Chairman of the USAMO. He has also authored a large number of books on the topic of problem solving and Olympiad style mathematics.
Titu Andreescu
1993- 2006 Coach for the USA IMO team
Photo: US team, at IMO, 2002, in Scotland >>>>
1993- 2006 Coach for the USA IMO team
In 2006 he established a math camp for talented middle and high school students, AwesomeMath
Year-round program AMY and a mathematical circle, Metroplex.
In 2008 he founded Math Rocks! Program for kids with very high-level math skills.
Titu Andreescu Inequality
𝒙𝟏𝟐
𝒚𝟏 +
𝒙𝟐𝟐
𝒚𝟐 + … +
𝒙𝒏𝟐
𝒚𝒏≥( 𝒙𝟏+𝒙𝟐+ …+𝒙)
𝟐
𝒚𝟏+ 𝒚𝟐+ ...𝒚𝒏 𝒂𝒊 𝒓𝒆𝒂𝒍 𝒏𝒖𝒎𝒃𝒆𝒓𝒔, 𝒙𝒊 > 𝟎, 𝒏 ≥ 𝟐
Proof:
• From Cauchy-Buniakovski - Schwartz inequality 𝑎𝑖
2 ∙ 𝑏𝑖2 ≥ ( 𝑎𝑖𝑏𝑖
𝑛𝑖=1 )2 ∙𝑛
𝑖=1𝑛𝑖=1
• By replacing 𝑎𝑖 =𝑥𝑖
𝑦𝑖 and 𝑏𝑖 = 𝑦𝑖 we obtain:
• 𝑥𝑖2
𝑦𝑖∙ 𝑦𝑖 ≥ ( 𝑥𝑖
𝑛𝑖=1 )2 ∙𝑛
𝑖=1𝑛𝑖=1
• And we divide by the second multiplier from left member.
“Learning maths, you learn to think. ” G. Moisil
Titu Andreescu
• After the success of USA team, U.S. journalists have called him "Bela Karoly of mathematics".
• Comparing with Romanian students, Titu Andreescu think American students "are not smarter, but more organized. Romanians are perhaps more creative, but Americans are more disciplined and work hard.
• "Know that American math Olympians are" well-rounded ", multilateral. Are good at sciences, sing - very well - an instrument, write well. Not nice, but well, "said U.S. professor.
Next Generation of Academicians? Romanians or abroad?
Romanian team at the International Mathematics Olympiad in Argentina, 2012