Initial point: Short presentation of Comenius project
Optimization and number theory
Optimization problems
Fractals
Napoleon’s problem
Dynamical systems
Curved space and new phenomena
Some open problems
Vladimir Georgiev and Bozhidar Velichkov,
University of Pisa
30.3.3013
Some open problems or proposal for math projects
Spring conference Borovetz 2013
Vladimir Georgiev and Bozhidar Velichkov, University of Pisa
Some open problems
Initial point: Short presentation of Comenius project
Optimization and number theory
Optimization problems
Fractals
Napoleon’s problem
Dynamical systems
Curved space and new phenomena
1
2
3
4
5
6
7
Initial point: Short presentation of Comenius project
Optimization and number theory
Diophantine game
Problem from harmonic analysis
Combinatorial inequalities
Optimization problems
Optimal packing
Moon problem
Fractals
Besicovitch set
Napoleon’s problem
Some generalizations of Napoleon’s theorem
Dynamical systems
Biliards
Curved space and new phenomena
Vladimir Georgiev and Bozhidar Velichkov, University of Pisa
Some open problems
Initial point: Short presentation of Comenius project
Optimization and number theory
Optimization problems
Fractals
Napoleon’s problem
Dynamical systems
Curved space and new phenomena
Title of the Comenius Project
Dynamical and creative Mathematics using ICT - Comenius project
Goal: to prepare concrete examples, hints and good practices in
preparation of future and in service teachers to develop creativity
of their pupils.
Vladimir Georgiev and Bozhidar Velichkov, University of Pisa
Some open problems
Initial point: Short presentation of Comenius project
Optimization and number theory
Optimization problems
Fractals
Napoleon’s problem
Dynamical systems
Curved space and new phenomena
Partners:
Italy, University of Pisa
Austria, University of Vienna,
Denmark: VIA University College - Bachelor Programme in
Teacher Education, Aarhus
Bulgaria: Institute of Mathematics and Informatics, Bulgarian
Academy of Sciences, Sofia
Slovakia: Constantine the Philosopher University Nitra, Nitra
Iceland: University of Iceland - School of Education, Reykjavik
Vladimir Georgiev and Bozhidar Velichkov, University of Pisa
Some open problems
Initial point: Short presentation of Comenius project
Optimization and number theory
Optimization problems
Fractals
Napoleon’s problem
Dynamical systems
Curved space and new phenomena
Diophantine game
Problem from harmonic analysis
Combinatorial inequalities
Diophantine game
During the work of the Math Lab in Pisa a specific game with
Flash script was prepared. The game is connected with the
following Diophantine equation
Bx = A + Cy ,
(1)
where A, B, C are natural numbers that we choose before starting
the game, while x, y can be interpreted by two buttons ( the green
on Figure 2 and the red one on the same Figure).
Vladimir Georgiev and Bozhidar Velichkov, University of Pisa
Some open problems
Initial point: Short presentation of Comenius project
Optimization and number theory
Optimization problems
Fractals
Napoleon’s problem
Dynamical systems
Curved space and new phenomena
Diophantine game
Problem from harmonic analysis
Combinatorial inequalities
Diophantine game
Vladimir Georgiev and Bozhidar Velichkov, University of Pisa
Some open problems
Initial point: Short presentation of Comenius project
Optimization and number theory
Optimization problems
Fractals
Napoleon’s problem
Dynamical systems
Curved space and new phenomena
Diophantine game
Problem from harmonic analysis
Combinatorial inequalities
Diophantine game
There are several questions that have been discussed in the lab:
find triples A, B, C such that the Diophantine equation has no
solution;
find a necessary and sufficient conditions so the the
Diophantine equation has at least one solution;
find an algorithm to win or to find optimal x + y .
Vladimir Georgiev and Bozhidar Velichkov, University of Pisa
Some open problems
Initial point: Short presentation of Comenius project
Optimization and number theory
Optimization problems
Fractals
Napoleon’s problem
Dynamical systems
Curved space and new phenomena
Diophantine game
Problem from harmonic analysis
Combinatorial inequalities
Problem from harmonic analysis
If R 2 = |k|2 , k ∈ Z2 , there are at most
C
log R
log log R
many points on the circle of radius R
Grosswald, E., Representations of integers as sums of squares
Springer- Verlag (1985).
Davenport H. Analytic methods for Diophantine equations and
Diophantine inequalities, 2005.
Vladimir Georgiev and Bozhidar Velichkov, University of Pisa
Some open problems
Initial point: Short presentation of Comenius project
Optimization and number theory
Optimization problems
Fractals
Napoleon’s problem
Dynamical systems
Curved space and new phenomena
Diophantine game
Problem from harmonic analysis
Combinatorial inequalities
Combinatorial inequalities
Consider the sequence ηk (L) defined as follows.
ηk (1) = k, k ≥ 1 η1 (L) =
1
, L≥1
L
(2)
and for any integer k ≥ 1 and any couple of integers k1 , k2 ≥ 1
such that k1 + k2 = k, we have
ηk (L) =
L
X
j=0
ηk1 (j)ηk2 (L − j), ηk (0) = 1.
Vladimir Georgiev and Bozhidar Velichkov, University of Pisa
Some open problems
(3)
Initial point: Short presentation of Comenius project
Optimization and number theory
Optimization problems
Fractals
Napoleon’s problem
Dynamical systems
Curved space and new phenomena
Diophantine game
Problem from harmonic analysis
Combinatorial inequalities
Combinatorial inequalities
One can find
η2 (L) =
L
X
j=0
η1 (j)η1 (L − j),
or explicitly,
η2 (1) = 2, η2 (2) = 2, η2 (3) =
5
.
3
For L ≥ 2 we have the identities


L−1
L−1
1
2 2 X 1 
2 X
= +
.
η2 (L) = +
L
j(L − j)
L L
j
j=1
Vladimir Georgiev and Bozhidar Velichkov, University of Pisa
j=1
Some open problems
(4)
Initial point: Short presentation of Comenius project
Optimization and number theory
Optimization problems
Fractals
Napoleon’s problem
Dynamical systems
Curved space and new phenomena
Diophantine game
Problem from harmonic analysis
Combinatorial inequalities
Combinatorial inequalities
Therefore applying the estimate
m≥k ≥1⇒
m
X
1
j=k
j
≤
m
1
,
+ log
k
k
(5)
we get
2(2 + log(L − 1))
.
L
Task 1: Find estimate similar to (6) for η3 (L), · · · ηk (L).
Task 2: Try to define ηp (L) for p > 1 real number.
η2 (L) ≤
Vladimir Georgiev and Bozhidar Velichkov, University of Pisa
Some open problems
(6)
Initial point: Short presentation of Comenius project
Optimization and number theory
Optimization problems
Fractals
Napoleon’s problem
Dynamical systems
Curved space and new phenomena
Optimal packing
Moon problem
Optimal packing
Problem
Problem 4 (Istituto Statale ’C. Lorenzini’, Pescia, Italy) A box
contains 48 cylindrical cans ordered in 8 rows so that each row
contains 6 cans. Show that one can put 50 cans in the same box.
Task: See if 50 is optimal solution.
Vladimir Georgiev and Bozhidar Velichkov, University of Pisa
Some open problems
Initial point: Short presentation of Comenius project
Optimization and number theory
Optimization problems
Fractals
Napoleon’s problem
Dynamical systems
Curved space and new phenomena
Optimal packing
Moon problem
Moon problem
Problem
The satellite THETA of the society GoogleMoon is equipped with
digital camera having image with a 20 degree angle of view. To
make qualitative photos the satellite must circle around the Moon
keeping constant altitude equal to the radius R=1738 km of the
Moon and the digital camera must be oriented all the time in the
direction of the centre of the Moon. Find the minimal length of
the path of the satellite allowing to cover (with the images of the
digital camera) the whole surface of the Moon.
Remarks: If the cone of the camera has image with a 20 degree
angle of view, then the cone of the camera is a cone with angle of
semiaperture 10 degree.
Vladimir Georgiev and Bozhidar Velichkov, University of Pisa
Some open problems
Initial point: Short presentation of Comenius project
Optimization and number theory
Optimization problems
Fractals
Napoleon’s problem
Dynamical systems
Curved space and new phenomena
Optimal packing
Moon problem
Moon problem
One can see that this problem is similar to the problem known by
researchers in Pisa University. The open problem treated the
square on the plane, while the problem of the team of ”Dini” looks
for the case of a sphere. Therefore, the real life problem lead to
deep real math problem!
The difficulty of the problem posed the question how the teams
could solve the problem, since they have no deep math tools, but
even the specialists with heavy math preparation could not solve it.
The team of Brescia proposed an interesting simulation using
orange.
Vladimir Georgiev and Bozhidar Velichkov, University of Pisa
Some open problems
Initial point: Short presentation of Comenius project
Optimization and number theory
Optimization problems
Fractals
Napoleon’s problem
Dynamical systems
Curved space and new phenomena
Optimal packing
Moon problem
Moon problem
Figure: Shortest broken such that ε− neighbourhood covers the square.
Vladimir Georgiev and Bozhidar Velichkov, University of Pisa
Some open problems
Initial point: Short presentation of Comenius project
Optimization and number theory
Optimization problems
Fractals
Napoleon’s problem
Dynamical systems
Curved space and new phenomena
Besicovitch set
Besicovitch set
On the plane (2D space) there exists a set B so that:
(i) is compact,
(ii) has measure zero,
(iii) contains a translate of every unit line segment.
In the space (3 D space) is impossible to find a compact set such
that i) ii) and iii) are satisfied.
The fractal dimension of the Besicovitch set is 2.
E. Stein Real analysis, Princeton 2005.
Vladimir Georgiev and Bozhidar Velichkov, University of Pisa
Some open problems
Initial point: Short presentation of Comenius project
Optimization and number theory
Optimization problems
Fractals
Napoleon’s problem
Dynamical systems
Curved space and new phenomena
Some generalizations of Napoleon’s theorem
The following statement (known as Napoleon’s theorem) is closely
connected with the Fermat theorem.
Theorem 1
(Napoleon’s Theorem) On each side of a triangle an exterior
equilateral triangle is constructed. Show that the centers of the
three thus obtained equilateral triangles are vertices of an
equilateral triangle.
Vladimir Georgiev and Bozhidar Velichkov, University of Pisa
Some open problems
Initial point: Short presentation of Comenius project
Optimization and number theory
Optimization problems
Fractals
Napoleon’s problem
Dynamical systems
Curved space and new phenomena
Some generalizations of Napoleon’s theorem
Napoleon’s theorem
Vladimir Georgiev and Bozhidar Velichkov, University of Pisa
Some open problems
Initial point: Short presentation of Comenius project
Optimization and number theory
Optimization problems
Fractals
Napoleon’s problem
Dynamical systems
Curved space and new phenomena
Some generalizations of Napoleon’s theorem
Our interest to study Napoleon’s theorem is partially motivated by
the fact that it is not very popular among Italian Math professors.
Although, it is difficult to explain this phenomenon, one can easily
predict ( and this was indeed verified in practice) that Math
students and consequently future math teachers have no any idea
about this beautiful math statement.
Vladimir Georgiev and Bozhidar Velichkov, University of Pisa
Some open problems
Initial point: Short presentation of Comenius project
Optimization and number theory
Optimization problems
Fractals
Napoleon’s problem
Dynamical systems
Curved space and new phenomena
Some generalizations of Napoleon’s theorem
A well - known generalization of Napoleon’s theorem is the
following one
D. Wells, You Are a Mathematician, John Wiley & Sons, NY, 1981.
For an arbitrary △ ABC , three exterior points A1 , B1 , C1 are
constructed such that
△ ABC1 ∼△ BCA1 ∼ CAB1 .
Then the centroids of these triangles are vertices of a triangle
similar to them.
Actually it’s not even necessary to consider the centroids.
Vladimir Georgiev and Bozhidar Velichkov, University of Pisa
Some open problems
Initial point: Short presentation of Comenius project
Optimization and number theory
Optimization problems
Fractals
Napoleon’s problem
Dynamical systems
Curved space and new phenomena
Some generalizations of Napoleon’s theorem
For arbitrary △ ABC take three external points A1 , B1 , C1 such that
∢AC1 B + ∢BA1 C + ∢CB1 A = 3600 .
Then △ A1 B1 C1 is similar to a triangle with angles
∢C1 AB + ∢B1 AC , ∢C1 BA + ∢A1 BC , ∢A1 CB + ∢B1 CA.
One can see the references (
D. Wells, You Are a Mathematician, John Wiley & Sons, NY, 1981.
S. B. Gray, Generalizing the Petr-Douglas-Neumann Theorem on
N-Gons, The American Mathematical Monthly, 110(3) (2003), p.
210 – 227.
for the proof of this interesting fact.
Vladimir Georgiev and Bozhidar Velichkov, University of Pisa
Some open problems
Initial point: Short presentation of Comenius project
Optimization and number theory
Optimization problems
Fractals
Napoleon’s problem
Dynamical systems
Curved space and new phenomena
Some generalizations of Napoleon’s theorem
Napoleon's Theorem quadrilaterals - GeoGebra Dynamic Worksheet
Page 1 of 1
Napoleon's Theorem quadrilaterals
Vladimir, Created with GeoGebra
Geogebra application: Quadrilaterals and Napoleon’s theorem
Another generalization is due to Steve Gray
S. B. Gray, Generalizing the Petr-Douglas-Neumann Theorem on
N-Gons, The American Mathematical Monthly, 110(3) (2003), p.
210 – 227.
Vladimir Georgiev and Bozhidar Velichkov, University of Pisa
Some open problems
Initial point: Short presentation of Comenius project
Optimization and number theory
Optimization problems
Fractals
Napoleon’s problem
Dynamical systems
Curved space and new phenomena
Some generalizations of Napoleon’s theorem
This time, the construction starts with an arbitrary n−gon and one
considers the n− gon formed by the centers of the regular n−gons
constructed externally on its sides. As it was shown in
S. B. Gray, Generalizing the Petr-Douglas-Neumann Theorem on
N-Gons, The American Mathematical Monthly, 110(3) (2003), p.
210 – 227.
after repeating this construction n − 2 times one obtains a regular
n−gon.
Having in mind Gray’s generalization of Napoleon theorem, one
could ask if it is possible to obtain a regular n− gon after less that
n − 2 steps. For example taking n = 4 and using Geogebra one can
see that in general it’s impossible.
Vladimir Georgiev and Bozhidar Velichkov, University of Pisa
Some open problems
Initial point: Short presentation of Comenius project
Optimization and number theory
Optimization problems
Fractals
Napoleon’s problem
Dynamical systems
Curved space and new phenomena
Some generalizations of Napoleon’s theorem
Hence, it is natural to ask for which n−gons one can obtain a
regular n− gon after the first step. We do not know the answer to
this question for arbitrary n, but for the case n = 4 it is given in
the next exercise.
Exercise 2
On the sides of a quadrilateral external squares are constructed.
Prove that:
(a) The centers of the squares are vertices of a quadrilateral with
perpendicular diagonals of equal length.
(b) The quadrilateral in (a) is a square if and only if the initial
quadrilateral is a parallelogram.
Vladimir Georgiev and Bozhidar Velichkov, University of Pisa
Some open problems
Initial point: Short presentation of Comenius project
Optimization and number theory
Optimization problems
Fractals
Napoleon’s problem
Dynamical systems
Curved space and new phenomena
Some generalizations of Napoleon’s theorem
We propose to the reader to prove the following generalization of
Napoleon’s theorem:
Exercise 3
On the sides of a non-equilateral triangle three regular n−gons are
constructed externally to the triangle. Prove that
their centers are vertices of an equilateral triangle iff n = 3. (7)
Vladimir Georgiev and Bozhidar Velichkov, University of Pisa
Some open problems
Initial point: Short presentation of Comenius project
Optimization and number theory
Optimization problems
Fractals
Napoleon’s problem
Dynamical systems
Curved space and new phenomena
Biliards
Billiards as dynamical systems
Figure: Billiard table
Vladimir Georgiev and Bozhidar Velichkov, University of Pisa
Some open problems
Initial point: Short presentation of Comenius project
Optimization and number theory
Optimization problems
Fractals
Napoleon’s problem
Dynamical systems
Curved space and new phenomena
Biliards
Billiards as dynamical systems
A caustic of a plane billiard is a curve such that if a trajectory is
tangent to it, then it again becomes tangent to it after every
reflection.
The first result is the following optical property of ellipses.
Lemma
A ray of light, emanating from one focus, comes to another focus
after a reflection in the ellipse. Said otherwise, the segments, that
join a point of an ellipse with its foci, make equal angles with the
ellipse.
Vladimir Georgiev and Bozhidar Velichkov, University of Pisa
Some open problems
Initial point: Short presentation of Comenius project
Optimization and number theory
Optimization problems
Fractals
Napoleon’s problem
Dynamical systems
Curved space and new phenomena
Biliards
Periodic triangles
Periodic triangle is any billiard trajectory such that S3 = S0 , i.e.
they are periodic trajectories.
Task 1 Periodic triangles starting at fixed point S0 have the
following optimal property: minimization of the triangles with
vertex at S0 inscribed in the given ellipse.
S.Tabachnikov, Geometry and Billiards, Students Mathematica
Library, (2005).
Vladimir Georgiev and Bozhidar Velichkov, University of Pisa
Some open problems
Initial point: Short presentation of Comenius project
Optimization and number theory
Optimization problems
Fractals
Napoleon’s problem
Dynamical systems
Curved space and new phenomena
Biliards
If A0 is ANY point on the ellipse e, then there exists a unique
periodic triangle △ A0 A1 A2 having constant perimeter, i.e. the
perimeter is independent of position of the point A0 on the ellipse
e!!!
Reference mathoverflow:
Difficult3 task If a convex curve satisfies the property that periodic
triangles have constant perimeter, then the curve is ellipse.
Vladimir Georgiev and Bozhidar Velichkov, University of Pisa
Some open problems
Initial point: Short presentation of Comenius project
Optimization and number theory
Optimization problems
Fractals
Napoleon’s problem
Dynamical systems
Curved space and new phenomena
Study started with Davide Catania and Bozhidar Velichkov
Consider the semiplane
M = R × (2M, ∞),
equipped with the Schwarzschild metric having the form
g = F (r ) dt 2 − F (r )−1 dr 2 .
Vladimir Georgiev and Bozhidar Velichkov, University of Pisa
Some open problems
(8)
Initial point: Short presentation of Comenius project
Optimization and number theory
Optimization problems
Fractals
Napoleon’s problem
Dynamical systems
Curved space and new phenomena
Here
2M
,
r
the constant M > 0 has the interpretation of a mass.
F (r ) = 1 −
Vladimir Georgiev and Bozhidar Velichkov, University of Pisa
Some open problems
(9)
Initial point: Short presentation of Comenius project
Optimization and number theory
Optimization problems
Fractals
Napoleon’s problem
Dynamical systems
Curved space and new phenomena
The D’Alembert operator associated with the metric g is
1
F
g =
∂t2 − 2 ∂r (r 2 F )∂r .
F
r
Vladimir Georgiev and Bozhidar Velichkov, University of Pisa
Some open problems
Initial point: Short presentation of Comenius project
Optimization and number theory
Optimization problems
Fractals
Napoleon’s problem
Dynamical systems
Curved space and new phenomena
Our goal is to study the long–time behavior of solutions to the
corresponding Cauchy problem for the semilinear wave equation
g u = |u|p
Vladimir Georgiev and Bozhidar Velichkov, University of Pisa
in R × (2M, ∞).
Some open problems
(10)
Initial point: Short presentation of Comenius project
Optimization and number theory
Optimization problems
Fractals
Napoleon’s problem
Dynamical systems
Curved space and new phenomena
It is well–known
F. John,(1979) Blow–up of solutions of nonlinear wave equations
in three–space dimensions, Manuscripta Mathematica.
F. John, (1981) Blow–up for quasi–linear wave equations in three
space dimensions, Comm. Pure Appl. Math.
√
that there exists a critical value p0 = 1 + 2, such that the
Cauchy problem for (10) admits a global small data solution
provided p > p0 (n). For subcritical values of p ≤ p0 , a blow–up
phenomenon in the flat background is manifested.
Vladimir Georgiev and Bozhidar Velichkov, University of Pisa
Some open problems
Initial point: Short presentation of Comenius project
Optimization and number theory
Optimization problems
Fractals
Napoleon’s problem
Dynamical systems
Curved space and new phenomena
Our first goal is to study the semilinear wave equation in the
presence of the Schwarzschild
metric and to show a blow–up result
√
for 1 < p < 1 + 2. A natural idea here is to adapt the approach
of F. John to the semilinear problem (10) in the flat metric and to
show the blow–up for some subcritical values of p. This approach
meets the essential difficulty that there is no simple explicit
representation of the corresponding fundamental solution to the
D’Alambert operator in the Schwarzschild metric.
Vladimir Georgiev and Bozhidar Velichkov, University of Pisa
Some open problems
Initial point: Short presentation of Comenius project
Optimization and number theory
Optimization problems
Fractals
Napoleon’s problem
Dynamical systems
Curved space and new phenomena
In order to reduce the case of a radially symmetric wave equation
in the Schwarzschild metric to the case of a 1–dimensional wave
equation with suitable potential we use the Regge–Wheeler
coordinate
s(r ) = r + 2M log(r − 2M).
(11)
We can rewrite equation (10) as
∂t2 u − ∂s2 u −
2F
∂s u = F φ(u),
r (s)
where
F = F (s) = 1 −
2M
r (s)
and r (s) is the function inverse to (11).
Vladimir Georgiev and Bozhidar Velichkov, University of Pisa
Some open problems
(12)
Initial point: Short presentation of Comenius project
Optimization and number theory
Optimization problems
Fractals
Napoleon’s problem
Dynamical systems
Curved space and new phenomena
Making further the substitution
u(t, s) =
v (t, s)
,
r (s)
we obtain the semilinear Cauchy problem:
(
[∂tt − ∂ss +W (s)] v = f (s)|v |p ,
v (0, s) = v0 (s), ∂t v (0, s) = v1 (s),
where
W (s) =
2MF
,
r3
Vladimir Georgiev and Bozhidar Velichkov, University of Pisa
s ∈ R,
f (s) = Fr (s)1−p .
Some open problems
(13)
(14)
Initial point: Short presentation of Comenius project
Optimization and number theory
Optimization problems
Fractals
Napoleon’s problem
Dynamical systems
Curved space and new phenomena
√
Our main result asserts that for any p, 1 < p < 1 + 2, the
solution with small initial data blows up in finite time. provided the
initial data are supported far away from the black hole.
Vladimir Georgiev and Bozhidar Velichkov, University of Pisa
Some open problems