sábado, 4 de abril de 2015

Vito Volterra's work on pathologic functions

In the 1880s the research on the theory of integration was focused mainly on the properties of infinite sets. The development of nowhere dense sets with positive outer content, known nowadays as nowhere dense sets with positive measure, allowed the construction of general functions with the purpose of extending Riemann's definition of integral. Vito Volterra provided, in 1881, an example of a differentiable function $F$ whose derivative $F'$ is bounded but not Riemann integrable. In this article we present and discuss Volterra's example. To read more about this follow the link below:

Link to journal: Vito Volterra's function

domingo, 29 de marzo de 2015

Möbius transformations with stereographic projections

A Möbius transformation of the plane is a rational function of the form
$$f(z) = \frac{a z + b}{c z + d}$$
of one complex variable $z$. Here the coefficients $a, b, c, d$ are complex numbers satisfying $ad - bc\neq 0.$

Geometrically, a Möbius transformation can be obtained by stereographic projection of the complex plane onto an admissible sphere in $\mathbb R^3$, followed by a rigid motion of the sphere in $\mathbb R^3$ which maps it to another admissible sphere, followed by stereographic projection back to the plane. 

The following applet shows the stereographic projection representing different Möbius transformations: 

Link: http://tube.geogebra.org/student/m839839


Inversion in the complex plane

Inversion using stereographic projection

A Möbius transformation is a combination of dilatation, inversion, translation, and rotation.

Rotation, inversion and translation in the plane

Rotation, inversion, and translation using stereographic projection

The above applet was made based on the work of D. N. Arnold and J. Rogness.

Further reading:

Arnold, D. N. & Rogness, J. (2008).  Möbius transformations revealed. Notices of the AMS. 55, 10: pp. 1226-1231. 

sábado, 28 de febrero de 2015

Roots of complex numbers

Consider $z=a+ib$ a nonzero complex number. The number $z$ can be written in polar form as
\[z=r(\cos \theta +i \sin \theta)\]
where $r=\sqrt{a^2+b^2}$ and $\theta$ is the angle, in radians, from the positive $x$-axis to the ray connecting the origin to the point $z$.

Now, de Moivre's formula establishes that if $z=r(\cos \theta +i\sin \theta)$ and $n$ is a positive integer, then
\[z^n=r^n(\cos n\theta+i\sin n\theta).\]
Let $w$ be a complex number. Using de Moivre's formula will help us to solve the equation $z^n=w$ for $z$ when $w$ is given. Suppose that $w=r(\cos \theta +i\sin \theta)$ and $z=\rho (\cos \psi +i\sin \psi)$. Then de Moivre's formula gives $z^n=\rho^n(\cos n\psi+i\sin n\psi)$. It follows that $\rho^n=r=|w|$ by uniqueness of the polar representation and $n\psi = \theta +k(2\pi)$, where $k$ is some integer. Thus
\[z=\sqrt[n]{r}\left[\cos\left(\frac{\theta}{n}+\frac{2k\pi}{n}\right)+i\sin\left(\frac{\theta}{n}+\frac{2k\pi}{n}\right) \right]\]
Each value of $k=0,1,2,\ldots ,n-1$ gives a different value of $z$. Any other value of $k$ merely repeats one of the values of $z$ corresponding to $k=0,1,2,\ldots ,n-1$. Thus there are exactly $n$th roots of a nonzero complex number.

The complex number $z$ can also be written in exponential form as
because $e^{i\theta}=\cos \theta +i \sin \theta$.

Thus, the $n$th roots of a nonzero complex number $z$ can also be expressed as
where $k=0, 1, 2, \ldots , n-1$.

In the following applet you can see a geometrical representation of the $n$th roots for a family of complex numbers. Change the values of the real and imaginary parts of $z$ and the $n$th root. Some examples are:
  1. Re(z)=1/2, Im(z)=1.72, n=3; 
  2. Re(z)=sqrt(2), Im(z)=pi, n=7.

External link to GeoGebra applet: http://tube.geogebra.org/student/m298919

viernes, 2 de enero de 2015

Foundations of mathematics, Supplementary notes


This notes are meant to be a supplementary material for the  course Mathematical Foundations and are constantly updated. If you've found a typo or a factual error, by all means let me know: 


Supplementary Material by Juancarlos Ponce

sábado, 25 de octubre de 2014

Relative velocity: Boat problems

Problem 1. 

A river  flows due East at a speed of 1.3 metres per second. A girl in a rowing boat, who can row at 0.4 metres per second in still water, starts from a point on the South bank and steers due North. The boat is also blown by a wind with speed 0.6 metres per second from a direction of N20ºE.

Figure 1: The red arrows represent the velocities of the boat (b), wind (w) and flow (r).

  1. Find the resultant velocity of the boat and its magnitude.
  2. If the river has a constant width of 10 metres, how long does it take the girl to cross the river, and how far upstream or downstream has she then travelled?

Problem 2. 

A river  flows due West at a speed of 2.5 metres per second and has a constant width of 1 km. You want to cross the river from point A (South) to a point B (North) directly opposite with a motor boat that can manage to a speed of  5 metres per second.

  1. If you head out pointing your boat at an angle of 90 degrees to the bank. How long does it take to get from point A to point B?
  2. After crossing the river you realised that it took  longer than expected. In what direction should you point you motor boat in order to reduce the time to cross the river? How long will it take you to get from point A to point B? Is it a better time?

Applet GeoGebra

The following applet shows a representation of the problem 2, considering that the boat starts from a point A. It also shows the velocities (vectors) and their magnitudes (speeds) of the boat and current.

  1. Move the sliders to change the magnitude and direction of vectors.
  2. Click the 'Start' button to activate the motion of the boat.
  3. Click the 'Reset' button to put back the boat to its original position.
  4. You can also change the width of the river. Chose a number between 5 and 1000.
  5. All velocities can be considered either as metres per second or km per second. 

Open this applet in an external window: Relative Velocity: Boat Problem