Hi, this a part of my work in the course “Measure theory and integration” at the University of Calgary. I discuss the concept of Hausdorff dimension and its application to fractals. You can download the pdf file here
Conversion between objects I would like to begin this article with a question. Is there a way to turn bread into sugar? The answer from biology is yes. $$\textrm{Bread} \xrightarrow{\text{chewing}} \textrm{Smaller pieces of bread}$$ $$\textrm{Smaller pieces of bread} \xrightarrow{\text{enzyme}} \textrm{Sugar}$$ The question of whether or not there is a way is different from the question of whether or not the way is practical. A practicality question may ask whether or not a person can do this. The answer is still yes. A person can chew bread with their teeth and amylose in their saliva can break down the starch in the bread into disaccharide sugar. However, if we ask whether or not a tree can do this, the answer may be different. Obviously, a tree cannot chew bread. A tree might have another way to turn bread into sugar, but it is not the same way as a human does. ...
A wavefunction is a mathematical object which represents a quantum state. Many would have been introduced to wavefunctions from Schrödinger’s cat or the double-slit experiment. However, there is a more straightforward way to present the concept of wavefunction without resorting to the complex historical development of physics. Often, the wavefunction is introduced by not knowing exactly where a particle is or its state. This is inaccurate because wavefunction is about something other than the lack of knowledge but the nature of quantum state and measurement. ...