Mathematics Puzzle: Climbing up a Staircase

Here’s a short little puzzle I was asked at a job interview some time ago.
Suppose you have a staircase with n steps. You wish to climb up the staircase. At each point you either climb up one step, or make a small leap and jump up two steps - just like a normal human being […]

Mathematics Puzzle: Adding Random Numbers

I have no intention of turning this blog into a math blog, but every now and then I run into interesting mathematical puzzles. My two favorite fields are combinatorics and probability, and this puzzle belongs to one of them:
Suppose you start adding up random numbers chosen from the interval [0,1]. How many numbers, on the […]

Wavelets: Analyzing Scale

Physics is littered with transforms: Fourier, Radon, Wavelet, Laplace, Hilbert and many more. The idea behind each transform is - almost exclusively - to represent data in a way which makes it easy to “see” hidden structures. For instance, the Fourier transform allows you to see if there is any periodicity within your signal quite […]

Google’s PageRank (Sort of) Explained

In this post we’ll take a look at the algorithm which defines Google’s pagerank (PR) analysis. A webpage’s PR is a number between 0 and 10 that Google uses to estimate the usefulness of that page. For example, CNN.com has a PR of 9. A “typical” web site might have a PR of 5. Pages […]

Neural Networks Tutorial, Part #1

This is the first part of a set of postings on neural networks (NNs). NN are constructs that attempt to mimic our brain’s activity to a degree. NNs have become so common that it’s useful to know what people are talking about, so I’ve decided to write a brief tutorial - in several parts - […]

Shearing your Data with a Fourier Transform

Here’s a neat little trick that has helped me out a few times.
It often happens, when acquiring 2D data, that the resulting data set seems slanted, or sheared in some way. This can be the result of hardware error or some other artifact in your system (it happens to me all the time when acquiring […]

The Mathematics of Picture Taking

or: The Logic Behind Convolutions
When I was taught about convolutions in my undergraduate degree I found them somewhat puzzling. Somehow they always got tied up with the Fourier Transform and the convolution theorem. But any convolution really has nothing to do with Fourier Transforms. They actually have a lot to do with the science of […]

On Factors of 2-pi in the Fourier Transform

Every physicist is exposed during his or her undergraduate degree to the Fourier Transform, defined as:

However, a source of great confusion is that of factors. Different books define the Fourier Transform using varying factors in front of the integral:

Are those definitions valid, and if so, how can we settle the conflict between them?
The […]