Grant: Discovery Grants Program - Individual

A radix representation for an n-dimensional vector is analogous to our usual positional, representations of real numbers, such as base 10 or base 2. In the standard base 10 representation, a number, such as 567, has a ones digit (7), a tens digit (6), and hundreds digit (5), and so on. This means that 567 is equal to 7 times 1, plus 6 times 10, plus 5 times 100. Each of these places comes from a power of 10 (1 is 10 raised to the power 0, 10 is 10 raised to the power 1, and 100 is 10 raised to the power 2). Thus 10 is called the base, also known as the radix. The digits are chosen from a given set. For example, in the standard base 10 representation, digits can be 0, 1, 2, 3, 4, 5, 6, 7, 8, or 9. In the standard base 2 representation, digits can be 0 or 1. In higher dimensions, the base (radix) is an expanding matrix, A, and the digits are vectors instead of numbers. We still have an A to the power 0 digit, an A to the power 1 digit, an A to the power 2, and so on. Now, however, when we take a sum of powers of the base times digits, we get vectors instead of numbers. This representation ties together two exciting new areas of mathematics: wavelet theory and fractals. Wavelet theory is concerned with the decomposition of functions or signals to different levels of resolution, or scales, and has many applications in signal processing, data compression, digital image analysis, and related fields. Many fractals are generated by applying the same set of functions repeatedly, which gives them the well-known property of self-similarity at different scales. Fractals can be used to model a variety of natural phenomena, and can also be used in applications such as digital image analysis. Both wavelet theory and fractals study mathematical objects (functions or sets) that exhibit self-similarity at different scales, and radix representations helps us study these areas by breaking vectors or numbers down by scale (the ones level, the tens level, and so on). In bringing together these two different areas of mathematics, we will use what is known about one topic to learn more about each of the other topics.