DEEP Neural Networks

show how the brain is wired

## About the Author

I have a master’s degree in computer science and a Ph.D. in mathematics from Purdue University, where I worked under the direction of Casper Goffman. I worked at Hughes Aircraft Company in design and analysis of space communication systems, including communication and imaging satellites, the Strategic Defense Initiative, and the manned space station. I have also taught at public and private colleges and universities in the United States and Asia. I am now retired. My undergraduate work was in pre-med, and I recently returned to my interest in biology by considering unsolved problems in neuroscience. The question of how synaptic connections are organized to process information led to the logic circuits presented in this website.

Besides neuroscience and its applications in electronic logic circuits, I enjoy reading, mostly about science; playing badminton and go; hiking and traveling; and spending time with friends and family.

__fractal__properties of random processes. Fractals are irregularly shaped objects whose dimensions are not necessarily whole numbers. Fractals have attracted intense interest in recent years because many natural objects – such as clouds, trees, mountains, and coastlines – can be modeled with fractals. For example, an online search for “fractal mountain” returned 12,200,000 results. One such computer-generated surface is illustrated below.

As an example of one of my results on fractals, I found the fractal dimension of a mountain’s surface modeled by Brownian motion. To put the problem in perspective, the mountain’s surface is contained in a three-dimensional space so its dimension cannot be more than three. Since it arises above a two-dimensional spherical surface (defined by latitude and longitude), its dimension must be at least two. I proved that the dimension of any such randomly generated surface is exactly two and a half with probability one [1]. (This result is included in a Wikipedia list of illustrated fractals. Word search for “Brownian surface” __here__.)

Another way to measure the dimension of the mountain is by its variation. Roughly speaking,* the variation is the amount of vertical “wiggle.” It turns out mountains wiggle a great deal, regardless of whether the wiggle is measured by its "strong" variation or its "weak" variation. I proved that a randomly generated mountain’s strong and weak variation dimensions are both four with probability one [2]. For other objects the two types of variation can be quite different. I showed by example how different a function's strong variation can be from its weak variation [3].

A famous problem in probability is best known as “The Drunk and the Lamppost.” If a drunk starts from a lamppost and wanders randomly,** will he ever return to the lamppost? He will, in a finite length of time, with probability one. This means he will continue returning again and again forever, and he will also repeatedly travel to every other lamppost on a plane surface. A drunken fly, however, will fly off to infinity with probability one. More precisely, for any arbitrary distance, a random wanderer in three dimensions will eventually reach that distance from the starting point and never again come any closer.

The original proof of the "drunken fly" theorem - and the proofs of other Brownian path properties - depended on a powerful theorem of Paul Lévy. I discovered new proofs for these well-known theorems that did not require Lévy's theorem, and this made it possible to extend the theorems to the general case in higher dimensions [4].

* What exactly is variation? It's fairly simple. For example, to find the (simple) *strong variation* of the price of a certain stock you’re interested in, have your stockbroker find how much its price changes each day for a month and add up all the changes. Now do the same thing with price changes for each hour, then do it again for each minute, etc. As you use smaller and smaller time intervals, the sum of the price changes will get closer and closer to (or will reach) some number. That number is the strong variation of the stock price for the month. To get the *weak variation*, use the difference of the highest and lowest prices during each time interval instead of the change in price.

** For a really random walk, you toss a coin to decide whether to take a step forward or backward. A second coin determines a step to the right or left. Repeat. The fly has a third coin to fly up or down one step. For Brownian motion, the coins are essentially tossed infinitely fast.

- Yoder, L. The Hausdorff dimensions of the graph and range of N-parameter Brownian motion in d-space. The Annals of Probability 3 (1975), 169-171.
- Yoder, L. Variation of multiparameter Brownian motion. Proceedings of the American Mathematical Society 46 (1974), 302-309.
- Yoder, L. Functions with different strong and weak φ-variation. Proceedings of the American Mathematical Society 56 (1976), 211-216.
- Yoder, L. Transience, density, and point recurrence of multiparameter Brownian motion. Indiana University Mathematics Journal 24 (1975), 607-611.

For publications related to this website, see the Related Articles page.