John
H. Hubbard, Cornell University Many systems governed by simple differential equations exhibit extremely complicated behavior. This is in particular the case for a forced damped pendulum, governed by the differential equation x'' + a x' sin(x) = b cos wt where a represents friction, b is the amplitude of the forcing term and w is the frequency. We will examine the behavior of this system for a = .1, b = w = 1, though the results are valid for a fairly large region in parameter space. In particular, we will show that given any desired sequence of gyrations, there is a an initial position and velocity such that the solution specified by these initial conditions will undergo the required behavior. The proof of this result requires isolating a Smale horseshoe in an appropriate Poincaré section; we will demonstrate how appropriate software can help do this. We will also show that the pendulum has an attracting periodic oscillation. If you color the plane of initial positions and velocities according to how many times the pendulum goes "over the top" before settling down to this oscillation, you obtain an amazingly complicated coloring. We will illustrate how this coloring is obtained, and show that it has the Wada property: every point in the boundary of one color is also in the boundary of infinitely many other colors. The results above are published in The Forced Damped Pendulum: Chaos, Complication and Control, American Mathematical Monthly 106, Number 8 (October 1999), pp. 741 - 758.
Ricardo
Cortez, Tulane University This talk will be about mathematical models that describe the spread of infections. Imagine an isolated population of individuals who are susceptible to a specific infection, like a flu, and suppose that a small number of individuals become infected. The result could be that the number of infected individuals decreases in time and the disease is eliminated. On the other hand, it may happen that the number of infected individuals grows rapidly, leading to an epidemic. I will present simple models of this situation and discuss the conditions for an epidemic to occur, how long it would take for the number of infected individuals to decrease, and the end result.
M.
Cristina Villalobos, University of Texas at El Paso Newton's method is a fundamental technique for approximating solutions of nonlinear equations. However, it is often not fully appreciated that the method can produce significantly different behavior when applied to equivalent systems. In this talk, we investigate differences in local and global behavior of two well-known methods for constrained optimization: the Newton logarithmic barrier function method and the Newton primal-dual interior-point method. As we shall show, these two methods can be viewed as applying Newton's method to two different but equivalent systems. Through theoretical analysis and numerical experimentation, we show the Newton primal-dual method performs more effectively.
William
Y. Vélez, University of Arizona Problems dealing with the set of integers have intrigued mankind for centuries. Problems are so easily stated, yet sometimes their solutions hide from us for decades or centuries. A solution to Fermat’s Last Theorem took 350 years to obtain. The attempts to solve this problem, and many others, have enriched the mathematical landscape. I this talk, I will present several problems in Elementary Number Theory that I worked on during my academic career. Let me mention one such problem. Suppose that you are to take a looooonnnnnng, loooooooonnnnnnng walk. You are going to take a walk on stepping stones that are each one unit apart, and there are infinitely many such stepping stones. The idea is to walk in such a way that each stepping stone is walked on exactly once. Of course, you could start on the first stepping stone and simply take a unit step each time. This would do it, but how boring. Let me impose the following restriction on your walking. You are to keep track of the size of each step that you take. You can walk forwards and you can walk backwards. In your pocket, you have a notebook that has the natural numbers 1, 2, 3, … on a long list. Each time that you take a step, you cross out the step size and you cannot use that step size again in your walk. So, here is the first question. Can you design an infinite walk (that is, an algorithm for describing an infinite walk) so that you manage to walk on each stepping stone and you also cross off each integer on your list? Let me get you started. Suppose that you start at stepping stone #1. You take a step to the right onto stepping stone #2. You just took a step of size 1. You can never take a step of size 1 again. You cannot step to #3, because that would require a step of size 1, however you could jump over to stepping stone #4. That would be a step of size 2. So, now you have used steps of size 1 and 2. What could be your next move? You can’t walk backwards to stepping stone #3, since that would be a step of size 1. The next steeping stone could be #7, that would use a step of size 3. However, notice something. We can now jump back to stepping stone #3, a jump of size 4, then jump to steeping stone #8, a jump of size 5, which is allowed. Here are the first few terms in the sequence of stepping stones and step sizes. Sequence of Stepping Stones:
1 2 4 7 3 8 14 5 12 Of course, you would have to be superman to carry out this walk since the step sizes are getting really big. Let’s consider a related problem that would be easier to walk. Suppose that you can only take steps of size 6, 10, or 15 units. Can you design a walk, remember that you can walk backwards and forwards, so that you step on each stepping stone, using only steps of size 6, 10, 15?
John
B. Little, College of Holy Cross A signal is a quantity that varies with time (or space, or any other independent variable or set of variables) -- for example a speech or audio waveform, the NASDAQ Composite Stock Index, or an image. A digital signal is obtained from a continuous function of a continuous variable (called an "analog" signal -- for instance, the actual sound waveform produced by a musical instrument) by sampling at a discrete set of times and quantizing the values obtained. Working with signals in digital form has many advantages and has become a common technique with the development of powerful hardware for computations on and storage of digital information. Digital signal processing (DSP) is the study of methods for analyzing, extracting information from, and performing "filtering" operations on digital signals (for example removing high- or low-frequency components). Time-limited digital signals can be "encoded" by polynomials via the so-called z-transform, and this is in turn very closely related to the discrete Fourier transform. So there are many points of contact between operations in DSP and techniques from computational algebra. In this talk we will give an introduction to basic DSP concepts, a very brief overview of Gröbner bases for those who have not studied this topic, and show how Gröbner bases can be used to solve a design problem for a particular type of digital filter.
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