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Mathematica Q&A: Excluding Points from Plots

Got questions about Mathematica? The Wolfram Blog has answers! Each week, we'll answer a selected question from users around the web. You can submit your question directly to the Q&A Team. For our first post in this new series of Mathematica Q&A articles, we're going to address a very frequently asked question about plotting in Mathematica. How can I control the appearance of discontinuities in a plot? The short answer is, use the options Exclusions and ExclusionsStyle! Let's see how they work. By default, Plot shows the function 1 ⁄ sin(x) with lines joining its discontinuities:
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Innovating Interactive Web Publishing with Wolfram Demonstrations

Today we are pleased to announce an exciting new phase in the development of the Wolfram Demonstrations Project. In addition to its slick new design and structure, allowing for more intuitive navigation, the website now features a groundbreaking technology that takes interactivity on the web to a whole new level. This technology tightly integrates live computations into the web browser, making interaction with Demonstrations a fluent part of the online experience. Powered by a new web browser plugin, each Demonstration’s dynamic interface is now an element of the web page, similar to text, images, or videos, and yet is so much more than the typical inert content. Sliders, buttons, 3D graphic manipulations, color palettes, and the rest of the Wolfram interactive arsenal are now at your fingertips. They provide dynamic access to Mathematica’s universal engine spanning vast areas of pure and applied math, image processing, finance, control systems, wavelet analysis, and much more.
Leading Edge

Stabilized n-Link Pendulum

In the previous post in this series, we looked at how to model a stabilized inverted pendulum using the control systems design features in Mathematica 8. We were quickly able to simulate a linearly controlled cart-and-pendulum system, and show that it is stable against some fairly large perturbations. But what about a double (or triple or quadruple… ) pendulum? A general n-link pendulum is depicted below. In this post we'll see how to derive the equations of motions for this system, find out whether we can stabilize it with a linear control scheme, and produce some animations of the results.
Education & Academic

Optimizing Financial Modeling with Mathematica

On January 25 and 27 in Chicago and New York, respectively, Wolfram, in conjunction with NVIDIA, hosted a seminar themed "Optimizing Financial Modeling" to showcase how Mathematica and CUDA can be applied within the financial industry. Full presentations and a white paper on CUDA programming with Mathematica are available for download on the seminar page. Dr. Phillip Zecher, Chief Risk Officer of EQA Partners, detailed how Mathematica is used in every facet of his firm's operation, and NVIDIA's Senior CUDA Consultant John Ashley explained how CUDA programming is changing financial computation. My talk concerned Mathematica 8's broad functionality for finance. Each capability is deserving of a full seminar unto itself, so because of the sheer number of topics and functions, I was only able to briefly touch on a few examples from each category. A full list of financial tools in Mathematica is available in the online documentation. The following TabView presents an overview of the new financial functions:
Education & Academic

An Educator’s Story: Creating Immersive Teaching Environments with Mathematica

Paul Abbott, a faculty member in the School of Physics at The University of Western Australia, wants to teach his students a tool that they can use to tackle real-world problems—not only in his physics and mathematics courses, but throughout their studies and into their professional careers. For him, Mathematica is that tool. Abbott uses Mathematica to build all of his courseware, from lecture slide shows and assignments to quizzes and exams. His students use Mathematica to visualize surfaces, explore concepts using interactive examples, hypothesize results, and check their work. He says Mathematica is an "immersive environment" that helps his students reach a higher level of understanding.
Design & Visualization

Retreat from Blenheim

When last seen in the whereabouts of the Marlborough Maze, I was slinking off stage left, having been upstaged by Jon McCloone and his mix of image processing and graph theory alchemy. In a comment on my post, Jaebum Jung showed similar methods. Me, I only wanted to compute a bunch of distances from the entrance, then walk the maze. But I was not at that time able to show which was the shortest path, or even to prune off the dead ends. I'm over that lapse now. In this post I will provide brief Mathematica code to take the grid of maze pathway distances that I computed, and get the hopeless paths to melt away. Technically this is referred to as a retraction—not in the sense of an apology, but, rather, topology.
Education & Academic

Breaking Secret Codes with Mathematica

Mathematica can make you feel like a computational superman. Armed with that attitude and some schoolboy knowledge of cryptography, I turned my attention to cipher breaking this week, only to discover buried kryptonite. The weakness of ciphers (where you swap every occurrence of a particular letter in your message with the same different letter) is that they don't change the patterns of letters. The simplest attack that exploits this fact is frequency analysis. The most common letter in English is "e", and so it follows that the most common character in an encoded message (assuming the message is written in English) will correspond to "e". And so on through the alphabet. Mary Queen of Scots famously lost her head when Queen Elizabeth's spymaster broke Mary's cipher using frequency analysis. I figured that if sixteenth century spies could do it by hand, I should be able to automate it in Mathematica in about 10 minutes.
Leading Edge

Stabilized Inverted Pendulum

Can you balance a ruler upright on the palm of your hand? If I concentrate, I can just barely manage it by constantly reacting to the small wobbles of the ruler. This challenge is analogous to a classic problem in the field of control systems design: stabilizing an upside-down (“inverted") pendulum. One of the best things about Mathematica is that it makes specialist areas like control systems accessible to non-specialists. This lets you freely combine and develop new ideas without needing to be an expert in everything. It also makes Mathematica a great platform for learning and exploring new areas. Using the new control systems features (one of several new application areas integrated into Mathematica 8), I've been experimenting with models of stabilized inverted pendulums. I'm no expert in control theory, but you'll see that one doesn't need to be.
Education & Academic

The Benefits of Free-Form Input for Precollege Teachers

After talking with community college educators recently at the national AMATYC conference in Boston, I'm reminded, once again, that time is the most valuable commodity in a teaching setting. It takes time to plan a lesson for students, time to refine this lesson such that it has the most impact, and time to plan what technology will accompany a lesson and how to guide students through the process of using that technology. Any wrinkles with using the technology will greatly distract students from the course concept at hand. As a concrete example, community college faculty are used to explaining to students the four menus, and roughly eight steps, to visualize a function and its derivative using a calculator, which is a significant time investment. (The examples are from my own TI calculator I've kept all these years.) It seems that most community college educators know how powerful and useful Mathematica can be to support lectures or individual student projects. But this year, more than anything else, we talked about how Mathematica 8's new free-form input will reduce or eliminate a teacher's preparation time and will help students who are new users access Mathematica's powerful functionality immediately.