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Dialogue: 0,0:00:02.86,0:00:06.98,Default,,0000,0000,0000,,Hi this is Liz Bradley, I'm a Professor\Nin the Computer Science department
Dialogue: 0,0:00:07.78,0:00:11.49,Default,,0000,0000,0000,,at the University of Colorado at Boulder\Nand also on the external faculty of the
Dialogue: 0,0:00:11.65,0:00:17.10,Default,,0000,0000,0000,,Santa Fe Institute. My research interests\Nare in nonlinear dynamics and chaos and
Dialogue: 0,0:00:17.13,0:00:21.13,Default,,0000,0000,0000,,in artificial intelligence, and I'm going\Nto be your guide during this course on
Dialogue: 0,0:00:21.29,0:00:25.29,Default,,0000,0000,0000,,nonlinear dynamics and chaos. Here's an\Nexample of a nonlinear dynamical system.
Dialogue: 0,0:00:26.95,0:00:30.95,Default,,0000,0000,0000,,It's a double pendulum. Two pieces of\Naluminium and four ball bearings. Even
Dialogue: 0,0:00:31.09,0:00:36.00,Default,,0000,0000,0000,,though the system is physically very\Nsimple, it's behavior is very complicated.
Dialogue: 0,0:00:41.100,0:00:45.100,Default,,0000,0000,0000,,Moreover, this system is sensitively\Ndependent on initial conditions. If I
Dialogue: 0,0:00:46.09,0:00:51.31,Default,,0000,0000,0000,,started here, or here, the future evolution\Nof the behavior will be very different.
Dialogue: 0,0:00:59.80,0:01:03.80,Default,,0000,0000,0000,,Even though the behavior of that device is\Nvery very complicated, there are some very
Dialogue: 0,0:01:03.80,0:01:07.80,Default,,0000,0000,0000,,strong patterns in that behavior, and the\Ntandem of those patterns and the sensitivity
Dialogue: 0,0:01:07.80,0:01:13.57,Default,,0000,0000,0000,,is the hallmark of chaos. Now there's \Nlots of words on this slide that we'll get
Dialogue: 0,0:01:13.57,0:01:17.02,Default,,0000,0000,0000,,into over the next ten weeks. I'll just\Ngive you some highlights here.
Dialogue: 0,0:01:17.02,0:01:21.02,Default,,0000,0000,0000,,A deterministic system is one that is not\Nrandom. Cause and effect are linked and
Dialogue: 0,0:01:21.02,0:01:23.84,Default,,0000,0000,0000,,the current state determines the future\Nstate.
Dialogue: 0,0:01:23.84,0:01:27.84,Default,,0000,0000,0000,,A dynamic system (or a dynamical system),\Neither are fine, is a system that evolves with time
Dialogue: 0,9:59:59.99,9:59:59.99,Default,,0000,0000,0000,,A nonlinear system is one where the\Nrelationships between the variables that
Dialogue: 0,9:59:59.99,9:59:59.99,Default,,0000,0000,0000,,matter are not linear. An example of a non\Nlinear system is the gas gauge in a car,
Dialogue: 0,9:59:59.99,9:59:59.99,Default,,0000,0000,0000,,at least in my car, where I fill up the\Ntank, and then I drive a hundred miles and
Dialogue: 0,9:59:59.99,9:59:59.99,Default,,0000,0000,0000,,the needle barely moves. And then I drive \Nanother hundred miles and the needle.
Dialogue: 0,9:59:59.99,9:59:59.99,Default,,0000,0000,0000,,plummets. That's a nonlinear relationship \Nbetween the level of gas in the tank
Dialogue: 0,9:59:59.99,9:59:59.99,Default,,0000,0000,0000,,and the position of the needle. Now non\Nlinear dynamics and chaos are not rare.
Dialogue: 0,9:59:59.99,9:59:59.99,Default,,0000,0000,0000,,Of all the systems in the universe that\Nevolves with time, that's the outer
Dialogue: 0,9:59:59.99,9:59:59.99,Default,,0000,0000,0000,,ellipse in this Venn diagram, the vast\Nmajority of them are nonlinear.
Dialogue: 0,9:59:59.99,9:59:59.99,Default,,0000,0000,0000,,Indeed a famous mathematician refers to\Nthe study of nonlinear dynamics as the
Dialogue: 0,9:59:59.99,9:59:59.99,Default,,0000,0000,0000,,study of non-elephant animals. Now this is\Nsomewhat problematic, because the
Dialogue: 0,9:59:59.99,9:59:59.99,Default,,0000,0000,0000,,traditional training that we get in\Nscience, engineering and mathematics uses
Dialogue: 0,9:59:59.99,9:59:59.99,Default,,0000,0000,0000,,the assumption of linearity, and that's\Nonly a very small part of the picture.
Dialogue: 0,9:59:59.99,9:59:59.99,Default,,0000,0000,0000,,Now looking at the inner two ellipses on\Nthis Venn diagram conveys the point that
Dialogue: 0,9:59:59.99,9:59:59.99,Default,,0000,0000,0000,,the majority of nonlinear systems are\Nchaotic, and so that's gonna play a big
Dialogue: 0,9:59:59.99,9:59:59.99,Default,,0000,0000,0000,,role in this course. And the equations\Nthat describe chaotic systems cannot be
Dialogue: 0,9:59:59.99,9:59:59.99,Default,,0000,0000,0000,,solved analytically, that is with a paper\Nand pencil, rather we have to solve them
Dialogue: 0,9:59:59.99,9:59:59.99,Default,,0000,0000,0000,,with computers. And that is a large part\Nof what distinguishes this course on
Dialogue: 0,9:59:59.99,9:59:59.99,Default,,0000,0000,0000,,nonlinear dynamics and chaos from most\Nother courses on this topic area,
Dialogue: 0,9:59:59.99,9:59:59.99,Default,,0000,0000,0000,,including Steve Strogatz's great lectures\Nwhich are on the web, and the courses on
Dialogue: 0,9:59:59.99,9:59:59.99,Default,,0000,0000,0000,,the complexity explorer website about this\Ntopic. We will focus not only on the
Dialogue: 0,9:59:59.99,9:59:59.99,Default,,0000,0000,0000,,mathematics, but also on the role of\Ncomputation in the field. In this field,
Dialogue: 0,9:59:59.99,9:59:59.99,Default,,0000,0000,0000,,the computer is the lab instrument. This\Nis experimental mathematics. And that's
Dialogue: 0,9:59:59.99,9:59:59.99,Default,,0000,0000,0000,,actually why the field of nonlinear\Ndynamics only took off four decades ago
Dialogue: 0,9:59:59.99,9:59:59.99,Default,,0000,0000,0000,,Before that, there weren't computers to \Nhelp us solve the equations. Now to
Dialogue: 0,9:59:59.99,9:59:59.99,Default,,0000,0000,0000,,succeed in this course, you'll need to\Nunderstand the notion of a derivative,
Dialogue: 0,9:59:59.99,9:59:59.99,Default,,0000,0000,0000,,because dynamical systems are about change\Nwith time, and derivatives are the
Dialogue: 0,9:59:59.99,9:59:59.99,Default,,0000,0000,0000,,mathematics of change with time. You'll\Nalso need to be able to write simple
Dialogue: 0,9:59:59.99,9:59:59.99,Default,,0000,0000,0000,,computer programs. Basically, to translate\Nsimple mathematics formulas into code, run
Dialogue: 0,9:59:59.99,9:59:59.99,Default,,0000,0000,0000,,them, and plot the results, say on the\Naxis of x versus t. There is no required
Dialogue: 0,9:59:59.99,9:59:59.99,Default,,0000,0000,0000,,computer language. You can use\Nwhichever programming language you want.
Dialogue: 0,9:59:59.99,9:59:59.99,Default,,0000,0000,0000,,And you're not gonna turn in your code in\Nthis course. We're interested in the
Dialogue: 0,9:59:59.99,9:59:59.99,Default,,0000,0000,0000,,results that come out of it. You'll also\Nneed to know about basic classical
Dialogue: 0,9:59:59.99,9:59:59.99,Default,,0000,0000,0000,,mechanics, the stuff that you get in first\Nsemester physics, like pendulums and
Dialogue: 0,9:59:59.99,9:59:59.99,Default,,0000,0000,0000,,masses on springs, and bodies pulling on\Neach other, with GmM over r-squared kinds
Dialogue: 0,9:59:59.99,9:59:59.99,Default,,0000,0000,0000,,of forces. Speaking of GmM over r-squared,\Nyou may have seen this movie in the promo
Dialogue: 0,9:59:59.99,9:59:59.99,Default,,0000,0000,0000,,video that I made. This is movie taken by \Na camera on the Cassidy spacecraft as it
Dialogue: 0,9:59:59.99,9:59:59.99,Default,,0000,0000,0000,,flew by Saturn's moon, Hyperion. Hyperion\Nis a very unusual shape and as a result of
Dialogue: 0,9:59:59.99,9:59:59.99,Default,,0000,0000,0000,,that shape, it tumbles chaotically.\NThere's also chaos on how planets move
Dialogue: 0,9:59:59.99,9:59:59.99,Default,,0000,0000,0000,,through space, not just how they tumble.\NYou may remember from Physics, that the
Dialogue: 0,9:59:59.99,9:59:59.99,Default,,0000,0000,0000,,solutions in those cases can only be conic\Nsections, ellipses, parabolas and
Dialogue: 0,9:59:59.99,9:59:59.99,Default,,0000,0000,0000,,hyperbolas. As we will see, systems with\Nthree or more bodies can be chaotic. Now
Dialogue: 0,9:59:59.99,9:59:59.99,Default,,0000,0000,0000,,think about it, how many bodies are there\Nin the solar system: lots more than two.
Dialogue: 0,9:59:59.99,9:59:59.99,Default,,0000,0000,0000,,Indeed several hundred years, the King of\NSweden issued the challenge of a large
Dialogue: 0,9:59:59.99,9:59:59.99,Default,,0000,0000,0000,,cash prize to the person who could prove\Nwhether or not the solar system was stable
Dialogue: 0,9:59:59.99,9:59:59.99,Default,,0000,0000,0000,,in the long term, and that prize was never\Nclaimed. But the answer appeared in the
Dialogue: 0,9:59:59.99,9:59:59.99,Default,,0000,0000,0000,,1980s. Indeed the solar system is chaotic,\Nalthough it is stable in a sense and we'll
Dialogue: 0,9:59:59.99,9:59:59.99,Default,,0000,0000,0000,,get back to that. So just some brief\Nhistory of our field, it really dates back
Dialogue: 0,9:59:59.99,9:59:59.99,Default,,0000,0000,0000,,to Henri Poincare in the late 1800s. But\Nit really got going in the 1960s with Ed
Dialogue: 0,9:59:59.99,9:59:59.99,Default,,0000,0000,0000,,Lorentz's paper, called Deterministic Non\Nperiodic Flow. Lorentz was the first
Dialogue: 0,9:59:59.99,9:59:59.99,Default,,0000,0000,0000,,person to recognize the patterns of chaos\Nand the sensitivity of the evolution of
Dialogue: 0,9:59:59.99,9:59:59.99,Default,,0000,0000,0000,,the system, within the context of those \Npatterns. In the 70s, this paper by Li and
Dialogue: 0,9:59:59.99,9:59:59.99,Default,,0000,0000,0000,,Yorke was the first to use the word\N"chaos" in conjunction with this behavior.
Dialogue: 0,9:59:59.99,9:59:59.99,Default,,0000,0000,0000,,In the late 70s and 80s, the chaos cabal\Nat the University of California at Santa
Dialogue: 0,9:59:59.99,9:59:59.99,Default,,0000,0000,0000,,Cruz, got very interested in nonlinear\Ndynamics, and one of the problems that
Dialogue: 0,9:59:59.99,9:59:59.99,Default,,0000,0000,0000,,they approached it with was trying to beat\Nroulette, that is, modelling the path of a
Dialogue: 0,9:59:59.99,9:59:59.99,Default,,0000,0000,0000,,ball on a roulette wheel, and using that\Ninformation to advantage. After this,
Dialogue: 0,9:59:59.99,9:59:59.99,Default,,0000,0000,0000,,things really took off. And I should say,\Nof course, that I'm only cherry-picking a
Dialogue: 0,9:59:59.99,9:59:59.99,Default,,0000,0000,0000,,very small number of examples by lots of\Nsmart people in a very active field.
Dialogue: 0,9:59:59.99,9:59:59.99,Default,,0000,0000,0000,,Nonlinear dynamics turns up all over the\Nplace. Imagine an eddy in a creek, so a
Dialogue: 0,9:59:59.99,9:59:59.99,Default,,0000,0000,0000,,patch of swirling water on the surface of\Na creek or a river, you can imagine
Dialogue: 0,9:59:59.99,9:59:59.99,Default,,0000,0000,0000,,dropping a wood chip in that patch of\Nwater and watching its path from above,
Dialogue: 0,9:59:59.99,9:59:59.99,Default,,0000,0000,0000,,perhaps with a camera, and then dropping\Nanother wood chip in that eddy at a
Dialogue: 0,9:59:59.99,9:59:59.99,Default,,0000,0000,0000,,slightly different point, and watching its\Npath. Those paths, they will trace out
Dialogue: 0,9:59:59.99,9:59:59.99,Default,,0000,0000,0000,,the patches of swirling water in that eddy\Nin different order, but if you did a time\N
Dialogue: 0,9:59:59.99,9:59:59.99,Default,,0000,0000,0000,,lapse photograph of their paths, they\Nwould both trace out the same eddy.
Dialogue: 0,9:59:59.99,9:59:59.99,Default,,0000,0000,0000,,Weather is nonlinear and chaotic. You may \Nhave heard of the butterfly effect.
Dialogue: 0,9:59:59.99,9:59:59.99,Default,,0000,0000,0000,,A butterfly flapping its wings setting off\Na hurricane a week later, a thousand miles
Dialogue: 0,9:59:59.99,9:59:59.99,Default,,0000,0000,0000,,away. Again, small change, large effect, \Nsensitive dependence on initial conditions
Dialogue: 0,9:59:59.99,9:59:59.99,Default,,0000,0000,0000,,Marine invertebrates actually make use of\Nchaotic mixing in the water around them
Dialogue: 0,9:59:59.99,9:59:59.99,Default,,0000,0000,0000,,during spawning, and I'm interested in\Nexploiting chaotic mixing to design better
Dialogue: 0,9:59:59.99,9:59:59.99,Default,,0000,0000,0000,,fuel injectors in cars. Nonlinear and\Nchaotic dynamics also turns up in driven
Dialogue: 0,9:59:59.99,9:59:59.99,Default,,0000,0000,0000,,nonlinear oscillators, like the pendulum\Nthat I showed you, like the human heart
Dialogue: 0,9:59:59.99,9:59:59.99,Default,,0000,0000,0000,,which is normally kind of mostly periodic\Nbut, can go into a chaotic state called
Dialogue: 0,9:59:59.99,9:59:59.99,Default,,0000,0000,0000,,ventricular fibrillation and as you saw\Nwith the example of Hyperion, there's a
Dialogue: 0,9:59:59.99,9:59:59.99,Default,,0000,0000,0000,,lot of nonlinear and chaotic dynamics in\Nclassical mechanics ranging from the three
Dialogue: 0,9:59:59.99,9:59:59.99,Default,,0000,0000,0000,,body problem and how black holes move\Naround each other. And nonlinear and
Dialogue: 0,9:59:59.99,9:59:59.99,Default,,0000,0000,0000,,chaotic dynamics turns up in lots and lots\Nof other fields, including, certainly,
Dialogue: 0,9:59:59.99,9:59:59.99,Default,,0000,0000,0000,,things that you are interested in. So as I\Nhope you can see, nonlinear and chaotic
Dialogue: 0,9:59:59.99,9:59:59.99,Default,,0000,0000,0000,,dynamics are not an academic oddity. They\Nare widespread, and they are fascinating,
Dialogue: 0,9:59:59.99,9:59:59.99,Default,,0000,0000,0000,,and I hope that you will get infected by\Nsome of that fascination over the course
Dialogue: 0,9:59:59.99,9:59:59.99,Default,,0000,0000,0000,,of the next ten weeks. There are other\Nfascinating courses on the Complexity
Dialogue: 0,9:59:59.99,9:59:59.99,Default,,0000,0000,0000,,Explorer website including Dave Feldman's\Ncourse on the same topic area that only
Dialogue: 0,9:59:59.99,9:59:59.99,Default,,0000,0000,0000,,assumes knowledge of high school algebra,\Nand Melanie Mitchell's wonderful course on
Dialogue: 0,9:59:59.99,9:59:59.99,Default,,0000,0000,0000,,complexity. The difference between\Ncomplexity and chaos actually bears a
Dialogue: 0,9:59:59.99,9:59:59.99,Default,,0000,0000,0000,,little bit of explanation. Put perhaps too\Nsimply, you can think of chaos as
Dialogue: 0,9:59:59.99,9:59:59.99,Default,,0000,0000,0000,,complicated behavior from simple systems,\Nlike my pendulum. And you can think of\N
Dialogue: 0,9:59:59.99,9:59:59.99,Default,,0000,0000,0000,,complexity science as addressing systems\Nthat are very complicated but have simple
Dialogue: 0,9:59:59.99,9:59:59.99,Default,,0000,0000,0000,,behavior. Again, that is too pat but the\Nidea is generally right. So, a thousand
Dialogue: 0,9:59:59.99,9:59:59.99,Default,,0000,0000,0000,,fish forming a single school. Now, some\Nlogistics. There are several thousand of
Dialogue: 0,9:59:59.99,9:59:59.99,Default,,0000,0000,0000,,you and one of me. We have an email\Naddress for this course but it can very
Dialogue: 0,9:59:59.99,9:59:59.99,Default,,0000,0000,0000,,rapidly get overwhelmed. Please do not use\Nmy own personal email address, or that of
Dialogue: 0,9:59:59.99,9:59:59.99,Default,,0000,0000,0000,,the TA, for course-related communications.\NThat thousands-to-one ratios is one of the
Dialogue: 0,9:59:59.99,9:59:59.99,Default,,0000,0000,0000,,major issues with MOOCs like this one.\NPart of the way we plan to work around
Dialogue: 0,9:59:59.99,9:59:59.99,Default,,0000,0000,0000,,that is with an electronic forum. This is\Nnot just to take a load off the course
Dialogue: 0,9:59:59.99,9:59:59.99,Default,,0000,0000,0000,,staff, it's also to solve one of the other\Nproblems with MOOCs, which is, instead of
Dialogue: 0,9:59:59.99,9:59:59.99,Default,,0000,0000,0000,,being in a traditional classroom, everyone\Ntaking this course is working by themselves
Dialogue: 0,9:59:59.99,9:59:59.99,Default,,0000,0000,0000,,all over the world in all sorts of time\Nzones. And we hope to use the forum to
Dialogue: 0,9:59:59.99,9:59:59.99,Default,,0000,0000,0000,,help with that. So if you've a question,\Nlook on the forum. Someone else may have
Dialogue: 0,9:59:59.99,9:59:59.99,Default,,0000,0000,0000,,posted that question already. If not,\Npost it yourself. If someone has posted an
Dialogue: 0,9:59:59.99,9:59:59.99,Default,,0000,0000,0000,,answer, look at that answer. If you see a\Nquestion that you know the answer to, or
Dialogue: 0,9:59:59.99,9:59:59.99,Default,,0000,0000,0000,,you think you do, offer your answer. I'll\Nalso use the forum, by the way, to post
Dialogue: 0,9:59:59.99,9:59:59.99,Default,,0000,0000,0000,,announcements, like there's a bug in the \Nproblem set, or I've just posted a whole
Dialogue: 0,9:59:59.99,9:59:59.99,Default,,0000,0000,0000,,new unit, or, the New York Times has an\Narticle about the stuff I just talked about.
Dialogue: 0,9:59:59.99,9:59:59.99,Default,,0000,0000,0000,,I'll also post discussion questions and \Nanswers for topics that may interest some
Dialogue: 0,9:59:59.99,9:59:59.99,Default,,0000,0000,0000,,people in the course, if somebody wants to\Ngo deeper into something or sideways along
Dialogue: 0,9:59:59.99,9:59:59.99,Default,,0000,0000,0000,,a tangent, that's where the forum can \Nplay a role.
Dialogue: 0,9:59:59.99,9:59:59.99,Default,,0000,0000,0000,,Here's another piece of technology that\Ncan help.
Dialogue: 0,9:59:59.99,9:59:59.99,Default,,0000,0000,0000,,There's no textbooks for this course. I'm\Npulling together material from many many
Dialogue: 0,9:59:59.99,9:59:59.99,Default,,0000,0000,0000,,different sources, including a substantial\Namount from my own work, papers that I've
Dialogue: 0,9:59:59.99,9:59:59.99,Default,,0000,0000,0000,,read, talks that I've heard at conferences\Nand so on and so forth.
Dialogue: 0,9:59:59.99,9:59:59.99,Default,,0000,0000,0000,,These video lectures are short, self-\Ncontained summaries of each topic. I use
Dialogue: 0,9:59:59.99,9:59:59.99,Default,,0000,0000,0000,,the Supplementary Materials page to\Nsupplement those summaries. So if you want
Dialogue: 0,9:59:59.99,9:59:59.99,Default,,0000,0000,0000,,to dig more deeply into something I\Nmentioned, or you'd like some background
Dialogue: 0,9:59:59.99,9:59:59.99,Default,,0000,0000,0000,,material, or, you wanna read the original \Npaper that I mentioned. This is where you
Dialogue: 0,9:59:59.99,9:59:59.99,Default,,0000,0000,0000,,should look. In the next segment of this\Ncourse, we'll start digging into some
Dialogue: 0,9:59:59.99,9:59:59.99,Default,,0000,0000,0000,,ideas and mathematics and plots and\Ncomputer examples. Most of my video
Dialogue: 0,9:59:59.99,9:59:59.99,Default,,0000,0000,0000,,lectures, by the way, are not quite as long\Nas this one. We had a lot to cover today.
Dialogue: 0,9:59:59.99,9:59:59.99,Default,,0000,0000,0000,,And there will be a short quiz after most\Nof my video lectures, a way for you to
Dialogue: 0,9:59:59.99,9:59:59.99,Default,,0000,0000,0000,,rote test your understanding of the\Nmaterial. Those will not be graded. At the
Dialogue: 0,9:59:59.99,9:59:59.99,Default,,0000,0000,0000,,end of each unit, of which there are ten,\Nthere will be a unit test. Those are
Dialogue: 0,9:59:59.99,9:59:59.99,Default,,0000,0000,0000,,graded electronically, and that grade will\Nbe the basis of your eligibility for a
Dialogue: 0,9:59:59.99,9:59:59.99,Default,,0000,0000,0000,,certificate of completion for this course,\Nif you want one. Some of you may not want
Dialogue: 0,9:59:59.99,9:59:59.99,Default,,0000,0000,0000,,a certificate. You may just wanna watch\Nthe lectures, and that's absolutely fine.
Dialogue: 0,9:59:59.99,9:59:59.99,Default,,0000,0000,0000,,This is all here on offer for you to use\Nin the way that best suits you.
Dialogue: 0,9:59:59.99,9:59:59.99,Default,,0000,0000,0000,,A word about computers. Functional\Ncomputer literacy is a prerequisite for
Dialogue: 0,9:59:59.99,9:59:59.99,Default,,0000,0000,0000,,this course. If you can't program, you're\Nnot gonna be able to write the programs
Dialogue: 0,9:59:59.99,9:59:59.99,Default,,0000,0000,0000,,that you will need to explore in the\Nhomework. Now, I've designed the course so
Dialogue: 0,9:59:59.99,9:59:59.99,Default,,0000,0000,0000,,that you can still pass it without doing\Nthat and you can still get a flavor of the
Dialogue: 0,9:59:59.99,9:59:59.99,Default,,0000,0000,0000,,concepts. But to get the full experience,\Nyou really do need to be able to do the \N
Dialogue: 0,9:59:59.99,9:59:59.99,Default,,0000,0000,0000,,homework. And there will be problems on\Neach exam that depend on your having done
Dialogue: 0,9:59:59.99,9:59:59.99,Default,,0000,0000,0000,,the programming for the homework for that\Nunit. You're welcome to use any computer
Dialogue: 0,9:59:59.99,9:59:59.99,Default,,0000,0000,0000,,programming language that you wish, modern\Ncomputer programming languages are all
Dialogue: 0,9:59:59.99,9:59:59.99,Default,,0000,0000,0000,,Turing equivalents, so it shouldn't matter\Nwhat you use. What's gonna matter is what
Dialogue: 0,9:59:59.99,9:59:59.99,Default,,0000,0000,0000,,comes out of your code, not the how well\Ncommented it is or what style it has.
Dialogue: 0,9:59:59.99,9:59:59.99,Default,,0000,0000,0000,,We're interested in what comes out and\Nthat's what we'll be looking for in the
Dialogue: 0,9:59:59.99,9:59:59.99,Default,,0000,0000,0000,,exams and the quizzes. Another related and\Nimportant point, there are thousands of
Dialogue: 0,9:59:59.99,9:59:59.99,Default,,0000,0000,0000,,you, and among the thousands of you, there\Nare going to be dozens of favorite
Dialogue: 0,9:59:59.99,9:59:59.99,Default,,0000,0000,0000,,programming languages, so there's no way\Nthat we'll be able to help you debug your
Dialogue: 0,9:59:59.99,9:59:59.99,Default,,0000,0000,0000,,code. You can post on the forum, and your\Nclassmates will help you. Please do not
Dialogue: 0,9:59:59.99,9:59:59.99,Default,,0000,0000,0000,,just post entire solutions on the forum\Nand ask, "Where's the bug?" We have
Dialogue: 0,9:59:59.99,9:59:59.99,Default,,0000,0000,0000,,chosen Matlab as the program in which we\Nwill post our solutions, because it's
Dialogue: 0,9:59:59.99,9:59:59.99,Default,,0000,0000,0000,,pretty widespread and pretty simple. It's\Na good lingua franca for that purpose. If
Dialogue: 0,9:59:59.99,9:59:59.99,Default,,0000,0000,0000,,you've never encountered Matlab, you may\Nwant to look over one of the many
Dialogue: 0,9:59:59.99,9:59:59.99,Default,,0000,0000,0000,,tutorials that are available on the web\Nfor the basic syntax for that language so
Dialogue: 0,9:59:59.99,9:59:59.99,Default,,0000,0000,0000,,that you can understand our solutions.