Basics: Discrete vs Continuous | ScienceBlogs
To be a bit more precise, we say that a function $ f(x) $ is continuous at a point a when we can make the value of $ f(x) $ become close to f(a) by taking x close to. The preference relation ≿ ≿ on X is continuous if it is preserved under limits. That is, for any As you note, a closed set is a set that contains all its limit points. We recall some definitions on open and closed maps. In topology an open map is a function between two topological spaces which maps open.
If you're reading this Abby, I don't think I ever really thanked you for that blow to the head in the lecture hall at Lorree. Tags Log in to post comments More like this Iterated Function Systems and Attractors Most of the fractals that I've written about so far - including all of the L-system fractals - are examples of something called iterated function systems.
Speaking informally, an iterated function system is one where you have a transformation function which you apply repeatedly. At the time, the vote was narrowly in favor of topology, with graph theory as a very close second. We're pretty much done with category theory, so it's topology time! Answering those questions, and thinking more about the answers while sitting on the train during my commute, I realized that I left out some important things that were clear to me from thinking about… Chaos One mathematical topic that I find fascinating, but which I've never had a chance to study formally is chaos.
I've been sort of non-motivated about blog-writing lately due to so many demands on my time, which has left me feeling somewhat guilty towards those of you who follow this blog. So I… Good place, historically, to bring up Zeno's Paradox?
The the connections between Difference Equations and Differential Equations? By Jonathan Vos Post not verified on 01 Mar permalink The defining quality of [the real numbers] is that given any two numbers, you can always find another number between them - in fact, you can always find an infinite set of numbers between them.
What separates the real numbers -- the continuum -- is that it's complete. That is, every sequence of real numbers that "should" converge because the elements get closer and closer together actually does converge to a real number.
Equivalently, every set of real numbers bounded above has a least upper bound. That's the defining quality of the real number system. By John Armstrong not verified on 01 Mar permalink Speaking of diffyq, you might be able to get an interesting article about how the emphasis in that subject has changed due to the computer revolution.
I know that all math teaching has, but I find the disparity between new and old diffyq books especially striking. I taught myself ordinary differential equations from a book written, I believe, in the early nineteen sixties that I found in a secondhand store surrounded by romance novels and old almanacs.
Much later, in the early nineteen eighties, I got out of the Navy and decided to take a math course to restart my mind. Diffyq I was the only course that fit my time constraints and that I was able to talk my way past the prereqs to be allowed to take it.
The textbook seemed like an entirely different subject. The techniques taught, the uses demonstrated, the amount of emphasis on numerical analysis and the use of tables, and above all the use of graphics were completely alien to my memories of the old secondhand book. Incidentally, on the final day I went to the professor's office to get my grade and saw a copy of that old book I had learned from on his bookshelf. Looking at the author's name on the spine, I suddenly realized for the first time that my teacher had written that old book!
Log in to post comments By clem not verified on 01 Mar permalink Perhaps your problem with diffEQs wasn't psychological, but the way your brains intuition about mathematics works. I'm kinda the opposite to you, I have good intuition about continuous things, but discrete mathematics and graph theory largely escapes me.
You may well have discovered how to use your talents in one area, to understand the other. Log in to post comments By bigTom not verified on 01 Mar permalink I once bought myself a slim book on linear difference equations. I found that I could cruise right through it, because I'd already learned how to solve linear differential equations in undergrad; the methods were exactly the same.
Log in to post comments By Canuckistani not verified on 01 Mar permalink Reading your story makes me think I ought to look into diff eq again.Access 2016 - Relationships - How To Create One To Many Relationship in Database Between Two Tables
On the other had, we spent a few days on recurrence relations in a discrete math class I took, and I think that I can find the closed form for any recurrence relation you care to name. Perhaps if I go at it with the knowledge that I have of recurrence relations, it'll stick.
- Continuous linear operator
- Continuous maps that are not closed or not open
I can always hope, anyway. By Sopoforic not verified on 01 Mar permalink Hmm By Jesse not verified on 01 Mar permalink For the sake of non-math readers, I think it is important to point out the existence of many connections between discrete and continuous math, just to avoid giving the idea of two disconnected big branches. In math, no theory is an island, and as your anecdote with diffEQs shows, there are many bridges, often wide, highly trafficked and productive bridges.
Sometimes you just borrow an intuition from the other side e. I chose discrete math because I liked it much more, but soon discovered that I'd never get rid of the continuum: By dileffante not verified on 01 Mar permalink The rationals are not complete, as John strongly notes, but I think that there may be an interesting issue here. The rationals are not what we mean when we talk of 'continuous mathematics', but are they not continuous? Prima facie they seem to be continuous, in MarkCC's sense whence they would be fine for most topology; or rather some countable set of the not-too-weird reals would.
Where they seem to fail to be continuous, is in having gaps where the irrational numbers are. But then, it is the presence of so many real numbers uncountably many that leads us to the Banach-Tarski paradox, which indicates that the real number line violates some of our intuitions about continuity. And what if actual, physical continua exist, and actually happen to have an abstract structure that includes infinitesimal quantitities e.
Then even the real number line would have gaps in it. In short, although completeness is important, lesser forms of closure could be adequate for much continuous mathematics, and continuity may not be either conceptually or actually the same as completeness.
Seeing the curve grow was wonderful and you could be as accurate or rough as required. Now, with all tech-drawing done on computers, this isn't taught anymore. Discrete mathematics only entered my life when I had to start doing some computing.
It is still a confusion to me: On that view, asking whether the rationals are continuous is like asking whether they're one to one. I was an EE undergrad and didn't give a rat's tail about differential equations, either. Similarly, I didn't care much about control theory, laplace transforms, etc. However, I liked electromagnetic theory, which makes me think that classes that I didn't like had everything to do with the instructor and not the subject. Maybe I wouldn't have liked lambda calculus either if I hadn't learned it myself and been taught by a jackass.
Banach-Tarski is a problem with our notions of set theory. Particularly, it's connected with the acceptance of the Axiom of Choice. However, you're setting up a straw man. Let's go back to the basics. Yes, you can define "nonstandard analyses", but these go radically beyond the real numbers, and their topology such as it is is horrendous.
Continuous linear operator - Wikipedia
You can't even probe it with sequences anymore! Readers may "Look inside this book" at Amazon. Basar EditorA. Modeling and Analysis of Synchronized Systems: Here's a relevant link: A continuum has to be compact. And the defining characteristic of R is that is complete, ordered, and a field. You need all three. Complex numbers are a complete field. Surreals for non-standard analysis aren't a field.
Rationals are an ordered field, but not complete. Log in to post comments By Douglas not verified on 02 Mar permalink Interesting story. I'm an undergraduate math major and the university I attend allows math majors to really specialize in a particular mathematical area we have an entire faculty devoted to math rather than a department as in most universities. Any way your story about DEs and RRs is interesting because a few semesters ago I was taking a DE course since I was thinking about majoring in Applied Math which is a math-physics focused degree but I did horribly in the DE class an important prerequist at the same time I was taking an introductory class in combinatorics and we were doing RRs and I also noticed the connection between DEs and RRs.
Unfortunately it didn't help me a lot because while I did a lot better in the combinatorics class the RR part of the class was one of my weakest.
Douglas Thank you for the reference in your Post of March 2, The Time Scales Calculus Figure 1. Y Koide, 'Universal texture of quark and lepton mass matrices with an extended flavor symmetry', [one of many papers] Log in to post comments By Doug not verified on 02 Mar permalink The Banach-Tarski paradox is not just a problem for set theory, John, not if set theory is our way of defining the real number line. The trouble with the Axiom of Choice is that it is not particularly troubling, if a realistic view of sets is taken.
Not unless we have a potential infinity, but for the standard real number line we do not.
And of course, if we are considering mathematical continua that might be physically instantiated e. The Axiom of Choice just says, in effect, that we already have all the subsets of what we have got already. It only seems weird when it is added to a first-order and finitely-axiomatised set theory that captures at best only part of the structure of the real number line; and only then because it is telling us more about the real number line. The Axiom of Choice was being assumed implicitly by mathematicians long before it was formulated: In short, the Banach-Tarski paradox really does show that the standard real number line is an implausible continuum it is a paradox when we think of the real numbers as the structure of a physically plausible continuum; it is simply a theorem within standard real analysis.
As for nonstandard analysis, since I did not mention it that must be your straw man! The "smooth" sort were implicitly implied; but I recon that both of those kinds are unrealistic. I hope I don't sound pedantic or bitchy.
I just think that MarkCC's highlighting of density was justified. But why worry about physical manifestations of nonstandard analysis in the midst of the debate about whther there are actually physical manifestations of the continuum? Or exactly PI, since the universe is not Euclidean? The connection between Physics and Math is somewhat slippery His mention of density of the reals demonstrates the point he's trying to make, but he's needlessly written something that is just plain false in the process.
Nobody would say that the rational numbers form a continuum, and they share that property with the reals. What sets continua apart is completeness. By John Armstrong not verified on 03 Mar permalink My earlier comment was apparently deemed unsuitable. It noted the error involved in asking of the rationals whether they are continuous. Perhaps our host, having made a similar error in the past, still feels such talk makes sense. Continuity Thus far in the course, the functions we have considered have been continuous.
To say it in plain words, this means that we can draw the graph without lifting our pens like the graph on the right.
To be a bit more precise, we say that a function is continuous at a point a when we can make the value of become close to f a by taking x close to a. We will write this as If this condition is not satisfied, we say that the function f is discontinuous at a. This situation could arise in several different ways.
Consider the price of parking at a parking meter for a length of time t. Here we suspect that the integer values of t are discontinuities of the function since we could not draw this graph without picking up the pen at these points. However, we cannot force the function to be close to 4 by taking values of t close to 1. Notice that, for a function like this, our usual methods from calculus would not be applicable.
Semi-continuity - Wikipedia
That is, if we wanted to find the maximum value on some interval, we would not be able to find it by looking for critical points. Consider the function and remember that the graph looks like: Here the function is not defined at the points and near these points, the function becomes both arbitrarily large and arbitrarily small. Since the function is not defined at these points, it cannot be continuous. Again, if this function arose in a situation which we wanted to optimize, we would have to be careful when applying our usual methods from calculus.
There are some situations which present us with a function which has an "unusual" point in fact, we'll see an example of this later on. Here is an example: Again, if we were to apply the methods we have from calculus to find the maxima or minima of this function, we would have to take this special point into consideration. Mathematicians have made an extensive study of discontinuities and found that they arise in many forms.
In practice, however, these are the principle types you are likely to encounter. Differentiability We have earlier seen functions which have points at which the function is not differentiable. An easy example is the absolute value function which is not differentiable at the origin. Notice that this function has a minimum value at the origin, yet we could not find this value as the critical point of the function since the derivative is not defined there remember that a critical point is a point where the derivative is defined and zero.
A similar example would be the function. Notice that which shows that the derivative does not exist at.