Friday, August 24, 2007

Advertising: Anyone Can Take My Randomized Algorithm Class

Next semester, I'm teaching my class on randomized algorithms and probabilistic analysis, based on the Mitzenmacher/Upfal book. Harvard, like many universities, has an "Extension School", and I offer my courses through the distance education program. Basically, my lectures get taped and put online, I put the assignments online, and you or anyone who pays the Extension fee can take the course. While I'm sure my teaching performance on video is reminiscent of say an early Mel Gibson or perhaps a Moe Howard, I personally still don't think distance education is as good as "being there" by any means (at least, not yet...). But it offers an opportunity that people may not otherwise have.

The course is taught at the level of an introductory graduate class, meant for non-theorists as well as theorists. These days, who doesn't need to know randomized algorithms and probabilistic analysis? If you know someone, for example in industry, who might like to take such a course, here's the link to the bare-bones syllabus.


Anonymous said...

Will the lectures be freely viewable online, or will they be restricted to paying Extension students?

Michael Mitzenmacher said...

The lectures will be restricted. After the first lecture or two, you'll need a password.

Anonymous said...

There are a lot of excellent lectures freely available online. That's why I would feel kind of stupid paying money for watching your lectures.

Michael Mitzenmacher said...

Feel free to watch those lectures. I was talking about "taking a course". You get credit, as well as some instruction beyond the lectures. That's what you're paying for.

Anonymous said...

Typo on your syllabus: remove "soon to be published".

Oz said...
This comment has been removed by the author.
Anonymous said...

CSCI E-300 Randomized Analysis (12884)
(Print version)
Michael Mitzenmacher, PhD, Gordon McKay Professor of Computer Science, Harvard University.
Course tuition: noncredit and graduate credit $1,975.
Fall term
Online only, beginning Sept. 19. See Distance Education. Optional sections to be arranged.

This advanced course focuses on randomized algorithms and probabilistic analysis of algorithms. Topics include Chernoff Bounds, Markov Chains, the probabilistic method, and hashing. The recorded lectures are from the Harvard School of Engineering and Applied Sciences course Computer Science 223. Prerequisites: MATH E-104, CSCI E-119 or E-124, basic probability. Students must view sample online lectures before they register. (4 credits)

Anonymous said...

For anonymous 5:35pm, are there really video lectures available for this kind of course material? (I'm asking seriously.) I know that, for example, some of MIT's OpenCourseWare classes include video, but most of them do not. I would be interested in identifying CS class sites that include video.

For Michael, I'd love to take this course but unfortunately I don't think I can afford it.

Anonymous said...

M. Mitzenmacher said:
'Feel free to watch those lectures. I was talking about "taking a course". You get credit, as well as some instruction beyond the lectures. That's what you're paying for'

Well if that is the case, why not let anyone watch your lectures for free, and only make those who want instructions and credit pay?

Michael Mitzenmacher said...

You'd have to ask the extension shcool -- not me -- since they're the ones doing the taping and making it available. My guess would be that they would tell you that it costs money to do these things, and that they think the "intellectual property" contained in the course is an item of non-trivial value, and hence they charge for it. But do feel free to ask them.

If you're asking me why don't I personally tape my own lectures and make them freely available, the answer would be that I don't have the time or resources. Also, technically, I don't believe I "own" my lectures -- Harvard does -- although they'd probably give me some leeway if I asked. Are you offering to come tape my lectures, professionally edit them, and host them on a server for the public? If so, perhaps we can talk.

Also, while I don't currently make my lectures free, I did write a textbook based on them, which you can purchase for an insanely low price of under $40 on Amazon. Now that is a great deal.

While this has been amusing, I think now I'll stop answering the silly troll who wonders why everything in life isn't free...

Anonymous said...

OCW is free.

Anonymous said...

OCW hosts video lectures for only a small fraction of their courses and I can't think of any other universities that offer their lectures for free.

Honestly, it is difficult to find distance courses that are advanced in nature. Columbia, USC and Stanford are places that I can think of but they are a lot more expensive. Most of the courses there are $4000+ if you include all the components of the fees.

Michael, I am seriously thinking of taking up this course. I would be further grateful if you could use your aforementioned "leeway" to pursue Harvard into offering Rabin's crypto & algos and other advanced courses through extension.

P.S. I certainly support the idea of free lectures though. If not anything else, it will greatly help students of developing countries who are as talented and motivated but 100000 rupees for a course is just not an option (You are right, I am originally from India)

Anonymous said...

The argument for making the video freely available appears to be based on the premise that one could learn more readily and more efficiently from a video rather than the text. In the interest of playing devil's advocate, I'd like to challenge this premise.

I've noticed that it is quite common for students in more advanced undergrad courses to skip lectures and instead try to pick up the material on their own, possibly via the course textbook.

This raises the question: for the talented and motivated, how much value does the video really add? (This is orthogonal to the question of, is the video is readily there, why not just make it freely available?)

Michael Mitzenmacher said...

Harvard Extension has offered some courses for "free" to the public -- at least, I know of one, CS-1, Understanding Computers and the Internet. (The class was a top-rated podcast.) But even there, there's an economic motivation, I'm sure -- I'd assume the idea is to get people interested by making the first class free, and hope they sign up for more.

OCW is free, but it's also heavily subsidized by various foundations. (Check their web pages.) And it's an experiment. Just taping lectures and putting them online is currently an expensive proposition. MIT will have to find the money to make it ongoing concern from somewhere. Providing infrastructure for an actual course costs even more. OCW just puts stuff on the Web, which is a good thing, but it's not the same as a course.

ilikeit -- I'm sure the Extension school would LOVE to get Rabin's course. I'm sure they've asked. But for Rabin, I'm sure it seems like annoying extra work for little payoff. If the course isn't offered via extension, my guess would be it's because he's not interested in doing it. (I do it because I'm good friends with Henry Leitner, who runs the CS program in the extension school -- I've known him since I was an undergrad, when he gave me summer teaching jobs. And as a poor junior faculty member, the extra bit they pay for it was nice.)

theory grad studnet -- There are definitely students out there who don't need courses to learn things. But I think they're the exception. Most people, I think, learn much better with the structure that a course provides.

Anonymous said...

I'm not saying that you should provide the videos on the web but blaming this on costs and saying that OCW is subsidized is somewhat hypocritical.

Harvard is also a non-for-profit organization that is heavily subsidized by many people. In fact, it's one of the richest non-for-profit foundations in the world (endowment second only to the vatican).

Anonymous said...

For theory grad student: When I was back in college, there were certain classes for which I skipped many of the lectures -- when the teacher was particularly bad and the course material was relatively easy. As the difficulty level goes up, though, so does the importance of the instructor.

When trying to learn stuff on my own, I've wasted countless hours struggling with hard-to-comprehend texts, even with things as basic as just trying to parse through the author's notation. This is the kind of thing that gets cleared up in 5 minutes of "face time" with an instructor, even if it's via video. Add to that the psychological boost of getting the material from a human being (again, even if it's through video), and I think there is no contest. Of course, that's just me; I realize other people may prefer other studying strategies.

Back during the dot-com boom, there was an Internet entrepreneur (I forget his name now) who had a grand plan for starting a free online university. He was going to use a couple hundred million of his personal fortune to fund it. The plan fell apart when his company went belly-up and the value of his stock shares disappeared with it.

Anonymous said...

OCW is not just an experiment. It will only keep growing and getting richer. And MIT is not the only school that has OCW. I suggest you check out
Also for those who appreciate free lectures, here's another site worth mentioning:

"Just taping lectures and putting them online is currently an expensive proposition."

Why is it expensive? What makes it expensive?

"OCW just puts stuff on the Web, which is a good thing, but it's not the same as a course."

What's the difference? Graded assignments and exams? If that's about it, then I don't think that's worth 2K. Many OCW courses include solution for them anyways.

Michael Mitzenmacher said...

Anonymouses above --

I really am afraid you don't know what you're talking about.

Please go to

Check out, for example, OUR STORY. You'll see how many people are involved in making OCW work at MIT, as well as the extensive Foundation support and partner support. This was a challenging, well-organized effort; I salute it. But to somehow think that this sort of stuff should automatically be free is absurd. Clearly, it requires a lot of funding, a lot of people time and support, and a lot of technical infrastructure. It's not surprising to me that MIT was one of the first places (if not the first place) to put something of this magnitude together, with its ties to appropriate partner companies. However, if you think that this will automatically continue to be free in the same way for the long term without it somehow generating some sort of "return" to MIT -- whether in monetary terms in from donations and grants or in PR terms -- you're deluding yourself.

I know that Harvard, too, is building up its Web presence in various ways. Perhaps someday my lectures will all be freely posted in a similar fashion. It's not like I object to letting knowledge be free -- again, check out my (very affordable) book. Arguing that Harvard has lots of money so it should pay to put all its lectures online to the general public, though, is not really an argument. Bill Gates has lots of money -- still more than Harvard, last I checked -- why shouldn't he pay for it? (And actually, if he donated the money with the stipulation that it was explicitly for putting lectures online, my guess is Harvard would take it and do it.) I'm happy to discuss issues with people on the blog, but please try to develop a richer argument.

If you don't think the course is worth $2K, that's fine. Don't sign up. I don't get the $2K, the Extension School does, and I don't know how they decide prices. Oddly, they didn't ask me. I recognize it's a non-trivial amount of money. Please feel free to send them a note to complain.

In many cases, employers pay for such courses as part of ongoing employee training. This was the audience I thought might be most interested, as if you read carefully, I suggested at the end of the post. I'm still hopeful some might hope to sign up.

Anonymous said...

The title is a bit misleading: "Anyone can take my randomized algorithms class". It should be: "Anyone who pays $1975 can take my randomized algorithms class."

Richard said...

For what it's worth...

I took CSCI S-124 from Professor Mitzenmacher a year or so ago. He is truly an excellent professor and I'm afraid that the trolls that are publishing comments aren't aware of how incredibly useful his course has been for me.

If you think you can learn Algorithms from a book or a video you got off of the Internet for free, be my guest. I paid the fee for his class and it was worth every penny. (BTW, I got a B- ; it's a HARD class)

If he ever offers his "End of the Wire" class again I'll probably take that too.

Don't underestimate the value of having access to someone who can help you understand a difficult concept.

Anonymous said...

I made the comment about Harvard's wealth. I just want to clarify I don't think Harvard should put the course free on the web just because it's rich. In fact, I don't think you have to put the course on the web period. I am just not buying the argument that Harvard can't spend the roughly $100 per lecture to do so unless the only person richer than Harvard would divest some money from treating Malaria to pay for this.

Personally I think that it's fine to charge $2K for the course. However, it would be nice if Harvard charged a vastly reduced price (say $100) from people in developing countries.

It's just important to remember that we're not in a for-profit business. Many people work very hard to put excellent content free on the web (Michael is an example with his papers, others such as Victor Shoup have put online complete books, MIT with OCW is another example). Rather than looking for ulterior motives such as PR or money, you can just acknowledge that they are doing this for the greater good.

Anonymous said...

"Why is it expensive? What makes it expensive?"

I still don't see how putting video lectures online is an expensive proposition.

I'm sure 2K is a suitable price for people who are getting credit and want to submit assignments bla bla. But for others who just want to learn the material and don't care about the credit (for exp, graduate students) this is clearly an absurd deal.

Also about OCW staying free... I'm sure they'll find a way to get a return but it will certainly not be making people pay money. Some people see the greater good in offering knowledge for free.

But then again.. i'm a silly troll.. so i probably don't realize the DSL connection i am using costs me money and after all, OCW is not free.

Anonymous said...

Oooooh Bill Gates should pay for everything since he has more money than Harvard.. Wonderful argument..

Bill Gates gives many things of his own for free (like money for instance). One would expect Harvard to be more generous in terms of education than Bill Gates, since Harvard is a university and Gates is a businessman.

Michael Mitzenmacher said...

anonymous's above --

1) Again, putting video online is still a non-trivial proposition in terms of expense. I'm sure it's decreasing over time, but realistically I'm sure it's much more than the $100 per hour people are quoting. If the Extension school wasn't taping my class, I doubt Harvard would at all (that is, for just the students who attend), since the class will likely be smaller than their threshold. I know Harvard would like to tape more classes, but is resource-limited.

2) The Bill Gates argument was to point out how inane your argument was, which seemed to be Harvard has money so it should do this. It wasn't a serious argument for Bill Gates to pay for it. Your sarcastic comeback suggests that your missed this realization.

3) Harvard does a great deal for the community and for education -- again, suggesting that it doesn't is not a particularly compelling argument to anyone who is informed. Besides it's numerous libraries and museums, it does offer content online. Check out,
for example, and such programs appear to be expanding. The problem appears to be the fact that the class you want isn't freely available, not that Harvard is somehow evil or not fulfilling a larger educational mission.

Michael Mitzenmacher said...

Richard --

Glad to hear you liked the class, and I hope you'll take further classes (with me, or at the extension school).

Richard brings up a further truth-in-advertising issue: my classes are hard. For the undergraduate algorithms class, roughly 1/2 the people that sign up drop along the way (often quite quickly), when they realize that perhaps they didn't have the time or background to take the course.

Thanks Richard!

Anonymous said...

How hidebound are you? I'm not gonna reply to the rest of the comments.

"The problem appears to be the fact that the class you want isn't freely available, not that Harvard is somehow evil or not fulfilling a larger educational mission."

Nobody is saying Harvard is evil, etc. Harvard is great but that doesn't mean people can't have the opinion that it should offer your class for free (At least the videos). I mean.. they're gonna tape your lectures anyway right? So why not put them on youtube or something for people who don't want the credit but just want to see your lecture?

I don't understand why you felt the need to relentlessly defend Harvard's decision to charge 2K for your class. Either you overestimate the cost/effort, or you take yourself too seriously and think your lectures would value less if they were for free.

Richard said...


Anonymous said...

My personal opinion is that Harvard should be charging for the video lectures --- heck, Michael should even get a cut --- but I understand that not everyone agrees.

I have a question, though, for those who think the lectures should be free: do you also think that anyone off the street should be able to walk in (without paying) and take any class at Harvard they like? If so, what exactly is tuition paying for? (Gym access?) If not, please account for the difference between videos and live lectures.

Anonymous said...

do you also think that anyone off the street should be able to walk in (without paying) and take any class at Harvard they like?

No that's not the same. Their presence in the class might be disturbing/overcrowding etc.

If so, what exactly is tuition paying for?

You pay tuition for credit, not for the course content. Otherwise an hour of gym and an hour of randomized algorithms would be the same.

Anonymous said...


The fact that people are disappointed your lectures won't be on the web is a compliment. I don't understand why you need to insult them and call them "trolls" rather than just explaining that you'd love to have your lectures on the web but it's not up to you.

Anonymous said...

To the person who asked why pay tuition if courses will be available on the web, note that tuition is a very minor part of the budget of a university like Harvard, whose main sources of income are donations from private people, grants from government foundations and tax exemption.

You pay tuition for an overall undergraduate experience, including courses and interaction with faculty and other students. This cannot be duplicated by watching a videotaped lecture on the web.

Michael Mitzenmacher said...

I am, of course, complimented that people would like to see my lectures. I may have mistakenly thought it was the same anonymous in the first few comments (anonymous's, please number yourselves, or have someone explain to me how to get numbered comments in blogger...); somehow, I didn't read:

There are a lot of excellent lectures freely available online. That's why I would feel kind of stupid paying money for watching your lectures.

as a compliment, and labeled the resulting behavior as "troll-like".

Richard seems to agree.

Anonymous said...

Perhaps that comment of the first anonymous was "too raw" but the truth is many people share a similar point of view. Like your book, the course is a bargain for those who take non-degree classes etc. But those people made a commitment to take those courses and pay the money because here is some sort of short term return for them or their company pays for it not them.

On the other hand many people who are just looking for quality content online to improve themselves will not find the benefit of paying 2K out of their pocket regardless of how good the lecturer is.

It's really not nice or inviting to insult people who leave genuine comments on your blog.

Anonymous said...


I figured out it might be a good idea to inject some "inside" info about OCW based on my personal experience.

First: as kurt points out, there are the slides/lecture notes, and there are the tapes.

1) On the slides/lecture notes front, OCW is doing very well, I think. Most faculty are already using electronic media for lecturing, so the material is out there. The main leftover issues are packaging, and "lawyer-proofing" the material (i.e., making sure that there is no use of unauthorized copyrighted stuff). I believe that most of the work is in the last part, although this is likely going to change, as the faculty are learning not to use copyrighted material in the lectures (unless it is really necessary).

Here is an illustrative example. In the algorithms class, for the intro lecture to randomized algorithms, I decided it would be nice to start by "tossing" a virtual coin. So, I lifted a photo of a coin from the web, and included the animation in the lecture slides.
A few months later, I got an email from the nice people at OCW, who were wondering if it was really absolutely necessary to use a photo of a silver coin depicting a Roman Emperor, or if a non-copyrighted image of a coin would do instead. After replacing the coin by a good old quarter, the lecture was ready to go.

Altogether, OCW-ing the material is not particularly time consuming, and the overhead is likely to decrease as the lecturers are learning which material (not) to use. Also, given that so many courses are already OCW-ed, one can now simply keep updating the material, which takes even less time and resources.

2) Taped lectures is a tougher part, mostly because someone has to record and edit them, which is not as easy as it sounds. Also, the instant karma of "offering quality education to the developing world" is somewhat hard to realize if people have to download multi-megabyte files over the web. On the other hand, I would conjecture that more than 50% of the students prefer well-prepared notes to videotaped lectures (at least this seems to be true for my lectures...hmm). This is especially true if the notes look like
So maybe we can live without the taped lectures, after all.


Anonymous said...

Dear Mike and other readers,

Mike was saying that he doesnot believe he "owns" his course, so videotaping it might not be "legal".

On the same line of thought, I'm wondering if we "own" our lecture notes?

Afterall, we use our working hours to compose the notes, the (respective) department computing resources to type/edit/compile them, etc.

Is there any specific law at Harvard or other places which forbids putting notes online for free?

If notes are ok, I suppose video lectures are ok too. You can make an argument that lectures are put there for students who miss the class, or for students who want to review the lectures again, etc.

Michael Mitzenmacher said...


It's been a while since I looked it over, but the relevant Harvard rules appear in the
Harvard greybook
. I suppose it's open to interpretation, but it seems to me that putting notes on-line is actively encouraged; self-publishing a video of my lecture that includes for example the appearance of Harvard students is a big no-no; making a video of my lectures myself in front of a blackboard without an audience and either selling it or just putting it online would be perfectly fine. (Though I'd still ask permission first.)

For example, Harvard has no involvement that I can remember when I wrote my book, which I think is pretty standard.

Anonymous said...

Mike can you elaborate on that? That's kind of interesting. Why does Harvard have a problem showing their students?

What exactly is the dilemma?


Michael Mitzenmacher said...

I'd have to check the regs, but there's some sort of presumption of confidentiality for students. It would be considered a breach if, for example, I just passed around a list of names of students taking my course (though perhaps the important thing is really any additional information associated with the student names). Extending this, showing a student is in the class by having them appear on videotape is (I would think) to considered questionable unless some kind of permission is gathered first.

But perhaps I've exaggerating the intent of the policy.

Anonymous said...

New anonymous here,

>>But perhaps I've exaggerating the intent of the policy.
I don't think you are exaggerating the intent of the policy. As a student, I recall hearing of this exact same policy. Also, as a student, I would not want my picture (or a recording of a dumb question I may ask after not reviewing last lecture's note) to be there for the viewing pleasure of millions. Thanks :)

2. I took CS124 as well, and it was an excellent course. I'm hoping to take E300- hopefully this semester, if I can arrange to find the time. I feel the class would be beneficial to me - but anonymous #1 might disagree. Fine. He doesn't have to take the course.

3. I fail to understand the point behind the 'this should be free' post. I certainly can support the 'I wish it were free' argument; I often feel that way in Borders and grocery stores. Wishing is free. And you know what? I wish I didn't have to pay either. For anything, actually.
The point is that the leap to 'it -should- be free' is not a reasonable one. You don't have a claim on the time and intellectual property of Michael (nor Harvard), and saying they 'should' make a class you are interested in free does not strengthen your position.

4. I recently looked at the OCW Randomized algorithms notes; they did not help me much. The supplied notes for this particular course looked to me like my own study notes - they are great if you already know the material, baffling if you do not. I feel I need either a book or a lecture to make heads or tails (cough) out of those notes alone.
Another point is that, without taking a course with assignments, deadlines, etc, I don't think I (or most students, actually) prioritize any specific learning enough. I could have bought several books on randomized algorithms before, read them, and went through the exercises, possibly posted questions on random board online. I didn't. I don't know any other student who was able to spend the same time and attention on learning an academic topic of interest as they did on an actual coursework, assignment, thesis, etc. I suspect it's human nature.

"Real knowledge takes effort, for everything else there's Wikipedia"

// YP

alex said...

Two points:

1. It may be that its quite costly to film and edit a lecture. However, once this has already been done for the extension school, the marginal cost of releasing the lecture publicly is close to 0.

2. I think there is something to the argument that Harvard has an obligation to do this. After all, isn't dissemination of knowledge part of its mission? Harvard is not simply a corporation dedicated to churning out its product (i.e. competent graduates); its a nonprofit with some quite lofty goals. From the harvard webpage, I notice that back in the founding days the existence of the college was justified as to "...advance Learning and perpetuate it to Posterity..."

Michael Mitzenmacher said...


1. Please keep in mind you need to distinguish Harvard and the Harvard Extension School. Harvard does not tape my lectures at all; the class, honestly, would be too small for the group that tapes Harvard classes. (They have more classes to be taped than they can handle.) Harvard Extension does tape my lectures precisely so they can make it available as a Distance Education course. For them, it's part of a cost-benefit calculation.

The marginal cost of releasing the lecture (to the Extension School) is debatable. If it keeps people from signing up in the future, the marginal cost is actually quite high. If, instead, it encouraged people to sign up for the course, then the marginal cost might even be negative -- it would be advertising! Again, not my decision, so I don't know what into the thinking behind it, but your simply stating "the marginal cost is close to 0" isn't really a compelling argument to me.

2. You're ignoring issues that have been raised elsewhere -- issues like the concern about copyright violation, etc., that affect the decision of whether to make things public. There are people at Harvard looking at how to make things available electronically, and the cost-benefit tradeoffs. (One tradeoff nobody in the comments is considering -- several lecturers, like myself, are happy with student having access to the lectures AFTER they are given, for review or if they miss a lecture, but don't actually want students to have access to them BEFORE, as they'll be more likely to skip lecture... anyhow, there's a whole host of issues that people here aren't even thinking about.) Just saying "Harvard should do this, it fits the educational mission" isn't really a good argument, in my mind. There's lots of things that fit into Harvard's long-term educational mission, and it can't do all of them. I know electronic dissemination is being considered, but Harvard (naturally) is a somewhat conservative place; it will take time for them to figure out if and how best to do it, in a way that satisfies the faculty and is consistent with everything else the university does.

alex said...

Just saying "Harvard should do this, it fits the educational mission" isn't really a good argument, in my mind. There's lots of things that fit into Harvard's long-term educational mission...

This is true, but I think we can say that this particular initiative will tremendously advance Harvard's educational mission. Harvard's main impact is on the several thousand students it graduates every year. Making student videos available would have an impact for students everywhere throughout the globe. Imagine a modern-day galois who is rejected from the top academic institutions of our day; or imagine anyone else who wants more than his education offers him. Its hard not to get overexcited about the potential impact of this initiative.

The marginal cost of releasing the lecture (to the Extension School) is debatable. If it keeps people from signing up in the future, the marginal cost is actually quite high.

Much of the previous discussion has focused on the physical cost of filming and editing, and this was meant in that vein. Anyway, I think its fairly safe to adopt an extremely cynical model of human behavior: the degrees offered by harvard extension school are the real reason people pay for its courses, rather than the knowledge dispensed.

Sven Seuken said...

A late but perhaps still relevant comment on ilikeit's post from August 25:

I can't think of any other universities that offer their lectures for free.

The University of Freiburg (Germany) makes all of their recorded CS lectures available to the public. Most of the advanced lectures are available in English (with a strong German accent though ;-) and I like their recording technology:

Obviously, Freiburg is not Harvard or Stanford ;-) but hey - that's where I got my undergraduate education from and I think it was quite good. If you are interested in watching a good CS course for free that's certainly one place to do it. They don't offer "Randomized Algorithms" so there is no direct competition with Michael's class ;-)

I don't know what the per lecture costs of those recordings were and whether their server set-up is scalable.

I remain intersted in this discussion about "free video lectures" and "free education" in general. Very interesting discussion...

Anonymous said...

still unable to watch the lecture video. it sucks

Anonymous said...

I agree, especially after Stanford has taken the commendable stance of offering the world access to its computer science video lectures. The mental barrier to offering something comparable should have diminished. I have the book, but would love to see Professor Mitzenmacher lecture about the material.