Medical Researcher Rediscovers Integration
parallel_prankster writes "I find this paper very amusing. From the abstract: 'To develop a mathematical model for the determination of total areas under curves from various metabolic studies.' Hint! If you replace phrases like 'curves from metabolic studies' with just 'curves,' then you'll note that Dr. Tai rediscovered the rectangle method of approximating an integral. (Actually, Dr. Tai rediscovered the trapezoidal rule.). Apparently this is called 'Tai's Model.'"
No, Tai.
Wait 'til the next paper comes out about Simpson's Rule. That'll really rock the medical community!
This Article 1. doi: 10.2337/diacare.17.2.152 Diabetes Care February 1994 vol. 17 no. 2 152-154
We were all warned a long time ago that MS products sucked, remember the Magic 8 Ball said, "Outlook not so good"
While boat-builders use Simpson's rule on hull surfaces to estimate the displacement...with a slide rule and a sharp pencil.
Oh, but they're trained in Union apprenticeship programs and so could not *possibly* be as bright or talented or well-trained as a Doctor who went to University. And see? This Doctor has a publication! He must deserve 10X the salary of a boat builder.
OK, can we now agree that no thought is unique and that no idea is so big that no-one will think it again? (If this could be agreed upon we could solve a lot of problems with patents and copyright.)
ABSTRACT:
Method for dissipation of influenza symptoms through prolong dietary restriction versus current methods of hypercaloric intake treatment of cold virus carriers.
t
First, does anyone have a link to the actual article? TFS only seems to include an abstract. Second, this was published in 1994. Third, while it may simply seem that the author is rediscovering integration, the field of numerical integration is actually a rather rich one. It's all well and good to say "take an antiderivate and evaluate at the endpoints", but for a function that is found experimentally this is essentially nonsense. While the submitter here claims that this article is simply rediscovering the trapezoid rule, there's actually no such evidence given in the Abstract--algorithms for determining how big of rectangles/trapezoids/etc to use in your calculations is actually an active area of research (albeit usually for the multidimensional case) and it is possible that this researcher did actually discover a better algorithm for deciding how to do the numerical approximations.
before you blame medical science.
Economists constantly rediscover mathematics and tag their name in front of long-known mathematical things. And then, they collect a Nobel Prize for it.
And then, they use it for investing your retirement savings, in .com stock or CDOs. At the same time, they pay themselves at lot of gratification and bonus. And then, you are very surprised that your money is gone.
Sure, it's math that has been known by math and physics types for centuries, but what is truly impressive is that a medical researcher, in other words someone who, if they still remember any math is chemical math or statistical math oriented actually managed to handle a topic such as this.
What I think is most odd about this is that no-one in his peer review group noticed that this is actually relatively trivial calculus. My nephew has recently applied to study medicine in the university and I was more than a little surprised that he wasn't required during his undergraduate studies to obtain a classical scientific education. In fact, the only non-chemistry oriented science he was required to take was "Physics 1" and he wasn't required to take calculus at all. I'm not even sure how you can teach a physics course without calculus, but they appeared to be happy with nothing more than "pre-calc" style topics covering basic derivatives.
I believe what makes this impressive even though he could have Googled the topic quite easily is that it shows a small shift towards educating medical researchers in sciences which demand precision. It wasn't until quite recently (during the span of my life at least) that engineers who can in fact apply science started working closely with theoretical scientists (such as medical researchers) to devise actual solutions to problems.
If the gap is closed further then eventually, a new breed of medical research may come about who is educated in both medicine AND math and technology. Then we may start solving problems much more rapidly. I'm sure there is such a thing somewhere, but those guys, instead of publishing and bragging are probably doing silly little things like actually solving problems and don't have time for that.
The first link is even more amusing than the paper itself. Look at the number of citations the paper received!!! I mean, WTF???
Nothing spoils the joy of having an original idea more than discovering it's actually a basic concept of another discipline.
As the OP is probably already aware of, it was recently blogged about by John D. Cook in two entries:
1. 'Three surprises with the trapezoid rule' on Dec 2, 2010:
http://www.johndcook.com/blog/2010/12/02/three-surprises-with-the-trapezoid-rule/
2. 'You can be a hero with a simple idea' on Dec 3, 2010:
http://www.johndcook.com/blog/2010/12/03/you-can-be-a-hero-with-a-simple-idea/
Tai's model is obviously doing well its field, it has 38 citations with the last being in 2010.
Not if he had 1.21 Gigawatts first.
http://virtualize.wordpress.com/
Did it ever occur to anyone that the author is nothing more than a publication troll, seeing what exactly he can get away with? It's possible that the joke's on the journal, not the author.
I don't know what kind of academic curriculum a student could choose these days that would permit them to pursue a career in medical research without ever having learned basic calculus at SOME point. I mean, when I was in high school, having taken AP Calculus AB was more or less a requirement for applying to almost any reasonably competitive four-year university. How do you enter a pre-med program without even knowing what an integral or derivative is? It seems completely implausible to me, given how competitive these programs have become. Moreover, that this author somehow thought it novel to estimate the area under a curve via trapezoidal approximation is not nearly as bewildering as the fact that they should have had the basic research skills to find that their "discovery" amounted to something that is regularly taught to high school kids. To me, that's the real scandal--that someone who can write a journal article doesn't know or care to look for prior research.
About 40 papers supposedly reference this one.
Of course, I can't read them, because they're behind a paywall. The rights to the paper are owned by the American Diabetes Association, which supports something called the "Washington DC Principles for Free Access to Science". This is a lobbying group against free access to scientific publications. They've been fighting open publication since 1994. Here's their latest output, opposition to the Federal Research Public Access Act, which would force all Government-funded research papers onto public servers.
Life scientists don't get the same calculus we get as engineers.
This summer I helped a MD discover that factorials yield largish integers. At first I thought he was mocking me but it turned out that he really was serious.
Turns out that MD's are ordinary mortals after all.
Things that are ridiculous about this paper:
1) The man names the method after himself. I can see the smug look on his face when he figured out how to integrate, and decided to name his newfound discovery after himself. That's a big no no in science.
2) It's been cited 137 times since it was published. Most recently in June. That means that there has been ~137 people that cited it without seeing that it's just an integral.
3) It completely reaffirms the whole stereotype of the premedical student memorizing everything they need to get into medicine but understanding nothing.
...19Mar07...
First timothy can't even distinguish between Wikipedia and WikiLeaks now he's posting crap thats years old?
What about the fact that the story actually helped a lot of people! It was not worthless, at least.
When God goes to war, He drops big bangs.
You may laught at this, but you find the same thing in all fields. Programming language designers are writing papers on decades old language features, user interface researchers are getting lots of citations for decades old ideas or gimmicks from scifi movies, and theoretical computer science authors are woefully ignorant of statistics and machine learning. Mathematicians and physicists aren't immune either.
Maths was ahead of other sciences, in the sense that the maths for a certain breakthrough was already there, but simply ignored.
The one needed by Einstein was already there since decades, but no physicist was aware of it!
Anyway, the bottom line of the story is that every BS/MS should include a calculus course.
Maybe Computers will never be as intelligent as Humans.
For sure they won't ever become so stupid. [VR-1988]
One of the papers that cites Tai's: http://www.lexjansen.com/wuss/2004/posters/c_post_the_sas_calculations_.pdf It includes a formula Tai 'invented' (quotes in the paper) and acknowledges that it is the trapezoid rule. I can't find Tai's full paper of course, but this article shows that Tai frighteningly might have been serious about his discovery, but also that at least some MDs took calculus.
As a physics grad student, I TA a LOT of life-science, pre-med students for introductory physics. In these courses, calculus is not necessary. Considering how horrific an average student performs when confronted a problem requiring more than 3 lines of algebra manipulations, I would not be surprised if there's a statistic somewhere more than half of MDs cannot do first-year college level math. I also tutored people taking the MCAT, again, calculus not necessary.
The only possible interpretation of any research whatever in the 'social sciences' is: some do, some don't
got a published article with a lot of citations in a high impact factor journal.
I'm sure he gives a shit what you think about it.
The derivation of the trapezoidal rule is like 3 lines of algebra.
This happens all the time. And, it is a good thing. It shows that one method of one field can be applied in another. Nothing new there.
To be honest, I don't think parallel_prankster or many others realize that how many scientific ideas come about. The 'pure mathematics' in many instances had very 'unclean' background, firmly rooted in applied enigmas.
This Tai guy had no reason to look into the trapezoidal first, and see if that could have been applied, his discovery was to see that there was a pattern in the first place.
Then, and this IS good, someone else saw that these two phenomena are the same. Excellent.
Then, that low life flip tomato at http://fliptomato.wordpress.com/ "postgraduate, expatriate physics student" who has "the utmost respect for the people, places, and groups that I write about, otherwise I wouldn’t write about them." makes fun Tai... What a f*cked-up "student"...
anyone?!
Tai's article was printed in February of 1994. An author comment printed in the October 1994 issue is titled "Tai's formula is the trapezoidal rule." I don't have full text access to either, but the title of the followup is not encouraging.
> Method for dissipation of influenza symptoms through prolong dietary restriction versus current methods of hypercaloric intake treatment of cold virus carriers.
You keep using that abstract. I do not think it means what you think it means.
-- IANAL, this isn't legal advice, and definitely isn't legal advice for you. Also, Squee!
I figured that by using only one and zero, we should be able to build what I like to call a "central processing unit" and "random access memory".
Unfortunately, at best, I don't see more than a dozen of machines built on this design to be used worldwide.
A. They cut out the plot and weigh the piece of paper. Then compare this with the weight of a piece of paper of known area.
You can read the PDF of the source, for $38.00. If anyone does, could you tell us what you find?
A correction was published in the same journal that year. Doesn't explain how it got past peer review, but it was corrected quickly.
"Tai's formula is the trapezoidal rule" http://www.ncbi.nlm.nih.gov/pubmed/7677819
This doctor may be a perfect math newbie, but the schocking part is that other searchers have read his paper, and they validated his work !
Even worse : it got published in the end !
Theory: All odd numbers above 1 are prime.
Proofs by discipline:
Philosopher: 3 is prime, 5 is prime, 7 is prime, therefore by induction all odd numbers are prime.
Physicist: 3 is prime, 5 is prime, 7 is prime, 9 is experimental error, 11 is prime...
Computer Scientist: 3 is prime, 3 is prime, 3 is prime, 3 is prime, 3 is prime...
Engineer: 3 is prime, 5 is prime, 7 is prime, 9 is prime, 11 is prime...
Statistician: In the same of odd numbers: 3, 5, 11, 13, and 29 they are all prime so all odd number are prime.
Artist: 1 is prime, 2 is prime, 3 is prime, 4 is prime...
... the one in which a psychologist manages to publish a paper in Science because he knows what the integral over a Gaussian looks like when plotted: http://dx.doi.org/10.1126/science.1144073
A rectangle is indeed a trapezoid I believe.
So... what's the story?
Actually the headline should say 'Slashdotter Rediscovers Paper from 1994 '
Actually the headline should say 'Slashdotter Rediscovers Paper from 1994 '
exactly... it's been a running gag in the biology department of our university probably ever since it came out back then
And then, they use it for investing your retirement savings, in .com stock or CDOs. At the same time, they pay themselves at lot of gratification and bonus. And then, you are very surprised that your money is gone.
Inflammatory generalizations - presented as gospel truth - are not "Insightful."
Now it can be claimed that Slashdot also cited the article in question.
An integral requires that you know a formula that describes the curve. I think (can only see the abstract) this paper deals with measurement curves from lab tests. Other techniques apply there. I don't know if dr. Tai's technique was an important new development, but I do know that this slashdot item is bogus.
This isn't as stupid as it sounds, because up to the 1980s spectrometers and chromatographs had pen-and-paper plotters, not personal computers for data recording. Numerical integration would've been a waste of time without a computer.
Slashdot reader rediscovers blog entry from mars 2007.
And you also have to realize that in humanistic education they also use some rules that you actually can find when you look at them closely have a corresponding algorithm in mathematics - but the humanistic people do have a much more complicated method for doing something simple because they have a hard time to understand mathematics.
If builders built buildings the way programmers wrote programs, then the first woodpecker would destroy civilization.
EPIC FAIL !
Without a formula, you CAN'T do integration, and must rely on a numerical technique. What he's 'invented' here is the trapezoidal rule.
You are aware that the trapezoidal rule is simply an approximation technique for a definite integral, right? QED it is integration via a numerical technique.
Well if he tries to patent it, we can probably find some prior art from 4 centuries back or so...
I don't know what kind of academic curriculum a student could choose these days that would permit them to pursue a career in medical research without ever having learned basic calculus at SOME point.
I think you waaaaay overestimate medicine, doctors, and pre-med. My training is in chemistry, and it was shocking when I was an undergrad the lengths that the pre-meds would go to avoid actually learning anything. They worked their asses off at *memorizing*, but actual learning seemed to be something they weren't interested in. They also had a pathological fear and loathing of hard classes that weren't especially prone to being memorized. You know, like Calculus. I think the doctor who is well-versed in mathematical techniques to be the exception, not the rule.
Worse, they also seem to punt on difficult classes in the sciences, as well. They are essentially required to take the big-boy version of Organic Chemistry, as it is used as a weed-out class by med school admissions. And they memorize their way through that, by and large. However, they will do anything in their power to avoid classes like Physical Chemistry. At my college, nearly all of them opted to take a dumbed-down semester version of it that was nearly devoid of math instead of the full-year course that required Calculus.
So the next time you see an article published that claims an effect that would violate some basic tenet of physics, just remember - the ones publishing are probably the brains of the bunch. The real idiots are the ones working on you.
The story is one of the problem of overspecialisation. This is a very good example, because it's a very basic principle in mathematics that someone sufficiently advanced in the field of medicine to be publishing research papers. It's a problem all over academia, however. Pick up a journal from a distantly related field and you'll be pretty much guaranteed to see a paper inventing or discovering something that everyone in your field has known about for decades.
I am TheRaven on Soylent News
It is not only that they don't understand mathematics. They also learn from their teachers that mathematical models are useless, and distrust anybody that uses math on their research.
Rethinking email
In this paper, http://www.lexjansen.com/wuss/2004/posters/c_post_the_sas_calculations_.pdf the authors mocks tai claims and explains it is the trapezoidal rule
If my lucky what?
Tai would have invented integration if he had just kept making those rectangles and triangles smaller and smaller. Such a shame that he couldn't see it through to realize a great innovation. Why doesn't our government support these researchers adequately? How many other creations that could touch our daily lives have been lost?
It was a shame when infoseek died. Somehow, though I'm sure it would be awful by today's standards, I fondly recall it being the greatest thing ever made.
My wife is a direct descendant of a "famous chemist" from the 1800's. A guy named Peter Waage, we was the primary theorist for developing "The Law of Mass Action". Because of this, I have wondered if he would ever achieved fame in modern times as the review process is so rapid now that it's very likely that from the day a theory is posted for review, it can take hours or even minutes for major holes to be blown in the theories.
Oddly what has me most disturbed about this isn't the fame, but that papers are withdrawn too quickly these days. There has been tremendous numbers of discoveries made based on people mulling over the errors in other peoples work. I often wonder if this accelerated process is sending more theories with possible merit (no matter how wrong they may be in their initial form) straight to the trash bin. In the past there when papers had to be shared and distributed in ways that required ships and horses to move the information around. Ideas had time to incubate a bit further before either being rejected or revised.
On thing I am happy about though is that search engines have evolved so dramatically, that from a research perspective we "lesser minds" (including me hehe) are able to find people brighter, smarter, more experienced etc... to slap the hell out of us and call us stupid when we're wrong at an amazing rate. One of the authors of x264 just slapped me silly the other day for suggesting I waste my time on implementing a feature that is already in the encoder, though in a different form. I might have done 90% of the work before finding out that it was already in there.
I don't know whether I should be awed that we have discovered something like this hundreds/thousands of years ago or whether I should be frustrated that people take this knowledge for granted. We should be able to build upon all the things we discovered for progress, not digress centuries into the past.
A lot of people here have defended Dr. Tai, stating that it's not important he know calculus, or that a lot of people rediscover methods of mathematics. But the real problem is he didn't ask anyone about it or seek any kind of opinion on his "new" mathematical technique before publication. Any 2nd year math or physics major could have told him this, or anyone who took and remembered calculus in high school. Any one of his peers that took calculus could have told him.
He is rightly labeled an idiot because only an idiot would think he discovered a new method in a field that was not his specialty and publish not only without asking an expert in that field, but without asking anyone with even basic knowledge of that field.
This sentence no verb.
.... this sounds so familiar... in the 1990's, one group inside Siemens discovered that contacts made of little carbon blocks can be used in CT scanners to transfer current and data from x-ray tube and detector (part of gantry that is moving around patient) to stationary part of gantry/scanner.
After proudly presenting that at internal meeting, one guy said: ".... but we have been using it for decades in trains.... for the same purpose..."
Actually the headline should say 'Slashdotter Rediscovers Paper from 1994 '
Shouldn't there also be a reference to the patent and the year it was granted?
I should add that it's a very difficult problem to solve. In general, people need a lot of specialised knowledge to make a valuable contribution to a specific field. Acquiring the same level of knowledge of multiple fields would take many years. That said, it's always worth spending time with people outside your own discipline. Richard Hamming, for example, claimed that he always had lunch with the physicists or chemists in his group, rather than with other mathematicians, and often provided or gained new insights into problems by approaching them from an unusual direction.
I am TheRaven on Soylent News
most of us don't have analytical scale on 1 tonne marble block like I had access to in college I think I'll count squares on graph paper instead, thanks
Different places call it different things, so year 11 for me and others near me now but in the middle of the night I don't have a clue what it's called in the USA so "second last year" it is so people like you can understand. It's not a big deal to remember the very simplest parts of something you learnt even if it was more than a couple of decades ago, what it your problem? You still remember how to ride a bike don't you?
Besides, it was only a bit over ten years ago that I had to remind some engineering students about it. It's easy high school stuff which is why there is an article making fun of the guy writing the paper.
Robot mind? Talk to a decent lawyer (as in one that can read some of the cases a century ago with a lot of latin terminology) and you'll find that almost photographic memory "robot mind" which would be really useful to have.
Anyway, back on point, a paper is supposed to be written and reviewed in a professional manner and not in complete isolation.
How about the Tai-square value as a measurement of ignorance of widely-known prior art in science and mathematics?
Some mornings it's hardly worth chewing through the restraints to get out of bed.
The story is one of the problem of overspecialisation. This is a very good example, because it's a very basic principle in mathematics that someone sufficiently advanced in the field of medicine to be publishing research papers. It's a problem all over academia, however. Pick up a journal from a distantly related field and you'll be pretty much guaranteed to see a paper inventing or discovering something that everyone in your field has known about for decades.
Not new?
It's new to me!
TRWTF, IMHO, is that Tai's article is cited almost 40 times. I'd like to think it was meant as an April Fool's joke and got published too soon (in February).
You'd be surprise how many academic papers cite other papers based on keyword matching and one-line sentence citations only.
Wait. You're paying supplies and insurance in after-tax income? Dude, you need a better tax advisor. Or a course in honesty.
Apart from the loans, you don't have much of a point. Just look at other countries and you'll see what I mean. It's not that unusual for doctors in other countries to make half or less what we pay our doctors.
He has a point if his comments are specific to the deplorable state of student finances in the US. One can easily accumulate 30-40K in student loans just to get a BS and MS in an engineering discipline. More for an MBA. For medical students it can get worse. This is assuming going to a university that is both local and public. That was my case, and I raked enough student loans that took a substantial chunk of my salary for the first 10 years of my career. Now it is worse.
And it can get much worse if you want to pursue a certain quality of post-grad education (or a career choice) and your only options are to study 1) in a private university (what my sister had to do); or 2) a non-local university (worse if it has to be a non-local, private one.)
As I sometimes look at the German model of education with envy and admiration, I dread for my poor baby daughter for the time when she goes to college. A substantial chunk of my salary will have to an Ed. fund for her future college expenses sans she has to sell an eye and an ovary just to get a 4-year college degree.
It's really f*up here in the US. How the hell we stay afloat as an industrialized country is still beyond me.
Always better to have people derive concepts from first principles.
I don't have access to the full text, but some of the titles of the replies look amusing:
TAI FORMULA IS THE TRAPEZOIDAL-RULE, MONACO JH, ANDERSON RL, Diabetes Care, 17, 1224
DETERMINATION OF THE AREA UNDER A CURVE, BENDER R, Diabetes Care, 17, 1223
COMMENTS OF TAI MATHEMATIC MODEL, SHANNON AG, OWENS DR, Diabetes Care, 17, 1223
Anybody have text from these?
The best way to predict the future is to invent it.
Title: A simple algorithm for factoring large numbers
Abstract:
The inability to factor a number into its prime components is the basis of most encryption schemes that are used in practice today. In this paper we show a simple way of factoring numbers composed of two primes. Our method is deterministic and based on exhaustive search and guaranteed to find the right solution. Our idea is simple, given a number x we search all possible prime numbers n for which x/n is also prime. For example, the number 15 can be found by only two searches, 15/2 and 15/3, which is 5. Our initial results show that this approach can be generalized to factoring very large numbers as well, rendering all encryption mechanisms useless. However, our method is computationally expensive and may require a long time to find the solution, especially with our current inefficient Matlab implementation. We expect significant acceleration by using the C programming language, and by using parallel platforms, like GPGPU.
I recall hearing about two people that shared a nobel price, because they discovered that something in physics where the same as some pure math issue. This simply by having someone math researcher walk in on a physics lecture or something.
Basically there are to many ivory towers, and not enough bridges between them.
Hell, a astronomer suggested that volcanic activity could be linked to the moon (in much the same way as tides). Volcanologist basically told him to get lost as he was not qualified to speak on the subject. This even tho he have been able to use his theory to predict volcanic activity.
comment first, facts later. http://chem.tufts.edu/AnswersInScience/RelativityofWrong.htm
"The well know Tai's method can in most cases be improved by fitting a quadratic curve under each section of the curve and then summing the respective areas below the quadratic curves. I call this method S. Impson's Rule."
After this I would publish improvements every three months, fitting higher and higher degree polynomials. Then: Profit!
My UID is prime. Hah!
But I think that most software engineering isn't engineering. Though I love the way the word gets thrown around sanitation engineer, domestic engineer, network engineer, petroleum engineer. it just cheapens the whole field.
Storm
p.s. yes petroleum engineers have an engineering degree, but it's still fun to tease them.
Spread some weed out (roughly uniformly).. cut the graph under the paper, roll it and smoke it. write down the number of ideas you get about cow powered space flights. compare with weed on a known area of paper. repeat for reducing error.
the problem is that most MDs aren't real scientists
MDs aren't supposed to be scientists, any more than JDs are. MD is a professional degree, indicating that they've learned everything that the certifying organization says they need to know to be an MD.
Now that's not to say that some (many?) MDs don't think that they're scientists.
I've worked with a few MDs in research, and it is a truly painful experience. Most of them have a fundamental lack of grasp of basic scientific ideas (the scientific method, controls, etc) and the results of their "research" are facepalm inducing. But as medical doctors, they're fine. They memorized all of the symptoms for whatever disease and can identify it just fine. I couldn't do that.
(Of course I'm not too happy that they were paid five times more than me during this because HR likes MD more than PhD.)
If you want a vision of the future, imagine a youtube comments section scrolling - forever.
I'll admit doctor's don't make "a lot of money" when they drive 15 year old cars and live in the ghetto like I do. Everything is relative.
I was at the American Medical informatics Association meeting in November. The conference presenters in the sessions I attended were divided into three major groups: Health Care Providers, Management and Software Developers (including academic folks). In general, at least one group in the audience had a contemptuous "no shit", "no way" or "who cares" response to what was, in fact, clever, original work (even though it did replicate some basic tenet of another discipline.) And that was the response if it was a completely correct conclusion! That sort of mutual contempt between disciplines only ensures a continuous, wasteful cycle of "reinventing the wheel"
MY method is to generate variables uniformly on the rectangle [0,T]*[0, max(f(t)]. Let N be the number of uniform variables and n be the number of points (x,y) such that y f(x). Then n/N is the approximate area under the curve. Anonymous Coward's method compares favorably to Tai's method...
Actually, some of the folks I know who worked early NASA efforts (Mercury-Apollo) did exactly this [weighing graph paper] as a means of integrating functions. Indeed, the graph paper they used was spec'd to have uniform density to within a specified tolerance - so that variations in thickness, etc. didn't affect the integral.
...added the words "on a computer" and patented it. Then it would have been novel, in-obvious, and valuable.
Innocent people shouldn't be forced to pay for inferior software development.
--"Code Complete" Microsoft Press
Last time I checked, one had to take freshman calculus to get in to med school -- at which point the good Dr. should have been exposed to things like the trapezoid rule. But what's really scary is that the journal is (I believe) peer-reviewed -- so the reviewers missed it too.
Dear mods, Anonymous Gem.