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Math Science

Ramanujian's Deathbed Problem Cracked 205

Posted by kdawson
from the mock-theta-soup dept.
Jake's Mom sends word of the serendipitous solution to a decades-old mathematical mystery. Researchers from the University of Wisconsin have unraveled a major number theory puzzle left at the death of one of the twentieth century's greatest mathematicians, Srinivasa Ramanujan. From the press release: "Mathematicians have finally laid to rest the legendary mystery surrounding an elusive group of numerical expressions known as the 'mock theta functions.' Number theorists have struggled to understand the functions ever since... Ramanujan first alluded to them in a letter written [to G. H. Hardy] on his deathbed, in 1920. Now, using mathematical techniques that emerged well after Ramanujan's death, two number theorists at the University of Wisconsin-Madison have pieced together an explanatory framework that for the first time illustrates what mock theta functions are, and exactly how to derive them."
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Ramanujian's Deathbed Problem Cracked

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  • Good job! (Score:5, Insightful)

    by UbuntuDupe (970646) * on Monday February 26, 2007 @11:09PM (#18162980) Journal
    The summary didn't refer to Ramanujan as "the Indian math guy" [slashdot.org] this time! Great work! (Don't ask how I remember that one.)

    Although, it could do with one less "i" ...
  • Spelling error (Score:4, Informative)

    by kraemate (1065878) on Monday February 26, 2007 @11:11PM (#18163000)
    Spell error in story title! Its Ramanujan, without the 'i'.
  • by TubeSteak (669689) on Monday February 26, 2007 @11:11PM (#18163002) Journal

    Now, using mathematical techniques that emerged well after Ramanujan's death, two number theorists at the University of Wisconsin-Madison have pieced together an explanatory framework that for the first time illustrates what mock theta functions are, and exactly how to derive them.

    There's gotta be a Scientology joke in there somewhere
  • by UnknownSoldier (67820) on Monday February 26, 2007 @11:12PM (#18163006)
    Since the article STILL doesn't define what a mock theta func is, what is, and how can it be applied?

    Guess the wiki [wikipedia.org] still needs to be updated

    There is (as yet) no generally accepted abstract definition of a mock theta function; Ramanujan's own definition of the term is notoriously obscure.


    --
      "I want to work in Theory -- everything works in Theory!" -- John Cash, id
  • by pyite (140350) on Monday February 26, 2007 @11:13PM (#18163010)
    Ramanujan was so amazing. His work on integer partitions was enough to be revolutionary, yet he hardly stopped there--all before dying at such a young age.

    • He was indeed. My favorite (ok, only) story about him is:

      I remember once going to see him when he was lying ill at Putney. I had ridden in taxi cab number 1729 and remarked that the number seemed to me rather a dull one, and that I hoped it was not an unfavorable omen. "No," he replied, "it is a very interesting number; it is the smallest number expressible as the sum of two cubes in two different ways."
      - Godfrey H. Hardy (1877-1947)
      • by pyite (140350)
        I believe 1729 was mentioned in the movie Proof [imdb.com] which contained not enough math to make the average math geek enjoy and just enough math to make the average girlfriend annoyed. Still an awesome story about Ramanujan though :-)

  • Ramanujan (Score:5, Insightful)

    by theurge14 (820596) on Monday February 26, 2007 @11:30PM (#18163118)
    From what I've read about Ramanujan, what I still can't understand is how a guy from a poor background with little to no formal schooling is able to just sit around and write in a notebook and come up with the equations he did. I just have to wonder what it was in nature that made him so more adapted to mathematics than the rest of us mere mortal humans. This guy was on a completely different level. Mozart comes to mind when I think of him.
    • Re:Ramanujan (Score:5, Interesting)

      by teetam (584150) on Tuesday February 27, 2007 @12:10AM (#18163326) Homepage
      He was poor and from a poor country, but he did go to school and learn math there. He just happened to be fascinated by it and continued to work on it, neglecting everything else. He obviously also had a knack for math. That has nothing to do with poor or rich.

      Math, being theoritical, does not require a lot of external resources (like laboratories etc.)
      • Re: (Score:3, Informative)

        by Anonymous Coward
        Not quite.
        He did not have advanced learning in math.
        even though he went to school, in the end he was so enamored with maths that he stopped studying everything else, which cost him high. He was unable to get through to college. Thus, his knowledge was limited and was from primarily two books he found in the library.

        Hardy once even mentioned that his greatest regret was that Ramanujan did not have the higher learning that would have avoided him rediscovering many - many theories. On one count, 1/3 of his dis
        • Re:Ramanujan (Score:5, Insightful)

          by MrBoombasticfantasti (593721) on Tuesday February 27, 2007 @02:32AM (#18163962)
          Still, that means that 2/3 of his discoveries are new and original!

          Might it be that education structures the mind to follow the known paths? Perhaps by not knowing the 'usual' solutions, you can come up with a more elegant and deep solution?

          • When approaching a problem, I often avoid learning too much about how others have done it before I try to think of a way to solve it. Sometimes I come up with the same thing as widely accepted practice, and sometimes I come up with something that's significantly worse than widely accepted practice, and sometimes I come up with something better. That latter is much less likely to happen if I fully educate myself on other people's solutions first.

            I do not think education is bad, and I'm not anti-intellect

      • Re:Ramanujan (Score:4, Informative)

        by ezzthetic (976321) on Tuesday February 27, 2007 @12:53AM (#18163524)
        He won prizes at school for his maths prowess, and went to university on a scholarship. He lost the scholarhip due to his obsessive inability to do other aspects of the curiculum that were not maths related, or which were offensive to his Brahman beliefs. There was never any doubt that he was mathematically gifted, and his mother promoted him intensively. There seems to be a myth that he was an illiterate peasant who happened to stumble on a maths book came from, but I don't know where it came from.
        • Re: (Score:3, Informative)

          by bogjobber (880402)

          There seems to be a myth that he was an illiterate peasant who happened to stumble on a maths book came from, but I don't know where it came from.

          Ramanujan is mentioned in the movie Good Will Hunting [wikipedia.org] and that is how he is presented. That's the first time I heard of him. I'm sure people just use that myth because it's not too far from the truth and makes a much better story.

      • Re: (Score:3, Informative)

        by theurge14 (820596)
        Quoted from Hardy "So the real tragedy of Ramanujan was not his early death at the age of 32, but that in his most formative years, he did not receive proper training, and so a significant part of his work was rediscovery..."

        And yes there were instances during his life when he struggled for money, even to eat.

        I'm not saying rich or poor makes you smart. I'm saying being poor tends to keep you from being discovered by the rest of us. The immense contributions of Ramanujan could have been lost to us all if
        • Re:Ramanujan (Score:5, Insightful)

          by rxmd (205533) on Tuesday February 27, 2007 @08:55AM (#18165984) Homepage

          Quoted from Hardy "So the real tragedy of Ramanujan was not his early death at the age of 32, but that in his most formative years, he did not receive proper training, and so a significant part of his work was rediscovery..."


          At the same time, Hardy acknowledged that "on the other hand he would have been less of a Ramanujan, and more of a European professor, and the loss might have been greater than the gain." (From Hardy's article in "The American Mathematical Monthly" 44.3 (1937), p. 137-155.)
    • by ookabooka (731013)
      Mozart comes to mind when I think of him.

      How? Mozart was very privileged. When his father saw a little musical talent in him, he threw plenty of resources to develop that talent including "instruction in clavier, violin, and organ." Wiki-link. [wikipedia.org] This was all at the age of 3 mind you, one has to take into consideration the amount that you can condition a human being to excel in a certain area if you train them from such an early age.
    • Re:Ramanujan (Score:5, Insightful)

      by OldManAndTheC++ (723450) on Tuesday February 27, 2007 @01:23AM (#18163732)
      It's sad to think that geniuses may languish among the world's millions of underprivileged children who lack access to education. When you think of the potential impact of a single person of the caliber of Mozart, Ramanujan, etc., our civilization could be missing out on some truly wonderful things.
      • Re: (Score:3, Funny)

        by phasm42 (588479)

        It's sad to think that geniuses may languish among the world's millions of underprivileged children who lack access to education. When you think of the potential impact of a single person of the caliber of Mozart, Ramanujan, etc., our civilization could be missing out on some truly wonderful things.

        Yes, but think of all the Hitlers we're also missing out on. We'd better play it safe and hold them down. Besides, they only discover things I don't care about or don't want to hear.

      • by dpilot (134227)
        But just think of the things that American society's lopsided distribution HAS produced - like Paris Hilton.
      • Whenever someone says something like this (which is pretty frequently) it occurs to me, it's sad to think that human beings languish among the world's millions of underprivileged children...

        But I guess the one-in-a-million starving, uneducated supergeniuses would affect us more :-P

    • I just have to wonder what it was in nature that made him so more adapted to mathematics than the rest of us mere mortal humans
      A little bit of talent, a lot of desire.
    • Re:Ramanujan (Score:4, Informative)

      by The Cydonian (603441) on Tuesday February 27, 2007 @05:53AM (#18164848) Homepage Journal

      Ramanujan's family was NOT poor. His father was among the first rung of urban middle-class professionals, who've just moved from their villages as (colonial) India's cities started expanding, finding employment as a minor clerk somewhere. His mother was very educated, and often sang in the local temple, thus earning some petty, but useful, cash in the process.

      They weren't well-off, but they weren't poor either. Ramanujan had no absolutely pressure whatsoever to find an actual job while he was sitting in the verandah of his Sarangapani Street house, and writing his fantastical proofs in that mystical notebook of his. (In fact, he got married while he was jobless, a prospect that is unimaginable even in still-arranged-marriage-friendly contemporary India).

    • I've often thought that the grammatical structures of the various languages, profoundly affects the way people think. There has always seemed to be an unusually high number of Indian mathematicians, German/Scotts engineers, Irish poets, etc. IS this just playing into stereotypes, or does the way that words are strung together affect the thought patterns, making it easier to do, or create certain things. Clearly there are some economic issues - to create math, all one needs is a paper and pencil, but I th
    • by blueskies (525815)
      I saw a documentary about him once.

      He was in and out of foster homes throughout childhood. He ended up taking up menial jobs at a local university to be around math. But he lost his job when he almost went to jail for assault and battery and hitting a cop. But luckily one of the professors at the university noticed he solved some "unsolvable" problem on a chalkboard and got him paroled.

      He skips out on his parole, but Robin williams saves him by telling him it isn't his fault and he ends up driving cross-
  • Both TFA and wiki mention that these functions keep cropping up in real world problems from chemistry and physics.

    So... uh, which ones?

    See, this is why I switched majors from physics. Any time I look at an infinite series, my head starts to hurt.
    • Re:Real World Uses? (Score:4, Informative)

      by arlo5724 (172574) <jacobw56@gmai3.14159l.com minus pi> on Tuesday February 27, 2007 @12:10AM (#18163322)
      To answer this very loosely, parts of these functions are bounded by geodesics with cusps at the corners, and this means that any geodesic structure of this type (certain types of chemical structures and a slew of phenomena in relativistic physics) can be partly described by those pieces of these functions and that it is possible that these functions represent a certain type of generalization for these structures, allowing scientists to better describe some existing structures with similar modular forms and even some that exist only in thought.
  • Ken Ono's seminar (Score:5, Informative)

    by alpha_foobar (820088) on Monday February 26, 2007 @11:55PM (#18163250) Homepage Journal
    It appears that Ken is holding a seminar at UW on March 29 2007 (http://math.uwyo.edu/DEPTCOLLOQ.asp#Mar%2029). We will probably have to wait until then for any details.
    • by mihalis (28146)

      Don't forget :

      I have a discovered a truly marvelous demonstration of this proposition beneath your current threshold

      The next truly marvelous demonstration will be ready soon, but subscribers can beat the rush and see it early!

  • by phreakv6 (760152) <.moc.liamg. .ta. .6vkaerhp.> on Tuesday February 27, 2007 @12:54AM (#18163538) Homepage
    not totally offtopic but i would like to recommend this amazing book [amazon.com] (the man who knew infinity) to anyone interested in reading his biography. its one of the best biographies i've ever read.
    • Ramanujan is the only person I've ever considered as an idol. That book is the reason. I've been to countless places after I first read the book; I still carry my dog-ear-ed copy wherever I go.

      In fact, I think I'll re-re-read it again tonight; always good to look back on your heroes' stories and see where you are since you first read about them. (Not far away, I'm afraid, in my case).

  • Disappointing (Score:5, Insightful)

    by grimdawg (954902) on Tuesday February 27, 2007 @12:56AM (#18163570)
    As a young mathematician-in-training (just finished my undergrad degree), it disappoints me to see the kind of coverage the maths community gets.

    It takes a near-century-old problem to be solved to pop a maths story on slashdot - and TFA holds no details. To get on any kind of mainstream news, the Poincare conjecture needs to be solved, and then we get "Perelman proved a rabbit was a sphere".

    Mathematics at universities worldwide is being dumbed down for the pursuit of the cashed-up Engineering student. Mathematicians get no kind of acclaim for their work - even compared to other 'unglamourous' pursuits. People these days don't seem to appreciate the debt they owe to mathematics.

    What's it going to take for mathematicians to get some mainstream coverage? A sex scandal?
    • by Nicky G (859089) on Tuesday February 27, 2007 @01:18AM (#18163708)
      Sex scandal? Uh, yeah... don't hold your breath.
      • by jd (1658) <imipak@nOSPam.yahoo.com> on Tuesday February 27, 2007 @02:52AM (#18164018) Homepage Journal
        It would have to be imaginary, or complex. But that's a bit of a tangent from the point. The TFA is obtuse, cos() it doesn't exp()lain anything much. It would seem that the Slashdot crowd are caught on it Hooke, line and sinker, though. Of course, any maths problem is as easy as Pi, if you use sufficiently advanced techniques. However, if the problem cannot be differentiated meaningfully, can it be integral?
      • by ezzzD55J (697465)
        Haha, I wish I hadn't just wasted my mod points rating other comments funny :)
      • Mathematicians have sex, too. There's fair evidence that the causes which led to Galois' death in a duel qualify as a sex scandal.
    • by Anonymous Coward on Tuesday February 27, 2007 @03:11AM (#18164084)
      Ease of understanding & teaching.

      I really think the reason why a lot of people are bewildered with math (& thus ignore it) is that they were never really able to approach it properly. Mathematics has a tendency in university to not explain itself properly. Things that I found rather simple in the end were just never explained clearly, were incomplete explanations, assumed you knew & understood concepts from other, unrelated courses, or were given "pseudo-explinations" that kind-of explained something but not properly, giving potential incorrect understandings that could be disastrous later (think high school math).

      The entire cutter mentality that math classes can tend to be in university don't help much either (what is probably the #1 reason why people drop their hard science/engineering/comp sci courses?? Probably MATH!)

      Once I figured whatever a concept really meant in math, I realized reading the textbook after the fact (sometimes several courses later) they use terms and concepts that aren't explained at all or they use really obtuse english sentences while simply defined symbolic language could easily show the concept. Actually most of it I found rather simple & clear in the end once I got to understand it but found that the textbook just explained it, badly or with huge gaps in their explinations.

      • by muecksteiner (102093) on Tuesday February 27, 2007 @05:27AM (#18164714)
        You have a very good point about math generally not being taught as well as it could be.

        Not in the sense that the curricula should be dumbed down in any way - this would not work out well in the long run.

        But there definitely is a streak of the beloved "if it was hard to code, it should be hard to understand" mentality to be found in mathematics.

        Introductory math courses at universities usually do not have concepts of such bewildering complexity on the curriculum, that they should be considered to be as "hard" as they turn out to be for everyone.

        However, they still are the bane of undergrads everywhere, and sometimes I wonder if the obtuseness of these courses is not just an in-joke perpetrated by the mathematicians.

        If you are not smart enough to "get it" in the arcane way the stuff is being presented, you woul not hack it further down the road anyway - at least not in pure math, and they are not inclined to have pity on anyone who could not have gone down that road in the first place.

        Or so the reasoning might go, when mathematicians are amongst themselves... :-)

        Note that the remarks in this posting mostly apply to the teaching of the kind of "working math" that an engineer might use, which (to put it mildly) can still be pretty involved in terms of complexity, but always has a goal-oriented quality to it that pure math does not necessarily share. This residual "grounding in reality" usually makes the teaching of even advanced concepts much easier - a potential bonus that (at least in my opinion) is not used nearly as often as it could be.

        A.
      • by Flyboy Connor (741764) on Tuesday February 27, 2007 @06:56AM (#18165140)

        I can relate to that. I studied math at a famous university for a couple of years before I dropped out. Here are some of the things I remember:

        We started with over 100 students in the first year. By the third year, the number had dropped to less than 10 students. Half of those dropped out later. The professors were proud of this fact.

        Each lecture took three hours, with one fifteen minute break. You were only allowed to ask questions in the last 15 minutes of the lecture.

        Professors only took the trouble to learn students' names when they entered third-year courses.

        I once wrote a research paper for one of the professors for a first-year course. In the very last paragraph of the paper I wrote a little joke. The paper was marked "A", then the "A" was crossed out, "C-" written below it, with an arrow pointing to the joke.

        Math students had access to the faculty mainframe (this was in the early 1980s), but did not get instruction on how to use it, as opposed to physics students. The reasoning was that math students either should not need computers for their work, or should be smart enough to figure it all out by studying the manuals.

        Professors often supplied example excercises. Students were encouraged to make these excercises and supply their answers to the professor. However, these answers were NEVER corrected, so that after a while students simply did not bother anymore.

        Professors were notorious for not preparing lectures, and working out examples as they were going along, often failing to prove what they wanted to prove. One particularly telling incident was when a professor was working out a complex proof, starting at the top left of one of the two four-piece blackboards in the hall, and chalking down, very fast, formula after formula. I was trying to follow his proof, but, of course, was always several lines behind. But I thought I did understand it, and was approaching to where he was. When he was at the bottom-right of the second blackboard, he paused, and kept staring at the last line he had written, muttering to himself. While I was approaching this last line (making lots of notes, because OF COURSE these proofs weren't in the textbooks or anything), he started scanning back. After doing this for about five minutes, he suddenly walked over to the first board again, changed a plus into a minus in one of the first lines, then made lots of changes in the rest of what he had written, and finally wrote "Q.E.D." at the bottom-right. Then he closed the blackboards and sent us on our way.

        Through this experience I thought I simply was not good enough at math. But when I switched to computer science, where math courses were taught by computer scientists, I passed with flying colours, usually as the best of the class. Not because the courses were easier, but because they were taught better.

      • Actually most of it I found rather simple & clear in the end once I got to understand it but found that the textbook just explained it, badly or with huge gaps in their explinations.

        Who gave you the right to criticize communication skills, Mr. Random Comma Guy?
      • There is a certain amount of truth to the claim that math is taught badly. Explanations are often not clear, and there is a lot of emphasis on proofs in preference to understanding. This is especially true of pure math as contrasted with applied math. However,
        • some things are just hard to learn and only seem easy in retrospect.
        • People learn differently. The words that bring clarity to me may be muddy to you, and vice versa.
        • The words of someone who knows the subject thoroughly will be very different from th
      • Re: (Score:3, Insightful)

        by Ibag (101144)
        Teaching mathematics is difficult. Many people can only think about things in terms of concrete examples, but even when math is trying to generalize a real world concept, it generally does so by using abstract looking definitions (which are made using precise terms, often employing a symbolic language). People generally don't care about these abstract ideas, though. They either want to know what a concept "means" or exactly how to use it. Often times, there aren't any good examples that illustrate exact
    • by Kopretinka (97408)

      What's it going to take for mathematicians to get some mainstream coverage? A sex scandal?
      If you guys ever applied your results, and didn't leave that to the physicists, computer scientists, economists etc., you might get some recognition. Heck, give me a good usable logics framework, I'll apply it to the Semantic Web and I'll mention your name in every interview when I'm famous for making SemWeb work.
    • by RAMMS+EIN (578166)
      ``Mathematics at universities worldwide is being dumbed down for the pursuit of the cashed-up Engineering student.''

      That's the inevitable backlash from exaggerating the importance of math to other fields. Speaking for computer science, there is a lot of CS that can be done without good math knowledge. If you make math a required subject, you will either hamper (otherwise) good computer scientists in getting degrees, or you will have to dumb down the math curriculum. I claim that both of these are happening.
    • If you are in math for fame and fortune, you should really switch over to the business college. Preferably, the marketing dept.
    • by CodeShark (17400)
      How about we start with a young mathemetician explaining why in the heck some of these unsolved mathematical problems even matter. Another poster started to explain something about how certain functions had to do with "cusps" on geodesic structures, but his explanation made about as mud.


      I can just about guarantee that if you can explain why this solution matters, your post will go to +5 on the moderation scale.

    • by mdielmann (514750)

      To get on any kind of mainstream news, the Poincare conjecture needs to be solved, and then we get "Perelman proved a rabbit was a sphere".
      Even worse, it's totally untrue! Don't believe me? Feed a rabbit a string. After a while it will hang out of both ends. It's clearly a deformed torus, not a deformed sphere. This is the exact opposite of what Perelman proved!

      Take that, non-mathematical pundits and biologically unaware mathematicians!
    • by geekoid (135745)
      As long as you keep saying 'maths ' it's not going to get any better.
  • by sankyuu (847178) on Tuesday February 27, 2007 @04:19AM (#18164344) Journal

    Wisconsin-Madison have pieced together an explanatory framework that for the first time illustrates what mock theta functions are, and exactly how to derive them.

    I resent that mockery, you insensitive... oh, I thought you said deride.
  • by dino213b (949816) on Tuesday February 27, 2007 @07:31AM (#18165316)
    It's easy - just have a good writer release a few Bible Code books.

    - Bible code for children
    - Bible code for dummies
    - Bible code howto
  • by d0n quix0te (304783) on Tuesday February 27, 2007 @07:37AM (#18165358)
    India has had a long standing history in mathematics much of which predates that in the Islamo-christian tradition.

    Formal mathematical schooling among Brahmins was particularly important among people in Tamil Nadu and Kerala, two of the sea-faring communities in India. Ramanujan belonged to the Iyengar tradition of mathematics (although many people related Iyengars to Yoga...) from Tamil Nadu.

    Among other contributions of Indian mathematics include

    Pre-ACE

    The decimal system and the number zero
    Inductive reasoning and the inductive method
    Fractions
    Equations
    Mathematical tables
    Binomial theorem
    Pythogorean theorem
    Area calculations
    Conic sections
    Irrational numbers
    Boolean Logic
    Null Sets
    Transformations and recursions
    Number theory
    Trignometry
    Formal language and grammar theory

    Post ACE (pre renaissance)

    Cubic and Quartic Equations
    Pi as an infinite series
    Geometric and Harmonic series
    Series theory
    Permutations and combinations
    Cardinal numbers
    Transfinite numbers
    Set theory
    Fibonnacci series
    Derivative
    Rolles theorem
    Differentiation
    Limits
    Differential and integral calculus (predating Leibnitz and Newton by 200 years) ......
    For a laundry list see

    http://en.wikipedia.org/wiki/Indian_mathematics [wikipedia.org]

    Some of these brahmanic schools were far more advanced than European schools. Ramanujan had good schooling from a tradition steeped in mathematics. He was Europe's first direct exposure (as opposed to published books that were translated) to Indian mathematics hence the cult status.

    Imagine a Narayana Pandit or a Chitrabhanu from the Kerala schools in Europe in 1500 AD spouting Calculus and Reimann's theorem (two well known theorems in India at that time)... they too would have been declared as geniuses.

    -S

    • by torokun (148213)
      What in the world does ACE mean?
      • ACE = After Christian Era
        BCE = Before Christian Era

        Basically a politically correct way of saying A.D (Anno Dominus = Year of the Lord) since not everyone believes in the christian "Lord"

        -S
    • by ideonode (163753)
      A recent In Our Time radio broadcast [bbc.co.uk] covered the impact of Indians on the history of mathematics. Rather interesting listening, as are all In Our Times.
    • by kestasjk (933987) *
      What are we supposed to get out of this? Who cares what nationality this guy was?

      Isn't it insulting to say "I know what you're thinking, but Indians aren't stupid! They're as smart as we are! Look at all this stuff they've done!" ?
      Or maybe it's "Look at all these great Indians! I'm an Indian just like them, therefore I'm one of them!" Except being a good mathematician has nothing to do with your race, there's just not that big of a difference between races.

      If this guy had a beard would you post all t
      • Re: (Score:3, Insightful)

        by ChrisMaple (607946)
        Some nationalities (and more importantly, some cultures) have a history of making contributions to various aspects of civilization out of proportion to their numbers. It is both interesting to find these correlations and important to find cause-and-effect relations if they exist. Getting annoyed because people point them out, and flaming them, is not a contribution.
  • Here is a link to a more informative article: http://www.telegraphindia.com/1050411/asp/knowhow/ story_4560152.asp [telegraphindia.com]
  • Did you notice the Google ads on this page? "Sympathy gifts," "Living wills," "Estate planning." I looked aroung the page to figure out why Google picked these ads. It must be because of the word "deathbed" in the article!

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