What Is Research?

November 16, 2008

Wikipedia — side-effects

In a recent blog post, Nicholas Carr talked about the “centripetal web” — the increasing concentration and dominance of a few sites that seem to suck in links, attention and traffic. Carr says something interesting:

Wikipedia provides a great example of the formative power of the web’s centripetal force. The popular online encyclopedia is less the “sum” of human knowledge (a ridiculous idea to begin with) than the black hole of human knowledge. At heart a vast exercise in cut-and-paste paraphrasing (it explicitly bans original thinking), Wikipedia first sucks in content from other sites, then it sucks in links, then it sucks in search results, then it sucks in readers. One of the untold stories of Wikipedia is the way it has siphoned traffic from small, specialist sites, even though those sites often have better information about the topics they cover. Wikipedia articles have become the default external link for many creators of web content, not because Wikipedia is the best source but because it’s the best-known source and, generally, it’s “good enough.” Wikipedia is the lazy man’s link, and we’re all lazy men, except for those of us who are lazy women.

This is an important and oft-overlooked point: when saying whether something is good or bad, we need to look not just at the benefit it provides, but also at the opportunity cost. In the case of Wikipedia, there is at least some opportunity cost: people seeking those answers may well have gone to the “specialist sites” instead of to Wikipedia.

Of course, it’s possible to argue that specialist sites of the required quality do not exist, but it can again be argued, in a counter-response, that specialist sites would have existed in greater number and greater quality if Wikipedia didn’t exist, or at any rate, if Wikipedia weren’t so much of a default. It might be argued, for instance, that of all the free labor donated to Wikipedia, at least a fraction of it could have gone into developing and improving existing “specialist sites”. As I described in another blog post, the very structure of Wikipedia creates strong disincentives for competition.

Wikipedia, Mathworld and Planetmath

In 2003, at a time when I was in high school and used a dial-up to connect to the Internet, I was delighted to find a wonderful resource called Mathworld. I devoured Mathworld for all the hundreds of triangle centers it contained information on, and I eagerly awaited the expansion of Mathworld in other areas where it didn’t have much content. I was on a dial-up connection, so I saved many of the pages for referencing offline.

Later, in 2004, I discovered Planetmath. It wasn’t as beautifully done as Mathworld (Planetmath relies on a large contributor pool with little editorial control, as opposed to Mathworld, that has a small central team headed by Eric Weisstein that vets every entry before publication). But, perhaps because of less vetting and fewer editing restrictions, Planetmath had entries on many of the topics where Mathworld lacked entries. I found myself using both these resources, and was appreciative of the strengths and weaknesses of both models.

A litte later in the year, I discovered Wikipedia. At the time, Wikipedia was fresh and young — some of the policies such as notability and verifiability had not been formulated in their current form, and many of the issues Wikipedia currently faces were non-existent. Wikipedia’s model was even more skewed towards ease of editing. It didn’t have the production quality looks of Mathworld or the friendly fontfaces of Planetmath, but the page structure and category structure was pretty nice. Yet another addition to my repository, I thought.

Today, Wikipedia stands as one of the most dominant websites (it is ranked 8 in the Alexa rankings, for instance). More important, Wikipedia enjoyed steady growth both in contributions and usage until 2007 (contribution dropped a little in 2008). Planetmath and Mathworld, that fit Nicholas Carr’s description of “specialist sites”, on the other hand, haven’t grown that visibly. They haven’t floundered either — they continue to be at least as good as they were four years ago, and they continue to attract similar amounts of traffic. But there’s this nagging feeling I get that Wikipedia really did steal the thunder — in the absence of Wikipedia, there would have been more contributions to these sites, and more usage of these sites.

The relation between Wikipedia and Planetmath is of particular note. In 2004, Wikipedia wasn’t great when it came to math articles — a lot of expansion needed to be done to make it competitive. Planetmath released all of its articles under the GNU Free Documentation License — the same license as Wikipedia. Basically, this meant that Wikipedia could copy Planetmath articles as long as the Wikipedia article acknowledged the Planetmath article as its source. Not surprisingly, many of the Planetmath articles on topics that Wikipedia didn’t have were copied. Of course, the Planetmath page was linked to, but we know where the subsequent action involved with “developing” the articles happened — Wikipedia.

Interestingly, Wikipedia acknowledged its debt to Planetmath — at some point in time, the donations page of Wikipedia suggested donating to Planetmath, a resource Wikipedia credited for helping it get started with its mathematics articles (I cannot locate this page now, but it is possibly still lying around somewhere). Planetmath, on its part, introduced unobtrusive Google ads in the top left column — an indicator that it is perhaps not receiving enough donations.

Now, most of the mathematics students I meet are aware of Mathworld and Planetmath and look these up when using the Internet — they haven’t given up these resources in favor of Wikipedia. But they, like me, started using the Internet at a time when Wikipedia was not in a position of dominance. Will new generations of Internet users be totally unaware of the existence of specialist sites for mathematics? Will there be no interest in developing and improving such sites, for fear that the existence of an all-encompassing behemothing “encyclopedia” renders such efforts irrelevant? It is hard to say.

(Note: I, for one, am exploring the possibility of new kinds of mathematics reference resources, using the same underlying software that powers Wikipedia (the MediaWiki software). For instance, I’ve started the Group properties wiki).

The link-juice to Wikipedia

As Nick Carr pointed out in his post:

Wikipedia articles have become the default external link for many creators of web content, not because Wikipedia is the best source but because it’s the best-known source and, generally, it’s “good enough.” Wikipedia is the lazy man’s link, and we’re all lazy men, except for those of us who are lazy women.

In other words, Wikipedia isn’t winning its link-juice through the merit of its entries; it is winning links through its prominence and dominance and through people’s laziness or inability to find alternative resources. Link-juice has two consequences. The direct consequence is that the more people link to something, the more it gets found out by human surfers. The indirect consequence is that Google PageRank and other search engine ranking algorithms make intensive use of the link structure of the web, so a large number of incoming links increases the rank of a page. This is a self-reinforcing loop: the more people link to Wikipedia, the higher Wikipedia pages rank in searches, and the higher Wikipedia pages rank in searches, the more likely it is that people using web searches to find linkable resources will link to the Wikipedia article.

To add to this, external links from Wikipedia articles are ignored by search engines, based on Wikipedia’s settings. This is ostensibly a move to avoid spam links, but it makes Wikipedia a sucker of link-juice as far as search engine ranking is concerned.

In addition, the way people link to Wikipedia is also interesting. Often, links to Wikipedia articles do not include, in the anchor text, any information that the link goes to the Wikipedia article. Rather, the anchor text simply gives the article name. This sends the message to readers that the article on wikipedia is the first place to look something up.

Even experienced and respected bloggers do this. For instance, Terence Tao, a former medalist at the International Mathematical Olympiad and a mathematician famous for having settled a conjecture regarding primes in arithmetic progressions, links copiously to Wikipedia in his blog posts. To be fair, he also links to articles on Planetmath, and papers on the ArXiV in cases where these resources offer better information than the Wikipedia article. Nonetheless, the copious linking suggests that it is likely that not every link to a Wikipedia article is based on the Wikipedia article genuinely being the best resource on the web for that content.

What can we do about it?

Ignoring a strong centripetal influence, such as an all-encompassing knowledge source, does not make us less immune to its pull. There is a strong temptation to use Wikipedia as a “first source” for information. To counter this pull, it is important to be both understanding of the causes behind it and critical of its inevitability.

The success of a quick reference resource like Wikipedia stems from many factors, but two noteworthy among them are desire to learn and grow and laziness. Our curiosity/desire to learn leads us to look for new information, and our laziness prevents us from exerting undue effort in that regard. Wikipedia capitalizes on both our curiosity/desire to learn and grow and laziness in its readers (quick and dirty access to lots of stuff immediately), contributors (easy edit-this-page), linkers (satisfying reader curiosity by providing web links, but using Wikipedia instead of others thanks to laziness). Wikipedia is what I call a “pinpoint resource” — something that provides one-stop access to very specific queries over a large range of possibilities very quickly.

For something to complete with Wikipedia, it must cater to these fundamental attributes. It must be quick to use, provide quality information, and encourage exploration without making things too hard. It must be modular and easily pinpointable. This doesn’t necessarily mean that everything should be modular and easily pinpointable — there are other niches that don’t compete with Wikipedia. But to compete for the “quick-and-dirty” vote, a site has to offer at least some of what Wikipedia offers.

Of course, one of the questions that arises naturally at this point is: isn’t Wikipedia “good enough” to satisfy passing curiosities? I agree that there is usually no harm in using Wikipedia — when compared with ignoring one’s curiosity. But I emphatically disagree with the idea that we cannot do better with dealing with the passing curiosities and desires people have to learn new stuff and teach others, than funnel it through Wikipedia. Passing curiosities can form the basis of enduring and useful investigations, and the kind of resource people turn to at first can determine how the initial curiosity develops. For this reason, if Wikipedia is siphoning off attention from specialist sites that do a better job, not just of providing the facts, but of fostering curiosity and inviting exploration, then there is a loss at some level.


November 15, 2008

Small fry or big fish?

Filed under: Uncategorized — vipulnaik @ 11:17 pm

How do we acquire the practice necessary to become perfect? This is a very general question, and I’m considering the question with regard to mathematical skill. Suppose my aim is to become a mathematics researcher. How do I acquire the practice necessary to do mathematical research?

In this blog post, I consider a specific trade-off: is it better to develop practice and intuition by considering a large number of simple problems, or simple things, or is it better to go after some of the big fish?

I’m personally in the “small fry” camp, and I’ll explain my reasons here.

Building a balanced repository of experience

As I mentioned in a previous post, the main advantage of experience is the presence of a large repository of knowledge that allows for more efficient pattern identification. I’ve been studying group theory for many years now, and thus, when confronted with a question in group theory, I am likely to either have seen the question before or at least have some meaningful closely related past experiences. Within a few years, by which time I should hopefully have explored more of the subject, I should be even better at tackling new questions in the subject.

A large repository of experience depends on knowing a lot of small facts here and there. These facts are connected in different ways. By tackling the small fry, either randomly or systematically, I am likely to cover many of these small facts. If I concentrate on the big fish, I may get to know very well all the small stuff that leads to that big fish, but many other things may have poor foundations.

Here’s an analogy. Suppose I want to explore the city of Chennai (Chennai is an Indian city, formerly known as Madras). One approach (the big fish approach) may be to identify a particularly difficult-to-locate spot in the city, and decide to reach that spot, with no help whatsoever. So I start walking around the streets of Chennai, going into some blind alleys and getting stuck at times, but I soon find my way and reach my destination. I go through a lot of parts of the city but my eyes are always seeking the destination point. Another approach would be to explore a new street each day. I might do this with an explicit ordering of the streets to explore, or I might do it in a pseudo-random way: each time I pick a new street that is slightly beyond the area I am currently familiar with. In the big fish approach, I might get to know the streets that lead to my destination very well, and I may also get to know very well the streets that misled me for a long time. In the small fry approach, I have a little knowledge of a much larger number of streets, but there is no overarching organizing framework to my knowledge, and no single goal.

The thrust of my argument is that the big fish approach leads to a less balanced and comprehensive repository of experience, as opposed to the small fry approach, leading to less preparedness for later research life. This is particularly important keeping in mind that most of us aren’t great at predicting what research problems we will work on a few years from now — so having a broader base makes more sense.

An argument for big fish: a more authentic research experience

There are at least a few ways in which the big fish approach seems more appealing. Because there are bigger fruits and bigger fish at the end, it can be more motivating and inspiring than simply doing a random collection of things on different days. I don’t disagree that big fish can be more exiciting to fish for, and juicier and larger fruits can be more exciting to reach for. In some cases, the greater excitement of something bigger can make up for the lack of breadth that may result from chasing it too hard.

But it is a mistake to look down upon, or sneer at, the tackling of small problems that aren’t aligned towards a specific big goal. In a sense, tackling a host of small problems without an overarching agenda is harder and more challenging than going out after a clearly defined problem. This is analogous to the fact that it may be a greater indication of inner strength to wander aimlessly rather than stride briskly and purposefully. At the same time, tackling small problems can be more rewarding, because it reduces the extent of commitment to a particular big problem and increases the amount of serendipity.

My final argument is that it is more efficient and less risky to consider and tackle a large number of small problems, or even settle wrinkles in many little definitions, than to try to prove big things. Just as we’re taught to diversify monetary investments in order to get a better average rate of return and be less prone to extreme risks, diversifying the problems being worked on is a good strategy against ruin. Some might view this as a “thinking small” attitude, citing people such as Andrew Wiles and John Nash who tackled and successfully solved hard problems. But there are a lot of people who tackled hard problems and did not solve them — and when you start out, you don’t really have an idea which camp you’re in (if you’re really really sure you can get the big fish, reading this blog post isn’t going to change your mind).

How do small fry and big fish compare with the theory versus practice divide?

There is a dichotomy between the theory builders and problem solvers in mathematics (something I alluded to earlier). Theory-building, a la Grothendieck, involves building general theories, while problem solving tackles specific problems.

The dichotomy between small and big is, as far as I understand, largely independent of the dichotomy between theory-building and problem-solving. Both theory-building and problem-solving can be done in minor incremental steps as well as in major, directed steps. Andrew Wiles, for instance, wanted to solve a problem (the so-called Fermat’s last theorem) and spent years doing that — his intention wasn’t to solve a theory. On the other hand, most problem-solvers are tackling separate isolated problems without the aim of making it to the national newspapers. Similarly, some theory-builders like Grothendieck seek to alter the foundations of geometry and mathematics. Others add in a few definitions here and there, introduce new symbol calculi or formalisms, and adapt past ideas to increase the strength of existing theories.

The difference between theory-building and problem-solving possibly lies with the inherent risks associated. With reasonable levels of rigor having entered mathematics, few published mathematical results have errors. Theory-builders, who are working incrementally based on what is known, are less likely to develop wrong theories, but run greater risks of being irrelevant. Problem-solvers, who are working on problems that others have identified as important, are more likely to do relevant work, but they are also more likely to not get anywhere or not succeed at all.

Can small fry lead to big fish, and vice versa?

Can a person chasing small fry end up netting the big fish? Can people chasing the big fish end up getting good at all the small stuff?

Paradoxically, it seems that the less efficient one is at chasing the big fish, the more one may learn about the small stuff. This follows from the I learn more when I do it wrong phenomenon, and is conditional to having a continued (and misplaced) sense of optimism on getting it right the next time. Chasing big fish, specially those totally out of reach, may therefore be an appealing strategy to learning more small stuff through self-deception.

Can a person chasing small stuff land a big fish? This is unlikely, and at any rate, a person chasing small stuff is unlikely to have the multiple insights needed to land the big fish. Nonetheless, the person may, without aiming to do so, develop some incremental insights that make the big fish look a little smaller for other people. Thus, even while a single individual who decides not to try for the big stuff foregoes the opportunity to hit it big, the mathematical community as a whole may not be adversely impacted in terms of the number of big problems it gets solved.

Big fish — later or earlier in life?

It would be folly for me to argue that people who spend many years tackling big problems are doing a disservice to mathematics by spending their time inefficiently. Tackling the big fish has positive externalities beyond the mathematical value it creates. First, it generates buzz about mathematics outside the mathematics community, and provides meat to popular math writers who can help entice more people to the subject. It is hard to entice kids into math by telling them that they can do a little more stuff every day and become cogs in the mathematical wheel. Big conjectures carry the romance of jackpots of lottery tickets.

Second, it makes the mathematical community bolder and braver and more confident of its abilities when a long-standing conjecture is resolved. Apart from the specific techniques developed to solve the conjecture, the idea that conjectures that have withstood assault for so long have yielded to perseverance and hardwork speaks to that ideal we so often want to believe in and yet keep doubting: “There is nothing that fails to yield to intelligence, hardwork, and sheer perseverance.”

Third, and perhaps most importantly, it saves other less talented people the agony of trying to prove the conjecture. With Wiles having settled Fermat’s last theorem, there are fewer people spending hours trying to settle it in the hope of winning fame.

Nonetheless, the question remains: when trying to build one’s research skills and abilities, is it a good idea to tackle relatively bigger fish? Here, I think the answer is no. Bigger fish may be incorporated as further inputs for random exploration, but a systematic attempt to go after a big fish is likely to lead nowhere.

Intuition in research

Filed under: Thinking and research — vipulnaik @ 10:06 pm

I recently read a couple of books by Gary Klein: Sources of power and The Power of Intuition. Klein is a decision researcher who started work in the 1980s studying high-stakes high-pressure decision-making. His research team began by studying how firefighters make high-stakes decision in the face of severe time constraints, incomplete information, high pressure, and unclear goals. The research team found that the firefighters rarely compared multiple options. Rather, when faced with a particular situation, the experienced firefighters typically found an immediate first response, simulated it mentally, and executed it if the simulation seemed to work fine.

While there were certain situations where the firefighters rejected one course of action and selected another, Klein found that the courses of action were usually considered sequentially: the first course of action was simulated mentally, and if the firefighter sensed it to be good, he or she executed it. If the course of action didn’t seem good, another course of action was simulated. The hallmark of experienced firefighters was their ability to pick out a good first option and execute it.

According to Klein’s book, his findings contradicted the formal decision-making strategies considered good by decision researchers at the time. Many decision researchers warned people against using their intuitions, which were prone to being misleading, and to instead consider multiple options and compare them across multiple dimensions. In recent years, there has been greater acknowledgement of the infeasibility of comparing multiple options as well as the advantages of strengthening intuition in order to pick good first options.

Through studies of firefighters, marines, NICU nurses, and many other decision makers, Klein came up with a model he called the Recognition-Primed Decision (RPD) model. The core idea is that the repository of experience that a worker builds creates certain templates, and when the worker is thrust into a new situation, the new situation is matched against these templates. If a good match is obtained, the course of action suitable for that template is tried. The rich repository of experience helps with the initial recognition of the appropriate template, with the mental simulation that follows, and with collecting feedback once a course of action has been sought.

For instance, a firefighter, over the years, becomes sensitive to different cues such as the smell, floor temperature, room temperature, way in which the fire is spreading, and numerous other small indicators. By gauging these cues, the firefighter subconsciously develops a “story” around how the fire developed and what the priority should be (rescue people, douse the flames, call for more help). Similarly, nurses in intensive care units for children (The NICU nurses) develop a repository of experiences on subtle cues as well as combinations of cues that ill children provide. An experienced nurse can thus size up a situation based on the many cues he or she (usually, she) sees, and develop a story that immediately suggests a next course of action.

The emphasis is thus not on analysis but on building a story, and judging the story by how well it fits the facts, where the extent of the fit is determined by feedback from past experience.

Do similar principles apply to research?

In terms of speed, research is the opposite of firefighting. For a firefighter, the situation is live and demands immediate action, with high stakes and usually very immediate feedback about success (either the flames get doused or they don’t). Research, on the other hand, is a slow process, with very little riding on decisions made on the spur of the moment and very rare opportunities for instantaneous feedback. If the research problem I pick is too hard for me, I don’t get to know the consequences and feel the pain for quite a bit of time. I might suffer the delusion that I am making steady progress on the problem and figure out that it is too hard after several years of trying.

Given the obvious differences, it is natural to be suspicious of an assertion models that help with high-stakes decision-making are prima facie suitable for researchers. However, I make the case here that intuition is important in research, albeit in a different way.

In general, intuition is important in situations where either the information explicitly and clearly available is inadequate, or the effort needed to process all this information is infeasibly high. In a firefighting situation, the information available is inadequate at the time the decision needs to be made, even though the story usually does become clearer in a short while. When working on a research problem, we again have a gap: the information available (as to whether or not I should work on the problem) is inadequate at the time I start work on the problem, though it is likely to become more clear once I have worked on the problem. The difference is in the time scale. But in both cases, there is inadequate information at the time of decision-making.

Strengthening one’s intuitions

Klein’s book on the power of intuition offers many concrete suggestions on strengthening intuitions. Klein begins with the (obvious) observation that practice improves intuition. More importantly, he identifies two aspects: frequency of exposure to situations, and feedback that helps in correct model-building. Frequency alone is not enough.

Of course, there are obvious problems with providing practice for emergency situations — the kind that shouldn’t occur anyway. How do novice doctors get practice in performing critical surgical procedures and making critical medical diagnoses, without risking the lives of patients more than necessary? Atul Gawande, a Massachussetts surgeon, discusses these issues in his bestselling book Complications, where he points out that doctors have a learning curve, and this necessitates that some patients receive substandard care because they get treated by residents and new doctors rather than more experienced ones.

Similarly, how do firefighters get experience fighting fires? Again, this experience is provided through apprenticeship and observing other firefighters — the less experienced firefighter accompanies the more experienced firefighter and sees the more experienced firefighter make the critical decisions while offering support.

Apprenticeship is one approach, but it may be expensive, and it is best complemented with other approaches that are less expensive. Other approaches (some of which are mentioned in Klein’s book) include simulation and training exercises, where some of the features of the real experience are captured through simulation. An example is flight simulation. In addition, there is the crucial aspect of experienced people documenting their experiences, and sharing these experiences with others, so that the many little nuggets of wisdom get passed on.

Now let’s move from high-stakes, instantaneous decision-making to the world of research. The same ideas seem to apply: newer researchers generally have less experience, and they learn more through apprenticeship and through interaction with more experienced people. The timescales are, of course, very different. A Ph.D., which is like an apprenticeship under a guide (the thesis advisor) may take anywhere between three and eight years. Even after completing a Ph.D., researchers generally tend to work under the guidance and tutelage of more experienced people before fleshing it out totally on their own.

So the same questions seem to apply: what are the skills that experienced researchers have, that their less experienced counterparts may lack? And how can new researchers pick up these skills faster?

The importance of metacognition

Klein claims that people with experience and expertise in a topic not only have a richer repository of experience to draw upon — they are also capable of analysis of their own thoughts on the subject. In other words, they not only know more of the territory they are exploring, they also know more about their own tendencies in exploring the territory. For instance, an experienced athlete not only knows how to run long distances, but is also aware of how it will impact his or her mood and energy levels, and can thus plan ahead accordingly. Awareness of their own proclivities thus helps experts plan around as well as exploit their human strengths and weaknesses.

Thus, a person who has just learned a subject, say group theory, and doesn’t have much experience with it, may not be able to distinguish between two problems in group theory in terms of their level of difficulty without attempting the two problems. A more experienced person may be able to look at both problems and get a feel as to which one is likely to be harder, even without attempting either problem. This comes from a stronger intuitive grasp of the territory, including its highs and lows.

Of course, there are times when expertise of this sort can be misleading, because it may lead the more experienced person to be less adventurous in exploring some things based on false preconceptions. This needs to be watched out for.

The upshot is that there are certain skills: a clearer understanding of the specific territory of the subject being researched, as well as a broader intuitive feel of the subject that helps one decide what direction to go in. The question is: can these skills be developed faster? Are there any obvious ways of doing this?

Some concrete suggestions

The goal is that new researchers should get a sense of how to think about specific problems, be better at metacognition (understanding their own thought processes) and get a broader intuitive idea about the direction that certain approaches will lead to. This should not be done at the cost of diversity in thinking — new researchers should not be handed down the right way of thinking about something. I discuss here some interesting suggestions to help improve the intuitions of new researchers.

  • The two-problem faceoff: This is a game of sorts, where two problems are presented to the novice researcher. Neither of these problems is trivial; one of them, however, is easy (i.e., it can be solved by the researcher) and the other is hard (i.e., it either requires a lot of ingenuity or some new machinery). The researcher has to decide which of the problems to try, and try that, and succeed.

    This faceoff game has some interesting aspects. First, researchers are forced to develop their intuitions not just on how to solve a problem, but also on how to pick a problem to solve among a collection of two. Thus, researchers are forced to consider metacognitive questions: which path should I tread?, and to look ahead and predict what will happen. Second, it may actually turn out that the purportedly harder problem is the one the researcher picks and actually does solve (perhaps coming up with an easier solution). Perhaps this indicates that the new researcher is particularly good with problems of that kind.

  • What came first?: Here, a researcher is presented with two proofs of the same theorem, both of which arose at different historical points, and is asked to do a comparison: which proof came first? Which one is more useful? Which one would be the kind of proof you’d come up with?
  • Spotting relations, thinking creatively: New researchers should constantly confront reflective questions regarding different aspects of the work they are doing or learning about. For instance: what does the statement of this result tell me? What does the structure of proof tell me? Are there corollaries of the statement? Are there other related statements? Are there other statements whose proof follows the same structure? Can the proof idea be transferred to a totally new subject? Can I come up with similar-sounding statements that are false?

I have been exploring some of these possibilities for spotting relations and encouraging reflective thinking that builds intuitions for some time. I’ve implemented some of these ideas in the structure I’m using for the Group properties wiki. For instance, see this page about a property of normal subgroups, or this page about a cute fact regarding unions of two subgroups of a group.

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