Готовый JWH-018 высшего качества – заказать на catchop.ru ! Абсолютно к каждому (крупному или мелкому) заказу мы прикрепляем пробник одной нашей продукции БЕСПЛАТНО! При желании получить пробник конкретного продукта, пишите его название в примечании к заказу! ]]>

It’s actually quite a good moment to bring the topic up as the Tricki really is about to appear now. And there are various labelling systems such as the one you propose. More on this soon.

]]>Sorry for bringing this topic up.

One idea would be to find someone willing to extend mediawiki (maybe there’s such an extension) allowing you to add labels to each trick, plenty of labels. Say you have a trick on an inequality dealing with limits often being used in real analysis (say). Then one could label it as “analysis”, “real analysis”, “limit” and “inequality”.

So if a person isn’t sure about what trick he or she is looking for (or even if there is some sort of trick that could help out with the problem), then he or she could specify some labels to search for and getting a list of tricks related to these labels. And who knows, there might be something there to help out solving the problem!

]]>As noted, the naming of tricks is a vexing problem. One solution is to **not worry** about the names!

Rather, **link** the trick *to and from* the **applications**. Think of how one identifies unnamed concepts in general: one *describes* them and gives *context*. This allows one to *find* a trick: “oh, it’s the trick you use to show that the uniform limit of continuous functions is continuous – let’s look up uniform limit!” (in this case, the epsilon/3 trick) (yes, I’m eliding the assumption of uniform continuity/local compactness). Similarly, if one thinks: “In Theorem Y, one uses the same trick one uses in Theorem X”, one can look up theorem X to link to this same trick.

Some form of this is already present in Wikipedia: obstruction theory is both a specific term in algebraic topology and geometric topology, and a general trick for defining invariants (usually cohomological) for various problems – and it’s also v. similar to the Hasse principal and sheaf cohomology, though these tend to go by “local-global” problems, not “obstruction theory”.

Ultimately, while good names are useful, they are just a tag: concepts are defined by their content and their connections.

]]>Have you considered Wikibooks? The Mathematics bookshelf could certainly accommodate this project, and there’s no setup required.

In your desiderata, the key conflict is between **collaboration** and **quality control,** and these are simply **irreconcilable:** 2 extremes that work in their way are the anarchy of Wikipedia or the order of a collection of essays, as in PCM.

The problem is that the burden of reviewing every change is onerous, and that authors come and go (or have time for a few years, then not after).

I would encourage using a public, freely licensed wiki, like Wikibooks, and **trusting that wikis work** – the quality is surprisingly good, because it’s trivial to roll back vandalism, and it’s rare for vandals to care enough to vandalize obscure subjects.

You may consider, if you are not familiar with it, the history of Nupedia, which tried such a peer-reviewed system of high quality articles, and got nowhere (rather, not very far). Especially for so varied a topic as “mathematical tricks”, allowing all and sundry to contribute can very much help.

Further, using an existing project (notably Wikipedia & co.) attracts contributors – wouldn’t it be great for Wikipedia entries on various topics to link to the tricks used, within the same project?

To address quality concerns, taking a particular revision and blessing it as “good quality”, as in the

Wikipedia Version 1.0 project or mathematics assessment, works quite well.

So I’d urge you to consider a publicly editable wiki (esp. Wikibooks) – they really work far better than one may expect.

Best,

Nils

I just happen to find a question for that tricki wiki:

Whether is there a common-shared refference system for that tricki wiki? Similar to that of Mathscinet of ams math review

It will be a basic instrument for a mathematical website.

]]>Knol

MathWeb

MyWikiBiz

PlanetMath

ProofWiki

Jon Awbrey

]]>Quality control requires collaboration among responsible contributors who know the subject matter and who respect the knowledge and diverse points of view that other bring to bear on the subject. Wikipedia has nothing to do with that.

The wikioid projects of my acquaintance that approach these goals are Knol, MathWeb, MyWikiBiz, PlanetMath, and ProofWiki.

Jon Awbrey

]]>gets read somehow 😀

· First thing I thought about the subject: we NEED to get organized! In particular, we need to open a forum about the Tricki: the planification and organization of such a feat can’t be done from zero-level, we must keep adding layers of data and information distribution until we arrive to the Wiki, and it seems to me that the first step should be a forum (blogs were the 0-step).

· About the difficult and fascinating subject of naming and identifying every trick (btw, this should be not a thread, but a whole subforum on the forum!):

a) I suggest we “impose” the official name by being the de facto most authoritative source for math tricks (as it happens with OEIS or Wikipedia). Once we have launched the project and having such skilled, expert, famous people on it, authority will follow fast. People will tend to use the names we give to them just because the rest of the people will want to look at them at our Wiki (here, when I say “we” and “our” I refer to everyone who wants to collaborate, of course!)

b) The name should be composed of (at least) three components:

1) An informal, catchy name.

This would be the best known name for the trick, and would methaphorically refer to its use or main characteristics, or to any circumstances around it or its first apparition, like theorems tend to be named “lately” (hairy ball theorem, sandwich theorem, rising sun lemma, egregius theorem, etc).

The word “TRICK” (or any other to be decide) should be the last name, just to let people know what we are talking about.

An invented example: The “1-is-not-so-trivial” trick -> In your main FORMULA [specialized term to be defined within the WIKI], substitute 1 for any fancy SUBFORMULA that does equal to 1, like sin^2(x)+cos^2(x), or partitions of unity, and REARRANGE the main formula (i.e. separate in “components”, simplify, glue, etc).

2) A technical name within an a priori specified naming rule system (to be defined yet) that already implies some categorization, something like the naming of new biological species. For example: FORMULA SUBSTITUTING+ (where Formularity refers to the fact that the trick is about formulas, Substituting to the obvious fact, and the “+” means that the substitution actually gets more complex, and not less (-)).

Another invented categorization family name could be Equation DifferentiatingN (take your equation and differentiate it N times)

3) A series number. Just to catalogue the trick and make internal database work with it (but also to have it uniquely identified!)

c) To solve the search problem, a multitag system looks like the most reasonable system to me, but the selection of the tags is critical. I think we should have two kinds of tag:

predetermined distinguished tags, and configurably non-distinguished tags. The first ones would be created with the project (always subject to suggestions and changes) and would be things like:

WHERE (in which branchs of math)

IN (in which type of structures)

FROM (hypothesis and elements you need to have for the trick to make sense; i.e. the trick starting point)

TO (the generic result you accomplish by applying the trick)

WITH (“the how”, technical terms that describe key parts of the process)

NAME (informal name)

CATEGORY (formal name),

SN (series number)

AUTHOR (the creator or best-known user of the trick, if it’s known)

REFERENCE (well-kown book, paper or source where it’s used in an important manner)

USES (well-known uses of this trick, theorems where it is important, etc)

The non-distinguished tags would all fall to the same category OTHERS, as a list, just like the tags in YouTube.

Not all the tags need to be filled, and some can be multivalued.

An example:

SN 0000

NAME “The connection trick”

CATEGORY Set Identificating Connected

WHERE Topology, Algebraic Topology

IN Topological spaces, Manifolds

FROM Connected set, subset

TO Subset is Whole Set

WITH Connected Sets Open-Closed Characterization

USES

OTHER Standard, Basic

Did you know what trick it was? The informal description (which I think it should be the first thing you get after the tags) would be “In a connected set C there are no subsets that are open and closed at the same time, other than the empty set and C. Therefore, if you want to prove that your subset D of your connected set S is in fact the whole S, just prove that S is open, closed, and non-empty”

·Other basic tricks I thought just know (please forgive their triviality, I’m just a math student yet):

* Smart application of Hölder inequality

* Partitions of unity

* Linearization in the axioms non-associative algebras

* Zorn’s Lemma and, similarly, Ascending Chain Condition and Maximal Condition for noetherian modules

* If in a Normed Space you want to prove that x is 0, just prove that its norm is 0

* If you want to observe the ideals that contain I, then study A/I. If you want to study the ideals contained in the prime P, study the localization A_P.

* If you are working in a bound that involves exp(x) maybe you just need to work with (1+x). In general, if you are working in a bound that involves an analytic function, it may suffice to work with the first n terms of its Taylor Series (where n=0,1,…,5).

Well, I adore to give more ideas and to get involved with this great project, but we definitely need a forum! (If it isn’t implemented yet!). The other way all this effort will eventually come to waste…

Regards!

Jose Brox

In abstract settings the determinant trick doesn’t show up explicitly so much; instead one appeals to something like Nakayama’s lemma. Or one can simply define “integral over R” to mean finitely generated as a module over R. But if you’re doing explicit computations then the determinant trick is still handy to know.

As for integrals attackable by Feynman’s method, there is an excellent collection of examples in the Wikipedia article on differentiating under the integral sign. However, it’s a little dry, and could benefit from a better presentation, that helps students figure out how to come up with these kinds of arguments themselves. The (sin x)/x example would be a good one to start with. Clearly, the x in the denominator is annoying, so one wants to cancel it out. One might first try differentiating (sin xt)/x with respect to t, but this introduces convergence problems. That’s why one chooses exp(-xt) (sin x)/x, so that you still cancel out the x in the denominator, but now you needn’t worry about convergence (since exp(-xt) goes to zero so fast).

While we’re on the topic, here are two tricks that I would like someone to explain to me. The first trick is the use of partitions of unity. Some basic local-to-global arguments on manifolds can be proved using partitions of unity, while others require more subtle methods such as sheaves and cohomology. When can one expect a partition-of-unity argument to work and why?

The second trick is one I encountered years ago when taking a course in algebraic number theory from Goro Shimura. Shimura frequently used the following lemma. Let F be a number field and let J be its ring of integers. Given any integral ideal A and any fractional ideal X, there exists a nonzero element z in F such that zX is an integral ideal and zX + A = J. This lemma can be used to give quick proofs of a number of results, e.g., that every fractional ideal can be written in the form xJ + yJ for some x and y. However, I never fully understood why this lemma was so useful or when one could expect it to be invoked.

]]>Incidentally, your idea of a meta-trick is one that we already plan should be a major feature of the Tricki. As well as “first-level articles” that describe specific methods for attacking specific kinds of problems, we will strongly encourage higher-level articles that help you to classify your problem appropriately and then direct you to relevant Tricki pages. I will have much more to say about this in my post on the subject.

Meanwhile, I’d love to know what the determinant trick is that you talk about, and would also like to see a few examples of integrals that can be attacked by Feynman’s method. If you were prepared to write a couple of articles that could be included in the Tricki the moment it is launched, that would be great. And if you are someone else reading this comment and have a trick that might conceivably make a suitable article, then it does make a suitable article. If you haven’t got time to polish it up into the perfect article, write an imperfect one and let others add to it. (I’m hoping that there will be a facility for explaining in what way an article could be improved — e.g. by the addition of more examples, or a clearer discussion of exactly when the trick can be used.)

]]>The first thing that came to my mind was the “determinant trick” (so called by Atiyah and Macdonald) that is so useful in commutative algebra.

Combinatorics is so full of tricks that systematic compilation of them has already begun. Nowadays a surprising amount of combinatorics boils down to a single “meta-trick”: Recast your problem so that a certain bag of tricks may apply, and pull out that bag of tricks. For example: generate an integer sequence and look it up in Sloane; or recast the problem probabilistically and pull out Alon and Spencer; or write down a recurrence and pull out Wilf’s generatingfunctionology; or write down a determinant and pull out Krattenthaler’s Advanced Determinant Calculus; or write down a hypergeometric sum and fire up Zeilberger’s Maple packages.

Chern allegedly used to say, “When in doubt, differentiate.” Feynman was fond of evaluating integrals by introducing an extra parameter just so that he could differentiate with respect to that parameter. I’d love to see more examples of Chern’s principle.

Gian-Carlo Rota once said that every mathematician, even Hilbert, has only a few tricks. The beautiful book “Making Transcendence Transparent” by Burger and Tubbs illustrates how this can be. They show how all those amazing theorems in transcendental number theory boil down to the fact that there is no integer between 0 and 1.

These examples make me think that a “tricki” might work best if thought of as an incubator—a place to collect isolated, unnamed, pre-theoretical tricks until enough patterns emerge that a book-length exposition can be created. Inexplicit expertise then turns into explicit knowledge that can be routinely consulted by everybody.

]]>The site permit and encourage the user organize the items of wiki, to set up his own “book” of Tricks. Then the tricks in this book will show a deeper comprehension, according to the level of its organizer.

Hence, besides the bookmark system, the new-comer has another choice to get familar with the materials of this trick-wiki site, by studying the books of different old users, some of whom will be his teachers in reality (school or university), some of whom will be very talent mathematicians.

That reminds me the Chinese old classical book “Book Of Change” (written time is BC), it is indeed a book of combination of “tricks”, not of math, but of the experience of politics and social science.

Now by a web-site, we can expect many of “Book Of Change” for math can be set up. That would be a really sharp power for the improvement of math.

]]>This problem is really difficult, and I believe that the solution will be a new type of online Bookmark, which appears its prototype in del.icio.us and diigo (still not enough for a math wiki).

Here is some of my thoughts on this problem:

How to organize data by a new type of bookmark?

Bookmark=label+clip+alerts

Here:

I. label view=Groups/Folders view+Tag Clouds view+RoadMap view

II. clip=highlights texts×comments (diigo call comments as sticky notes)

III. alerts=rss reader×calendar reminder×searching alerts(like google alerts)

I will explain the item I. label view in a little detail:

labels (tags) is very important, both for bookmarking the website and organizing the personal files on his computer.

There are some suggestions for the “view pattern” of labels (tags)

I-1 Group view=Classical Folders view+Share Function（determine the sharing authority of members）。

Once users set some labels/tags as the “folders” or even “groups”, define their “hierarchies” structure, the user will be able to see the recursive nested view about these folders type of labels/tags. Yahoo bookmarks now has enable such function, but they do not understand that, “Folders” is just another view of labels/tags, — the same named Folder and Tag are different in Yahoo bookmark. But this is unnessary for our brain. And diigo realize the sharing function of the group, but diigo do not realize groups have the hierarchies structure. And Diigo also donot realize “groups” is just another view of labels, — the same named group and Tag are different in diigo.

Further, such hierarchies structure will satisfy the theory of psychology. That is, once you tags a file with “2.1.1” (for example), then this file will(should) be automatically tagged with the upper tags in the hierarchies, such as “2.1” and “2” . This is the ecnomical philosophy. That is once the user tag the item with subGroups/subfolders, then the item is tagged automatically with the fatherGroups/fatherfolders.

I-2. Tag clouds View=Matrix Pattern for the tags,

Tag clouds is now a very popular type of online bookmark. I suggest that, besides the clouds itself, there are two choices line of tags, first, the horizonal line (on the top) is same as that of temporary “tag clouds”, which interprete the “intersection” function of tags.

Second, another line, the verticle line (maybe lie beside the cloud) could be considered, which will interprete the “union” function of tags. Explicitly speaking, when you click the tags in the clouds, these tags will first appear in the verticle line, and the showing files are

related to the “union” of these tags. i.e. this file maybe belong to the first tags in the verticle line, and that file maybe belong to the second tags in the verticle line.

And then, the horizonal line will show the tags which has the intersection with that “unions”. Once the user click the tags in the horizonal line, the “intersection” function will be applied, just like what we do now.

I-3. RoadMap view of Tags: to indicate the logic structure of tags, especially useful for academic and universities.

]]>