most recent 30 from http://mathoverflow.net 2013-05-22T09:50:40Z http://mathoverflow.net/feeds/user/812 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/1977/why-is-the-gradient-normal Kim Greene 2009-10-22T23:34:35Z 2013-03-20T16:51:13Z

<p>This is a somewhat long discussion so bear with me. There is a theorem that I have always been curious about from an intuitive standpoint. This is an issue that has been glossed over in most textbooks that I have read. Quoting Wikipedia, The theorem is:</p> <p>"The gradient of a function at a point is perpendicular to the level set of f at that point."</p> <p><a href="http://en.wikipedia.org/wiki/Level_set#Level_sets_versus_the_gradient" rel="nofollow">http://en.wikipedia.org/wiki/Level_set#Level_sets_versus_the_gradient</a></p> <p>I understand the Wikipedia article's proof, which is the standard way of looking at things, but I see the proof as somewhat magical. It gives a symbolic reason for why the theorem is true without giving much geometric intuition.</p> <p>The gradient gives the direction of largest increase so it sort of makes sense that a curve that is perpendicular would be constant. Alas, this seems to be backwards reasoning. Having already noticed that the gradient is the direction of greatest increase, we can deduce that going in a direction perpendicular to it would be the slowest increase. But we can't really reason that this slowest increase is zero nor can we argue that going in a direction perpendicular to a constant direction would give us a direction of greatest increase.</p> <p>I would also appreciate some connection of this intuition to Lagrange Multipliers which is another somewhat magical theorem for me. I understand it because the algebra works out but what's going on geometrically? <a href="http://en.wikipedia.org/wiki/Lagrange_multipliers" rel="nofollow">http://en.wikipedia.org/wiki/Lagrange_multipliers</a></p> <p>Finally, what does this say intuitively about the generalization where we are looking to: maximize f(x,y) where g(x,y) > c</p> <p>I have always struggled to find the correct internal model that would encapsulate these ideas.</p>

http://mathoverflow.net/questions/5892/what-is-convolution-intuitively Kim Greene 2009-11-18T01:23:35Z 2012-10-27T00:23:21Z

<p>If random variable $X$ has a probability distribution of $f(x)$ and random variable $Y$ has a probability distribution $g(x)$ then $(f*g)(x)$, the convolution of $f$ and $g$, is the probability distribution of $X+Y$. This is the only intuition I have for what convolution means.</p> <p>Are there any other intuitive models for the process of convolution?</p>

http://mathoverflow.net/questions/3653/how-do-i-iterate-over-binary-trees Kim Greene 2009-11-01T09:29:02Z 2012-04-05T19:58:02Z

<p>Suppose I have n-1 distinguishable labels for internal nodes A={a1, a2, ..., an-1} and n distinguishable labels for leaves B={b1,b2, ..., bn} with A and B disjoint. What is the best way to iterate over all possible binary trees if I label without replacement?</p>

http://mathoverflow.net/questions/1722/free-high-quality-mathematical-writing-online Kim Greene 2009-10-21T21:17:26Z 2012-01-24T21:25:23Z

<p>I often use the internet to find resources for learning new mathematics and due to an explosion in online activity, there is always plenty to find. Many of these turn out to be somewhat unreadable because of writing quality, organization or presentation.</p> <p>I recently found out that "The Elements of Statistical Learning' by Hastie, Tibshirani and Friedman was available free online: <a href="http://www-stat.stanford.edu/~tibs/ElemStatLearn/" rel="nofollow">http://www-stat.stanford.edu/~tibs/ElemStatLearn/</a> . It is a really well written book at a high technical level. Moreover, this is the second edition which means the book has already gone through quite a few levels of editing.</p> <p>I was quite amazed to see a resource like this available free online.</p> <p>Now, my question is, are there more resources like this? Are there free mathematics books that have it all: well-written, well-illustrated, properly typeset and so on?</p> <p>Now, on the one hand, I have been saying 'book' but I am sure that good mathematical writing online is not limited to just books. On the other hand, I definitely don't mean the typical journal article. It's hard to come up with good criteria on this score, but I am talking about writing that is reasonably lengthy, addresses several topics and whose purpose is essentially pedagogical.</p> <p>If so, I'd love to hear about them. Please suggest just one resource per comment so we can vote them up and provide a link!</p>

http://mathoverflow.net/questions/5109/when-do-binomial-distributions-occur Kim Greene 2009-11-11T21:52:55Z 2011-10-13T15:15:02Z

<p>A binomial distribution is the distribution of the number of successes of n independent, identical Bernoulli trials. What happens when the trials are dependent and the Bernoulli trials are not identical by which I mean that the probability of success from trial to trial varies? How identical and close to independent do they have to be before we see something that resembles a binomial distribution?</p>

http://mathoverflow.net/questions/3951/memorizing-theorems Kim Greene 2009-11-03T16:22:38Z 2011-08-10T17:35:52Z

<p>I always have trouble memorizing theorems. Does anybody have any good tips?</p>

http://mathoverflow.net/questions/3134/whats-your-favorite-equation-formula-identity-or-inequality Kim Greene 2009-10-28T20:19:50Z 2010-08-22T16:14:57Z

<p>Certain formulas I really enjoy looking at like the <a href="http://en.wikipedia.org/wiki/Euler_Maclaurin" rel="nofollow">Euler-Maclaurin formula</a> or the <a href="http://en.wikipedia.org/wiki/Leibniz_integral_rule" rel="nofollow">Leibniz integral rule</a>. What's your favorite equation, formula, identity or inequality?</p>

http://mathoverflow.net/questions/6982/thorough-introduction-to-singular-value-decomposition Kim Greene 2009-11-27T21:05:52Z 2010-07-27T05:13:56Z

<p>Can you suggest a book that has a thorough introduction to Singular Value Decomposition?</p>

http://mathoverflow.net/questions/2437/is-there-an-image-for-you-that-epitomizes-mathematics Kim Greene 2009-10-25T07:30:25Z 2010-07-22T13:42:28Z

<p>Can you think of an image, whether technical or nontechnical, available for viewing online that says a lot about what you think mathematics or a particular field of mathematics is all about?</p> <p>For instance, some look at Hokusai's "Great Wave" as evoking a notion of fractals. <a href="http://en.wikipedia.org/wiki/File:Great_Wave_off_Kanagawa2.jpg" rel="nofollow">http://en.wikipedia.org/wiki/File:Great_Wave_off_Kanagawa2.jpg</a></p> <p>There is some interesting discussion of this here, <a href="http://www.squarecirclez.com/blog/math-in-art-hokusais-the-wave/595" rel="nofollow">http://www.squarecirclez.com/blog/math-in-art-hokusais-the-wave/595</a></p>

http://mathoverflow.net/questions/6139/how-can-i-learn-about-doing-linear-algebra-with-trace-diagrams Kim Greene 2009-11-19T17:24:28Z 2010-03-15T03:05:12Z

<p>There is a <a href="http://en.wikipedia.org/wiki/Trace_diagram" rel="nofollow">wikipedia article</a>. There is <a href="http://arxiv.org/abs/0910.1362" rel="nofollow">a paper by Elisha Peterson</a>. I tried reading these but they don't seem to click for me.</p> <p>Are there books or other resources for learning how to do linear algebra with trace diagrams?</p>

http://mathoverflow.net/questions/6984/introduction-to-structural-equation-modeling Kim Greene 2009-11-27T21:10:30Z 2009-11-27T23:27:14Z

<p>Can you suggest an introduction to structural equation modeling for math majors and mathematicians?</p>

http://mathoverflow.net/questions/5740/what-are-some-interesting-ways-of-making-new-metrics-out-of-old-metrics Kim Greene 2009-11-16T21:51:28Z 2009-11-20T11:03:01Z

<p>If $d(x,y)$ and $e(x,y)$ are metrics then $d(x,y)+e(x,y)$ and $\frac{d(x,y)}{1+d(x,y)}$ are metrics.</p> <p>If $d_i(x,y)$ for $i=1,\dots,n$ are metrics then so is $\sqrt{\sum_{i=1}^n{d_i^2(x,y)}}$</p> <p>Are there other interesting ways of constructing new metrics from old metrics?</p>

http://mathoverflow.net/questions/6009/whats-an-efficient-way-to-calculate-covariance-for-a-large-data-set Kim Greene 2009-11-18T20:19:45Z 2009-11-19T19:26:02Z

<p>What is the best algorithm for computing covariance that would be accurate for a large number of values like 100,000 or more?</p>

http://mathoverflow.net/questions/6002/finding-correlation-in-large-data-non-numeric-sets/6011#6011 Kim Greene 2009-11-18T20:32:17Z 2009-11-18T20:32:17Z

<p>I think sex can be written as a yes/no question. Female = 1. Male = 0.</p> <p>You can extend this to eye color. Blue eyes: yes/no. Green eyes: yes/no. Brown eyes: yes/no. One variable like eye color becomes three variables that are either 0 or 1. This gives you a lot of variables but you can do numerical things with them.</p> <p>Sometimes it makes sense to look at these variables as the probability of being male, having green eyes or whatever.</p>

http://mathoverflow.net/questions/5853/blackboard-rendering-of-math-fonts/5854#5854 Kim Greene 2009-11-17T19:04:46Z 2009-11-17T19:04:46Z

<p>I have never heard of anybody teaching how to write on a blackboard. There are a lot of books about calligraphy. You could try finding a calligraphy book with a font close to Fraktur.</p> <p>By the way, you can find tips on how to write greek letters: <a href="http://www.ibiblio.org/koine/greek/lessons/alphabet.html" rel="nofollow">http://www.ibiblio.org/koine/greek/lessons/alphabet.html</a></p>

http://mathoverflow.net/questions/4807/which-magazines-should-i-read/4809#4809 Kim Greene 2009-11-10T03:48:00Z 2009-11-10T16:00:46Z

<p>Collected answers from Kim Greene and one from Gerald Edgar:</p> <ol> <li><p>I hear Mathematical Intelligencer is good but I have never read it.</p></li> <li><p>I have heard that reading things that you don't completely understand is good for mathematicians so I also recommend the Notices of the AMS.</p></li> <li><p><em><a href="http://plus.maths.org/issue52/index.html" rel="nofollow">Plus </a></em> is a math themed magazine. I feel doubtful it would prepare you for graduate school in any way.</p></li> <li><p>College Mathematics Journal </p></li> <li><p>Mathematics Magazine</p></li> <li><p>American Mathematical Monthly</p></li> </ol>

http://mathoverflow.net/questions/4835/introduction-to-wavelets/4873#4873 Kim Greene 2009-11-10T15:01:53Z 2009-11-10T15:01:53Z

<p>Real Analysis with an Introduction to Wavelets and Applications by Hong, Wang and Gardner</p>

http://mathoverflow.net/questions/4835/introduction-to-wavelets/4871#4871 Kim Greene 2009-11-10T14:52:23Z 2009-11-10T14:52:23Z

<p>A Primer on Wavelets and Their Scientific Applications by James S. Walker</p>

http://mathoverflow.net/questions/4807/which-magazines-should-i-read/4869#4869 Kim Greene 2009-11-10T14:46:44Z 2009-11-10T14:46:44Z

<p>American Mathematical Monthly</p>

http://mathoverflow.net/questions/4807/which-magazines-should-i-read/4868#4868 Kim Greene 2009-11-10T14:45:28Z 2009-11-10T14:45:28Z

<p>Mathematics Magazine</p>

http://mathoverflow.net/questions/4807/which-magazines-should-i-read/4822#4822 Kim Greene 2009-11-10T06:55:36Z 2009-11-10T06:55:36Z

<p><em><a href="http://plus.maths.org/issue52/index.html" rel="nofollow">Plus </a></em> is a math themed magazine. I feel doubtful it would prepare you for graduate school in any way.</p>

http://mathoverflow.net/questions/4807/which-magazines-should-i-read/4808#4808 Kim Greene 2009-11-10T03:38:29Z 2009-11-10T03:38:29Z

<p>I hear Mathematical Intelligencer is good but I have never read it.</p>

http://mathoverflow.net/questions/4156/why-do-branches-of-math-vary-in-proof-styles-and-what-category-are-different-bran Kim Greene 2009-11-04T22:38:04Z 2009-11-05T02:34:57Z

<p>Some branches of math seem to have reasoning which is more global. There is a lot of efficiency in the proofs because the reasoning transfers easily between proofs. For other branches of math, a lot of truths seem to be more local. The proofs tend to have lots of sub-cases and exceptions. There are fewer general principles. Does anybody know why branches of math vary like this? Can you place different branches of math on this scale from being dominated by more ad hoc to being dominated by less ad hoc proofs?</p>

http://mathoverflow.net/questions/1812/learning-new-mathematics Kim Greene 2009-10-22T04:45:32Z 2009-11-04T15:16:51Z

<p>While many of us have had the experience of learning mathematics informally by osmosis or more formally in classes, there are times when we have to sit down and systematically learn, without the benefit of a class, large amounts of mathematics. For instance, there might be a technique that we need from a field we are not familiar with.</p> <p>When you find yourself in such a situation, what are your best tricks for teaching yourself new mathematics?</p>

http://mathoverflow.net/questions/3973/what-should-be-offered-in-undergraduate-mathematics-thats-currently-not-or-isn/3981#3981 Kim Greene 2009-11-03T18:06:00Z 2009-11-03T18:06:00Z

<p>Programming. I think it varies a lot from department to department but some places seem to do a bad job of teaching programming and it can be a really important skill.</p>

http://mathoverflow.net/questions/3973/what-should-be-offered-in-undergraduate-mathematics-thats-currently-not-or-isn/3978#3978 Kim Greene 2009-11-03T18:03:25Z 2009-11-03T18:03:25Z

<p>Traditional Statistics. Many biology majors end up knowing more statistics than many mathematics majors which I think is a weird state of affairs.</p>

http://mathoverflow.net/questions/3973/what-should-be-offered-in-undergraduate-mathematics-thats-currently-not-or-isn/3977#3977 Kim Greene 2009-11-03T17:59:26Z 2009-11-03T17:59:26Z

<p>Bayesian Statistics. I think it's more useful in many practical situations than traditional statistics.</p>

http://mathoverflow.net/questions/3863/what-functions-are-not-represented-by-their-power-series Kim Greene 2009-11-02T23:13:45Z 2009-11-03T00:50:58Z

<p>Some functions are not represented by their power series even when they are continuous and have all the necessary derivatives. What's the best characterization of these functions? Explanations at any level are welcome.</p>

http://mathoverflow.net/questions/3079/most-helpful-heuristic/3082#3082 Kim Greene 2009-10-28T16:21:38Z 2009-10-28T16:21:38Z

<p>Rather than suggest one heuristic, I thought I would point out that the Tricki (<a href="http://www.tricki.org" rel="nofollow">http://www.tricki.org</a>) is a repository of useful heuristics.</p>

http://mathoverflow.net/questions/2506/which-computer-algebra-system-should-i-be-using-to-solve-large-systems-of-sparse/2512#2512 Kim Greene 2009-10-25T21:11:18Z 2009-10-25T21:11:18Z

<p>Matlab might be a better choice. Be sure to use the sparse matrix functionality rather than just regular matrices. I haven't used Matlab for this purpose but I've seen people using it for large systems.</p> <p>Also, have you been using sparse matrices in Mathematica?</p> <p><a href="http://reference.wolfram.com/mathematica/howto/WorkWithSparseMatrices.html" rel="nofollow">http://reference.wolfram.com/mathematica/howto/WorkWithSparseMatrices.html</a></p> <p>This might improve the performance.</p>

http://mathoverflow.net/questions/7120/too-old-for-advanced-mathematics Kim Greene 2009-11-29T08:54:56Z 2009-11-29T08:54:56Z

There is a question about mathematicians that learned mathematics at a late age. <a href="http://mathoverflow.net/questions/3591/mathematicians-who-were-late-learners" rel="nofollow" title="mathematicians who were late learners">mathoverflow.net/questions/3591/&hellip;</a>

http://mathoverflow.net/questions/6984/introduction-to-structural-equation-modeling/6997#6997 Kim Greene 2009-11-28T18:48:02Z 2009-11-28T18:48:02Z

Thanks. I will take a look.

http://mathoverflow.net/questions/6982/thorough-introduction-to-singular-value-decomposition/6983#6983 Kim Greene 2009-11-27T23:01:18Z 2009-11-27T23:01:18Z

Good. I think I know where to find a copy of this. Thanks.

http://mathoverflow.net/questions/6984/introduction-to-structural-equation-modeling Kim Greene 2009-11-27T22:19:50Z 2009-11-27T22:19:50Z

@Greg: I looked at several books on SEM before coming here and the ones I found are directed toward social scientists. I wanted to ask if there were ones for people in math. The social science is heavy and makes it hard to understand the math.

http://mathoverflow.net/questions/6009/whats-an-efficient-way-to-calculate-covariance-for-a-large-data-set Kim Greene 2009-11-18T20:26:07Z 2009-11-18T20:26:07Z

One quick note. I know some people will think this is basic but I have looked in a lot of places (numerical analysis texts, computational statistics texts and google scholar) and I've never seen it discussed. Doing the straightforward thing does not work for large datasets.

http://mathoverflow.net/questions/5892/what-is-convolution-intuitively Kim Greene 2009-11-18T01:35:45Z 2009-11-18T01:35:45Z

See my previous question: <a href="http://mathoverflow.net/questions/1977/why-is-the-gradient-normal" rel="nofollow" title="why is the gradient normal">mathoverflow.net/questions/1977/&hellip;</a>

http://mathoverflow.net/questions/5853/blackboard-rendering-of-math-fonts/5876#5876 Kim Greene 2009-11-18T01:32:36Z 2009-11-18T01:32:36Z

If you used a laptop that was also a tablet then you could write things like a blackboard. Less need to prepare before hand.

http://mathoverflow.net/questions/5740/what-are-some-interesting-ways-of-making-new-metrics-out-of-old-metrics/5748#5748 Kim Greene 2009-11-17T02:10:50Z 2009-11-17T02:10:50Z

Yes, I mean series.

http://mathoverflow.net/questions/5740/what-are-some-interesting-ways-of-making-new-metrics-out-of-old-metrics/5748#5748 Kim Greene 2009-11-16T22:49:52Z 2009-11-16T22:49:52Z

Does the choice of convergent sequence matter?

http://mathoverflow.net/questions/5741/what-is-the-svd-of-this-matrix Kim Greene 2009-11-16T22:07:56Z 2009-11-16T22:07:56Z

Bad question. Read the FAQ.

http://mathoverflow.net/questions/4807/which-magazines-should-i-read/4809#4809 Kim Greene 2009-11-10T16:19:10Z 2009-11-10T16:19:10Z

Gerald provided three. I split his into three to comply with one answer. This made them look like mine.

http://mathoverflow.net/questions/4807/which-magazines-should-i-read Kim Greene 2009-11-10T03:38:56Z 2009-11-10T03:38:56Z

I think this should be community wiki.

http://mathoverflow.net/questions/4733/how-platonistic-is-your-attitude-towards-mathematics Kim Greene 2009-11-09T16:22:36Z 2009-11-09T16:22:36Z

This should be community wiki.

http://mathoverflow.net/questions/4156/why-do-branches-of-math-vary-in-proof-styles-and-what-category-are-different-bran Kim Greene 2009-11-05T04:41:23Z 2009-11-05T04:41:23Z

@Theo: I do not think that it is any more subjective than a question about what is a 'good' book for learning analysis. I think this is a question with real consequences for people learning and some people find one style of math easier than the other. I also don't think there is evidence in the comments so far of big differences in how people are interpreting the question as one would expect if it was very subjective.

http://mathoverflow.net/questions/4156/why-do-branches-of-math-vary-in-proof-styles-and-what-category-are-different-bran Kim Greene 2009-11-05T00:32:27Z 2009-11-05T00:32:27Z

I edited it. I hope it sounds less biased now.