I have added another essay to my career advice page, inspired partly by some earlier blog discussion, entitled “Continually aim just beyond one’s current range“.
Also, there seems to be some demand for discussion here on topics not directly related to any of the posts. So I will experiment a little here and turn this post into an “open thread”, in which discussion on any topic is permitted (though, of course, I would continue to ask that all comments remain polite and constructive). If this experiment turns out well, I will try to initiate some further open threads at periodic intervals on this blog.
[Update, Sep 22: As you might have noticed, I am experimenting with wordpress tags, which have recently been decoupled from wordpress categories. I am still not sure exactly what the best way to use tags and categories are, and whether there should be any attempt at standardisation amongst the maths blogs; suggestions are of course welcome.]
What's new 
96 comments
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14 June, 2008 at 8:30 am
Terence Tao
Dear Beetle B.,
According to the wordpress.com FAQ entry at
http://faq.wordpress.com/2005/09/26/category-atom-feeds/
it seems that RSS feeds are available for each category by adding “/feed/” at the end of the URL for that category.
28 January, 2009 at 3:01 pm
Restriction question
Hello,
I have a question about your (Park City) survey on the restriction conjecture (this is the most appropriate thread I could find). In these notes you talk about the Restriction problem for the cone, sphere, and paraboloid. You endow each of these surfaces with what you call the canonical measure. In the case of the sphere its the surface measure, in the case of the paraboloid it is the Lebesgue measure under a pullback, yet in the case of the cone its the Lebesgue measure under pullback with an additional factor. In the case of the cone, you have a brief comment/exercise justifying this particular choice (because it preserves certain symmetries of the cone with respect to linear transformations). Also, these choices of measures seem to be the right ones for applications to the related (wave, Schrodinger, and Helmholtz) PDEs.
That said, however, I can’t see why these choices are canonical to the particular surfaces. For example, if I were to consider the restriction problem for an arbitrary surface in R^n is there a canonical choice of measure?
28 January, 2009 at 4:36 pm
Terence Tao
Not every surface has a canonical measure (other than surface measure, of course, although this measure depends on the inner product structure of the ambient space and so can be argued to not be totally canonical either). What distinguishes the sphere, paraboloid, and cone from other surfaces is that they are homogeneous spaces, i.e. they have the transitive action of a Lie group (the rotation group, Galilean group, and Lorentz group respectively), and thus come with a Haar measure which is well-defined up to constants.
29 January, 2009 at 8:58 am
a
What about using results whose proofs you don’t know/don’t understand?
Sometimes in, say, analytic number theory, people borrow results that some algebraic geometer promised is true, and it would take a lifetime to go and verify the proof. That’s very scary.
Why do we do mathematics? Is it:
1. to know: ie to have answers to any questions about the things you study
2. to understand: ie to be able to prove whatever you want about the things you study.
I guess most of us do math to gain a deep understanding of the mathematical objects we study; because we care about *those particular objects* (otherwise it becomes an exercise in logical deductions). But some of the most crucial facts about our favourite objects are ridiculously hard to prove, and it may take decades to properly understand them.
Of course, I like knowing the answers too. But all my life I always thought that whenever you make any mathematical statement, if someone tortures you enough, you should be able to write down the proof from first principles.
29 January, 2009 at 9:01 am
rcourant
Dear Terry,
What do you mean when you say you put things in a “black box?” For example in your most recent Analysis notes you said that it would be okay to put a particular concept in a “black box.”
29 January, 2009 at 9:06 am
superman
Is it possible that every 100-term AP in the primes is contained in a 101-term AP in the primes? The quantitative/density bounds aren’t enough for this, but perhaps there is a direct argument?
29 January, 2009 at 10:49 am
Terence Tao
Dear rcourant, a “black box” result is one for which the hypotheses and conclusion is known, but the proof is not (in analogy with black box systems). It relates to a’s question about whether one should feel comfortable using results for which one does not know the proof. As a temporary measure, there is nothing wrong with this, especially if the proof comes from a very different area of mathematics than the one that one is focusing on, but of course one would ideally like to understand all the mathematical results that go into a proof one is trying to give, in order to be more certain about the proof, and to avoid errors caused by misunderstanding a result that was being applied as a black box.
Of course, given how vast mathematics is, it is often not feasible to read the proof of every result one needs line-by-line, but if one can get to the point where one has a heuristic understanding of why the result is true, at least (e.g. one works out some key examples, or reasons by analogy with simpler results of a similar nature, etc.), this is often good enough. In many cases, one does not need the full strength of a difficult result in one’s applications, and a simpler special case of it can often be worked out more directly by hand, which is also a good exercise to help one understand the full result.
Dear superman: my theorem with Ben implies that the number of progressions of primes less than N of length 100 is asymptotically larger than
for some absolute constant c, whereas standard sieve theory methods show that the number of progressions if length 101 is less than 
for some other constant C. Letting N be large enough, we can conclude that not every prime 100-AP is part of a prime 101-AP.
8 February, 2009 at 11:28 pm
Stuart Anderson
I have been reading some of your comments on Perelman’s proof, and if I have understood correctly, among other things, he has made some progress in statistical mechanics. This called to mind an old problem I had read recently;
In a paper ‘An Introduction To Probability And Random Processes’ by Kenneth Baclawski and Gian-Carlo Rota (1979) http://www.ellerman.org/Davids-Stuff/Maths/Rota-Baclawski-Prob-Theory-79.pdf, the authors posed a problem;
“We have a rectangular carpet and an indefinite supply of perfect pennies. What is the probability that if we drop the pennies on the carpet at random no two of them will overlap? This problem is one of the most important problems in statistical mechanics. If we could answer it we would know, for example, why water boils at 100C, on the basis of purely atomic computations. Nothing is known about this problem.”
I’ve tried using my undergraduate calculus but didnt get far, perhaps reducing the problem to simpler one and using combinatoric arguments might work?
Do you or anyone else know if this has been solved, or if progress has been made on it?
15 June, 2009 at 1:23 am
janiceenberg
I’m new here on the forum, found it by searching google. I look forward to chatting about various topics with all of you.
21 June, 2009 at 8:35 pm
Anonymous
Could you give a reference for the reoccurring statement (on your blog) that the restriction conjecture implies the Kakeya conjecture? thanks.
22 June, 2009 at 8:28 pm
Terence Tao
Dear Anonymous,
One can find a proof in Wolff’s survey article “recent progress on the Kakeya problem”. I believe the first formal appearance of this claim is in a paper by Beckner, Carbery, Semmes, and Soria, though the idea really goes back to Fefferman’s famous disk multiplier problem.
24 June, 2009 at 4:15 pm
AP Question
I have two (related) questions regarding your work with Green on Primes in AP.
First, in the original paper as well as subsequent work (as well as on this blog and a note on your website) you have addressed numerous quantitative questions related to the theorem such as (1) how big can the first k term AP (in the primes) be and (2) how many k term AP’s are there less then N (which is sharp up to the constant!). However, as far as I have seen, you didn’t explicitly give bounds of the form, “any subset of P \cap [N] with density greater than F_k(N) must contain a k-term AP”, although maybe I just missed this. I did notice that Green obtained a bound of this form in his initial work on 3-term AP’s. Can bounds of this form be extracted from these methods or is it (perhaps) a casualty of moving to a more ergodic point of view?
Secondly, related to this last point, I was hopefuly you would say something about why you took the ergodic approach. Was this a matter of convenience, or is there a fundamental obstacle to taking a more Fourier analytic approach, consistent with Green’s original paper?
25 June, 2009 at 12:38 pm
Terence Tao
The arguments in my paper with Ben are quantitative (we use finitary analogues of ergodic theory rather than infinitary ones) do in principle give an effective bound for this quantity
. My guess is that it will look something like 
where 
is the 5-fold iterated logarithm.
The Fourier analytic approach eventually works (modulo the inverse conjecture for the Gowers norm, which is not yet fully resolved), and is the topic of several further papers of myself with Ben. The approach actually gives more precise information (in particular, it gives the right constant for the number of k-term progressions asymptotically), but is substantially lengthier.
4 November, 2009 at 11:14 am
Ardninam Lawarga
Hi Professor Tao and To Whom It May Concern:
I program for fun in various languages, mostly in C# language using WPF, Silverlight, ADO, ASP as the front end. Saw this blog on Forbes magazine and was curious to anybody reading this thread if they could suggest some cool math related programs to write. I have a few ideas but could use some more input.
I did see the suggestion by Prof. Tao for turning math classifications into a sort of periodic table. This is quite easy to do in ADO.NET as a database but the difficulty would be in having nice graphics… that is the gating factor in writing such an app. If anybody would supply me with the graphics I can write this type program fairly easily.
Here are some thoughts I have for programs that are math related, but I could use more input.
1) The Game of Life as expresed by British mathematician John Horton Conway in 1970. I wrote this already actually, and I mention it here as the class of idea I am interested in finding from anybody reading this.
2) Gambler’s Ruin problem–where you flip a coin from a given level and count how many iterations N it takes to go to zero, with 50% chance of going up one unit and 50% chance of going down– I wrote this as well in console mode (non-graphical output), and now I am going to work on expressing the output in Silverlight or WPF (which has outstanding graphics capabilities). What’s interesting is that ‘survivors’ of Gambler’s Ruin (candidates that did not touch zero in period M) for any period M (M > N) of iterations appear to be a sort of Poisson distribution. The longer you make M, at any given level, the more survivors you but it’s a nonlinear relationship it seems, like Poisson’s distribution perhaps. It’s also cool to graph the output and see how many candidates die in any interval despite reaching a very high level. This is sort of like the laws of entropy–you can actually defeat falling to zero for a very long time.
3) A Polya urn problem (http://en.wikipedia.org/wiki/Urn_problem) but for ants. I saw this in the book “Butterfly Economics” and it involves the following: given some ants who have the opportunity to visit two food sources, A and B (you can generalize this to any number though), and given three different outcomes (ant visits previous food source it visited, ant visits the food source of the first ant it meets returning from a food source, and ant visits randomly one of the two food sources), you graph the percentage of ants that are at any given food source at any time. The graph is nonlinear and depending on how you tweak the coefficients you can get jumps in the percentage of ants visiting any given pile–feed-forward behavior as they say in systems engineering, no pun intended.
Anyway I have a few other ideas such as graphing the output of various distributions like the broken stick distribution and the Galton–Watson process but the point of this was to underscore what kind of projects I’m looking for.
No time limit, any replies appreciated, as I just do this outside of my day job for fun.
Sincerely,
Ardninam Lawarga
email: yamun31781@mypacks.net
8 November, 2009 at 1:12 am
Two new science & physics sites for asking questions « episodic thoughts
[…] that rule, but at least there’s a great potential here! I hope influencial bloggers like Terry Tao, and Peter Woit (who asked if such site existed), can mention them: that would boost the […]
24 December, 2009 at 9:20 pm
marthafines
Merry Christmas to all… and to all a good night.
21 April, 2010 at 2:17 pm
Wayne Lewis
Professor Tao,
I have what may be silly question, but I have given up trying to figure it out and trying to find analysts who know the answer. Is Gamma(z) ever a negative real number if z is not a negative real number?
Aloha,
Wayne
21 April, 2010 at 8:49 pm
Terence Tao
Yes; this can be seen from Stirling’s approximation and the argument principle.
More generally, for these sorts of questions, I recommend Math Overflow,
http://mathoverflow.net/
27 September, 2010 at 8:35 am
HONG
Dear Professor Tao,
I am not sure this is the right place to ask the question about your paper “CALESON MEASURES, TREEES, EXTRAPOLATION, AND T(b) THEOREMS”. In Lemma 4.1(or Lemma 4.2)(Calderon-Zygmund decomposition for size), I can not check that your induction hypothesis can give the right estimation, i.e. your induction on the size of a tree can give the estimation on the corresponding maximal size for the same tree. I wonder maybe we can go back to the method in “Chopping big trees into little trees”.
Thank you very much for all the wonderful works, thought I am not sure you have the time to answer my question.
27 September, 2010 at 10:50 am
Terence Tao
I am sorry, I do not understand the question. No induction argument appears in the proofs of Lemmas 4.1 and 4.2 in our paper. To bound the maximal size of a on a tree, we can upper bound it by 2^n because we are assuming that a has maximal size at most 2^n on the larger collection P_n, and we can of course lower bound the maximal size by the size, which is at least 2^{n-1} by construction.
28 September, 2010 at 3:50 am
HONG
In Lemma 4.1 we select a tree T such that the size of a on the tree bigger than 2^{n-1} maximal with respect to set inclusion, but this can not ensure that the size of a on any convex subtree of T is bigger than 2^{n-1}, even if we are assuming the upper-bound for the size of a on any convex tree is 2^n. For instance, a tree T consisted of two tiles is selected most due to its top, the size of a on another tile can be very small.
Maybe, I’m making some dumb mistakes!
Thanks anyway!
28 September, 2010 at 7:42 am
Terence Tao
To ensure a lower bound on the maximal size of a tree T, it suffices to lower bound just a single sub-tree of T; one does not need to lower bound _all_ subtrees of T. (This is because we are lower-bounding a sup, rather than an inf.)
In this particular case, the entire tree T has size at least 2^{n-1}, and so the maximal size of T is also at least 2^{n-1}.
23 May, 2011 at 9:02 am
Ken
Dear Terence,
I have been fascinated with the greatest integer function for the past decade. I have read journal and internet articles on it, read books on number-theory, and did a lot of research on my own. Although I have accumulated a lot of material over ten years, good literature on the greatest integer function (aside from its basic properties and applications) is very scarce. My purpose for posting this entry is not to promote my research here, but to inquire about advanced properties and applications for the greatest integer function that I have missed. I have put this question to other professors, but none of them seemed to know very much about the greatest integer function.
Then again, I cannot ask you what I missed unless I show you what I have done. The Full Text page on my site has a short summary listing chapter objectives just below the link to the pdf. In addition to addressing topics that I missed, by reading the summary, you could also give me ideas for what chapter objectives could be pursued further. Thank you.
Your fan,
Kenneth Beitler
12 October, 2012 at 10:33 am
Anthony Morris
Hello there. I would greatly appreciate a comment on this original take on prime numbers – http://archive.org/details/PrimeNumbers_986
Thanks Ant
11 November, 2012 at 6:06 pm
Ahmed Roman
Dr. Tao,
I and a few of my colleagues have been thinking about the following problem and are not making much progress. The question is made up by one of us and we think it is rather interesting. Here is the question:
1) Find a function from the real line to itself such that the function sends the rational numbers to the rational numbers and such that the derivative of the function at q is not rational for some rational number q.
2) After thinking about problem 1, one might ask the following question:
Is every continuous function from the real line to itself which sends the rational numbers to the rational numbers locally rational i.e. can one approximate the function on any small neighborhood by some rational function(different neighborhoods requiring different rational functions of-course).
Any intuition or hints would be appreciated.
29 November, 2012 at 12:37 pm
Jeremy
Terry, I accidentally posted my email address in my name field (and vice versa) with this post: https://terrytao.wordpress.com/2011/04/07/the-blue-eyed-islanders-puzzle-repost/#comment-197446. Would you please fix this for me? Or just delete the post? I reposted with the correct information just after I noticed my mistake, so if you can fix the first post (preferable, since it has replies already) then you can delete the second. Otherwise, vice-versa.
Thanks!
[Done – T.]
27 January, 2013 at 5:07 pm
Roland
Hi there.
This is an open challenge for any mathematically-apt mind looking for entertainment. Presented in a math olympiad 25 years ago:
Some Pythagorean triangles (a²+b²=c²), coincidentally, have integer lengths. For example 3²+4²=5².
Prove that for any integer Pythagorean triangle, one of the side lengths is
always a multiple of 5.
The original challenge consists of answering this in less than 1 hour,
with an 8th grader’s toolset.
Needless to say: i lamely excused myself in the original situation and got zero points. 25 years later, i have studied something other than mathematics. But my hard drive still rattling…
Any idea, any one ???
17 February, 2014 at 3:24 am
matti
This fan site has spelt your name wrong
http://retrokiwi.spreadshirt.com/terry-teo-mens-t-shirt-colour-changeable-A1157786/customize/color/2
13 November, 2014 at 8:32 pm
Anonymous
Hi! I just wanted to share that I am an undergraduate student who has been reading through your blog this year (mostly not understanding, striving for understanding)… suffice to say that I really do not know who is who in math. I vaguely know that the Fields Medal existed but that’s all. Some friends and I were watching the Colbert Report this week and when your name came on the screen, I screamed, “I READ THAT GUY’S BLOG!!” I really appreciate that despite what I imagine must be an insanely busy teaching and research schedule, fame, etc that you still keep this blog full of interesting and exciting things–interesting even for those of us who are undergraduates and don’t understand very much quite yet.
7 October, 2017 at 11:37 am
Babak Farhang
I “discovered” the other day that any equation that is symmetric in 2 variables defines a function that is its own inverse (an involution). Thus, for example
f(x) = (b – x) / (ax + 1)
is an involutory function for any constants a, b because that solves
y + axy + x = b.
So I can see it’s easy to define such functions using “symmetric” equations, but it might not be always easy to construct a simpler expression for it. For example
sin(y) + sin(x) + a sin(xy) + b = 0
is harder. And it’s not always clear to me whether the y = f(x) form is a bijection (and perhaps other issues related to constructing a proper inverse.)
My question: Is there theory about or a general approach to involutory functions defined this way?
24 November, 2017 at 6:43 pm
Roshan Jayaraman
Dear Terry!
I’m a junior in high school and really enjoy math.
I’ve been going to math competitions from middle school, and aspire to do well in the AMC competitions and possibly purse math in university.
I love the beauty and artistry of math; the idea that from a set of axioms you can create this whole world of craziness is just fascinating to me!
From time to time, however, I face the following problem:
When I attempt a challenging math problem, don’t make much progress, and then go read the solution, a part of me feels discouraged.
The solution looks so clean and beautiful, and I feel that the answer should have been obvious to me in the first place (this especially happens in geometry lol).
After a while, I tend to fall into the thought pattern of ‘Do these problems come naturally to other mathletes? Why isn’t this coming naturally to me? Maybe I’m not good enough…’
So this leads up to my question:
As a seasoned mathematician, I reckon you’ve made mistakes in your career. How have you learned to deal with mistakes? Was there a time in your life where you would be angry with yourself for making little progress on a challenging problem, and if so how did you cope with that situation? And now at 42, do you still get frustrated when you make little progress on a challenging problem?
Hearing your perspective could really help me. When I’m struggling with a crazy Euclidian Geometry in Math Olympiads problem and am making 0 headway, maybe instead of getting frustrated, I’ll think ‘Terry Tao makes mistakes too. I shouldn’t be too hard on myself’ :).
Cheers, and hope you had a happy thanksgiving!
Thanks, Roshan
25 November, 2017 at 5:45 am
Andrew Krause
This is actually something *all* mathematicians do. Theorems, proofs, and calculations as written in papers or problem sets are often quite cleaned up from their initial conceptualizations. Unless you have worked in a particular field for some time (e.g. typically years), almost nothing will come naturally. One of the most famous 20th century mathematicians, John von Neumann, one told a graduate student, “Young man, you don’t understand mathematics; you just get used to it.” Terry has several posts discussing the importance of struggling in order to really understand mathematics, and working hard, even to understand things which look easy to others, or after the solution is revealed.
https://terrytao.wordpress.com/career-advice/ask-yourself-dumb-questions-%E2%80%93-and-answer-them/
22 September, 2018 at 1:52 pm
Carlos M
New Paradigm of Science
⁂
A simple way to explain E=mc² is: energy is matter times the square of the lightspeed. Time is space, and the relationship between both is that the effects of time are inversely proportional to the space; in a smaller space, the events have (relatively) more velocity. This shows us, that the reality beyond the limits of the known Universe extends itself endlessly, outwards the Cosmos and inwards the particles. With a fractal shape unfolded in several dimensions, and an pattern that allows astronomical objects/particles to desintegrate and replenish themselves in continuous cycles.
It’s like a spiral that extends itself from a centre towards the extremes endlessly where, alternating matter and energy in the space-time, gives place to endless realities with countless possibilities.
Events from our reality, like a drop of dew falling from a flower petal into the ground, affect not just the farthest star, but also at microcosmic levels, since they’re galactic groups interacting.
Reality is a fractal, without an apparent start or end. Looking close and very slowly at any type of molecule, we can see something similar to our galaxies. In turn, the suns and their planets, orbiting repeatedly with a fourdimensional effect, when seen from a remote point of view, give place to the atoms with their particles in the next level of reality, and so on, consecutive and endlessly.
In what it lasts one breath ours, in subatomic worlds occurs numerous events, and while our worlds existence passes, in the macrocosm instants only happen.
Reality
When zooming inside any type of matter and pausing the scene as necessary, it’s confirmed atoms and their subatomic particles are, actually, astronomical objects (among other things) orbiting continuously with great velocity (depending on the observer’s point of view), driving an effect that gives the matter its aspect. And those stars and planets are made of matter, and the atoms that make them are also sideral bodies in movement, made of more matter, etc., and this way reality keeps on perpetually.
And when zooming out towards the outer space, the celestial bodies and their planets continually orbiting across the time, they form the atoms and their subatomic particles that make a fraction of the macrocosm, and in turn this macrouniverse has astronomical objects that, when orbiting across the time, from another point of view gives place to atoms that make a fragment of another level of reality, etc. And this way the reality goes on, extending itself without interruptions.
The energy/mass across the space/time in any of its proportions, gives place to reality, a fractal of incalculable strands always moving itself (in continuous creation and destruction), extending itself incessantly outwards and inwards any point. With practically endless possibilities – within their respective rules – where innumerous imaginable possibilities and situations outside our thoughts fit. This Universe is a fragment in more than one infinity of infinities of infinite realities; it’s part of a bigger fragment, in an even larger one, in another even larger, etc. and the inverse. Unfolding itself in dimensions and planes, largely intangible for us.
The space/time is inversely proportional to itself, when the space is shrunk the (relative) speed of the events in this increase, and when the spaces is enlarged, the course of the matter in this [space] happens in longer periods of time.
:::
The planetary systems, galaxies, clusters and all other astronomical bodies, when they complete their trajectory towards the time – like a long-exposure photo – look like pictures made in a 3D spirograph. Deppending on the pattern type, the next level of matter they make has particular characteristics.
…
When moving the energy/matter in the space/time causes events, and realizing those events aren’t distinct outcomes, even then it’s probable that there are alternatives in other realities for any situation. Every action that exists is part of a whole and it is contemplated like that. The successes are now and always in the reality, forever happen – and stop happening-. There’s no past or future, just eternity.
…
There are other universes symmetrical to ours; since reality has a fractal nature, it has self-similarity, property where every little portion of the fractal can be seen as a smaller replica of the whole. In other words, in other scales there are parallel universes to ours, among many different ones.
https://rox1012101.tumblr.com/
https://discord.gg/hRa8g3
22 September, 2018 at 10:44 pm
roland5999
Thanks Carlos, quite poetic :) Anything tangible follows from this? Any bottom line w.r.t. math or physics – like suggestion to turn attention anywhere in particular, any new tricks or experiments for sake of a new tangible result or so? Otherwise, poetic clauses and thats all, but thanks anyway. cheers, r.
23 September, 2018 at 4:50 am
roland5999
Sorry Carlos, not wanted to sound
any harshness or judgement. I hear music by is there any libretto?? cheers, r.
19 February, 2019 at 10:56 am
TonyP
home page code or articles need fixing to many on the page
19 February, 2019 at 10:58 am
TonyP
on previous post html tags p /a /p
10 June, 2020 at 6:11 pm
But Maybe...
This open thread looks very old, so I’m unsure if you still check it. That said, I had a bizarre thought that I was hoping you’d be willing to entertain.
Is there any chance you’d someday write a blog post or two containing advice for the highly academically intelligent but socially less so? Possibly – dear God, forgive me – even romantic advice? I have Asperger’s, and I seem to get my heart torn out of my chest every other time I turn around, whether due to difficulty connecting with others generally, even including intelligent peers, or women specifically. I’d like to believe that the same faculties which can be applied to problem solving in general can also be applied to personal matters. You’re married. How the hell did you do it, man? What’s the one weird trick I need to adapt my general problem solving skills into social ones, and how has it possibly eluded me for so long?
21 June, 2020 at 7:15 am
High
Hi all,
, where 
and 
are the Flux Jacobians and 
is the adjoint state. 
and 
. For external flows (flow around an airfoil, for example) 
obeys dual characteristic b.c. at the far-field (whereby the outgoing characteristic components of the adjoint field are set to zero), while at solid walls the b.c. is of the form 
where 
is the normal vector to the surface and 
is a function that depends on the choice of cost function. Depending on the choice of flow regime and cost function, the adjoint field can have singularities at airfoil/wing trailing edges, along stagnation streamlines, along supersonic characteristics, etc, and I would like to try to find a systematic way to understand the structure of singularities in relation to the fluxJacobians 
and boundary conditions. For example, funny things may be expected at points where the Jacobians become singular, or where the flow velocity goes to zero, or at discontinuities of the flow. The final goal is to understand why the numerical solution of the above equations is strongly grid-dependent (and even divergent) near solid walls, and why such grid-dependence seems to be strongly correlated with the presence of singularities at the forward stagnation streamline and at the trailing edge / rear stagnation point. Be assured that mesh dependence of numerical solutions in the vicinity of singularities is to be expected. Here I am speaking of global mesh dependence of the numerical adjoint solution across the entire wing airfoil surface as described here:
I would like to find a reference that discusses, in certain generality, the properties of solutions of systems of first order linear pde with non-constant coefficients. I am essentially trying to find a way to “predict” the properties of solutions of systems of first order linear pde with non-constant coefficients from the coefficient matrix, if such an endeavour actually makes any sense (forgive me if it doesn’t).
I am actually studying the particular case of the steady, two-dimensional adjoint Euler equations,
Click to access EUCASS2019-0291.pdf
Cheers!
4 August, 2020 at 8:08 am
asemic horizon
Do you have advice for people who have had *some* mathematical education (I dropped out with a Master’s degree) and want to keep learning?
I don’t think I was ever “mathematician material” but I want to keep growing anyway. I imagine most failed mathematicians like me have an “intuition overreach” problem in tackling proofs correctly enough that proofs become a first-class object. But I’ve seen a proof articulate the structure of an entire field (Darboux’s theorem and Hamiltonian mechanics); I know there’s more to learn like this.
I know I do better where there are numerous head-cracking counterexamples that challenge intuition enough that definitions must be revisited again and again until the unwritten point comes out. More generally I learn better from exposition that doesn’t appeal to intuition (ex. lattices from meets and joins and absorption laws, rather than from partial orders) because it gives my excess imagination a harder kernel to chew. But those materials tend also to be more advanced re: prerequisites.
It’s very likely you’ve had students like me; or that other comment box denizens have had them (or *are* somewhat like me).
21 November, 2020 at 4:28 pm
Collatz conjecture
Hey, Terry, I think I have a solution to 3n+1, or, at least, a chunk of the puzzle.
Let’s just say:
After I found it, I had a feeling “welp, that’s it then”.
It scratched the “itch”, I’m free from the desire to solve it further. Take it for what it’s worth.
I don’t have any credentials, but I solved “Einstein’s puzzle about five houses” without writing anything down. Weird flex, I know, I hope it’s ok.
I don’t have time to check if this solution exists, because time is food where I live. I think it’s fair, though, to spend some time to offer you a choice, since you are a renown Collatz solver :D
Also, if it is the solution, I would’t want to spoil the “fun” for everyone, and just drop it on 4chan or reddit.
Or, maybe, I should, so that at least some people can be finally free from this 3n+1 “mind bug”?
What do you think Terry? What should I do?
Is this problem a curse, or a fun challenge for you?
Is the solution more important than the process of solving, or does it open new paths?
I certainly used “unconventional” methods in solving this. I reject “conventions”, made by people, who insist that “one is not prime” to have shorter proofs. Rules may have exceptions, Laws may not. I’m a bit sorry for this rant, but that’s how I feel.
If you are interested, email me. I’ll check this inbox periodically.
Also, as my email is hidden, writing to it about Collatz is a strong bayesian proof only of “someone has access to blog back end and can see comment emails”. Think of something, or we can think of something together, if you like such games. I’ll make no promises on when I’ll get back to you, feeding my family comes first.
This is a message only for Terry Tao. I shall not respond to this thread.
21 November, 2020 at 8:43 pm
Anonymous
He’s obviously not going to read it. Probably you should check it carefully and find the error.
28 November, 2020 at 6:50 am
Anonymous
Indeed, mr. Anonymous. Good point.
If OP is at least half as smart, as they claim to be, they would, probably:
– Find the error, and then remove all, some or none of their claim.
– Know the answers to their questions, and leave Terry Tao to be Terry Tao, who, probably, has other things to think about, than “of something”.
– Know that “fun”, “mind bug”, “curse” and “challenge” are not mutually exclusive terms.
– Have a shower when they have “the itch”, just in case :P
Who knows? Maybe they did it regardless, for their own reasons.
I mean, I didn’t see the solution posted anywhere…
15 April, 2022 at 7:08 am
Scott Adler
A mundane observation. In your Masterclass presentation you discussed what is the best time to tie the shoelace when rushing to catch a plane and used the example of 2 identical twins. One stopped just as before getting on the moving walkway and the other who stopped on the moving walkway with the latter then getting ahead of the other. The conclusion was reached that stopping while on the moving walkway was best. However if one stopped on the moving walkway while the other twin continued to walk and then stopped after getting off, he would be ahead and then the other twin would catch up to him, therefore there is really no best time. T =time not on the walkway , W=time on the walkway, L= time tying laces;
T+(W+L)+T is the total time for stopping before the walkway
T+W+ (T+L) is for tying after the walkway. The time is the same.
15 April, 2022 at 7:12 am
Scott Adler
Clarification W is the same in both cases so it is really (T+L)+W+T and T+W+(T+L).
1 February, 2024 at 7:10 am
Anonymous
This is a very basic question regarding the inverse of a function.
it is clear that this is not a one-to-one function; therefore unless we restrict the domain we cannot construct the inverse. But there is also another condition: if we try to construct the inverse at the neighbourhood of a point 
we must ensure that 
. So, for 
since 
we cannot write down the inverse function. Is this the same as saying “the inverse of 
is not analytic at 
?
If I look at
[Analyticity of the inverse does imply non-vanishing of the derivative (if the function is differentiable). On the other hand, it is possible to have a (non-analytic) inverse even when the derivative vanishes; consider for instance the function
. -T]