User franklin - MathOverflow

Answer by Franklin for Fiction books about mathematicians?

2012-07-10T17:10:04Z

Godel, Escher and Bach: An eternal golden braid?

Answer by Franklin for Proofs without words

2012-05-29T22:27:02Z

I just saw this proof, which is of course not mine.

link text

Hilbert-Samuel function and that of the irreducible components.

2012-04-17T16:20:48Z

How to obtain a relation between the Hilbert-Samuel function of the local ring at a point of a reduced, but not necessarily irreducible variety, and the Hilbert-Samuel functions of the corresponding local rings of its irreducible components?

More concrete. R, a regular local Noetherian ring, complete if you wish, I an ideal in R that is the intersection of some prime ideals I_k such that there are no embedded primes.

I am looking for a formula relating the HS function of R/I and those of R/I_k? The filtration is with respect to the maximal ideal of R.

Edit: Info: There is also an analogous formula for Hilbert functions (analogous to the associativity formula for multiplicity). Proposition 3.2 in Equimultiplicity and blowing up, by Herrmann, Ikeda and Orbanz. $H^{(i)}[\underline{x},a,M]=\sum_{p∈Assh(M/aM)}e(\underline{x},R/p)H^{(i)}[aR_p,M_p]$, where $M$ is finitely generated $R$-module, $a$ and ideal in $R$, and $\underline{x}$ a multiplicity system for $M/aM$. If I put $R$ as my ring $R/I$, $a$ as the maximal ideal, and $M:=R$, if I understood their definition of $Assh$ this only gives me information about those components $p$ having $dim(R/p)=dim(M)=dim(R)$.

Answer by Franklin for The role of the mean value theorem (MVT) in first-year calculus.

2010-07-11T17:47:25Z

I agree there are many problems in the approaches done in many of the calculus books used but I disagree about the mean value theorem (Lagrange theorem for me). It is the cornerstone of analysis. You probably have some treatment in mind or a whole list of them. Lagrange theorem have the combined power of Bolzano's theorem (continuity of the reals, for what is worth) and the notion of derivative. If you want to pass global info from the derivative to the function, the mean value theorem is the place to go. Of course one has to be clear that the problem is really about the "how" are things presented. After all MVT is equivalent to the continuity of the reals.

One trivial change that I always try to do is a simple change in the writing. Instead of writing the equation with the derivative isolated in one side write the function isolated (like a Taylor). Also with the definition of derivative. instead of writing the derivative in one side of the equation writing it inside the limit. That apparently unimportant change has as outcome that students grasp better the connection between them: generalize MVT to Taylor, derivative to differential, use MVT in application.

--more added after the first comment---

Oh, that's true. Nevertheless the two are related. Of course the question about teaching is not a well stated one. It depends on the goals of the course, i.e. what it is that you want your students to be able to do. One unavoidable goal in a calculus course is to study nice functions. Nowadays, this means continuous and differentiable functions (although a close look at most of the courses tells us that the class of interest is much narrower.)

The concept of derivative seems to be then, required, although I bet a good course can also be planned with the notion of power series instead (which seems to be returning to times before Newton-Leibntz but I am not so sure [ask Doron Zeilberger about it]). A less chocking approach is to put both concepts side by side. And MVT is a way of linking them link.

I have to say that the way programs evolve is by taking the old ones and doing little "improvements". It is true that in the scope of basic ordinary calculus course you can skip MVT only losing the possibility of asking problems like "prove that $|sin(x)|\leq|x|$".

But again, it is a problem of goals. It is also good to take into account that learning process works starting from the horizon of already acquired knowledge. Even if students at the end cannot even remember the statement or how to use it, it prepares them for further development. Or just remember, you the working mathematician, how many times (if not every time), you have gone to a conference in which you don't understand a thing, in which you only remember two or three names at the end. Now remember how many times just knowing that that name (or word) exists or that is related to some other name has opened a completely new road in your research. It works exactly the same for students, even though they are not doing research. I say this to point out that the validity of an element in a curriculum program should not only be judged by the ability of students to actually get it but also by the grounds it creates to build on top of it. Education is a process that involve not only teaching but also evaluating. Maybe it is better to look at how MVT is evaluated instead. If it is playing a role of a connective element then it is wrong to evaluate the skills of applying it (which involve both understanding and skill). Changing the evaluation method is a less destructive approach than elimination from the program. Uff, I have written too much. If I forgot to say something I will say it later.

Is PA consistent? do we know it?

2011-05-26T22:08:16Z

1) (By Goedel's) One can not prove, in PA, a formula that can be interpreted to express the consistency of PA. (Hopefully I said it right. Specialists correct me, please). 2) There are proofs (although for the purpose of this question I should putt it in quotations marks) of the consistency of PA.

The questions are: A) Is it the consistency of PA still a mathematical question that can be considered open? B) Is it a mathematical question? (To this I dare to say that it is a mathematical question. Goedel himself translated it into a specific formula, but then I have question C) C) Is it accepting the proofs of the consistency of PA as conclusive a mathematically justified act or an act of taking a philosophical stance?

Motivation: There is a discussion in the mailing list FOM (Foundations Of Mathematics) about this topic, in part motivated by this talk link text . I thought a discussion about this fundamental matter concerns most mathematicians and wanted to bring it to a wider audience.

Edit: It is simple. Either: 1) Consistency of PA is proved and has a proof (as claimed by some in FOM) as valid as any other theorem in math, or 2) On top of the existing proofs a philosophic choice is needed (which explains the length of the discussions in FOM, justifies closing this question but contradicts what is being claimed emphatically by some in FOM)

But you see. If 1) is the case then there is no need for the lengthy discussions and this is a concrete math question as any other, terminating with a proof.

........................................................................................

Edit 2: Thank you all. Although I had seen some of these arguments at FOM now I think I have my ideas more organized and I can make my question more concrete. I would like to try to put aside what involves 'believes'. In, I think, all the answers shown there has been this action entering the argument quite soon, e.g. In Chow's: (approx.) If you believe in the existence of the naturals then con(PA) follows. In Friedman's (approx.) If you believe in (About a dozen Basic axioms) + (1/n subsequences) then con(PA) follows.

I want to put aside that initial action because (1): It is a philosophical question and that is not what I want to discuss, (2): Because of: If I believe (propositional logic) + (p/-p) then I believe ... for example (everything you can say) and maybe (3): Because I, personally, don't do math to believe what I prove. When I show P->Q, in a sequence of self imposed constrained steps I don't do it with the purpose of showing that, and at the end I don't have a complete conviction that, Q is a property of whatever could be a real world. But that is just philosophy and philosophy allows for any sort of choices. That is why I want to put it aside, at least until the moment in which it is inevitably needed.

My question is: Is any of the systems that prove con(PA) a system that has itself been proven consistent?

Why to ask this question? Regardless of how your feelings are about the ontological nature of what you prove. We can say that, since an inconsistent system proves everything, a consistent system is a bit more interesting for not doing so. At least if it is because there is not always a proof in which you use modus ponens twice (after you have found p/-p) for everything that you want to prove.

I guess that also, to answer the question above, it should be clarified what to accept for a consistency proof. Let's leave it kind of open and just try to delay the need for a philosophic stance as much as possible.

How/where are semi-log resolutions used?

2010-07-23T13:53:40Z

In the paper, by János Kollár there is problem 19 (page 8). It is one more strict resolution. A resolution that leaves untouched the semi-simple-normal-crossings singularities of pairs.

My question is: How/where is that kind of resolution used/needed?

Quick definitions:

Pair: $(X,D)$ with $X$ algebraic variety and $D$ a Weil divisor on it.

Semi-simple-normal-crossings: A point in $X$ where $X$ is (locally) a union of coordinates hyperplanes and $D$ is given by intersecting $X$ with some of the other coordinate hyperplanes not contained in $X$.

Answer by Franklin for Most helpful math resources on the web

2010-10-07T00:47:39Z

http://www.mathjobs.org/jobs

Answer by Franklin for Good books on theory of distributions

2010-08-27T16:17:02Z

I liked Functional Analysis by Kosaku Yosida. It is book on functional analysis but oriented to get the applications of it to differential equations.

Answer by Franklin for Looking for applications of a nice result in linear algebra

2010-08-27T16:13:03Z

That result sits inside a wider set of results. Search for spectral theorem, functional calculus of linear operators.

Books could be Halmos, A Hilbert Space problem book if you also need to read more about linear operators in general I think in Conway's Functional Analysis there is also stuff about these results, together with an introduction to functional analysis.

Answer by Franklin for "In the sequel" - outdated mathematical jargon or precise technical term?

2010-07-28T23:39:10Z

From Online etymology dictionary

sequel: early 15c., "train of followers," from O.Fr. sequelle, from L.L. sequela "that which follows, result, consequence," from sequi "to follow," from PIE base *sekw- (cf. Skt. sacate "accompanies, follows," Avestan hacaiti, Gk. hepesthai "to follow," Lith. seku "to follow," L. secundus "second, the following," O.Ir. sechim "I follow"). Meaning "consequence" is attested from late 15c. Meaning "story that follows and continues another" first recorded 1510s.

Poll about your proof of resolution of singularities and a request for advice

2010-07-27T01:47:54Z

The questions first: What is the proof of resolution of singularities that you know?

Why am I asking?: There are a number of proofs of resolution of singularities of varieties over a field of characteristic zero, all with more or less similar flavor but different in technical details and in choices that the resolution algorithm allows us to make. When writing a proof that uses specific features of some of those details I can't stop being uneasy about assuming the reader read about the specific constructions elsewhere. I would like to know from MOers what proof you have seen and if you have a reason for the choice, if it was a choice, I would like to hear it too.

Maybe asking about what you know is too invasive. I am just asking for the proof that you happened to find in your way, even you have only read a few lines of it.

The purpose of the question: The conspicuous one. To get a sense, by a rough approximation and a small sample, of what proofs are more culturally known. Have a concrete feeling when sending a reader to find the details in other paper, either of feeling OK with it or of guilt.

It is a question about fashion, which also has its role in mathematics... and knowing what the fashion is is useful.

What details?: Although I had in mind a specific detail of the proofs I didn't mention it because it is not the only one that changes from proof to proof and because the result of the poll gives information about all of them. Examples are: the resolution invariant, the ways of making the local descriptions of the centers of blowings-up match to form a globally defined smooth subvariety, the ways of getting functoriality and the different meanings that functoriality can have...

(edit) Forgot the "request for advise": If you have would like to give advise about how you have dealt with similar situations and describe your example that is welcomed.

It is a community wiki question, so feel free to change what is said here if needed or if you want the poll to also give information about other questions that you would like to be answered. (or for correcting the English!)

Answer by Franklin for Normal intersections of submanifolds

2010-07-26T22:00:31Z

If I take M to be the filled torus $D^2\times S^1$ and $M_1$ and $M_2$ two circles (the two transversal generators of the torus $S^1\times S^1$) then when you can contract one of them a little so that they no longer intersect.

Uff, manyfolds. Then replace $D^2\times S^1$ by $S^3$ and keep the two circles.

Answer by Franklin for How much reading do you do before you attack a problem?

2010-07-23T04:21:42Z

When you know the definitions, of the elements of your problem, no doubt, start attacking the problem. Ever if you still don't know the definitions for the most general form of the problem and only for a simplified version of it the answer is the same. No better sense of what the problem is about than by putting your own hand on it. Even if it only serves to get to some conclusion that you could have easily read somewhere. It could even happen that you solve the problem by putting together two or three things that you read but more important in research than solving the problem is understanding it. Because after solving it, you need to find new problems to continue. You don't understand anything better than those that you do yourself. The above doesn't mean don't read. It means don't wait a second before start doing all you can yourself.

Answer by Franklin for Undergraduate approach to learning math

2010-07-21T00:17:28Z

Why are you taking courses? Don't take courses if you don't need it.

Some psychologists match children development to human history. We have a period in which we play with dirt, a period of wars and fights, periods of obscurantism, renaissance,... You can let yourself be guided by history (math history) to study math. The most important to study first are those topics appearing first in history, the classics. Take this in a broad sense, after learning about the problem of squaring the circle you can read the proofs of the irrationality of Pi right away without waiting a proportional time to the one human kind waited to know them.

Don't be too eager about the "hard core" courses. Most of the time, what is hard core about them is an overwhelming number of definitions to learn. eg. Much more useful than an advanced algebra course, in which you learn the (should I least some) huuuge number of definitions that they will give you, is to solve the same number of high school problems in algebra. If you let the definitions come in some osmotic-historical-like way that will be enough and you get a better grasp of them than after a year of being drowned with a list of definitions and theorems that most of them are exercises (and most of them exercises simpler than the ones you would be solving if using the time in a different way). A key point is that what is important is not "what" but "how". It doesn't matter if you run of swim, what is important is to either run of swim a lot and with the finest technique, to keep the muscles trained. It is the same, with math.

Courses serve as orientation and motivation. They tell you what is important (if it is being taught in the course it should be important then) in the area and motivates you to solve problems (they give you homework) but, there are alternatives to courses to find orientation and motivation. Reading courses. Many of your professor would be willing to point to some sections in some books and tell you some names of theorems and concepts that are important and from that you got all the orientation that a course can give you. Join two more friends take a book recommended (maybe by some professor), a book with lots of problems and sit down with a fork and a knife and eat it like a gourmet pizza, solve each and every single problem.

Answer by Franklin for [Simplex Algorithm] Optimality

2010-07-18T13:38:08Z

I think something should be wrong. If a vertex doesn't give you the maximum then there is a neighboring vertex which improves the objective function and Simplex moves to that position. Maybe you are missing some step in the procedure when you get that -30.

Answer by Franklin for Fatou's Lemma and the bounded convergence theorem.

2010-07-16T11:19:52Z

If $f_n$ is the characteristic function of the interval $[n,2n]$ then $f_n\rightarrow0$ but $\int f_n = n$ which does not tend to zero. You need even more than the boundedness. Adding that your space is of finite measure will help is you want to bound with a constant. Alternatively you can bound with an integrable function. Without a bound the example above can be even worst.

Answer by Franklin for Factoring and solving trinomials

2010-07-15T14:50:42Z

Factor where? Over the rationals? Take a look to Berlekamp algorith. http://en.wikipedia.org/wiki/Berlekamp%27s_algorithm

The right link is http://en.wikipedia.org/wiki/Berlekamp%E2%80%93Zassenhaus_algorithm to Berlekamp–Zassenhaus algorithm.

Answer by Franklin for Abstract Thought vs Calculation

2010-07-15T16:26:30Z

All I see here are calculations. It only changed the nature of the object which you calculate with and its relation to the the final goal. For this reason I still can not make clear sense of the question.

Answer by Franklin for Direct proof of irrationality?

2010-07-15T16:18:47Z

Rational numbers have finite continuous fractions.

$\sqrt{2}=1+1/(\sqrt{2}+1)=1+1/(2+1/(\sqrt{2}+1))=...$ Then the continuous fraction is not finite 1+1/2+1/2+1/2+...

The geometric proof (not the one in Wikipedia), the one that proves $\sqrt{2}$ is not commensurable with $1$ is also direct (and is essentially the same as the continuous fraction).

Answer by Franklin for The Fundamental Theorem of Calculus in Lebesgue Theory

2010-07-14T14:05:20Z

For every function $f$ with $f'$ integrable there is a function $g$ equal to $f$ everywhere but a point such that $\int_{a}^{b}g'dx=g(b)-g(a)$. Take g(x)=f(x) for x different from b and g(b)=\int_{a}^{b}f'dx+f(a).

Answer by Franklin for Collapsing contractible subsets of the two-disk.

2010-07-13T19:10:31Z

Imagine that the Cantor set is on one diameter and that $\Lambda$ consists of the vertical cords passing through the Cantor. After collapsing you get a space that have some points that removing them makes it disconnected. Therefore is not homeomorphic to the disc.

Answer by Franklin for Magic trick based on deep mathematics

2010-07-13T03:34:27Z

Two persons, A and B, perform this trick. The public (or one from the public) chooses two natural numbers and give A the sum and B the product. A and B will ask each other, alternatively, the only single question "Do you know the numbers?" answering only yes or no until both find the numbers. There is a strategy such that for any input and only doing this, A and B will manage to find the original numbers.

I have never seen magicians actually performing this, but is perfectly doable.

This was a problem in the shortlist of the proposed problem for some international mathematical olympiad. Unfortunately I don't remember which. If someone remembers or finds it. Tell us please. i would also like to know.

Answer by Franklin for Functoriality of Hironaka's resolution of singularities

2010-07-13T01:16:25Z

The original theorem proved by Hironaka was not functorial. That feature was added later.

Answer by Franklin for Theorems in Euclidean geometry with attractive proofs using more advanced methods

2010-07-11T15:56:07Z

Let's see. Most of the results of projective geometry are much harder to show by more elementary methods. I think the reason for this is that they rest more on the incidence axioms while the elementary methods play more with the metric properties.

An interesting question that arises for any of these examples is to detect why is that the case that the elementary proofs are harder.

A classical example is the constructibility of regular polyhedral with ruler and compass.

What strict resolutions of singularities are needed?

2010-07-11T21:19:46Z

Suppose we have a collection, $S$, of singularities types and consider a resolution of singularities (this is: a proper birrational morphism $Y\rightarrow X$ such that Y only contains singularities of the types in the collection $S$ and such that the map is an isomorphism over the points of $X$ with singularity types in $S$.).

For example, if $S$ consists only of the smooth points then $S$-strict resolution is just the standard resolution of singularities. If $S$ consists of smooth points and simple normal crossing points $S$-strict resolution exists. If $S$ consists of only the smooth points and normal crossings points then there is no $S$-strict resolution (in embedding dimension at least 3). As you can't resolve a pinch piont $(x^2+yz^2=0)$ without blowing up normal crossing points.

Have you come around situations in which it is needed strict resolutions (excluding the cases of the first two examples)?

Extension of question (I guess this should go here and no in a new question) It is exactly the same question but this time asking for resolutions that never involve blowing up a center that intersects the S-locus of the total transform.

eg: -)For S= the smooth points the usual resolution is still fine since it never need to blow-up smooth points. If S is the smooth points and the simple normal crossings singularities this is already not known (I think).

Again the question is: Have you come around situations in which it is needed strict resolutions in this sense (excluding the first example in this second part)?

Answer by Franklin for Proofs without words

2010-07-11T15:11:05Z

Let $0\leq x,y,z,t\leq1$ Prove that $x(1-y)+t(1-x)+z(1-t)+y(1-z)\leq 2$.

Draw a 1x1 square and mark in consecutive sides disjoint segments starting at the vertexes of lengths $x,y,z,t$. Joining the consecutive end points of the intervals that are not vertexes of the square form four triangles, the area of the triangles is the left hand side divided by 2, the area of the square is the right hand side divided by 2.

Answer by Franklin for Examples of common false beliefs in mathematics.

2010-07-11T15:00:32Z

The distinction between convergence and uniform convergence. It even got Cauchy in its time.

Does listing the prime factors always stop?

2010-07-11T13:10:02Z

Take a natural number's prime factors and list them increasingly and repeating them according to multiplicity. Concatenate their decimal (or in any base) representation to get a new number and repeat the process. Does this always end in a prime number for any input?

Answer by Franklin for Hironaka desingularisation theorem -- new proofs in literature?

2010-07-11T02:38:44Z

Bierstone, E., Milman, P.: A simple constructive proof of canonical resolution of singularities.

Answer by Franklin for Quick proofs of hard theorems

2010-07-11T01:49:31Z

Gelfand–Mazur theorem. "A complex Banach algebra, with unit 1, in which every nonzero element is invertible, is isometrically isomorphic to the complex numbers."

The proof is the one everybody knows.

Comment by Franklin

2013-05-02T22:00:39Z

Orthogonal bases are useful enough and perhaps you are happy with them not being of normalized vectors. You may run Gram-Schmidt without normalization and I think you don't need square roots of positive elements for that.

Comment by Franklin

2013-04-10T22:21:29Z

Think on the power series of exp(x) and exp(-x), and why exp(x)exp(-x)=1.

Comment by Franklin

2012-07-12T21:36:33Z

O_o Epitome of fiction: Alice in wonderland. GEB, described by his publishing company as "a metaphorical fugue on minds and machines in the spirit of Lewis Carroll". I always thought Godel was a mathematician. I think what you mean to say is that it is not a novel.

Comment by Franklin

2012-04-18T22:06:37Z

A friend told me: To take $k[[s,t]]\mapsto k[[x,y]]/(xy)$ by sending $s\mapsto x+y$ and $t\mapsto x^n$. The image is $R$ and the kernel seems to be $t(s^n-t)$, which gives us that $R=k[[s,t]]/(t(s^n-t))$. But this is a hypersurface singularity. The Hilbert-Samuel function is only going to depend on the order, in this case $2$. Perhaps that is not the ring you meant.

Comment by Franklin

2012-04-18T19:15:58Z

@Knutson. I don't understand the example. Is $R:=k[[x+y,x^n]]/(xy)$? How does one sees the two lines?

Comment by Franklin

2012-04-17T21:06:16Z

Not $a$ as my ideal, $R$ is already my $R/I$ and a the maximal ideal.

Comment by Franklin

2012-04-17T20:55:13Z

Perhaps I should put the comment above as info in the question.

Comment by Franklin

2012-04-17T20:54:28Z

There is also an analogous formula for Hilbert functions. Proposition 3.2 in Equimultiplicity and blowing up, by Herrmann, Ikeda and Orbanz. $H^{(i)}[\underline{x},a,M]=\sum_{p\in Assh(M/aM)} e(\underline{x},R/p)H^{(i)}[aR_p,M_p]$, where M is finitely generated R-module, a and ideal in R, and $\underline{x}$ a multiplicity system for M/aM. If I put R as my ring, a as my ideal I, and M:=R. But if I understood their definition of Assh this only gives me information about those components p having dim(R/p)=dim(M)=dim(R).

Comment by Franklin

2011-12-16T18:54:34Z

@Andreas: Yes, and more than truthful, they should be well thought. So when one of them answers 'no' it means that he has not enough information to deduce the two numbers. Yes, there are many ways of cheating if they agree to do so beforehand. But it is probably hard to agree saying the numbers in binary by only saying 'yes' or 'no' and asking if the other knows the numbers. I don't remember if the original wording of the problem had something to avoid cheating but the cool thing is that they can actually deduce the two numbers, fairly, just from the answers.

Comment by Franklin

2011-07-03T23:06:19Z

Böttcher, A.; Silbermann, B. (2006), Analysis of Toeplitz Operators, Springer Monographs in Mathematics (2nd ed.), Springer-Verlag, ISBN 9783540324348 .

Comment by Franklin

2011-06-05T07:11:26Z

@Schwede. What were you referring to in your comment? I would like to understand. What is the sense in which they are highly non-controllable?

Comment by Franklin

2011-06-03T03:06:05Z

wow! thank you.

Comment by Franklin

2011-06-03T00:59:42Z

and that effective can still mean a lot of iterations logic.pdmi.ras.ru/~grigorev/pub/…

Comment by Franklin

2011-06-02T21:42:16Z

ok. I needed 5 more characters to post it.

Comment by Franklin

2011-06-02T21:20:43Z

Thank you Prof.