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https://mathoverflow.net/questions/32765 | 18 | If functors are morphisms between categories, and natural transformations are morphisms between functors, what's a morphism between natural transformations? Is there ever a need for such a notion?
| https://mathoverflow.net/users/2592 | What's after natural transformations? | (Small) categories form what's called a 2-category, which is a structure that has objects, morphisms (functors), and morphisms between morphisms (natural transformations). There are also n-categories, which have a deeper morphisms structure. A google search will point you to a lot of references about n-categories. But ... | 13 | https://mathoverflow.net/users/4183 | 32767 | 21,271 |
https://mathoverflow.net/questions/32761 | 1 | Let $\mathbb{F}\_q$ be a finite field with characteristic $p$ and $p < q$ (i.e. not a prime field). Let $D\subseteq \mathbb{F}\_q$ be a some set with $|D|=n$. Find a non-empty subset $\{x\_1,\dots,x\_k\} \subseteq D$ such that $x\_1+\cdots+x\_k=s$ for some given $s\in\mathbb{F}\_q$. This is the definition of the subset... | https://mathoverflow.net/users/7692 | Number of subset sums | There are $\binom{n}{k}$ ways to choose $k$ elements from $n$ elements. If we consider their sum it is "expected" to be equal to every element of the field with the same probability. Hence we get $\frac{1}{q}\binom{n}{k}$ for the number of solutions of $x\_1 + x\_2 + \cdots + x\_k = s$. This is the first part of your q... | 5 | https://mathoverflow.net/users/7079 | 32768 | 21,272 |
https://mathoverflow.net/questions/32762 | 3 | This is a sequel to my previous question [colimits of spectral sequences](https://mathoverflow.net/questions/28972/colimits-of-spectral-sequences) .
I think I've found the answer in S.A. Mitchell's paper "Hypercohomology spectra and Thomason's descent theorem". There the author states a "colimit lemma" (page 42) for ... | https://mathoverflow.net/users/1246 | Convergence of right half-plane spectral sequence bounded on the right | It doesn't -- just filter any chain complex trivially, the resulting spectral sequence has vanishing $E\_2$ and obviously doesn't converge in any sense. So you have to look at how the spectral sequence is constructed. A very useful notion in this context is *conditional convergence*, which is probably satisfied in the ... | 5 | https://mathoverflow.net/users/4183 | 32769 | 21,273 |
https://mathoverflow.net/questions/32766 | 23 | Let $X$ be a complex normal projective variety.
**Is there any sufficient condition to guarantee the torsion-freeness of Picard group of $X$?**
One technique I sometimes use is following:
If $X$ can be represented by GIT quotient $Y//G$ for some projective variety $Y$ with well-known Picard group, then by using Kem... | https://mathoverflow.net/users/4643 | Torsion-freeness of Picard group | [EDIT: A previous version mistakenly argued that the fundamental group of *X* was responsible for torsion in the Picard group. I hope that this is correct now! Btw, there is probably a more direct way of arguing, but I cannot find one at the moment.]
The Picard group of *X* is torsion free if and only if the group ${... | 30 | https://mathoverflow.net/users/4344 | 32773 | 21,275 |
https://mathoverflow.net/questions/32787 | 22 | Given any finitely-presented group $G$, there are a few equivalent techniques for constructing smooth/PL 4-manifolds $M$ such that $\pi\_1 M$ is isomorphic to $G$. For most constructions of these 4-manifolds, they embed naturally in $S^5$ (as the boundary of regular neighbourhoods of $2$-complexes in $S^5$.)
Questio... | https://mathoverflow.net/users/1465 | Word problem for fundamental group of submanifolds of the 4-sphere | Update:
My memory was quite blurry about this when I originally answered.
See Gonzáles-Acuña, Gordon, Simon, ``Unsolvable problems about higher-dimensional knots and related groups,'' L’Enseignement Mathématique (2) 56 (2010), 143-171.
They prove that any finitely presented group is a subgroup of the fundamental... | 15 | https://mathoverflow.net/users/1335 | 32803 | 21,293 |
https://mathoverflow.net/questions/32782 | 4 | Under what conditions on a and b is there a distribution $f\_{a,b}$ such that the product $XY$ of two independent realizations $X$ and $Y$ from $f\_{a,b}$ has a Beta(a,b) distribution?
A standard result on deriving the distibution of the product of two variables indicates that the p.d.f. $f\_{a,b}$ needs to satisfy:
... | https://mathoverflow.net/users/7801 | "Square root" of Beta(a,b) distribution | A partial answer is that $f\_{a,b}$ exists for every positive integer $b$.
To see this, first recall that for every positive $s$ and $a$ the distribution Gamma$(s,a)$ has density proportional to $z^{s-1}e^{-az}$ on $z\ge0$ and that the sum of independent Gamma$(s,a)$ and Gamma$(t,a)$ is Gamma$(s+t,a)$.
If $b=1$, c... | 5 | https://mathoverflow.net/users/4661 | 32805 | 21,295 |
https://mathoverflow.net/questions/32791 | 3 | For $C,D$ small categories, and $f : C \to D$ a functor between them, there is a precomposition, or "inverse image", functor $f^\* = (-) \circ f : Set^D \to Set^C$. It has a left and a right adjoint. What are their definitions, and in particular what is the right adjoint $f\_\*$? I couldn't find a definition in terms o... | https://mathoverflow.net/users/7674 | How is the right adjoint $f_*$ to the inverse image functor $f^*$ described for functor categories $Set^C$, $Set^D$ and $f : C \to D$ | Given a functor $f:\mathcal{C}\to\mathcal{D}$ and any complete category $\mathcal{A}$ (e.g., take $\mathcal{A}=\text{Sets}$ to get the case you are asking about), there exists a right-adjoint $f\_{\ast}:[\mathcal{C},\mathcal{A}]\to[\mathcal{D},\mathcal{A}]$ to the "inverse image functor" $f^{\ast}$ and this is given by... | 7 | https://mathoverflow.net/users/6485 | 32808 | 21,298 |
https://mathoverflow.net/questions/32752 | 1 | I was reading a paper where I came across the following argument :
For any $x$ in $M$ and for a geodesic ball $B(x; \varepsilon)$ in a compact Riemannian
manifold $M$ with injectivity radius bigger than or equal to epsilon, and for any smooth eigenfunction $f$ of Laplacian on $M$, we have :
the square of $f(x)$ is... | https://mathoverflow.net/users/6953 | The comparison between the square of the functional value and the sum of squares of the $L^2$ norms of function and its Laplacian | You are working on a Riemann surface. That bit of information is rather important, as [Sobolev inequalites](http://en.wikipedia.org/wiki/Sobolev_inequality) depends rather much on the dimension of the space. The basic Sobolev inequality is
$$ \| f \|\_{L^q(\Omega)} \leq C (\| \partial f \|\_{L^p(\Omega)} + \| f\|\_{L^p... | 4 | https://mathoverflow.net/users/3948 | 32811 | 21,300 |
https://mathoverflow.net/questions/32788 | 3 | Given a map of topological spaces $f:X\rightarrow Y$. Assume, that $X$ has finite Lebesgue dimension. I am wondering, what dim$(f(X))$ might be. Of course, if $f$ is a homeomorphism onto its image, then it's just dim$(X)$. On the other hand there are the space filling curves, that show, that the dimension might increas... | https://mathoverflow.net/users/3969 | Lebesgue dimension of images | Some results from Engelking's dimension theory book:
If $f: X \mapsto Y$ is a closed, continuous and surjective function between normal spaces $X$ and $Y$, and $\forall y \in Y: | f^{-1}[{y}] | \le k$ for some integer $k \ge 1$, then $\dim(Y) \le \dim(X) + (k-1)$.
If $f: X \mapsto Y$ is an open, continuous and sur... | 6 | https://mathoverflow.net/users/2060 | 32813 | 21,301 |
https://mathoverflow.net/questions/32812 | 6 | Suppose we have a sequence, { f\_n }, of symplectic diffeomorphisms of R^{2n} converging to a function f. By f\_n converging to f I mean: f\_n converges to f in C^k(B\_r) for every r > 0, where B\_r is the ball centered at the origin of radius r. Suppose k > 0.
Question: Is f is diffeomorphism? Certainly, f is sympl... | https://mathoverflow.net/users/nan | Is the limit of symplectic diffeomophisms a diffeomorphism? | No, $f$ doesn't have to be invertible. It has to be injective, even in the volume-preserving category rather than the symplectic category. None of the singular values of $Df$ can go to 0, because some other singular value would go to $\infty$. This shows that it is locally injective, and then globally it is not possibl... | 8 | https://mathoverflow.net/users/1450 | 32816 | 21,303 |
https://mathoverflow.net/questions/32815 | 7 | A $n$-th degree polynomial has precisely $n$ roots. So it is natural to ask the question
how large ( and small) can be the measure of a set where a polynomial takes small values ?
This, and other interesting variation of this must have been studied in depth.
I would really appreciate any reference to the relevant ... | https://mathoverflow.net/users/6766 | How large (small) can be the measure of a set where a polynomial takes small values ? | There is first Polya's estimate that if $f$ is a monic polynomial, then
$$
|\{x\in \mathbb{R}:\quad |f(x)|\leq 2\}| \leq 4.
$$
A proof can be found in the book "Proofs from the book". One can obtain inequalities for non-monic polynomials by rescaling.
Second there is Cartan's lemma or estimate. It can for example be... | 8 | https://mathoverflow.net/users/3983 | 32818 | 21,304 |
https://mathoverflow.net/questions/32799 | 7 | Let *X* be a normed space and denote by *X\** the space of all bounded linear functionals on *X*. Take a linear subspace *G ≤ X\** which separates the elements of *X*, i.e., for each *x ∈ X*, there is an *f ∈ G* with *f(x) ≠ 0*. Denote by *B* the closed unit ball in *X*. Now consider a linear subspace *Y ≤ X*. The ques... | https://mathoverflow.net/users/7804 | If *Y* is weakly dense in *X*, is the unit ball in *Y* necessarily dense in the unit ball in *X*? | If I understand the question correctly, then maybe you are after special cases, as well as a general comment. So, as one of your examples suggests, one special case is to let G be a Banach space, considered as sitting inside its own bidual, and let $X=G^\*$. Thus G induces the usual weak\*-topology on X.
So an exampl... | 2 | https://mathoverflow.net/users/406 | 32831 | 21,311 |
https://mathoverflow.net/questions/32744 | 3 | I have the following discrete time dynamical system
$$ y(t+1) = y(t) + \frac{1}{1+\exp(z+ u f y(t))} ,\quad y(0)=0,$$
where $z$ is a real number $f$ and $u$ are non-negative reals. I know I have little hope of obtaining a closed form solution for this process. But, actually, for my application, a "better" solution invo... | https://mathoverflow.net/users/3812 | Limit of a discrete time dynamical system | The continuous model of the problem suggests that the limit does depend on $f$ (and $u$). More precisely, it depends on how fast the parameter $f$ is suppressed in the expression whose limit you are taking; behavior of $\lim y(nt, f\_n)/n$ will depend on the limit of $nf\_n$ as $n \to \infty$. The answer will be a func... | 3 | https://mathoverflow.net/users/6579 | 32839 | 21,317 |
https://mathoverflow.net/questions/32847 | 5 | Is there a non-projective flat module over a local ring?
Here I assume the ring is commutative with unit.
| https://mathoverflow.net/users/5292 | Is there a non-projective flat module over a local ring? | $\mathbb{Q}$ is flat over $\mathbb{Z}\_p$, but not projective.
| 20 | https://mathoverflow.net/users/6950 | 32850 | 21,325 |
https://mathoverflow.net/questions/32824 | 16 | Languages decidable by weak models of computation often have certain necessary characteristics, e.g. [the pumping lemma for regular languages](http://en.wikipedia.org/wiki/Pumping_lemma_for_regular_languages) or [the pumping lemma for context-free languages](http://en.wikipedia.org/wiki/Pumping_lemma_for_context-free_l... | https://mathoverflow.net/users/6950 | Structure theorems for Turing-decidable languages? | As far as I know, the answer is no. We don't have an argument not based (directly or indirectly) on digonalization for even proving there is a language outside Dec (The set of Turing decidable languages).
Characterizing a class means having criteria for showing when a language is in the class and when a language is n... | 8 | https://mathoverflow.net/users/7507 | 32856 | 21,329 |
https://mathoverflow.net/questions/32854 | 4 | I've been looking at some thick subcategories in $K^b(R-proj)$ (the homotopy category of bounded chain complexes of projective modules), and, as expected, I'm running into the question of when chain complexes split quite often. I'm wondering what sorts of useful criteria there are for determining when chain complexes s... | https://mathoverflow.net/users/6936 | When do chain complexes decompose as a direct sum? | One result that guarantees such a decomposition comes from looking at the homological support of such complexes (assuming that $R$ is commutative so we have a tensor product). The homological support of a complex $A$ is just the union of the supports of the $H^i(A)$ as $R$-modules. Then it is a result of Balmer, in the... | 5 | https://mathoverflow.net/users/310 | 32868 | 21,335 |
https://mathoverflow.net/questions/32862 | 1 | By using a regular hexagonal arrangement it is simple to fit 19 identical circles into a larger circle of five times the radius with no circles overlapping. This leaves an area equal to six smaller circles uncovered. Is it possible to rearrange the 19 circles to accommodate a twentieth circle of the same size into the ... | https://mathoverflow.net/users/7819 | Settling a circular argument: room for one more? | No, according to the information at <https://erich-friedman.github.io/packing/cirincir/>
| 4 | https://mathoverflow.net/users/3684 | 32869 | 21,336 |
https://mathoverflow.net/questions/32875 | 9 | **Background**
In [his beautifully short answer](https://mathoverflow.net/questions/30647/fibered-products-of-cyclic-groups/30656#30656) to [a previous question of mine](https://mathoverflow.net/questions/30647/fibered-products-of-cyclic-groups), Robin Chapman asserted the following.
>
> Let $m,n,r$ be natural nu... | https://mathoverflow.net/users/394 | Lifting units from modulus n to modulus mn. | This can be done in an elementary way using the Chinese remainder theorem.
First of all, note $m$ only appears in the conclusion in the context of the product
$mn$. For any common prime factor of $m$ and $n$ suck that prime's contribution to $m$ into $n$ instead, which changes the meaning of $m$ and $n$ but does not ... | 13 | https://mathoverflow.net/users/3272 | 32878 | 21,343 |
https://mathoverflow.net/questions/32891 | 16 | (This question is motivated by the reading of the article *Large numbers and unprovable theorems* by Joel Spencer, which can be found at <http://mathdl.maa.org/images/upload_library/22/Ford/Spencer669-675.pdf> and that I recommend).
We all know of the game where a card of a predefined size, say 3x5 cm, is given to ev... | https://mathoverflow.net/users/1234 | Finding the largest integer describable with a string of symbols of predefined length | My first remark is that if you allow players to pick their own theories, but only allow consistent theories to win, then you will not be able to compute the winner of the contest. The reason is that the consistency of a theory is not in general a computable question.
To see this, suppose that we had an algorithm tha... | 16 | https://mathoverflow.net/users/1946 | 32893 | 21,352 |
https://mathoverflow.net/questions/32098 | 12 | There are different ways of showing that a given sequence $a\_0,a\_1,a\_2,\dots$
of integers, say, is nonnegative. For example, one can show that $a\_n$ count
something, or express $a\_n$ as a (multiple) sum of obviously positive numbers.
Another recipe is manipulating with the corresponding generating series
$A(x)=\su... | https://mathoverflow.net/users/4953 | Positivity of sequences via generating series | It is impossible, and not just for rational functions. To see this, let's consider the coefficients $b\_n$ of $p(x) A(r(x))$ as functions of the $a\_n$, the coefficients of $A(x)$. Since $r(0) = 0$ (as it must be) we see that:
* Each $b\_n$ is a linear combination of $a\_0, \dots, a\_n$; i.e. we have an upper-triangu... | 3 | https://mathoverflow.net/users/6545 | 32902 | 21,359 |
https://mathoverflow.net/questions/32908 | 21 | Yesterday, in the short course on model theory I am currently teaching, I gave the following nice application of downward Lowenheim-Skolem which I found in W. Hodges *A Shorter Model Theory*:
Thm: Let $G$ be an infinite simple group, and let $\kappa$ be an infinite cardinal with $\kappa \leq |G|$. Then there exists a... | https://mathoverflow.net/users/1149 | Three questions on large simple groups and model theory | The class of simple groups isn't elementary. To see this, first note that if it were, then an ultraproduct of simple groups would be simple. But an ultraproduct of the finite alternating groups is clearly not simple. (An $n$-cycle cannot be expressed as a product of less than $n/3$ conjugates of $(1 2 3)$ and so an ult... | 21 | https://mathoverflow.net/users/4706 | 32909 | 21,364 |
https://mathoverflow.net/questions/32892 | 67 | In Ebbinghaus-Flum-Thomas's *Introduction to Mathematical Logic,* the following assertion is made:
If ZFC is consistent, then one can obtain a polynomial $P(x\_1, ..., x\_n)$ which has no roots in the integers. However, this cannot be proved (within ZFC).
So if $P$ has no roots, then mathematics (=ZFC, for now) can... | https://mathoverflow.net/users/344 | Does anyone know a polynomial whose lack of roots can't be proved? | In his Master's Thesis, Merlin Carl has computed a polynomial that is solvable in the integers iff ZFC is inconsistent. A joint paper with his advisor Boris Moroz on this subject can be found at <http://www.math.uni-bonn.de/people/carl/preprint.pdf>.
| 41 | https://mathoverflow.net/users/7743 | 32914 | 21,366 |
https://mathoverflow.net/questions/32920 | 3 | Does anyone have an idea how to prove the following? It is a step in the proof of some theorem in a book about gaussian processes.
Let $f\_n$ be an orthonormal sequence of gaussian variables. Consider $\sum\_{n \geq 1} a\_n f\_n$ and assume that it is convergent in $L^2$. Show that it converges also almost everywhere... | https://mathoverflow.net/users/7827 | Convergence of a series of orthonormal gaussian variables | You can assume $f\_n$ have mean zero, independence implies
$$E[ \sum\_{n=1}^{\infty} f\_n| \mathcal{A}m ] = \sum\_{n=1}^m f\_n $$
where $\mathcal{A}\_m$ is the sigma field generated by $f\_1,..,f\_m$
This means that $Z\_m=\sum\_{n=1}^mf\_n$ is a martingal with respect to $\mathcal{A}\_m$.
You then have to apply... | 7 | https://mathoverflow.net/users/6531 | 32924 | 21,372 |
https://mathoverflow.net/questions/32880 | 11 | Does anybody know who was Atle Selberg's advisor?
I find it interesting to know the advisor's impact on his students.
Unfortunately, in Selberg case, this information (even his advisor's name) seems to be nowhere to be found.
| https://mathoverflow.net/users/7820 | Selberg's advisor? | I'm a student at the university of Oslo, so I thought I'd have a go at this. I just talked to Erling Størmer (Carl's grandson) who is a professor emeritus here. He said that in practice Atle had no advisor. Of course someone must have signed the papers but he doesn't know who (I don't really see what difference it make... | 26 | https://mathoverflow.net/users/1123 | 32927 | 21,373 |
https://mathoverflow.net/questions/31810 | 32 | One common reason given for the circularity of manhole covers is that they can't fall through the manhole. For convex manhole covers, this property is equivalent to having [constant width](http://en.wikipedia.org/wiki/Curve_of_constant_width) — if you have different widths, just orient the cover so that the shorter wid... | https://mathoverflow.net/users/27 | Nonconvex manhole covers | Here is a construction of a polygon that cannot fall through the hole.
Begin with a regular $MN$-gon circumscribed around a unit circle, where $M\gg N\gg 1$ and $N$ is even. For every $M$th side, draw a segment of length 1 extending this side in the clockwise direction. Take a rectangular neighborhood of width $\vare... | 8 | https://mathoverflow.net/users/4354 | 32928 | 21,374 |
https://mathoverflow.net/questions/32925 | 6 | We all know that Gödel showed that there, in a formal system, are true statements that are non-provable (undecidable). In ZFC, there's Kaplansky's Conjecture, the Whitehead problem, etc.
We can also agree that we're sure to find more non-provable statements in ZFC. What I'm curious about is the following:
What is t... | https://mathoverflow.net/users/7607 | When is a statement provable? | Not all statements that are not provable in ZFC are "strong", if by strong you mean that
ZFC + the statement in question is stronger than ZFC in the sense that it implies the consistency of ZFC.
The typical example is the Continuum Hypothesis (CH). ZFC + CH is consistent iff ZFC is,
and in this sense, CH is not stro... | 9 | https://mathoverflow.net/users/7743 | 32929 | 21,375 |
https://mathoverflow.net/questions/32423 | 3 | All the definitions I have seen in the literature present the possibility of computing the $\epsilon$-pseudospectrum of a rectangular matrix $A^(m \* n)$ with $m < n$ but I have not seen any definitions for the case $m > n$, why ? is there a trivial answer ?
| https://mathoverflow.net/users/7728 | Pseudo-spectrum of a short and fat rectangular matrix ? | I think that in the "short and fat" case (more columns than rows), the $\epsilon$-pseudospectrum with $\epsilon>0$ is the whole complex plane. So you don't get much information out of pseudospectra.
In more detail: Suppose you define the $\epsilon$-pseudospectrum of a "short and fat" matrix $A$ as the set of all comp... | 3 | https://mathoverflow.net/users/2610 | 32936 | 21,380 |
https://mathoverflow.net/questions/32912 | 6 | There are two common ways to define a series-parallel graph (or 2-terminal series-parallel graph).
Definition 1
* start with K\_2 marking both vertices as terminals
* repeatedly join two smaller 2-terminal s/p graphs either in series or in parallel
Definition 2
* start with K\_2
* repeatedly replace a single ed... | https://mathoverflow.net/users/1492 | Where is it shown how to construct a decomposition tree for a series-parallel graph in linear time? | It is easy enough to build the tree from a definition 2 description. Here is a sketch:
Suppose that SP graph $G$ occurs from subdividing graph $G'$ at an edge $e$, either in series or in parallel. Recursively, let $T'$ be the tree for $G'$. Then $e$ is a leaf of $T'$. Add two children below that leaf. Mark the node $... | 9 | https://mathoverflow.net/users/297 | 32946 | 21,386 |
https://mathoverflow.net/questions/32965 | 1 | I have heard about KKM (Knaster-Kuratowski-Mazurkiewicz) theorem in nonlinear analysis and I am trying to use that for a theorem I am working on. The original paper (1929) is in German and all the references I have found on the Net mainly deal with generalizations of this theorem or present the theorem with a lot of co... | https://mathoverflow.net/users/6748 | Knaster-Kuratowski-Mazurkiewicz (KKM ) Thoerem | There is a [book](http://books.google.co.uk/books?id=h_DaTNA8mw8C&printsec=frontcover&dq=kkm+theory&hl=en&ei=zm5ITJvNN8n24ga9yZneDA&sa=X&oi=book_result&ct=result&resnum=1&ved=0CCwQ6AEwAA#v=onepage&q&f=false) devoted entirely to KKM theory and its applications. It contains
an outline of the classical KKM theorem as well... | 3 | https://mathoverflow.net/users/5371 | 32969 | 21,394 |
https://mathoverflow.net/questions/32954 | 8 | A recent question on the notion and notation of multiplicative integrals
( [What is the standard notation for a multiplicative integral?](https://mathoverflow.net/questions/32705/what-is-the-standard-notation-for-a-multiplicative-integral) ) induced me to play with the Riemann products of the Gamma function, in order t... | https://mathoverflow.net/users/6101 | Multiplicative integral of $\Gamma(x)$ | Here a start:
We have the [reflection formula](http://en.wikipedia.org/wiki/Gamma_function#Properties)
$$z! (1-z)! = \frac{\pi z (1-z)}{\sin (\pi z)}.$$
Taking $\log$'s,
$$\log (z!) + \log ((1-z)!) = \log \pi + \log z + \log (1-z) - \log \sin (\pi z).$$
Split our integral in half and rearrange it
$$\int\_0^1 \log (z... | 11 | https://mathoverflow.net/users/297 | 32979 | 21,401 |
https://mathoverflow.net/questions/32964 | 17 | The content of this note was the topic of a lecture by Günter Harder at the School on Automorphic Forms, Trieste 2000. The actual problem comes from the article
[A little bit of number theory](http://sunsite.ubc.ca/DigitalMathArchive/Langlands/pdf/bit-ps.pdf) by Langlands.
The problem is about a connection between t... | https://mathoverflow.net/users/3503 | Quaternary quadratic forms and Elliptic curves via Langlands? | Chapter 16 of Jacquet--Langlands is about the Jacquet--Langlands correspondence, which concerns the transfer of automorphic forms from quaternion algebras to the group $GL\_2$.
The modularity of the theta series that you write down is a (very) special case of this
correspondence.
But probably Langlands more had in m... | 21 | https://mathoverflow.net/users/2874 | 32981 | 21,403 |
https://mathoverflow.net/questions/32990 | 1 | We know that $A$ embeds into $A$\*\* (the double dual space of $A$ ). Is the following true? If $\Psi$ is in $A$\*\* and weak\* continuous, is there an element $a \in A$ such that $ \Psi$ is the evaluation functional at $a$? That is to say, $\Psi(f)=f(a)$ for any $ f \in A^{\*}$?
| https://mathoverflow.net/users/7360 | Double dual space of a C* algebra A | This has nothing to do with operator algebras as it is well-known that if $B$ is a linear topological space (local convexity and Hausdorff-ness hypothesis included), the topological dual of $B^{\ast}$ endowed with the wk\*-topology is $B$ itself, that is, every wk\*-continuous functional on $B^{\ast}$ is an evaluation ... | 8 | https://mathoverflow.net/users/2562 | 32993 | 21,410 |
https://mathoverflow.net/questions/32957 | 4 | Let $X$ be a circle that with one corner (i.e. think of a triangle where we smooth out two of the vertices). Now let us consider the topological torus $M \cong \mathbb{T}^n$ which is the product of $n$ copies of $X$. Note that $M$ contains $n$ distinct circles along which it is not smooth.
Finally, suppose we are gi... | https://mathoverflow.net/users/nan | Morse Theory on Non-smooth Manifolds | Answer to your question is negative if $n\ge 1$ ($n$ is a number of smooth circles and I suppose that there is non-smooth circles). Indeed, in that case your manifold is homeomorphic to torus $T^k$ and the homeomorphism $\varphi\colon M\to T^k$ could be taken smooth on the smooth part of $M$. The image of non-smooth pa... | 5 | https://mathoverflow.net/users/2823 | 33006 | 21,416 |
https://mathoverflow.net/questions/32966 | 3 | Let me refer to Jech's "Set Theory" Chap. 33 Determinacy:
"With each subset A of $\omega^\omega$ we associate the following game $G\_A$, played by two players I and II. First I chooses a natural number $a\_0$, then II chooses a natural number $b\_0$, then I chooses $a\_1$, then II chooses $b\_1$, and so on. The game ... | https://mathoverflow.net/users/6466 | Determinacy interchanging the roles of both players | The answer is no. The game $G'(A)$ you describe is just the same as the complementary game $G(A^c)$ from I's point of view. So the question is: if one of the players has a winning strategy in $G(A)$, does one of the players have a winning strategy in $G(A^c)$? Using AC one can construct a set $A$ for which this is not ... | 7 | https://mathoverflow.net/users/2436 | 33008 | 21,418 |
https://mathoverflow.net/questions/33010 | 4 | It's not difficult to find a function $f\colon \mathbb R \to \mathbb R$ such that its restriction to any (not trivial) interval is surjective. Does anyone know whether such a function is necessarily not (Lebesgue) measurable? I'm pretty sure this is the case, but I cannot prove it.
Here is an example of such an $f$. ... | https://mathoverflow.net/users/7845 | Real-valued function "so surjective" that should be non measurable | No. [Conway's base-13 function](http://en.wikipedia.org/wiki/Conway_base_13_function) is measurable.
See this [question](https://mathoverflow.net/questions/32641/is-conways-base-13-function-measurable).
| 6 | https://mathoverflow.net/users/1335 | 33012 | 21,420 |
https://mathoverflow.net/questions/32923 | 18 | I'm currently trying to understand the concepts and theory behind some of the common proof verifiers out there, but am not quite sure on the exact nature and construction of the sort of systems/proof calculi they use. Are they essentially based on higher-order logics that use Henkin semantics, or is there something mor... | https://mathoverflow.net/users/602 | How do proof verifiers work? |
>
> * What exactly is the role of type theory in creating higher-order logics? Same goes with category theory/model theory, which I believe is an alternative.
>
>
>
Don't think of type theory, categorical logic, and model theory as *alternatives* to one another. Each step on the progression forgets progressively... | 23 | https://mathoverflow.net/users/1610 | 33019 | 21,424 |
https://mathoverflow.net/questions/32757 | 8 | It is usual to mention theorems of the kind:
Th. Assume there is a proper class of Woodin cardinals, $\mathbb{P} $ is a partial order and $G \subseteq \mathbb{P}$ is V-generic, then $V \models \phi \iff V[G] \models \phi$
where $\phi$ is some set theoretic statement (like "the Strong Omega Conjecture holds"), as s... | https://mathoverflow.net/users/6466 | Tractability of forcing-invariant statements under large cardinals | Typically, generic absoluteness is a consequence of a stronger property, that in many cases is really the goal one is after. To explain this stronger property, let me begin by reviewing two important absoluteness results.
1) The first is **Mostowski's Absoluteness**. Suppose $\phi$ is $\Sigma^1\_1(a)$, where $a\in\om... | 14 | https://mathoverflow.net/users/6085 | 33022 | 21,426 |
https://mathoverflow.net/questions/33028 | 7 | Does ZF prove that there exists a set S such that S is not in the closure of {{s} : s in S} under at-most-countable unions?
| https://mathoverflow.net/users/nan | Can iterating countable unions give every set? (ZF) | Moti Gitik proved (assuming large cardinals) that all uncountable alephs can have cofinality ω. [*All uncountable cardinals can be singular*, Israel J. Math. 35 (1980), 61–88] I believe this may be the model you're looking for, but I don't know what happens to non-wellorderable sets in that model. According to the abst... | 9 | https://mathoverflow.net/users/2000 | 33032 | 21,434 |
https://mathoverflow.net/questions/33037 | 7 | Let $S$ = { $a^2b^3$ : $a, b \in \mathbb{Z}\_{>1}$ }.
Does there exist $n$ such that $n$, $n+1 \in S$?
Motivation: I was thinking about [Question on consecutive integers with similar prime factorizations](https://mathoverflow.net/questions/32412/question-on-consecutive-integers-with-similar-prime-factorizations), w... | https://mathoverflow.net/users/7434 | Are there consecutive integers of the form $a^2b^3$ where $a$, $b$ > 1? | As once remarked by Mahler, $x^2 - 8 y^2 = 1$ has infinitely many solutions with $27 | x$.
| 25 | https://mathoverflow.net/users/nan | 33044 | 21,440 |
https://mathoverflow.net/questions/33043 | 10 | Suppose I have a graph $G$ with vertex set $V$, edge set $E \subseteq {V \choose 2}$, a poistive integer $d$, and a weight function $w:E \to \mathbb{R}^{+}$. Is there a nice algorithmic way to decide if there is an assignment of vertices to points in Euclidean space, i.e. a function $f: V(G) \to \mathbb{R}^d$ such that... | https://mathoverflow.net/users/4558 | Algorithm for embedding a graph with metric constraints | Let's say the graph is complete, and has weights on edges that satisfy triangle inequality. If you want an isometric embedding (which your original question indicates), then there's a necessary and sufficient characterization: the squares of the distances must be of *negative type*: specifically, given the $D\_{ij} = d... | 12 | https://mathoverflow.net/users/972 | 33047 | 21,441 |
https://mathoverflow.net/questions/33036 | 3 | The reason I am wondering this is that all of the reductions from 3-SAT => quadratic programming (or similar NP-hard reductions) involve encoding the underlying NP-hard problem into feasibility testing. If you take out that trick, can you still find another way to encode it?
EDIT: Killed the duality stuff, don't know... | https://mathoverflow.net/users/4642 | Is quadratic programming still NP-hard if you have bounds and a feasible point? | No; I think what you're observing is a side effect of reducing from deicision problems; if you tried to encode an NP optimization problem, you'd end up using more than just the feasibility machinery.
Take MAX-CUT, with variables $x\_i\in\{-1,+1\}$ indicating taking a vertex or not, and
$W\in\mathbb{R}^{n\times n}$ a ... | 3 | https://mathoverflow.net/users/2621 | 33048 | 21,442 |
https://mathoverflow.net/questions/33049 | 0 | Let $D$ be the open unit disk, and $J$ a Jordan arc (that is, a homeomorphic copy of $[0, 1]$) that lies in $D$, except $J(0)$ lies on the boundary of $D$, say $J(0)=1$. I would like to see that $D\smallsetminus J([0, 1])$ is a path-connected topological space. Please help, if you can. Thanks!
| https://mathoverflow.net/users/7305 | A Jordan arc in the unit disk | It's certainly the case that $\mathbb{R}^2\setminus J$ is path connected.
So any two points in $D\setminus J$ are joined by a path in $\mathbb{R}^2$
missing $J$. If this path isn't in $D$ it hits the boundary of $J$
but then replace part of the path by an arc of the boundary of $D$.
Then one can replace this part of th... | 4 | https://mathoverflow.net/users/4213 | 33053 | 21,446 |
https://mathoverflow.net/questions/33021 | 28 | In pondering [this](https://mathoverflow.net/questions/32938/surfaces-in-mathbbp3-with-isolated-singularities) MO question and people's efforts to answer it, and recalling also something that I learned in my youth about using Morse theory ideas to prove some results of Lefschetz in the complex case, I seem to have lear... | https://mathoverflow.net/users/6666 | Birational invariants and fundamental groups | I would like to mention one more homotopy invariant of smooth projective varieties that is also a birational invariant. If $X$ is a smooth projective variety, then the torsion subgroup $T(X)$ of ${\rm H}^3(X,\mathbb{Z})$ is a birational invariant. This is explained in the beautiful paper "Some elementary examples of un... | 19 | https://mathoverflow.net/users/4344 | 33061 | 21,449 |
https://mathoverflow.net/questions/33058 | 18 | Recently the elliptic curve $E:y^2+y=x^3-x^2$ of conductor $11$ (which appears in [my answer](https://mathoverflow.net/questions/11747/galoisian-sets-of-prime-numbers/32939#32939)) became my favourite elliptic over $\bf Q$ because the associated modular form
$$
F=q\prod\_{n>0}(1-q^n)^2(1-q^{11n})^2
$$
is such a nice "$... | https://mathoverflow.net/users/2821 | Eta-products and modular elliptic curves | There is an exhaustive list in the paper [Y. Martin and K. Ono, Eta-Quotients and Elliptic
Curves, *Proc. Amer. Math Soc.* **125** (1997), no. 11, 3169--3176].
Suppose that $E\_N$ is an elliptic curve of conductor
$N$, then the corresponding $L$-series is assigned to the eta product
$$
\eta(a\tau)\eta(ab\tau)\eta(ac\t... | 21 | https://mathoverflow.net/users/4953 | 33063 | 21,450 |
https://mathoverflow.net/questions/32716 | 6 | The Cheeger constant of a finite graph measures the "bottleneckedness" of the graph, and is defined as:
$$h(G) := \min\Bigg\lbrace\frac{|\partial A|}{|A|} \Bigg| A\subset V, 0<|A|\leq \frac{|V|}{2} \Bigg\rbrace$$
Here $V$ is the vertex set of $G$ and $\partial A$ denotes the collection of all edges going from a ver... | https://mathoverflow.net/users/6015 | What is the Cheeger constant of a cubical subset of the cubic lattice? | The result (for 3 dimensions and I think easily generalises to any dimension) follows from Theorem 3 of the Bollobás and Leader paper. The theorem (in 3 dimensions) states that for any subset $A$ of the vertices $V$ of a cubical grid of side length $N$ with $|A|\leq\frac{N^3}{2}$ that $$|\partial A| \geq \min\_{r=1,2,3... | 2 | https://mathoverflow.net/users/6015 | 33073 | 21,457 |
https://mathoverflow.net/questions/33046 | 11 | From [Wikipedia](http://en.wikipedia.org/wiki/Oracle_machine) (bold emphasis at the end is mine):
>
> In complexity theory and computability theory, an oracle machine is an abstract machine used to study decision problems. It can be visualized as a Turing machine with a black box, called an oracle, which is able to... | https://mathoverflow.net/users/2592 | Aren't "oracle machines" unsound concepts? | Oracle machines are not "problematic". Let us consider oracle machines that have the halting problem as their oracle. Now, by "the halting problem" we mean the collection of all Turing machines (without oracle) that halt when started with an empty tape.
This can be decided by an oracle machine with the halting proble... | 17 | https://mathoverflow.net/users/7743 | 33075 | 21,459 |
https://mathoverflow.net/questions/33084 | 7 | I have heard sometimes that the only dimensions $k$ for which there exists a "*good*" smooth product $P:S^k\times S^k\to S^k$ are $k = 0,1,3,7$ (the above products corresponding to $\mathbb Z\_2, U(1)\subset\mathbb C$, the product of unit quaternions and of unit Cayley numbers).
I would like to ask for references abo... | https://mathoverflow.net/users/5628 | Are there good product rules on the $k$-sphere? | I guess that what you want to say is that for $k=0, 1,3, 7$ the sphere $S^k$ is a *H-space* (as in the comment).
Also, Lie groups are a word to look at for having ``good'' products (better than the above, this works for $S^k$ with $k= 0,1,3$ not $7$, see the comment below).
See Theorem 2.16 of the following [nice b... | 7 | https://mathoverflow.net/users/5753 | 33086 | 21,468 |
https://mathoverflow.net/questions/31308 | 3 | Apologies if my question is poorly phrased. I'm a computer scientist trying to teach myself about generalized functions. (Simple explanations are preferred. -- Thanks.)
One of the references I'm studying states that the space of Schwartz test functions of rapid decrease is the set of infinitely differentiable functio... | https://mathoverflow.net/users/7486 | Splines, harmonic analysis, singular integrals. | I believe I now have the answer to the question. The power of $\omega$ appear from the taylor expansion of $e^{i\omega.t\_j}$ (in section 2.3 of Kent and Mardia's paper)
Thanks.
(Apologies for the seeming bit of self promotion, but I've tagged this as the correct answer.)
| 1 | https://mathoverflow.net/users/7486 | 33087 | 21,469 |
https://mathoverflow.net/questions/33088 | 6 | How is entropy of a general probability measure defined?
| https://mathoverflow.net/users/7699 | Entropy of a general prob. measure | The Entropy of a function $f$ with respect to a measure $\mu$ is
$$ Ent\_{\mu}(f)=\int f \log f d\mu - \int f d\mu \log(\int f d\mu ) $$
The entropy of a probability distribution $P$ with respect to $\mu$ is given by $ Ent(\frac{dP }{d\mu })$. I a not aware of a general definition that would not implie a reference ... | 9 | https://mathoverflow.net/users/6531 | 33090 | 21,471 |
https://mathoverflow.net/questions/33096 | 28 | Hi everyone, the summer break is coming and I am thinking of reading something about mathematical logic. Could anyone please give me some reading materials on this subject?
| https://mathoverflow.net/users/3849 | Reading materials for mathematical logic | Here are a few suggestions (which depending on your background may be more or less useful):
1. [Logic and Structure](https://rads.stackoverflow.com/amzn/click/3540208798) by Dirk van Dalen. I have used this as a textbook when teaching mathematical logic and for that purpose it is decent. Some people find it a bit dry... | 27 | https://mathoverflow.net/users/6485 | 33102 | 21,479 |
https://mathoverflow.net/questions/33112 | 5 | Can anyone estimate N such that Prob( 0 is in the convex hull of $N$ points ) >= .95
for points uniformly scatterered in $[-1,1]^d$, $d = 2, 3, 4, 10$ ?
The application is nearest-neghbour interpolation:
given values $z\_j$ at sample points $X\_j$, and a query point $P$,
one chooses the $N$ $X\_j$ nearest to $P$ (... | https://mathoverflow.net/users/6749 | Estimate probability( 0 is in the convex hull of N random points ) ? | This is a classical and essentially geometric problem. In fact, *the answer does not depend
on the distribution of the points* (as long as the distribution is centrally symmetric).
The following result is due to Wendel ([link](http://www.ams.org/mathscinet/search/publdoc.html?arg3=&co4=AND&co5=AND&co6=AND&co7=AND&d... | 17 | https://mathoverflow.net/users/5371 | 33132 | 21,500 |
https://mathoverflow.net/questions/33103 | 7 | This problem arose when trying to understand the stack of twisted stable maps into a stack (specifically BG), as introduced by Dan Abramovich, Angelo Vistoli and several co-authors (Olsson, Graber, Corti,...). However, my specific question can be formulated without mentioning such beasts and I will do so. If anyone wan... | https://mathoverflow.net/users/1310 | Twisted curves, admissible covers, and an algebraic analogue of a specific monodromy computation | (written hurriedly, hope it is semi-helpful and -correct)
Let's say you're over some field k. If it's not separably closed, basechange until it is. The strictly completed local ring of a ramification point in your base P^1 is going to look like k[[t]]. Now remove that point to give yourself Spec k((t)). The restricti... | 6 | https://mathoverflow.net/users/431 | 33136 | 21,504 |
https://mathoverflow.net/questions/33138 | 7 | Hi all, sorry if this is a dumb question, I don't know much about von Neumann algebras except the definition and a few relevant facts I've managed to prove by myself so I expect the answer will turn out to be well known. Anyway, let $\mathcal{H}$ be a Hilbert space, and suppose that $P$ is a commuting set of self-adjoi... | https://mathoverflow.net/users/7842 | Question about von Neumann algebra generated by a complete algebra of projections | The answer "yes" follows from Theorem 2.8 of Bade's "[On Boolean algebras of projections and algebras of operators](http://www.ams.org/journals/tran/1955-080-02/S0002-9947-1955-0073954-0/home.html)," 1955, which is in the more general context of algebras of operators on a Banach space.
Bade had previously proven a l... | 6 | https://mathoverflow.net/users/1119 | 33144 | 21,511 |
https://mathoverflow.net/questions/33158 | 11 | Name a theorem T that has a proof based upon the truth of a conjecture C, and also has another proof based upon the falsehood of the same conjecture C, but for longtime has no known direct proof that is independent of C. For instance, it would be a claim that can be proved if P=NP, and can also be proved if P is differ... | https://mathoverflow.net/users/7874 | Use of Conjectures to Prove a Theorem | If you are asking for an example of such a method in action, then you have the theorem of Hecke-Deuring-Heilbronn that $h(D) \rightarrow \infty$ as $D \rightarrow \infty$, where $h(D)$ is the class number of the imaginary quadratic field with discriminant $D$.
The Hecke part is that the result is true if there are n... | 20 | https://mathoverflow.net/users/2938 | 33159 | 21,520 |
https://mathoverflow.net/questions/33162 | 14 | I don't know any number theory, so excuse me if the following notions have names that I'm not using.
For a positive natural number $n\in{\mathbb N}\_{\geq 1}$, define $Log(n)\in{\mathbb N}$ to be the ``total exponent" of $n$. That is, in the prime factorization of $n$ it is the total number of primes being multiplie... | https://mathoverflow.net/users/2811 | How divisible is the average integer? | Hopefully I've read all your notation correctly. If so, by playing (very) fast and loose with heuristics, I think your friend is right that the answer is 0.
Your function $Log(n)$ is the additive function $\Omega(n)$. According to the mathworld entry
<http://mathworld.wolfram.com/PrimeFactor.html>,
$\Omega(n)$ ha... | 15 | https://mathoverflow.net/users/35575 | 33164 | 21,521 |
https://mathoverflow.net/questions/28776 | 37 | A theorem of Hecke (discussed in [this question](https://mathoverflow.net/questions/12352))
shows that if $L$ is a number field, then the image of the
different $\mathcal D\_L$ in the ideal class group of $L$ is a square.
Hecke's proof, and all other proofs that I know, establish this essentially by
evaluating all qu... | https://mathoverflow.net/users/2874 | Does (the ideal class of) the different of a number field have a canonical square root? | The following example shows that, in its strongest form, the answer to Professor Emerton's question is no. This answer is essentially an elaboration on what is already in the comments.
Let $p \equiv q \equiv 5 \pmod 8$. Let $K/\mathbb{Q}$ be a cyclic
extension of degree four totally ramified at $p$ and $q$ and unrami... | 26 | https://mathoverflow.net/users/nan | 33165 | 21,522 |
https://mathoverflow.net/questions/33169 | 10 | Thm (Kronecker).- If all conjugates of an algebraic integer lie on the unit circle, then the integer is a root of unity.
Question: Can one provide a good effective version of this? That is: given that we have an algebraic integer alpha of degree <=d, can we show that alpha has a conjugate that is at least epsilon awa... | https://mathoverflow.net/users/398 | (Good) effective version of Kronecker's theorem? | If $M(\alpha)$ is the Mahler measure of $\alpha$, then the largest conjugate of $\alpha$
has absolute value at least
$$1 + \frac{\log(M(\alpha))}{d}.$$
Lehmer's conjecture implies that this at least $O(d^{-1})$ away from one.
Dobrowolski's lower bound for $M(\alpha)$ shows that there is a conjugate at least
$O(d^{... | 14 | https://mathoverflow.net/users/nan | 33170 | 21,526 |
https://mathoverflow.net/questions/33176 | 4 | Let $G := SO\_n(R)$ be equipped with the Euclidean metric on vectors of length $n^2$. Is it true that for any $\epsilon >0$, there is a finite subgroup of $G$ which intersects every metric ball of radius $< \epsilon$ in $G$?
| https://mathoverflow.net/users/7821 | Can SO_n(R) be approximated arbitrarily well using a discrete subgroup? | Generalizing Robin's answer to arbitrary $n$:
Jordan's theorem implies that for any $n$ there is an integer $J(n)$ such that the index of a normal abelian subgroup of a finite subgroup of $GL(n,\mathbf{C})$ and hence $SO(n)$ is $\leq J(n)$. A theorem by Boris Weisfeiler (based on the classification of finite simple g... | 11 | https://mathoverflow.net/users/2349 | 33182 | 21,531 |
https://mathoverflow.net/questions/33181 | 3 | Let $f:C\to C'$ be a functor, and let $A$ be a locally presentable, complete, and cocomplete category. Then according to the paper I'm reading, the pullback functor, $f^\*:A^{C'}\to A^C$ (given by precomposition with $f$), admits left and right adjoints $f\_!$ and $f\_\*$. It's clear that the proof of this fact follows... | https://mathoverflow.net/users/1353 | Probably easy: Why is f*:A^C'->A^C continuous and cocontinuous for any functor f:C->C'? | Whenever $A$ has all small (co)limits, small (co)limits in functor categories $A^C$ are pointwise: a cone $F : I \to A^C$ is a (co)limit iff each of its “components” or “partial evaluations” $F(c) : I \to A$ is a (co)limit.
But the property of being a *pointwise* (co)limit is obviously preserved by composition with $... | 5 | https://mathoverflow.net/users/2273 | 33185 | 21,534 |
https://mathoverflow.net/questions/33120 | 2 | Let $n > 1$ be a square-free natural number, which is fixed. The assertion to be proved is the following:
>
> Let $p$ run through primes. Then, $$\left( \frac{n}{p} \right)$$ is equally distributed between $1$ and $-1$.
>
>
>
The precise statement of which is to be made using the appropriate asymptotic express... | https://mathoverflow.net/users/2938 | Distribution of quadratic residues of a fixed number without using Dedekind zeta function | This answer summarizes the above discussion: Extend $p \mapsto \left( \frac{n}{p} \right)$ to a multiplicative function $\chi$ on the positive integers. By quadratic reciprocity, $\chi$ is periodic modulo $4n$, and it is multiplicative by construction, so it is a character. We know that $L(1, \chi) \neq 0$. Thus, $\lim... | 2 | https://mathoverflow.net/users/297 | 33189 | 21,535 |
https://mathoverflow.net/questions/33191 | 6 | Let $K$ be a field and $E$ be an elliptic curve defined over $K$. It well understood the $K$-points on $E$ forms an abelian group. What is the structure of this group?(Depending on char($K$)?) Is it a direct sum of some well known abelian groups such as $\mathbb{Z}/m\mathbb{Z}$?
| https://mathoverflow.net/users/4171 | Elliptic curves — general structure of the group | First case: Complex numbers. Over $\mathbb C$ the structure as an abstract group is $\mathbb S^1 \oplus \mathbb S^1$ where $\mathbb S^1$ is the circle, i.e., $\mathbb R/\mathbb Z$. This follows as Robin Chapman notes [below](https://mathoverflow.net/questions/33191/elliptic-curves-general-structure-of-the-group#comment... | 10 | https://mathoverflow.net/users/2938 | 33194 | 21,536 |
https://mathoverflow.net/questions/33198 | 5 | There is a well-known Quillen's localization sequence for (algebraic) K-theory: $\dots\to K\_p^Y(X)\to K\_p(X)\to K\_p(X-Y)\to \dots$, where $Y\to X$ is a closed embedding of schemes.
Now suppose that $X$ is regular (and excellent of finite dimension, if needed).
Another well-known fact is that (in this case) the rela... | https://mathoverflow.net/users/2191 | On $\gamma$-graded pieces of the localization sequence for G-theory (i.e. for K'-theory) | The basic reference is Soule's paper "Operations en K-theorie algebrique" (Can. J. Math. 37 (1985) 488-550). Essentially, there is a grading on $K'$-theory (with rational coefficients) enabling one to interpret the localisation sequence as the long exact sequence of motivic homology. The filtration comes from the $\gam... | 6 | https://mathoverflow.net/users/5480 | 33203 | 21,541 |
https://mathoverflow.net/questions/33188 | 7 | Background:
-----------
The limit of a functor $F:D^{op}\to C$ is an object $\lim F$ representing the functor $$\ell F(x):=\operatorname{Psh}\_D(\ast,C(x, F(\cdot))),$$ where $\*$ denotes the terminal presheaf on $D$. (Notice that $C(x, F(\cdot)))$ is a presheaf on $D$).
We can define the limit of a functor weight... | https://mathoverflow.net/users/1353 | A slick definition of the Kan extension? | Firstly, the $W$-weighted limit $\lim^W F$ is defined to be a representation of $\operatorname{Psh}\_D(W,C(-,F-))$; the definition you've given isn't even well-typed.
There is no difference at all in $\mathrm{Set}$-enriched category theory between a representation and a universal arrow -- each determines the other, a... | 10 | https://mathoverflow.net/users/4262 | 33206 | 21,543 |
https://mathoverflow.net/questions/33205 | 6 | Is it possible that $\mbox{Tor }^{r+1}(M,N)=0 \ \ \forall N$ yet $\mbox{proj. dim }M>r$?
What I do know is that if $(A,\mathfrak{m})$ is Noetherian local and $M$ is finitely generated over $A$ then $\mbox{Tor }^{r+1}(M,A/\mathfrak{m})=0 $ if and only if $\mbox{proj. dim }M\leq r$.
Generally speaking, is $\mbox{Tor ... | https://mathoverflow.net/users/5292 | Tor and projective dimension | Over the integers, the rational numbers Q are flat and so Tor^i(Q,M) = 0 for all M and all i>0. However Q is not projective so has projective dimension 1.
| 10 | https://mathoverflow.net/users/51 | 33207 | 21,544 |
https://mathoverflow.net/questions/33214 | 0 | Let $f \colon X \to Y$ be a morphism of schemes, where $X$ and $Y$ are separated integral Noetherian schemes. Does there necessarily exist a nonempty open affine $U \subset Y$ such that $f^{-1}(U)$ is affine? If $X \to Y$ is not dominant, then the answer is clearly yes (take $U$ to have empty inverse image). Even for d... | https://mathoverflow.net/users/5094 | Are morphisms of schemes generically affine | I think the answer is no. Take any field $k$, and consider the natural morphism $\mathbb P^n\_k \to spec(k)$, where $\mathbb P^n\_k$ is the $n$-dimensional projective space. Since $spec(k)$ is, as topological space, a single point and $\mathbb P^n\_k$ is not affine, your open subset cannot exist.
| 9 | https://mathoverflow.net/users/7845 | 33215 | 21,550 |
https://mathoverflow.net/questions/33160 | 1 | For the following wave equation
$
\frac{{\partial ^2 p}}{{\partial ^2 x}} + \frac{{\partial ^2 p}}{{\partial ^2 y}} = A\frac{{\partial ^2 p}}{{\partial ^2 t}} + B\frac{{\partial p}}{{\partial t}}
$
is there a way to show that there are boundary conditions at or near positive and negative infinity, for both non-zero... | https://mathoverflow.net/users/7875 | Boundary conditions of wave equation near infinity | First of all, can you be more precise in your question? You are asking about boundary conditions at infinity, and this might make sense, but... for what purpose? do you need a set of conditions that imply existence and uniqueness of a global solution? or, do you need to classify solutions of the standard Cauchy problem... | 3 | https://mathoverflow.net/users/7294 | 33221 | 21,555 |
https://mathoverflow.net/questions/33228 | 7 | I recently came across a family of infinite graphs (in the context of two-dimensional convexity) that don't have induced 4-paths (paths with 4 vertices).
Note that the complement of a 4-path is again a 4-path.
Clearly, every induced $n+1$-cycle contains an induced $n$-path.
Hence, by the Strong Perfect Graph The... | https://mathoverflow.net/users/7743 | Simple proof that these graphs are perfect | These P4-free graphs are also known as [cographs](http://en.wikipedia.org/wiki/Cograph). A simple proof of the perfectness of such graphs was given by Seinsche, *On a property of the class of n-colorable graphs*, J. Comb. Th. Ser. B 16 (1974), 191–193. [MR0337679](http://www.ams.org/mathscinet-getitem?mr=337679) The ke... | 11 | https://mathoverflow.net/users/2000 | 33231 | 21,559 |
https://mathoverflow.net/questions/29590 | 54 | Let $R$ be a ring. A notable theorem of **N. Jacobson** states that if the identity $x^{n}=x$ holds for every $x \in R$ and a fixed $n \geq 2$ then $R$ is a commutative ring.
The proof of the result for the cases $n=2, 3,4$ is the subject matter of several well-known exercises in **Herstein**'s *Topics in Algebra*. T... | https://mathoverflow.net/users/1593 | A condition that implies commutativity | For fixed $n \in \mathbb{N}$, Birkhoff's completeness theorem implies that such a proof must exist in the first-order equational theory of rings - as I mentioned here in a recent [post](https://mathoverflow.net/questions/30220/abstract-thought-vs-calculation/30273#30273). Many years ago Stan Burris told me that John La... | 32 | https://mathoverflow.net/users/6716 | 33234 | 21,562 |
https://mathoverflow.net/questions/33199 | 8 | In *Singular points of complex hypersurfaces*, John Milnor proves the following theorem:
Let $x \in V$ be a point on a variety $V$ in $\mathbb{R}^n$ or $\mathbb{C}^n$. Assume $x$ is either a smooth point or an isolated singularity. Let $D\_{\epsilon}$ be the closed $\epsilon$-ball about $x$, $S\_{\epsilon}$ its bound... | https://mathoverflow.net/users/5094 | Small neighborhoods of singularities on varieties | There is a good paper of Goresky, "[Triangulation of Stratified Objects](http://www.jstor.org/stable/2042563)", that I think reasonably quickly implies Milnor's result and its generalization to non-isolated singularities. The result is that any Whitney-stratified set, and in particular any algebraic variety in $\mathbb... | 9 | https://mathoverflow.net/users/1450 | 33241 | 21,567 |
https://mathoverflow.net/questions/28125 | 13 | (1) Is the definition of [flop](http://en.wikipedia.org/wiki/Flip_%28algebraic_geometry%29) given by Wikipedia the industry standard?
(2) Regardless of the answer to (1), when is it expected that a birational transformation gives rise to a derived equivalence?
References to places where precise conjectures are reco... | https://mathoverflow.net/users/nan | What is a flop (and when are they conjectured to give derived equivalences)? | I'm not any kind of expert on this stuff and I'm not sure what the current state of this conjecture is, but Kawamata has conjectures in [this paper](http://arxiv.org/PS_cache/math/pdf/0205/0205287v3.pdf) and [this paper](http://arxiv.org/PS_cache/math/pdf/0311/0311139v2.pdf) regarding when two birational varieties have... | 5 | https://mathoverflow.net/users/7756 | 33244 | 21,569 |
https://mathoverflow.net/questions/33230 | 5 | The complex matrix exponential of a Hermitian matrix is unitary: $e^{-iH} = U$. Is there a name or a characterization for matrices Q whose real exponential is stochastic: $e^{-Q} = S$?
| https://mathoverflow.net/users/756 | Matrices whose exponential is stochastic | A matrix $A$ such that $\exp(tA)$ is (right) stochastic for all $t > 0$ should be called a "generator of a semigroup of stochastic matrices" or an "infinitesimally stochastic matrix". Clearly, since $A=\lim\_{t\to0} (\exp(tA)-I)/t$, (i) the sum of the elements in each row of $A$ has to be 0, and (ii) all non-diagonal e... | 6 | https://mathoverflow.net/users/6101 | 33247 | 21,571 |
https://mathoverflow.net/questions/33250 | 9 | There's a workshop at MSRI in a couple months on the Kervaire invariant problem that I'd really like to attend. I saw Hopkins speak about it a while back without understanding much of the talk, but I'm thinking that maybe by now I'm more prepared to see what it's all about. At least, I'd like to be able to get somethin... | https://mathoverflow.net/users/303 | references / general idea of kervaire invariant problem | Here's a recent survey article by Victor Snaith:
<http://chucha.math.cinvestav.mx/morfismos/v13n2/arfsurveyMFMS.pdf>
(I think there's also a copy on the arxiv)
| 6 | https://mathoverflow.net/users/6646 | 33251 | 21,573 |
https://mathoverflow.net/questions/33257 | 6 | The article I have checked for Baker's theorem is [Waldschmidt's](http://www.math.jussieu.fr/~miw/articles/pdf/GDL/SemBxMSGB.pdf). But the article and the citations therein are from the time of '88. Question:
>
> What is the the strongest known lower bound for Baker's theorem on linear forms on logarithms?
>
>
> ... | https://mathoverflow.net/users/2938 | Strongest known version of Baker's theorem | There is a big difference between linear forms in *many* logarithms and in *two* (or three) logarithms. The first case is covered in the archimedean case by the work of E. Matveev; Matveev's original works are hard even to specialists but there is a very nice survey [Yu. Nesterenko, Linear forms in logarithms of ration... | 7 | https://mathoverflow.net/users/4953 | 33259 | 21,577 |
https://mathoverflow.net/questions/33226 | 1 | Let $a\_1,a\_2 \ldots a\_n$ be a real-valued sequense with $a\_i=O(N^{-1})$.
How do I estimate Discrete Fourier Transform (DFT) of this sequence?
$$\hat{a}\_{j}=\sum\_{r=1}^{K}a\_{r}\exp\Bigl(-2\pi ij\frac{r}{K}\Bigr),\qquad K=O(N^\alpha),\quad \alpha<1$$
Can I say that DFT sequence $\operatorname{Re}[\hat{a}\_{i}]... | https://mathoverflow.net/users/3589 | Estimation of DFT | The answer is no, if you mean an uniform bound in $j$. Here is the example:
Fix $j$ and define
$$
a\_r = \begin{cases} \frac{1}{N}, & Re(\exp(-2\pi i j r/ K)) \geq 0;\\\
0, & otherwise.\end{cases}
$$
It is than easy to estimate that the number of $a\_r = \frac{1}{N}$ is comparable to $K$. Even more is true, one has... | 4 | https://mathoverflow.net/users/3983 | 33263 | 21,580 |
https://mathoverflow.net/questions/33282 | 11 | Does there exist a set $A$ such that $A=\{A\}$ ?
Edit(Peter LL): Such sets are called Quine atoms.
[Naive set theory By Paul Richard Halmos](http://dc108.4shared.com/download/YNLXDssM/Halmos_-_Naive_set_theory.djvu?tsid=20100725-105252-871bd162) On page three, the same question is asked.
Using the usual set nota... | https://mathoverflow.net/users/5627 | Can we have A={A} ? | In standard set theory (ZF) this kind of set is forbidden because of the [axiom of foundation](http://en.wikipedia.org/wiki/Axiom_of_foundation).
There are alternative axiomatisations of set theory, some of which do
not have an equivalent of the axiom of foundation. This is called
non-well-founded set theory. See e.g... | 33 | https://mathoverflow.net/users/1310 | 33283 | 21,584 |
https://mathoverflow.net/questions/20940 | 12 | In EGA IV, Sec. 16, Grothendieck defines the sheaf of principal parts as follows: Let $f:X\rightarrow S$ be a morphism of schemes and $\Delta:X\rightarrow X\times\_S X$ the diagonal morphism associated to $f$. $\Delta$ is an immersion, so the corresponding morphism $\Delta^{-1}\mathcal{O}\_{X\times\_S X}\rightarrow\mat... | https://mathoverflow.net/users/259 | Sheaves of Principal parts | The statement holds in general if $f : X \to S$ is a morphism of locally ringed spaces. The fibred product of locally ringed spaces can be constructed explicitly without gluing constructions, and also restricts to the fibred product of schemes. See [this article](http://maddin.110mb.com/pdf/faserprodukte.pdf) (german; ... | 8 | https://mathoverflow.net/users/2841 | 33286 | 21,586 |
https://mathoverflow.net/questions/33265 | 55 | It is widely stated that Fermat wrote his famous note on sums of powers ("Fermat's last theorem") in, or around, 1637. How do we know the date, if the note was only discovered after his death, in 1665?
My interest in this stems from the fact that if this is true, we can be absolutely certain that whatever proof Ferma... | https://mathoverflow.net/users/4790 | How do we know that Fermat wrote his famous note in 1637? | Not only do we not know the date, we don't even know whether he wrote the remark at all.
For all we know it might have been invented by his son Samuel, who published his father's comments.
In his letters, Fermat never mentioned the general case at all, but quite often posed the problem of solving the cases $n=3$ and... | 102 | https://mathoverflow.net/users/3503 | 33289 | 21,589 |
https://mathoverflow.net/questions/33269 | 34 | For any number field $K$, the Fontaine-Mazur conjecture predicts that any potentially semistable $p$-adic representation of the absolute Galois group $G\_K$ of $K$ that is almost everywhere unramified comes from algebraic geometry (i.e., is a subquotient of the etale cohomology of some variety over $K$, up to Tate twis... | https://mathoverflow.net/users/6074 | Fontaine-Mazur for GL_1 | Let $\chi$ be a one-dimensional geometric (in the sense of FM) $p$-adic Galois representation of $G\_K$ and let $\psi$ be the Hecke character of $K$ associated to $\chi$ by class field theory. The fact that $\chi$ is de Rham (=pst) at all primes above $p$ imples that $\psi$ is an *algebraic* Hecke character. Generally,... | 20 | https://mathoverflow.net/users/1021 | 33296 | 21,593 |
https://mathoverflow.net/questions/33303 | 2 | Say we have $n$-gons $P$ and $Q$. Is there any necessary condition for $Q = f(P)$, for some linear transformation $f : \mathbb{R}^2 \to \mathbb{R}^2$?
Sorry if this is too elementary / general.
| https://mathoverflow.net/users/2503 | Linear transformation takes a polygon to another one. | Jesse Douglas studied linear transformations of polygons on the complex plane in 1930s. He proved, in particular, that a transformation $z\_i{}'=\sum\_{i=1}^na\_{ij}z\_j$ (all numbers are complex) will transform a polygon $\pi=(z\_1,\cdots,z\_n)$ into a polygon $\pi'=(z\_1{}',\cdots,z\_n{}')$ if, and only if, the matri... | 5 | https://mathoverflow.net/users/5371 | 33309 | 21,600 |
https://mathoverflow.net/questions/33292 | 10 | It is apparent that the abc conjecture is deeply related to Arakelov theory. In one direction, it is shown in S. Lang, "Introduction to Arakelov Theory", that a certain height inequality in Arakelov theory implies the abc conjecture. I am wondering about the other direction and precise implications. Are there such resu... | https://mathoverflow.net/users/2938 | Implications of the abc conjecture in Arakelov theory | ABC is equivalent to the conjectured height inequality that Lang (or more precisely Vojta, in an appendix to Lang's book, following the ABC appendix he wrote for his own book) uses. This is shown in several papers by van Frankenhuijsen.
<http://research.uvu.edu/machiel/papers/abcrvhi.pdf>
<http://research.uvu.edu/... | 8 | https://mathoverflow.net/users/6579 | 33316 | 21,604 |
https://mathoverflow.net/questions/33294 | 16 | Is it true that the cardinality of every maximal linearly independent subset of a finitely generated free module $A^{n}$ is equal to $n$ (not just at most $n$, but in fact $n$)? Here $A$ is a nonzero commutative ring. I know that it's true if $A$ is Noetherian or integral domain. I thought it was not true in general bu... | https://mathoverflow.net/users/5292 | Cardinal of maximal linearly independent subsets of a free module | I think I have a counter-example. Let $A$ be the ring of functions $f$ from $\mathbb{C}^2 \setminus (0,0) \to \mathbb{C}$ such there is a polynomial $\widetilde{f} \in \mathbb{C}[x,y]$ such that $\widetilde{f}(x,y)=f(x,y)$ for all but finitely many $(x,y)$ in $\mathbb{C}^2$.
Map $A$ into $A^2$ by $f \mapsto (fx, fy)$... | 19 | https://mathoverflow.net/users/297 | 33321 | 21,609 |
https://mathoverflow.net/questions/33223 | 2 | A Perfectly Matched Layer (PML) is an absorbing boundary condition (ABC) which can be used to approximate free-field conditions for the numerical solution of wave equation problems.
[PML note](http://www-math.mit.edu/~stevenj/18.369/pml.pdf)
The PML is normally applied to a PDE using the following transformation:
... | https://mathoverflow.net/users/7875 | Application of coordinate-stretching transformation for Perfectly Matched Layer | Now, I am not familiar with the PML method to categorically say that it is generally impossible to have an absorbing boundary layer with the second order transform that you seek. But I can say confidently that if you are looking from the point of view of the complex coordinate transformation, then you *cannot* have a t... | 1 | https://mathoverflow.net/users/3948 | 33337 | 21,622 |
https://mathoverflow.net/questions/33335 | 3 | Recently I discussed an experiment with a friend. Assume we start a random experiment. At first there is an array with size 100 000, all set to 0. We calculate at each round a random number modulo 2 and select one random position in that array. If the number in the array is 1, nothing is changed and otherwise the pre-c... | https://mathoverflow.net/users/6674 | Looking for a probability distribution | This is equivalent to (among other names) the Coupon Collector problem. Your are asking about the distribution of the number of coupons collected after $t$ steps, when the total number of possible coupons is $n$.
<http://en.wikipedia.org/wiki/Coupon_collector%27s_problem>
ADDED: this and related distributions are ... | 7 | https://mathoverflow.net/users/6579 | 33342 | 21,625 |
https://mathoverflow.net/questions/33312 | 16 | You notice a stop-light ahead of you and it is currently red. You can't run the red light, so you will have to brake, but braking wastes energy and you want to be as fuel efficient as possible. What braking strategy maximizes efficiency?
Let's set down some notation and move slowly toward a well-defined question. Sup... | https://mathoverflow.net/users/2811 | What braking strategy is most fuel-efficient? | I think the problem would have been more naturally stated in the context of bicycles. In any case, the answer is as follows:
You are looking for an optimal velocity function $v: [0, T] \to \mathbb{R}\_{\geq 0}$ satisfying some conditions. Each such function represents the strategy, "if the light is still red at time ... | 5 | https://mathoverflow.net/users/4658 | 33347 | 21,627 |
https://mathoverflow.net/questions/33322 | 10 | Background
----------
I am working through a particular result in a paper of Cherlin, Shelah, and Shi, and am satisfied that it follows from basic model theory material - but I'm stuck on one point in the background material.
In Hodges' "A Shorter Model Theory", Lemma 7.2.5 on page 191 seems to have an unused assu... | https://mathoverflow.net/users/4594 | Extra assumption in Hodges' lemma on the resultant of a first-order formula? | The result doesn't need the assumption that $T$ is an $\forall\_2$-theory, nor does it need that the formula $\phi(x)$ is existential. The relevant general result is that a structure $A$ has an extension satisfying a theory $T$ (in the same vocabulary) if and only if $A$ satisfies all the universal sentences that are p... | 8 | https://mathoverflow.net/users/6794 | 33350 | 21,628 |
https://mathoverflow.net/questions/33339 | 7 | Sorry for the vague title, I can't think of a better one that isn't overly long.
Suppose that $S$ is a commuting set of projection operators on a Hilbert space. I'll introduce the following notation: if $p \in S$, let $p^+ \equiv p$ and $p^- \equiv 1 - p$. Let $I \equiv ${$+, -$}. The projections are ordered by defin... | https://mathoverflow.net/users/7842 | Question about projections on a Hilbert space | Here's a counterexample. For each natural number $n$, let $D\_n$ be the set of those $x\in[0,1]$ whose binary expansion has a 1 in the $n$-th place. Then the operation of multiplication by the characteristic function of $D\_n$ is a projection operator $p\_n$ on $L^2[0,1]$. Let $S$ be the set of these operators $p\_n$; ... | 5 | https://mathoverflow.net/users/6794 | 33352 | 21,629 |
https://mathoverflow.net/questions/33353 | 2 | Suppose $a,b$ are two matrices (arbitrary for now), and I have a function defined on a space of matrices, $T(x) = a x b$. This function is a linear and bounded transform on the a finite dimensional vector space of matrices, so can be represented as a matrix. Say $x$ is $m$ by $n$, then you can write $x$ as column vecto... | https://mathoverflow.net/users/7895 | naming for the map $T = x \mapsto a x b$ | I'd go for the obvious *two-sided multiplication operator*. A Google search shows this has been used indeed.
| 2 | https://mathoverflow.net/users/6101 | 33361 | 21,632 |
https://mathoverflow.net/questions/33366 | 7 | You might think that the title is an overstatement of a well-known fact but it is the best title I can come up with for the wonders the intersection operator does in some fields of math.
Recently,(on summer vacation) I was studying one subject after another and after changing about three subjects, I began to notice t... | https://mathoverflow.net/users/5627 | The unprecedented success of the “intersection” operator | When property P is universal ($\forall ...$) it is likely to correspond to closed sets, and thus be preserved under intersection. Examples: axioms of a group, ring, field, directed graph; having symmetry under a given group.
However, if P is existential ($\exists ...$) it corresponds to open sets and is more likely ... | 15 | https://mathoverflow.net/users/6579 | 33373 | 21,635 |
https://mathoverflow.net/questions/33356 | 1 | Burnside's Lemma / Counting Formula says that the number of orbits of an action is equal to the average number of fixed points of the acting permutations. In my case, I'm particularly interested in the sizes of the orbits themselves for a particular action.
Is there a result as general as Burnside's Lemma but that de... | https://mathoverflow.net/users/2971 | Does Burnside's Lemma / Counting Formula have a Cousin? | Maybe you're after the [Orbit-Stabiliser Theorem](http://en.wikipedia.org/wiki/Group_action). Let $G$ be a group that acts on a set $X$ and let $x \in X$. Then $|G|=|G\_x||G(x)|$ where $G\_x$ is the stabliser of $x$ in $G$ and $G(x)$ is the orbit of $x$.
| 2 | https://mathoverflow.net/users/2264 | 33376 | 21,637 |
https://mathoverflow.net/questions/33370 | 3 | In derived algebraic geometry there are several different setting,i.e., sometimes we use $E\_{\infty}$ring,somethings we use dg-algebra,... It is for different situations. But could someone give some examples illustrating under what problem we use relevant "derived structure" ($E\_{\infty}$ ring, dg-algebra...)? Some m... | https://mathoverflow.net/users/2391 | different "derived structure" in derived algebraic geometry | Here are the two motivations I know of:
Number 1 comes from algebraic topology. The definitive reference is <http://www.math.harvard.edu/~lurie/papers/survey.pdf> , which explains it much better than I can. Very roughly, complex oriented cohomology theories are represented by E-oo rings and are classified by their gr... | 5 | https://mathoverflow.net/users/473 | 33378 | 21,639 |
https://mathoverflow.net/questions/33348 | 7 | Is there anywhere where I can read a complete proof in English of this theorem by Borel and Tits:
>
> Suppose that $G$ is a simple algebraic group over an infinite field $k$, and that $H$ is a subgroup of $G(k)$ containing the subgroup of $G(k)$ generated by the rational points of the unipotent radicals of the $k$-... | https://mathoverflow.net/users/7909 | theorem of Borel and Tits | This is too long for a comment, but like others who have commented I don't expect to find anything like a "complete proof" of the Borel-Tits theorem written in English. Borel and Tits have each written at times in French, English, German, but their
serious joint work has been in French and is not especially hard to fol... | 11 | https://mathoverflow.net/users/4231 | 33381 | 21,642 |
https://mathoverflow.net/questions/33374 | 9 | I am interested in finding a lower bound of the sum:
$$\sum\_{i=0}^d \left(\genfrac{}{}{0pt}{}{n}{i}\right)
\left(\genfrac{}{}{0pt}{}{m}{k-i}\right)$$
when $d < k$ (and assuming both $n\geq k$, $m\geq k$). When $d=k$ this sum is equal to
$$\left(\genfrac{}{}{0pt}{}{n+m}{k}\right) $$ (the Chu--Vandermonde identity). Wha... | https://mathoverflow.net/users/7917 | Partial sums of the Chu--Vandermonde identity | Carla, I don't think it is a good idea to give a detailed solution here.
My point is that this problem is pretty standard: if you look inside
[N.G. de Bruijn's *Asymptotic Methods in Analysis*](http://books.google.com/books?isbn=0486642216),
especially in the 3rd chapter, you will find that your sum (generically) falls... | 6 | https://mathoverflow.net/users/4953 | 33382 | 21,643 |
https://mathoverflow.net/questions/33242 | 5 | I have a "continuous" linear programming problem that involves maximizing a linear function over a curved convex space. In typical LP problems, the convex space is a polytope, but in this case the convex space is piecewise curved -- that is, it has faces, edges, and vertices, but the edges aren't straight and the faces... | https://mathoverflow.net/users/6996 | Continuous Linear Programming: Estimating a Solution | Even though the $f\_i$ are not polynomials I'll give the answer in that case because it is very nice and it seems like there is some interest. I have to stress in advance though that the answer exploits the resulting algebraic structure in a fundamental way, and so is unlikely to extend to the case when the $f\_i$ are ... | 6 | https://mathoverflow.net/users/5963 | 33391 | 21,649 |
https://mathoverflow.net/questions/33389 | 2 | Consider Schrödinger's *time-independent* equation
$$
-\frac{\hbar^2}{2m}\nabla^2\psi+V\psi=E\psi.
$$
In typical examples, the potential $V(x)$ has discontinuities, called *potential jumps*.
Outside these discontinuities of the potential, the wave function is required to be twice differentiable in order to solve Schr... | https://mathoverflow.net/users/1291 | Justification for the matching condition for the wave function at potential jumps. Why is it both restrictive enough and sufficiently general? | To answer your first question:
Actually the assumption is *not* that the wave function and its derivative are continuous. That follows from the Schrödinger equation once you make the assumption that the probability amplitude $\langle \psi|\psi\rangle$ remains finite. That is the physical assumption. This is discussed... | 8 | https://mathoverflow.net/users/394 | 33398 | 21,654 |
https://mathoverflow.net/questions/33035 | 3 | Let V be a vector space over Z/2, and let X be a subset of V. Is there an algorithm to find the largest possible subspace of V which doesn't intersect X? Is it NP complete?
| https://mathoverflow.net/users/5463 | Largest subspace that doesn't intersect a given set | The problem is NP hard. Here's a reduction to it from 4-colorability. Given a graph with vertex set $G$ [not $V$ because that's supposed to be a vector space] and edge set $E$, form a vector space $V$ over $\mathbb{Z}/2$ having $G$ as a basis. Identify each edge $e$ with the vector that is the difference of the two end... | 6 | https://mathoverflow.net/users/6794 | 33407 | 21,657 |
https://mathoverflow.net/questions/33360 | 1 | Let $T$ be a torus defined over a field $K$ of characteristic $p>0$. Suppose that $T$ acts (algebraically) on some vector space $V$ (over the same field $K$). Let $W$ be a subspace of $V$. Now consider $N\_T(W)$, the normalizer in $T$ of $W$. I would like to be sure that this is a closed connected subgroup of $T$; i.e.... | https://mathoverflow.net/users/801 | Tori acting on vector spaces | Your more specific question is not actually more specific. $\mathbb{G}\_m^n \ltimes \mathbb{G}\_a^n$ embeds in $GL\_{2n}$ as $\left( \begin{smallmatrix} T & U \\ 0 & 1 \end{smallmatrix} \right)$, where $U$ is embedded along the diagonal. If $U$ is any $T$-rep, we can embed $T \ltimes U$ into $\mathbb{G}\_m^n \ltimes \m... | 3 | https://mathoverflow.net/users/297 | 33412 | 21,659 |
https://mathoverflow.net/questions/33385 | 7 | Is there a functor that preserves all small limits but not a large one?
| https://mathoverflow.net/users/6414 | Preservation of limits | Let's try this. I'll use colimits, so take the opposite.
The class of all ordinals is ordered. Add one more element $\infty$ at the end, bigger than all of them. View this "large ordered set" as a large category $\cal C$. A small diagram in $\cal C$ has colimit $\infty$ if $\infty$ occurs in the diagram, and otherwi... | 18 | https://mathoverflow.net/users/6666 | 33414 | 21,660 |
https://mathoverflow.net/questions/32894 | 10 | Does every 4-regular graph contain a cycle of length (number of edges in the cycle) $1 (\mod3)$? Are there only finitely many exceptions?
I suspect such cycles exist for most 3-regular graphs but 4-regularity is enough for what I'm investigating.
| https://mathoverflow.net/users/2384 | Cycles of length 1(mod 3) in regular graphs | According to [this paper](http://dx.doi.org/10.1016/S0166-218X%2801%2900190-1), N. Dean et al. have shown that if a simple graph *G*
* is 2-connected,
* has minimum degree at least 3, and
* is not isomorphic to the Petersen graph,
then *G* contains a cycle of length 1 mod 3.
Now consider any 4-regular graph *H*. ... | 7 | https://mathoverflow.net/users/7170 | 33423 | 21,665 |
https://mathoverflow.net/questions/33427 | 15 | Lately my studies have been focusing on learning the machinery of K-Theory, and I thought that learning the Atiyah-Singer Index Theorem would be a good way to see K-Theory in action a bit and to learn a deep result on the way. From what I have read, there are a few methods of proof of Atiyah-Singer, one of which uses K... | https://mathoverflow.net/users/3664 | Roadmap to a proof of the Atiyah-Singer Index Theorem which uses K-Theory | The original paper by Atiyah and SInger at <http://www.jstor.org/stable/1970715> is as good as anything
| 16 | https://mathoverflow.net/users/51 | 33428 | 21,667 |
https://mathoverflow.net/questions/30877 | 6 | If I take $\mathcal{A}$ = coherent sheaves on $X$\* up to isomorphism, then there are two things I could do which come to mind.
The first is noticing that $(\mathcal{A},\oplus)$ is a monoid and subsequently applying Grothendieck's $K$-functor to it obtaining a group $K(\mathcal{A})$.
The second is to take the free ... | https://mathoverflow.net/users/3701 | How is K-theory defined for coherent sheaves? | The Grothendieck group on $X$ is defined (in your notation) as $K(\mathcal{B})$, where $\mathcal{B}$ is the monoid of f.g. locally free sheaves on $X$, (i.e. vector bundles on $X$.) The group $L(\mathcal{A})$ is usually denoted as $G(X)$, or $K'(X)$. If $X$ is smooth, the embedding $\mathcal{B} \subset \mathcal{A}$ ind... | 6 | https://mathoverflow.net/users/339 | 33444 | 21,676 |
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