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https://mathoverflow.net/questions/26302 | 13 | Given a smooth manifold *M*, the following procedures yield the differential graded algebra (Ω\*(M),*d*dR) of differential forms:
* Procedure 1 **(synthetic geometry)**.
For each *n*, consider the object of infinitesimal *n*-simplices in *M*. This is a ringed space whose underlying space is the diagonal *M* ⊂ *M**... | https://mathoverflow.net/users/5690 | Two fancy ways of defining differential forms: How does one show that they are equivalent? | I'm afraid I'm not familiar with your first construction, but a very close one is the Alexander complex/Alexander-Spanier cohomology, where we replace your first order jets along the diagonals by all jets along the diagonal (functions on the formal neighborhood). This complex and its identification with de Rham cohomol... | 7 | https://mathoverflow.net/users/582 | 26305 | 17,235 |
https://mathoverflow.net/questions/26306 | 6 | Let $X$ be a complex manifold (for example $\mathbb CP^n$), let $v$ be a holomorphic vector field on $X$, and let $F$ be a coherent sheaf (for example a vector bundle or a structure sheaf of a point). Then $v$ defines an element in $Ext^1(F,F)$. Indeed $v$ generates an action of $\mathbb C$ on $X$ and taking pull-backs... | https://mathoverflow.net/users/943 | Deformations of sheaves via automorphisms. How to express $Ext^1$? | Holomorphic vector fields give elements of the first Hochschild cohomology group of the structure sheaf $$HH^1(O\_X,O\_X)\simeq H^1(X,O\_X)\oplus H^0(X,T\_X)$$ (via an easy part of the Hochschild-Kostant-Rosenberg theorem).
On the other hand the Hochschild cohomology may be identified (or indeed defined as) self-Ext... | 7 | https://mathoverflow.net/users/582 | 26310 | 17,239 |
https://mathoverflow.net/questions/26317 | 8 | A monoid in the Category Cat is a strict monoidal category according to Wikipedia. Is it possible to weaken the monoid so that its realisation in Cat is a weak monoidal category? Do we shift up a dimension, and throw in a 2-morphism associator for the monoid satisfying the analogue of Mac Lane's Pentagon, and similarly... | https://mathoverflow.net/users/6408 | Is a weak monoidal category a monoid object in some category? | Yes, you can define a *pseudomonoid* in any monoidal 2-category, such that a pseudomonoid in the 2-category Cat is precisely a non-strict monoidal category. The definition of monoidal category, interpreted in terms of functors and natural isomorphisms, i.e. in terms of the 2-category Cat, tells you exactly how to defin... | 8 | https://mathoverflow.net/users/49 | 26322 | 17,245 |
https://mathoverflow.net/questions/16226 | 19 | **Background:** The Shannon sampling theorem states that a bandlimited function (an $L^2$ function whose Fourier transform has compact support) can be determined uniquely from sampling it an integer lattice. More precisely, if the Fourier transform\* $\hat{f}$ is supported in $[-\Omega/2, \Omega/2]$ (which we express b... | https://mathoverflow.net/users/344 | What kinds of sets can replace lattices in the Shannon sampling theorem? | One sufficient condition is that, if $|X\_k| <= L$ for some $L < 1/4$ then the sampling map would be an injection, as well as a bounded map from $B$ to $l^2$. This is known as Kadec's theorem.
See
Nonuniform Sampling and Reconstruction in Shift-Invariant Spaces,
Akram Aldroubi and Karlheinz Gröchenig,
SIAM Revi... | 8 | https://mathoverflow.net/users/1229 | 26333 | 17,248 |
https://mathoverflow.net/questions/26342 | 10 | This is motivated by a recent [question](https://mathoverflow.net/questions/26336/integer-valued-factorial-ratios) by Wadim.
The negative answer should be known, since t is very natural, in this case I would be happy to see any reference.
May Pafnuty Lvovich Chebyshev's approach to distribution of primes lead to P... | https://mathoverflow.net/users/4312 | Chebyshev's approach to the distribution of primes | Erdős and Diamond proved in [**1**] that Chebyshev could have achieved sharper bounds for the asymptotic behavior of the prime counting function. Nevertheless, their proof does not shed any light on the first question that you posed because they took the PNT for granted throughout their note.
References:
[**1**] **... | 8 | https://mathoverflow.net/users/1593 | 26344 | 17,254 |
https://mathoverflow.net/questions/26350 | 0 | I'm wondering (hoping) if an inequality is true. Please can anyone help me?
Let $V$ be a complex vector space $dim\_{\mathbb{C}}(V)=n$
with a hermitian scalar product $h$.
Let $v,a, b \in V$.
Is it true that
$(h(v,v)h(a,a)-{|h(v,a)|}^{2})(h(v,v)h(b,b)-{|h(v,b)|}^{2})\geq |(h(v,v)h(a,b)-h(a,v)\overline{h(b,v)}|^{... | https://mathoverflow.net/users/4971 | Linear algebra inequality | Yes. The case where $v=0$ is trivial so suppose $v\ne0$. Consider
the projection map from $V$ to the hyperplane orthogonal to $v$
and let $a'$ and $b'$ be the images of $a$ and $b$ under this projection.
Then your inequality reduces to
$$h(a',a')h(b',b')\ge\vert h(a',b') \vert^2,$$the Cauchy-Schwarz inequality.
| 4 | https://mathoverflow.net/users/4213 | 26353 | 17,260 |
https://mathoverflow.net/questions/26267 | 154 | While not a research-level math question, I'm sure this is a question of interest to many research-level mathematicians, whose expertise I seek.
At RIMS (in Kyoto) in 2005, they had the best white chalk I've seen anywhere. It's slightly larger than standard American chalk, harder, heavier, and most importantly covere... | https://mathoverflow.net/users/391 | Where to buy premium white chalk in the U.S., like they have at RIMS? | You can buy it online [rakuten link deleted]
**Edit (2020-7-10):** Hagoromo Bunku company has been shut down in March 2015 ([Berkeley Alumni Magazine article](https://alumni.berkeley.edu/california-magazine/fall-2019/chalk-market-where-mathematicians-go-get-good-stuff)). There were two successors to their product, be... | 65 | https://mathoverflow.net/users/36665 | 26356 | 17,263 |
https://mathoverflow.net/questions/26368 | 0 | Let G and H be finite groups. Let G' = [G,G] and H' = [H,H] be the corresponding derived groups (commutator subgroups) of G and H. I am looking for an example where G' is isomorphic to H' and G/G' is isomorphic to H/H' but G and H are not isomorphic.
| https://mathoverflow.net/users/4048 | Must finite groups with isomorphic commutators and quotients be isomorphic? | take $G=S\_n$ and $H=A\_n\times Z/2Z$ (here $S\_n$ is the symmetric group and $A\_n$ is the alternating group).
| 13 | https://mathoverflow.net/users/4158 | 26369 | 17,267 |
https://mathoverflow.net/questions/26364 | 4 | Let X be a Calabi-Yau 3-fold, and $D\subset X$, smooth,Fano divisor.
Since $K\_X=0$ we have $N\_D^X=K\_D$. I have seen the following fact in many papers:
By deforming X within Kahler moduli, we can shrink D to a point to get a singular 3-fold Y.
Here are my questions?
1-Why is that possible?(Why it is not possibl... | https://mathoverflow.net/users/5259 | Shrinking Fano surfaces to a point in Calabi-Yau 3-folds | **Corrected.** In the first version I stated that CY 3-fold containing Fano surfaces are unknown, this is not at all true as Mohammad pointed out. For example, one can take a product $E\times E\times E=X$ with $E$ an elliptic curve admitting a $\mathbb Z\_3$ action,
and then take crepant resolution of the quotient of d... | 5 | https://mathoverflow.net/users/943 | 26370 | 17,268 |
https://mathoverflow.net/questions/26338 | 8 | Are there explicit examples of triangulations of exotic 4-spheres?
| https://mathoverflow.net/users/5196 | Triangulations of exotic 4-spheres | Here is my comment expanded to answer form: The question of existence of exotic 4-spheres (i.e., the smooth Poincaré conjecture) is still open, and (according to [Wikipedia](http://en.wikipedia.org/wiki/Generalized_Poincare_conjecture)) the existence of exotic PL structures is equivalent to it. Therefore, the answer is... | 6 | https://mathoverflow.net/users/121 | 26375 | 17,271 |
https://mathoverflow.net/questions/25723 | 17 | This should be straightforward; I'm sorry if it's too much so. Can someone point me to a reference which computes the Dolbeault cohomology of the Hopf manifolds?
Motivation: I'd like to work through a concrete example of the Hodge decomposition theorem failing for non-Kähler manifolds. The textbook I have handy (Grif... | https://mathoverflow.net/users/2819 | Dolbeault cohomology of Hopf manifolds | Even though this question has an accepted answer, the answers so far are not complete or explicit. I kept working on this question, because I have been curious for a long time about the structure of Dolbeault complexes. First of all, the Frölicher spectral sequence does not directly reveal all of the non-Hodge informat... | 29 | https://mathoverflow.net/users/1450 | 26380 | 17,275 |
https://mathoverflow.net/questions/26363 | 5 | Suppose that $A$ is a semisimple Hopf algebra with a commutative character ring. Does it follow that $A$ is quasitriangular, i.e $\mathrm{Rep}(A)$ is a braided tensor category?
I think I 've seen this statement in a paper without a proof long time ago. It might be obvious although I don't see how to construct a braid... | https://mathoverflow.net/users/2805 | Semisimple Hopf algebras with commutative character ring | Sebastian,
No, it does not follow.
In [this paper (Example 6.14)](http://lanl.arxiv.org/abs/0704.0195) we proved that if a Tambara-Yamagami fusion category
admits a braiding then its dimension is a power of 2. Note that a Tambara-Yamagami
category
has a commutative Grothendieck ring. Hopf algebras whose representa... | 7 | https://mathoverflow.net/users/3011 | 26381 | 17,276 |
https://mathoverflow.net/questions/26300 | 8 | Recall the following theorem due to Burnside:Let $G$ be a finite group and let $V$ be its irreducible complex representation of dimension greater than 1, then the character
of this representation is $0$ on some element of $G$. Is this statement still correct
if $G$ is any compact Lie group? Thanks.
| https://mathoverflow.net/users/6277 | Is there a generalization of Burnside's theorem for compact Lie groups? | The answer is yes--use the Weyl character formula, for example.
See: [Patrick X. Gallagher, Zeros of group characters. Math. Z. Volume 87 (1965), Number 3.](http://www.springerlink.com/content/l2m14425071477x3/)
| 7 | https://mathoverflow.net/users/6355 | 26384 | 17,277 |
https://mathoverflow.net/questions/26382 | 5 | If we have an extension of bundles $0 \to E \to F \to G \to 0$ on $X$, then to show that this is the zero element in $Ext^1\_X(G,E)$, we need to show that this sequence splits. To produce a splitting is a concrete question in homological algebra. I understand that in general if we have an extension $ 0 \to E \to E\_1 \... | https://mathoverflow.net/users/6425 | Showing an Ext^2 element is zero | There is a simple recipe to show that a product of two $Ext^1$ is zero (and it is clear that any $Ext^2$ can be represented as such a product). Namely, let $0 \to E\_0 \to E\_{01} \to E\_1 \to 0$ and $0 \to E\_1 \to E\_{12} \to E\_2 \to 0$ are two exact triples.
The product of the corresponding elements $e\_{01} \in Ex... | 10 | https://mathoverflow.net/users/4428 | 26387 | 17,279 |
https://mathoverflow.net/questions/26374 | 10 | I would like to know something more than what is written on wikipedia <http://en.wikipedia.org/wiki/Euler_characteristic>
What would be some large (largest?) class of topological spaces for which $\chi$ is defined, so that all standard properties hold, for example that $\chi(X)=\chi(Y)+\chi(Z)$ if $X=Y \cup Z$, ($Y\c... | https://mathoverflow.net/users/943 | For which classes of topological spaces Euler characteristics is defined? | The answer to the question as it is stated is that there is probably no "largest" class of spaces for which the Euler characteristic makes sense.
The answer also depends on where you would like the Euler characteristic to take values. Here is the tautological answer (admittedly not a very exciting one): if you have a... | 15 | https://mathoverflow.net/users/2349 | 26393 | 17,281 |
https://mathoverflow.net/questions/26390 | 6 | The base field will be the field of complex numbers. I have a slightly technical problem concerning the resolution of singularities of a certain variety. Basically, I want to to know if it is projective.
Let $Y$ be a normal projective variety with only quotient singularities. Let $C$ be a smooth projective curve.
L... | https://mathoverflow.net/users/4333 | Is the desingularization of a normal variety with only quotient singularities projective | The following paper might be useful.
*Nagata*, "Existence theorems for nonprojective complete algebraic varieties", Illinois J. Math. 2 1958 490--498.
At the end of the paper Nagata says the following:
"Hironaka recently proved the following: If $V$ is a nonsingular projective variety of dimension not less than ... | 6 | https://mathoverflow.net/users/3521 | 26398 | 17,284 |
https://mathoverflow.net/questions/26392 | 1 | Let T be a true theory of arithmetic to which the incompleteness theorems apply. Consider two sentences in the language of T to be equivalent if they are provably equivalent over T. How many equivalence classes are there, under this relation, that contain a true-but-unprovable sentence?
| https://mathoverflow.net/users/1320 | What is cardinality of the set of true undecidable minimal sentences in a formal theory of aritmetic | **Note:** *The top part answers an old version of the question, which is now irrelevant.*
Given a axiomatizable theory T of arithmetic, the set of all statements independent of T is the complement of a computably enumerable set. When nonempty (e.g. when T extends PA) this set is countably infinite. (Trivially, if φ i... | 6 | https://mathoverflow.net/users/2000 | 26399 | 17,285 |
https://mathoverflow.net/questions/26391 | 8 | Let $X$ be a negatively curved symmetric space. In other words, $X$ is one of the four examples: a hyperbolic space, a complex hyperbolic space, a quaternionic hyperbolic space or the hyperbolic Cayley plane.
Let $B\subset X$ be a compact set (wlog, a large ball). Does there always exist a subgroup $\Gamma$ of the is... | https://mathoverflow.net/users/4354 | Are there arbitrarily sparse "lattices" in negatively curved symmetric spaces? | The examples are arithmetic groups, constructed in general by [Borel and Harish-Chandra](http://www.ams.org/mathscinet-getitem?mr=147566).
See also [Dave Witte Morris' preliminary book](http://people.uleth.ca/~dave.morris/books/IntroArithGroups.html).
However, examples in hyperbolic and complex hyperbolic spaces prob... | 11 | https://mathoverflow.net/users/1345 | 26400 | 17,286 |
https://mathoverflow.net/questions/26385 | 37 | It is easy to see that if $A\times B$ is homeomorphic to $A\times C$ for topological spaces $A$, $B$, $C$, then one may not conclude that $B$ and $C$ are homeomorphic (for example, take $C=B^2$, $A=B^{\infty}$). The question is: for which $A$ such conclusion is true? I saw long ago a problem that for $A=[0,1]$ it is no... | https://mathoverflow.net/users/4312 | When factors may be cancelled in homeomorphic products? | For $A=[0,1]$, let $B$ be the 2-torus with one hole and $C$ be the 2-disc with two holes.
The products $B\times[0,1]$ and $C\times[0,1]$ can be realized in $\mathbb R^3$: the former as a thickening of the torus, the latter in a trivial way. Each of these products is a handlebody bounded by the pretzel surface (the sp... | 40 | https://mathoverflow.net/users/4354 | 26404 | 17,288 |
https://mathoverflow.net/questions/26402 | 12 | I know [Chaitin's constant](http://en.wikipedia.org/wiki/Chaitin%2527s_constant) Ω is not computable (and therefore transcendental). Are there other specific, known noncomputable numbers? I am trying to understand what distinguishes a computable transcendental number, such as π, from a noncomputable transcendental numb... | https://mathoverflow.net/users/6094 | {transcendental numbers} \ {computable transcendental numbers} | Note: Answer is pending update per attached comments.
The difference, stated informally, is that that the non-computable transcendentals in their k-base digit representation are entirely random and non compressible. A computable transcendental, such as e, can be represented by a finite algorithmic description, such a... | 2 | https://mathoverflow.net/users/1320 | 26408 | 17,291 |
https://mathoverflow.net/questions/26409 | 8 | On page 164 of his book **Models of Peano Arithmetic**, Kaye states Friedman's Theorem:
Let $M{\vDash}PA$ be nonstandard and countable, let $a\in M$ and let $n\in {\mathbb N}$. Then there is a proper initial segment $I\subseteq\_c M$ ($\subseteq\_c$ means "cofinal in") containing $a$ such that $I\cong M$ and $I<\_{\S... | https://mathoverflow.net/users/2361 | models of PA which are isomorphic but not elementarily equivalent? | There is a major difference between *elementary equivalence* and *elementary embedding*. Moreover, in this case, the actual embedding is somewhat ambiguous. First, let me recap some often confused terminology.
Two models are *elementary equivalent* if they satisfy the same first-order sentences. Any two isomorphic mo... | 10 | https://mathoverflow.net/users/2000 | 26412 | 17,293 |
https://mathoverflow.net/questions/26411 | 11 | The Second Incompleteness Theorem says that if $T$ is a consistent (computably) axiomatizable theory which extends IΣ1, then $\mathrm{Con}(T)$ is not provable from $T$. By analogy with computability theory, the stronger theory $T + \mathrm{Con}(T)$ can be thought of as the "jump" of $T$. To abuse this analogy, I will u... | https://mathoverflow.net/users/2000 | Uniform solutions to Post's problem for axiomatizable theories | (Note: this has been rewritten to reflect the comments below).
The answer to #1 is basically yes, because the proof that the Lindenbaum algebra above T is atomless is completely constructive.
Start with a (consistent) theory T to which the second incompleteness theorem applies, which means that T + ~Con(T) is als... | 8 | https://mathoverflow.net/users/5442 | 26413 | 17,294 |
https://mathoverflow.net/questions/25968 | 11 | Let $R$ be a reduced curve singularity over an algebraically closed field $k$ and $\tilde{R}$ its integral closure in its total ring of fractions.
The $k$-dimension of $\tilde{R}/R$ is finite. If we assume $R$ is non-planar and Gorenstein, then **how small can this number be?**
The ring $R = k[[x,y,z]]/(xy = z^2, ... | https://mathoverflow.net/users/5337 | What is the most simple non-planar Gorenstein curve singularity? | I think Graham's answer already gave most of what you need to prove that $4$ is the smallest possible. Let $V$ be the integral closure of $R$, $n$ be the embedding dimension of $R$, and $e=e(R)$ be the multiplicity.
Claim: If $R=k[[x\_1,\cdots,x\_n]]/I$ is Gorenstein and $n$ is at least $3$, then $\dim\_k(V/R)\geq e... | 6 | https://mathoverflow.net/users/2083 | 26418 | 17,297 |
https://mathoverflow.net/questions/26405 | 7 | Let $M\_{n\times n}$ denote the set of $n\times n$ real matrices and let $GL\_n$ be the subgroup of invertible matrices. $GL\_n$ acts on $M\_{n\times n}$ smoothly by conjugation, which means that each conjugacy class (which is an orbit of this action) is an immersed submanifold of $M\_{n\times n}$. However, the action ... | https://mathoverflow.net/users/4622 | When is a conjugacy class of matrices an embedded submanifold? | If $G$ is an algebraic group (say over $\mathbb C$), acting on an algebraic variety $X$, the orbits are always locally closed smooth algebraic subvarieties of $X$. This is standard, and easy to prove. If $x$ is a point of $X$, and $G \to X$ is the morphism sending $g$ to $gx$, consider the orbit $Gx$ as a subset of its... | 5 | https://mathoverflow.net/users/4790 | 26428 | 17,303 |
https://mathoverflow.net/questions/26434 | 4 | Hi math people.
I'm in the process of analyzing some data that I collected through an experiment. The data are (somewhat) normally distributed and I represent the different data-sets using [boxplot](http://en.wikipedia.org/wiki/Box_plot), to provide an easy way of visually comparing the mean between the data-sets and... | https://mathoverflow.net/users/5357 | Why 1.5IQR whiskers in boxplot? | Three sigma has less relevance for asymmetric distributions. Using quartiles keeps it nonparametric. Regarding why not other quantiles of the distribution rather than 1.5\*IQR, you can follow some comments on this [thread](http://tolstoy.newcastle.edu.au/R/help/05/07/8210.html), which basically argues that you want to ... | 6 | https://mathoverflow.net/users/5282 | 26435 | 17,309 |
https://mathoverflow.net/questions/26438 | 7 | Are there any computer algebra systems (e.g. Macaulay2 og singular) that allows one to compute the Picard number (i.e. the rank of the Neron-Severi group) of a given variety?
| https://mathoverflow.net/users/3996 | Are there any sofware packages for computing Picard numbers? | A more basic question is whether there even exists an algorithm to compute this number.
I've wondered this for a long time, and I honestly don't know what to expect.
Any algorithm would have to be quite subtle. In the early 1980's Shioda had to work quite hard to construct explicit examples of surfaces in $\mathbb{P}^... | 11 | https://mathoverflow.net/users/4144 | 26441 | 17,312 |
https://mathoverflow.net/questions/26443 | 0 | The following combinatorial problem has bothered me quite a bit. I guess people smarter than me have given the problem some taught as the problem has obvious applications (e.g. to the Ising model), but I have not found any solution on the web (this might be because I don't know the proper terminology).
Anyways, here... | https://mathoverflow.net/users/4626 | Number of config. of a binary string invariant under cyclic permutation. | <http://en.wikipedia.org/wiki/Necklace_(combinatorics)> will get you started.
| 1 | https://mathoverflow.net/users/6153 | 26445 | 17,314 |
https://mathoverflow.net/questions/26421 | 4 | Is Young's inequality true for an arbitrary measure on $\mathbb R^d$? If so, where can I find a proof of it? In particular, where can I find the proof of the discrete version (i.e the version for $\ell^p$ spaces) of this inequality?
Here is the statement of the inequality (from Wikipedia):
Suppose $f$ is in $L^p(\m... | https://mathoverflow.net/users/1229 | Proof of Young's convolutions inequality for a general measure on $\mathbb R^d$ | Note that if your measure is not translation-invariant, the convolution product is not commutative.
This allows for a simple counterexample with $r=q=\infty$, $p=1$,
$d\mu(x)={\bf 1}\_{[0,1]}(x) dx$.
Define $f\*g(x)= \int f(x-s)g(s)d\mu(s) \ (\neq g\*f(x))$.
Take $g\equiv 1, \ f={\bf 1}\_{[1,2]}$.
This gives $|... | 8 | https://mathoverflow.net/users/6129 | 26448 | 17,316 |
https://mathoverflow.net/questions/26461 | 12 | Let $G=\mathbb{Z}/2\mathbb{Z}$. Let $f\colon L \to N$ be a smooth map of connected smooth **closed** $n$-dimensional manifolds such that the induced map
$$f^\* \colon H^\*(N,G) \to H^\*(L,G)$$
is an isomorphism.
**Question**: Are the pull back of the Stiefel-Whitney classes of the tangent bundle of $N$ the Stiefe... | https://mathoverflow.net/users/4500 | Are the stiefel-Whitney classes of the tangent bundle determined by the mod 2 cohomology? | The answer to the question is positive, due to Wu's formula. See e.g. Milnor-Stasheff, Characteristic classes, lemma 11.13 and theorem 11.14. In fact, all one needs to compute the Stiefel-Whitney classes of a smooth compact manifold (orientable or not) is the cohomology mod 2 (as an algebra) and the action of the Steen... | 20 | https://mathoverflow.net/users/2349 | 26469 | 17,328 |
https://mathoverflow.net/questions/26446 | 21 | I'm looking for references on the "algebraic geometry" side of complex analytis, i.e. on complex spaces, morphisms of those spaces, coherent sheaves, flat morphisms, direct image sheaves etc. A textbook would be nice, but every little helps.
Grauert and Remmert's "Coherent analytic sheaves" seems to contain what I wa... | https://mathoverflow.net/users/4054 | References for complex analytic geometry? | Two books that I like a lot:
1) Joseph Taylor's [Several complex variables with connections to algebraic geometry and Lie groups](http://books.google.com/books?id=i8lUNpZ379MC&printsec=frontcover&dq=Taylor,+Several+complex+variables&source=bl&ots=7MVn4Xlu9E&sig=Bwt3jHAJvq5LyuuW8TODoykm9eE&hl=en&ei=0KcCTIL6LsaqlAeEwt... | 10 | https://mathoverflow.net/users/439 | 26476 | 17,332 |
https://mathoverflow.net/questions/26467 | 2 | Has it been proven that maximal planar graphs are reconstructible?
It seems like an easy result, but I am unable to find it in the literature. Classes of planar graphs that I know are reconstructible are: maximal outerplanar (Manvel 1970), maximal minimally non-outerplanar (has a single interior vertex) (Kunni, Anni... | https://mathoverflow.net/users/4167 | Reconstructing Maximal Planar Graphs | You may want to look at papers by Fiorini and Lauri. I found the following reference, and I believe there is part II that does reconstruction. But I can't locate it in my bib file, and don't have access to MathSciNet at the moment. Fiorini and Lauri have many papers on other classes of planar graphs. Mostly on edge rec... | 1 | https://mathoverflow.net/users/6438 | 26477 | 17,333 |
https://mathoverflow.net/questions/26482 | 1 | Dear all,
here is a question that has been bothering me. It goes without saying that I would appreciate any help in answering it.
Let {I\_k} be a countable sequence of two sided closed ideals in a C\*-algebra (ring) and J be a two sided closed ideal in the same C\*-algebra (ring).
Then "intersection of {I\_k + J} =... | https://mathoverflow.net/users/6439 | Intersection of ideals in C*-algebra or even rings in general | In the most general form, for arbitrary ideals over rings, this
is false. In the ring $\mathbb{Z}$ let $I\_k$ be generated by $2^k$
and let $J$ be generated by $3$. Then $I\_k+J=\mathbb{Z}$
for all $k$ and so $\cap\_{k=1}^\infty(I\_k+J)=\mathbb{Z}$.
But $\bigcap\_{k=1}^\infty I\_k=\lbrace0\rbrace$ and so
$J+\bigcap\_{k... | 3 | https://mathoverflow.net/users/4213 | 26483 | 17,335 |
https://mathoverflow.net/questions/26473 | 1 | How many positive integer solutions does the equation x^2+y^2+z^2-xz-yz = 1 have? My guess is (1,0,1), (0,1,1) and (1,1,1). What is proof of that fact that there are none other?
| https://mathoverflow.net/users/6437 | Diophantine equation problem | To elaborate on Robin's suggestion, set $x=v+w,y=u+w,z=u+v$, and the equation becomes $$u^2+v^2+2w^2=1,$$ with $u=(y+z-x)/2,v=(x-y+z)/2,w=(x+y-z)/2$ being half-integers. Now a brute force run through the possibilities is feasible, since $|u|\leq 1$ and such.
For the quick answer, you can use Mathematica:
Reduce[x x... | 3 | https://mathoverflow.net/users/935 | 26484 | 17,336 |
https://mathoverflow.net/questions/26361 | 2 | Is the following claim true?
**Claim** Let $A, B\in C^{n\times n}$ with $rank(A)=rank(B)=r$. Then there exist nonsingular matrices $P\_1, P\_2, Q\_1, Q\_2$ such that
$$ Q\_1AP\_1=Q\_2BP\_2=\left(\begin{array}{cc}I\_{r}&0\\\
0&0
\end{array}\right)$$ and
$$Q\_1Q\_2^{-1}=\left(\begin{array}{cc}X\_1&0\\\
X\_2&X\_3
\end... | https://mathoverflow.net/users/3818 | A question on matrix decomposition. | The Gauss-algorithm tells you that you can find matrices $Q\_i$ and $P\_i$ satisfying the first line. For the second line you ask, how can one modify, say, the $Q\_i$ to make the second line come true? One thing you certainly can do, is changing $Q\_1$ by
$$\left(\begin{array}{cc} 1&x\\\ 0&1\end{array}\right)Q\_1$$
a... | 3 | https://mathoverflow.net/users/nan | 26486 | 17,338 |
https://mathoverflow.net/questions/26462 | 7 | Suppose you have the autonomous ordinary differential equation $dx(t)/dt = f(x(t))$ with $x: \mathbb{R} \to \mathbb{R}$ and the initial condition $x(0)=x\_0$. Then, assuming some regularity conditions, you get as solution the
flow $\Phi(x\_0,t):=x(t)$. To give a trivial example: If $f(x)=x$, then $\Phi(x\_0,t)=x\_0 \ex... | https://mathoverflow.net/users/6415 | Noninteger iterates of functions: How to get ODE from flow at a given time? | If g is a real-analytic function defined near x0 with g(x0) = x0 and 0 < λ ≠ 1 where λ := g'(x0), then Koenigs proved that there exists a real-analytic homeomorphism h defined near x0 such that hgh-1(x) = L(x) where
L(x) := x0 + λ(x-x0)
and h is unique up to a constant factor\*.
This allows g to be embedded in ... | 4 | https://mathoverflow.net/users/5484 | 26495 | 17,343 |
https://mathoverflow.net/questions/26474 | 3 | Suppose we have a complex vector field on $\mathbb{C}^n$ which is analytic and has $|DV| < L$ on ball $B\_r$ with radius r.
I would like to understand:
1) if there exists an analytic flow $\phi\_t(x)$ with complex time $t$ such that $\partial\_t \phi\_t(x)=v(\phi\_t(x))$
2) if such flow is analytic in both $t$ and ... | https://mathoverflow.net/users/6357 | Analytic ODE with complex time | You'll find relevant information in the book
Ordinary differential equations in the complex domain
By Einar Hille
<http://books.google.com/books?id=I1OR4t8UZCgC&printsec=frontcover&dq=Ordinary+Differential+Equations+in+the+Complex+Domain&ei=t8wCTPaXFZLKygT_9e24DA&cd=1#v=onepage&q&f=false>
Fixed point (iteration) re... | 3 | https://mathoverflow.net/users/5365 | 26502 | 17,348 |
https://mathoverflow.net/questions/26497 | 20 | Hi. From one of the forms of Hilbert's Nullstellensatz we know that all the maximal ideals in a polynomial ring $k[x\_1, \dots, x\_n]$ where $k$ is an algebraically closed field, are of the form $(x\_1 - a\_1, \dots , x\_n - a\_n)$. So that any maximal ideal in this case is generated by polynomials $g\_j \in k[x\_1, \d... | https://mathoverflow.net/users/4170 | Maximal Ideals in the ring k[x1,...,xn ] | The stronger version of the Nullstellensatz states that a maximal
ideal $I$ of $R=k[x\_1,\ldots,x\_n]$ is the kernel of a $k$-homomorphism
from $R$ to $L$ where $L/k$ is a finite extension. Let $a\_1,\ldots,a\_n$
be the images of $x\_1,\ldots,x\_n$ under such a homomorphism.
Then $a\_1$ has a minimal polynomial $m\_1$ ... | 17 | https://mathoverflow.net/users/4213 | 26503 | 17,349 |
https://mathoverflow.net/questions/26491 | 34 | This is probably common knowledge, alas I have to confess my ignorance.
In simpler more abstract language, does $\mathcal{O}\_K$ being simply connected (having trivial etale $\pi\_1$) imply $\mathcal{O}\_K=\mathbb{Z}$?
| https://mathoverflow.net/users/5756 | Is there a ring of integers except for Z, such that every extension of it is ramified? | Yes, there are examples and Minkowski's proof for ${\mathbf Q}$ can be adapted to find a few of them. Some examples of this kind among quadratic fields $F$, listed in increasing size of discriminant (in absolute value), are
$$
{\mathbf Q}(\sqrt{-3}), \ \ {\mathbf Q}(i), \ \ {\mathbf Q}(\sqrt{5}), \ \ {\mathbf Q}(\sqrt... | 50 | https://mathoverflow.net/users/3272 | 26504 | 17,350 |
https://mathoverflow.net/questions/26505 | 9 | It's easy enough to build Turing Machines that don't halt. But how complex can we make these? For example, suppose a machine has access to its state transition table and can write to it like a C program could point to its own code page in RAM and poke around. The motivation for the question should clear up the particul... | https://mathoverflow.net/users/6419 | What are the limits of non-halting? | Your question is about many things, but let me give an answer focused on just one interesting issue, the question of determining how long a program will run.
The [busy beaver](http://en.wikipedia.org/wiki/Busy_beaver)
function exactly measures how long programs of a given size
run before halting (among the ones that... | 8 | https://mathoverflow.net/users/1946 | 26509 | 17,353 |
https://mathoverflow.net/questions/26498 | 5 | While reading a paper, I came across the following peculiar condition:
Let $1 \rightarrow H \rightarrow G \rightarrow G/H \rightarrow 1$ be a short exact sequence, and let $H$ be abelian. We require that any automorphism, $\sigma$, of $G$ that preserves $H$ pointwise and such that $\sigma(g)H=gH$ (preserves cosets po... | https://mathoverflow.net/users/5309 | A condition on finite groups |
>
> Short answer: a typical example is G=SL(2,5), H = Z(G) = Z/2Z. If G/H and H are coprime and satisfy the condition, the G = G/H × H is quite dull.
>
>
>
I'll assume you find this interesting, and want to read about it:
Let G be a group (finite is good), H be an abelian normal subgroup of G, and Q be the quo... | 10 | https://mathoverflow.net/users/3710 | 26510 | 17,354 |
https://mathoverflow.net/questions/26518 | 5 | There are more general definitions, but for my purposes a **Lie algebroid** on a smooth manifold $X$ is a vector bundle $A \to X$, a map $\rho: A \to {\rm T}X$ of vector bundles over $X$, and a bracket $[,]$ making $\Gamma(A)$ into an $\mathbb R$-Lie algebra, such that $\rho$ induces is a map $\Gamma(A) \to \Gamma({\rm... | https://mathoverflow.net/users/78 | Do Lie algebroids pull back (along submersions)? | As your example suggests, the vector bundle pullback is perhaps not the right thing to consider when wanting defining the notion of a pullback Lie algebroid. You have to go one step further and take the pullback of $\phi^\*\rho: \phi^\* A \to \phi^\*TX$ along $d\phi: TY \to \phi^\*TX$. The resulting bundle $\phi^{\*\*}... | 6 | https://mathoverflow.net/users/2552 | 26521 | 17,361 |
https://mathoverflow.net/questions/26515 | 4 | An orthogonal projection is an Hermitian matrix $P$ such that $P^2=P$.
Denote $U^\*$ the conjugate transpose of a matrix $U$.
It can be easily shown that for two projections $P\_1$ and $P\_2$, there exists a unitary
$U$ such that both $UP\_1 U^\*$ and $UP\_2U^\*$ are block diagonal with blocks of size one
or two ... | https://mathoverflow.net/users/6442 | Simultaneous Block decomposition of a set of orthogonal projections | If $P$ is a projection then $I-2P$ is a reflection. Two reflections generate the dihedral
group and all irreducible representations of the dihedral group have dimension at most two.
This explains your observation about two projections.
But the alternating group $Alt(n)$ can be generated by three involutions when $n\g... | 8 | https://mathoverflow.net/users/1266 | 26523 | 17,363 |
https://mathoverflow.net/questions/26528 | 5 | In many papers about dynamical system, I found the word " ideal boundary". T don't know what is the definition of ideal boundary.
| https://mathoverflow.net/users/5093 | What is the definition of ideal boundary? | For a [Hadamard space](http://en.wikipedia.org/wiki/Hadamard_space) $X$ there are two kinds of ideal boundaries, the set $Bd(X)$ of horofunctions up to additive constants, and the set $X(\infty)$ of equivalence classes of rays. These two are homeomorphic by the correspondence:
>
> $$\gamma \text{ (a ray)}\to \text{... | 6 | https://mathoverflow.net/users/2384 | 26532 | 17,369 |
https://mathoverflow.net/questions/26540 | 7 | I read in "Letters to a young mathematician" that 4900 is the only square integer that is the sum of consecutive squares (I believe he meant by that "starting from 1", but maybe that's not even necessary). I did a quick run through with a python script and of course this seems totally devoid of a computational pattern.... | https://mathoverflow.net/users/429 | 4900, a particularly square number | This is a classical Diophantine equation (Mordell, *Diophantine Equations*, p. 258). Apart from n = 0, 1, -1, there is only the solution n = 24. Proofs by G. N. Watson (1919), W. Ljunggren (1952).
There is some history in <http://www.math.ubc.ca/~bennett/paper21.pdf> .
| 14 | https://mathoverflow.net/users/6153 | 26542 | 17,374 |
https://mathoverflow.net/questions/26558 | 4 | **Background**
I have searched a bit for the definition/constructions on how to "semi-localize" a scheme, but have been unsuccessful in finding a good reference; I apologize in advance if this topic has been covered in detail elsewhere (e.g. in a book or article) and would be happy for a reference!
This question aros... | https://mathoverflow.net/users/4235 | On semi-local schemes | Your construction number (1) seems completely natural and correct to me, for the following reason:
If $A$ is a (Noetherian, commutative, unital) ring and $I$ is an ideal of $A$, then the localaization of $A$ away from $I$ is $A\_I=S^{-1}A$, where
$$S=\{a\in A: a\mod{I}\mbox{ is not a zero divisor is }A/I\}$$
If $... | 2 | https://mathoverflow.net/users/5830 | 26562 | 17,385 |
https://mathoverflow.net/questions/26557 | 25 | I read that the primitive element theorem for fields was fundamental in expositions of Galois theory before Emil Artin reformulated the subject. What are the differences between pre and post-Artin Galois theory?
| https://mathoverflow.net/users/4692 | What was Galois theory like before Emil Artin? | [The development of Galois theory from Lagrange to Artin](https://doi.org/10.1007/BF00327219) by B. Melvin Kiernan, is a history of pre-Artin Galois theory.
| 20 | https://mathoverflow.net/users/2384 | 26563 | 17,386 |
https://mathoverflow.net/questions/26551 | 18 | Felix Klein, when discussing how the popularity of areas in mathematics rises and falls, mentions that in his youth Abelian functions were at the summit of mathematics, and that later on their popularity plummeted. I could hardly find anything on the net on Abelian functions and Wikipedia thinks that they are barely wo... | https://mathoverflow.net/users/4692 | Why were Abelian functions so important in the 19th century? | For a really detailed answer to your question, see *The Legacy of Niels Henrik
Abel*, edited by O.A. Laudal and R. Piene (Springer 2004). In particular, there is a long introductory article by Christian Houzel, most of which can be viewed [here](http://books.google.com.au/books?id=HiXwhBm42hcC&lpg=PA21&ots=ibJuErm5sc&d... | 13 | https://mathoverflow.net/users/1587 | 26565 | 17,387 |
https://mathoverflow.net/questions/26538 | 2 | I have some questions about the following exercise in Hartshorne (III.4.7):
Let $f \in k[x\_0,x\_1,x\_2]$ be a homogeneous polynomial of degree $d \geq 1$ and $f \neq 0$ and let $X$ be the closed subscheme of $\mathbb{P}^2\_k$ defined by $f$. Then $\dim H^0(X,\mathcal{O}\_X) = 1, \dim H^1(X,\mathcal{O}\_X) = (d-1)(d-... | https://mathoverflow.net/users/2841 | Hartshorne Exercise III.4.7 (cohomology of closed subschemes in $\mathbb{P}^2$) | You can compute the cohomology via the Koszul resolution. If $i:X \to {\mathbb P}^2\_k$ is the embedding then the triple $0 \to O\_{{\mathbb P}^{2}}(-d) \stackrel{f}\to O\_{{\mathbb P}^2} \to i\_\*O\_X \to 0$ is exact. So, you can compute $H^t(X,O\_X) = H^t({\mathbb P}^2\_k,i\_\*O\_X)$ using the long exact sequence ass... | 4 | https://mathoverflow.net/users/4428 | 26567 | 17,388 |
https://mathoverflow.net/questions/26492 | 5 | Is there any software that will compute cohomology of vector bundles (or just line bundles) on flag manifolds?
The only one I know of is Macaulay2, via the Schubert2 package, but it works with what it calls "abstract varieties", which are really just the intersection rings over $\mathbb{Q}$, so it's explicitly limit... | https://mathoverflow.net/users/460 | software for computations on flag varieties in arbitrary characteristic | To answer the original question explicitly, there seems to be no relevant software in prime characteristic. Nor is there any on the horizon, unless the theory developed so far becomes much more definitive. In the setting of flag varieties, general principles show that Euler characters in characteristic $p$ are the same... | 5 | https://mathoverflow.net/users/4231 | 26572 | 17,391 |
https://mathoverflow.net/questions/26555 | 12 | Hello,
Let $X$ and $Y$ be two smooth (probably projective) algebraic varieties defined over $\mathbf{C}$.
What is known in general about the (topological) space of holomorphic maps $\mathrm{Hol}(X(\mathbf{C}),Y(\mathbf{C}))$ from $X(\mathbf{C})$ to $Y(\mathbf{C})$?
In particular, I would be interested to know under... | https://mathoverflow.net/users/5239 | Are spaces of holomorphic maps manifolds? | Let $X,Y$ be analytic spaces with $X$ compact reduced and $Y$ arbitrary (i.e. maybe with nilpotents in its structure sheaf). Then Douady showed in his [thesis](http://archive.numdam.org/item/AIF_1966__16_1_1_0/)
that the set of holomorphic maps $Hol(X,Y)$ can be endowed with the structure of an analytic space whose und... | 11 | https://mathoverflow.net/users/450 | 26580 | 17,398 |
https://mathoverflow.net/questions/26537 | 7 | It is fundamental to topology that $\mathbb{R}$ is a connected topological space. However, all the topology books that I have ever looked in give the same proof. (the proof I am thinking of can be seen in Munkres's topology or Lee's Introduction to topological manifolds)
This seems strange to me, because for other fu... | https://mathoverflow.net/users/4002 | Connectedness and the real line | If you've already developed basic facts about compactness you can prove it this way:
Let $[0,1] = A \cup B$ with $A$ and $B$ closed and disjoint. Then since $A \times B$ is compact and the distance function is continuous, there is a pair $(a, b) \in A \times B$ at minimum distance. If that distance is zero, $A$ and $... | 42 | https://mathoverflow.net/users/644 | 26594 | 17,406 |
https://mathoverflow.net/questions/26586 | 10 | This question is on bounding the degree of the Todd class on a complex threefold.
Let $X$ be a smooth compact connected complex surface. Let $c\_i=c\_i(TX)$ be its $i$-th Chern class. Recall the following two facts. These allow one to bound the degree of the Todd class on a surface in terms of $c\_2$.
**1**. If $X... | https://mathoverflow.net/users/4333 | Can one bound the Todd class of a 3-dimensional variety polynomially in c_3 | The case of complex manifolds of higher dimension is very different from the case of complex surfaces. So the answer to the question about complex $3$ folds is no, there exists a real 6-dimensional simply connected manfiold with integrable complex structures $J\_m$ for all $m\in \mathbb Z^+$ such that $c\_1c\_2=48m$. T... | 18 | https://mathoverflow.net/users/943 | 26598 | 17,410 |
https://mathoverflow.net/questions/23601 | 18 | My question is based on the following vague belief, shared by many people: It should be possible to use von Neumann algebras in order to define the cohomology theory TMF (topological modular forms) in the same way one uses Hilbert spaces in order to define topological K-theory. More precisely, one expects hyperfinite t... | https://mathoverflow.net/users/5690 | Monoidal structures on von Neumann algebras | The category of von Neumann algebras W\* admits a variety of monoidal structures of three distinct flavors.
(1) W\* is complete and therefore you have a monoidal structure given by the categorical product.
(2a) W\* is cocomplete and therefore you also have a monoidal structure given by the categorical coproduct.
... | 6 | https://mathoverflow.net/users/402 | 26600 | 17,411 |
https://mathoverflow.net/questions/26602 | 2 | I have to say whether or not the following two separation logic statements are valid:
1. $ x \mapsto 3 \* y \mapsto 7 \Longrightarrow x \mapsto 3 \* true $
2. $ true \* x \mapsto 3 \Longrightarrow x \mapsto 3 $
Where $ x \mapsto 3 $ means x (in the stack) points to an abitrary memory location in the heap containing... | https://mathoverflow.net/users/5718 | Are these separation logic statements valid? | The trick with separation logic is that the formulas describe resources (heaps) and if a logical implication holds, then both sides of the implication are descriptions of the same heap.
True can correspond to any heap.
$h \* x\mapsto 3$ is almost always a larger heap than $x\mapsto 3$, which consists of just one el... | 1 | https://mathoverflow.net/users/2620 | 26603 | 17,412 |
https://mathoverflow.net/questions/26549 | 13 | Edwards, in his book "Divisor theory" says that Kronecker's methods are quite different to Dedekind's and those of today. Is there really much of a difference apart from Kronecker's methods being more constructive?
| https://mathoverflow.net/users/4692 | Is there much difference between Kronecker's and Dedekind's methods in algebraic number theory and commutative algebra? | You can find a nice description of Kronecker's approach in an article by Harley Flanders,
"The Meaning of the Form Calculus in Classical Ideal Theory" (Trans. AMS 95 (1960), 92--100). It is at JSTOR [here](https://www.jstor.org/stable/1993332?seq=1#metadata_info_tab_contents). I found that more to my tastes than Edward... | 23 | https://mathoverflow.net/users/3272 | 26628 | 17,426 |
https://mathoverflow.net/questions/946 | 5 | One of the basic tools in subfactors is the conditional expectation. If $N\subset M$ is a $II\_1$-subfactor (or an inclusion of finite factors), then there is a unique trace-preserving conditional expectation of $M$ onto $N$. This should be thought of as a (Banach space) projection of norm 1. In fact, it is the restric... | https://mathoverflow.net/users/351 | Operator Valued Weights | Since $T(m\_T)$ is a non-trivial two sided ideal in $N$, it follows from spectral calculus that $T(m\_T)$ contains a non-zero projection. $T(m\_T)$ then contains all subprojections, hence there exists $x \in m\_T$ such that $p = T(x)$ is a projection with trace Tr$(p) = 1/n$ for some natural number $n$.
We may assum... | 6 | https://mathoverflow.net/users/6460 | 26632 | 17,429 |
https://mathoverflow.net/questions/26626 | 4 | Is there any easy proof or/and reference for the following
**Proposition.** If a polynomial $f\in \mathbb{R}[x\_1,\dots,x\_n]$ with zero constant term has isolated local minimum in 0, then $|f|> C (x\_1^2+\dots +x\_n^2)^N$ in some neighborhood of 0, where
$\bullet$ $N$ depends only on degree of $f$ and $n$,
$\... | https://mathoverflow.net/users/4312 | minimum of polynomial | That there is some $C$ and $N$ for which this is satisfied is a case of Lojasiewicz's theorem, which says that if $f(x\_0) = 0$, then there is an open set containing $x\_0$ on which $|f(x)| \geq C|d(x,Z)|^N$ for some $C$ and $N$. Here $d(x,Z)$ denotes the distance from $x$ to the zero set $Z$ of $f(x)$. So in the isola... | 2 | https://mathoverflow.net/users/2944 | 26637 | 17,433 |
https://mathoverflow.net/questions/22748 | 8 | Hi,
I am looking at inclusion of discrete groups $H\subset G$ such that $H$ is abelian and $(hgh^{-1},h\in H)$ is infinite if $g\in G-H$. If you have this, $LH\subset LG$ is a maximal abelian subalgebra of a finite von Neumann algebra.
Suppose that $LH\subset LG$ is a Cartan subalgebra, i.e. the group of unitary of $LG... | https://mathoverflow.net/users/2045 | normalizer of algebras and groups | This is true, and in fact more has been shown in the recent preprint <http://arxiv.org/abs/1005.3049> of Fang, Gao, and Smith. One can also give the following alternative argument based on ideas of Popa:
If $LH \subset LG$ is a MASA then it follows from the condition $ ( hgh^{-1} \ | \ h \in H ) = \infty$ for all $g ... | 6 | https://mathoverflow.net/users/6460 | 26638 | 17,434 |
https://mathoverflow.net/questions/26601 | 0 | I have been struggling with this problem and hope someone could help. I am trying a variation of non-repetitive combination scenario. I can use the formula n!/r!x(n-r)! to find non-repetitive combinations of size "r" from "n" numbers. However, these combination have repeating elements.
For example:
I have 9 letters... | https://mathoverflow.net/users/6455 | Creating a combinations with unique sets | I have deleted what was here as it was based on a misunderstanding of the problem. My current understanding of the problem is that, given positive integers $k\le n$ one wants the largest number of $k$-element subsets of an $n$-element subset, no two intersecting in more than one element. This has indeed been studied as... | 1 | https://mathoverflow.net/users/3684 | 26639 | 17,435 |
https://mathoverflow.net/questions/26651 | 16 | Hi, everybody. I'm recently reading W.Bruns and J.Herzog's famous book-Cohen-Macaulay Rings. I personally believe that it would be perfect if the authors provide for readers more concrete examples. After reading the first two sections of this book, I have two questions.
1. Given a non-negative integer n, how can we c... | https://mathoverflow.net/users/5775 | Is there a simple method to test a local ring to be Cohen Macaulay? | Some interesting examples of Cohen-Macaulay but not Gorenstein rings:
1) Determinantal rings: Let $m\geq n\geq r>1$ be integers. Take $S=k[x\_{ij}]$ with $1\leq i\leq m, 1\leq j\leq n$ and $I$ be the ideal generated by all $r$ by $r$ minors. Then $R=S/I$ is CM, but is Gorenstein if and only if $m=n$. And $R$ has dime... | 13 | https://mathoverflow.net/users/2083 | 26654 | 17,445 |
https://mathoverflow.net/questions/26661 | 3 | Whenever I have seen unique factorization discussed, it is always with respect to the solution of diophantine equations; the equations are solved by splitting the equation into linear functions over a ring and then invoking unique factorization. But the discussions always give the impression that the failure of unique ... | https://mathoverflow.net/users/4692 | What goes wrong in a ring that does not have unique factorization? | I think a glance at any elementary number theory textbook will show you numerous topics which rely on uniqueness of factorization. The formulas for the number of divisors, the sum of the divisors, the Euler phi-function, all depend on unique factorization. Going up a level, the Euler product for the Riemann zeta functi... | 2 | https://mathoverflow.net/users/3684 | 26662 | 17,450 |
https://mathoverflow.net/questions/26643 | 2 | ZFCfin (ZFC without the axiom of infinity, plus its negation) is biinterpretable with Peano Arithmetic.
Each countable model of PA has what is known as its "standard system": the collection of sets of standard natural numbers which can be coded in that system. Usually the coding is via prime exponentiation: $n$ is in... | https://mathoverflow.net/users/2361 | "standard system" of a nonstandard model of PA, interpreted in ZFCfin? | Yes, there is a perfect agreement between the standard system of the nonstandard model of ZFCfin, such as a nonstandard version of HF, and the standard system of its corresponding model of PA. The standard systems are identical.
That is, because of the mutual interpretability, from any model of PA we may form a mode... | 4 | https://mathoverflow.net/users/1946 | 26670 | 17,456 |
https://mathoverflow.net/questions/26644 | 1 | For this question, I am in the smooth finite-dimensional category. All objects are $\mathcal C^\infty$ manifolds and all maps are smooth. Recall the notion of [groupoid](http://www.google.com/search?hl=&q=groupoid), and let's restrict our attention to those for which the source map (and hence also the target map) is a ... | https://mathoverflow.net/users/78 | Given an object in a Lie groupoid, does there exist a subgroupoid for which the object has no automorphisms but retains its equivalence class? | Consider the case of a group $K$ acting on $X$. Restricting to the orbit of the point $x$, by the orbit-stabilizer theorem $X$ is identified with $K/\text{Stab}(x)$. Your question seems to amount to asking whether the projection $K\times K/\text{Stab}(x)\to K/\text{Stab}(x)\times K/\text{Stab}(x)$ given by $(\gamma,p)\... | 1 | https://mathoverflow.net/users/250 | 26675 | 17,461 |
https://mathoverflow.net/questions/26665 | 4 | I come across the following problem in my study.
Consider in the real field. Let $ 0\le x\le1 $, $a\_1^2+a\_2^2=b\_1^2+b\_2^2=1$.Is it true
$ (a\_1b\_1+xa\_2b\_2)^2\le\left(\frac{(1-x)+(1+x)(a\_1b\_1+a\_2b\_2)}{(1+x)+(1-x)(a\_1b\_1+a\_2b\_2)}\right)^{2}(a\_1^2+xa\_{2}^{2})(b\_1^2+xb\_{2}^{2})$?
| https://mathoverflow.net/users/3818 | Another plausible inequality. | As Can Hang points out in his response, the inequality does not hold in general. Thanks to his comment to my own post, I stand corrected and claim the inequality is valid at least for the case
$$
a\_1b\_1+xa\_2b\_2\ge0
\qquad(\*)
$$
(and this seems to be a necessary condition as well).
Let me do some standard things.... | 10 | https://mathoverflow.net/users/4953 | 26677 | 17,462 |
https://mathoverflow.net/questions/26680 | 49 | Background
----------
Let $(X,x)$ be a pointed topological space. Then the fundamental group $\pi\_1(X,x)$ becomes a topological space: Endow the set of maps $S^1 \to X$ with the compact-open topology, endow the subset of maps mapping $1 \to x$ with the subspace topology, and finally use the quotient topology on $\pi... | https://mathoverflow.net/users/2841 | Fundamental group as topological group | **Update**: A bit of a digital paper chase led me, via [David Robert's thesis](http://ncatlab.org/davidroberts/show/HomePage) (note that in the latest version, it is Chapter 5, section 2 that is most relevant), to [this paper](http://arxiv.org/abs/0910.3685) on the arXiv. The last sentence of the abstract is:
>
> T... | 30 | https://mathoverflow.net/users/45 | 26684 | 17,468 |
https://mathoverflow.net/questions/26520 | 13 | Skip Garibaldi asks if there is an elementary proof of the following fact that "accidentally" fell out of some high-powered machinery he was working on.
Say that two matrices $A$ and $B$ over the rationals are *rationally congruent* if
there exists a nonsingular matrix $S$ over the rationals such that $S^t A S = B$.
... | https://mathoverflow.net/users/3106 | Rational congruence of binomial coefficient matrices | Wadim, isn't that 95% of the proof? First let me correct your first displayed equation (thanks to fherzig for pointing this out): It is not sufficient for the proof, but
$$
\sum\_{i=0}^{n-1}\binom{4n}{2i}P\_i(t)P\_i(s)
=\sum\_{i=0}^{n-1}\binom{4n}{2i+1}\hat P\_i(t)\hat P\_i(s)
$$
is, where $t$ and $s$ are two independe... | 10 | https://mathoverflow.net/users/2530 | 26693 | 17,474 |
https://mathoverflow.net/questions/26676 | 9 | The following are a collection of doubts, some of which shall have concrete answers while others may have not. Any kind of help will be welcome.
Reading Peter Smith's "Gödel Without (Too Many) Tears", particularly where he gives a nonstandard model of Q, I began wondering if the reason for the existence of nonstandar... | https://mathoverflow.net/users/6466 | Incompleteness and nonstandard models of arithmetic | Unfortunately, nonstandard models will survive any such attempt. This is guaranteed by the [Löwenheim-Skolem Theorem](http://en.wikipedia.org/wiki/L%C3%B6wenheim%E2%80%93Skolem_theorem) which says that if a countable first-order theory T has an infinite model then it has one of every infinite cardinality. Since an unco... | 16 | https://mathoverflow.net/users/2000 | 26697 | 17,476 |
https://mathoverflow.net/questions/26689 | 19 | Given an [integral equation](http://en.wikipedia.org/wiki/Integral_equation) is there always a differential equation which has the same (say smooth) solutions?
It seems like not but can one prove this in some example?
**Edit:** Naively I'm hoping for some algorithm which takes an integral equation and applies some op... | https://mathoverflow.net/users/745 | Can an integral equation always be rewritten as a differential equation? | While I second Deane's comment that the author should be a bit more specific about the kind of equations he is interested in, in general the answer is **no** for integral and, more broadly,*integro-differential* equations. However, the latter can be reduced to *functional-differential* equations rather than to purely d... | 8 | https://mathoverflow.net/users/2149 | 26705 | 17,481 |
https://mathoverflow.net/questions/26718 | 1 | Consider a simple queue model like the one described in <http://en.wikipedia.org/wiki/M/M/1_model>. The article states what the expected waiting time is before a request enters the queue.
Assuming that the actual queue length is unknown, does the expected value of "time left to wait" for a given request change over t... | https://mathoverflow.net/users/6476 | Does waiting in a queue change the expected time left to wait? | I am not entirely sure what you are asking but ...
The model you reference in the wiki article has a memory-less distribution for waiting time and inter-arrival time. Thus, the total time for a request to get processed is not dependent on time.
| 2 | https://mathoverflow.net/users/4660 | 26722 | 17,491 |
https://mathoverflow.net/questions/26318 | 18 | Wonder whether any of you know where it was that the following pearl of topology first appeared:
>
> Prove that at any instant of time you can find three isothermal points on the surface of the Earth that correspond to the vertices of an equilateral triangle.
>
>
>
According to [Léo Sauvé](https://mathoverflow... | https://mathoverflow.net/users/1593 | Points on a sphere | Let me just elaborate a little on the references that Charlie Frohman listed (so this isn't really a separate answer, but it's too long for a comment).
The theorem for equilateral triangles is due to S. Kakutani (1942 Annals). He stated it just for triangles formed by orthonormal bases for $\mathbb R^3$, but the argu... | 23 | https://mathoverflow.net/users/23571 | 26731 | 17,495 |
https://mathoverflow.net/questions/26727 | 3 | A. Aubry published a paper entitled "Les Logarithmes avant Neper" in the 1906 edition of L'Enseignement Mathématique signed simply "A. Aubry (Beaugency, Loiret)". Does anyone know what the A. stood for (perhaps Auguste?) or anything more about this A. Aubry?
The only other reference I've found is that it may have bee... | https://mathoverflow.net/users/4111 | Information about A. Aubry | I think it is Auguste Aubry. The *L'Enseignement Mathématique* volume is on archive.org, and there is an earlier paper in it on hyperbolic functions by Aubry. The material and location suggests a school teacher. There is more about him (I presume) in <http://www.univ-lille1.fr/bustl-grisemine/pdf/extheses/50416-1999-De... | 4 | https://mathoverflow.net/users/6153 | 26732 | 17,496 |
https://mathoverflow.net/questions/26599 | 11 | This is a question I have thought about and asked a number of people, but have never got an answer beyond "It should be true that..."
Given a finitely generated group $G$ (eg. a link group $G\_L:=\pi\_1(S^3-L)$ for a link $L$) and a finite group $H$ we want to count homomorphisms from $G$ to $H$. For link groups as a... | https://mathoverflow.net/users/6355 | complexity of counting homomorphisms | For $G$ a knot group, and for $H$ a dihedral group, there should be a simple algorithm for counting the number of homomorphisms. The meridians of $G$ normally generate, and are all conjugate, so they must be sent to conjugate elements in $H$. If they are sent to the cyclic subgroup of index 2, then the image is cyclic,... | 3 | https://mathoverflow.net/users/1345 | 26742 | 17,504 |
https://mathoverflow.net/questions/21657 | 4 | A stupid AG question: could singular (Zarisky) points be dense in a reduced (Noetherian) scheme $S$? If yes, which 'standard' restrictions on $S$ could ensure that this does not happen? For example, should one demand $S$ to be excellent? Will anything change if we want singular points not to be dense in any finite type... | https://mathoverflow.net/users/2191 | When singular points of a reduced scheme are not dense in it? | For examples with dense singular locus, see William J. Heinzer and Lawrence S. Levy: Domains of Dimension 1 with Infinitely Many Singular Maximal Ideals, Rocky Mountain J. Math. (2007), 203-214. Their examples are affine and noetherian.
| 3 | https://mathoverflow.net/users/3485 | 26756 | 17,514 |
https://mathoverflow.net/questions/26754 | 0 | I have two variables, $x$ and $y$, and I am using linear regression to predict $y$ from $x$ over a large set of subjects. There are multiple observations per subject.
I have tried several things:1. pool all observations together and fit one model to the entire dataset;
- fit separate models (same specification as (1)... | https://mathoverflow.net/users/6484 | Combining regressions | You should use hierarchical bayesian regression. Google search will provide lots of pointers.
| 0 | https://mathoverflow.net/users/4660 | 26757 | 17,515 |
https://mathoverflow.net/questions/26695 | 3 | We choose our category of spaces to be compactly generated weak Hausdorff spaces for convenience, denoted $CGWH$.
Questions:
1.) Is there any sort of slick argument to verify that CGWH with the Quillen model structure is a right-proper (closed) model category?
2.) If we give the following presentation of the mod... | https://mathoverflow.net/users/1353 | Slick verification of the model category axioms for Spaces and SSets with the q-model structure? | The Joyal and Tierney notes contain a combinatorial proof as Dan says. They are available [here](http://www.crm.cat/HigherCategories/tierney.pdf). (Wanted to post this in the comments, but it seems impossible to do without sufficient reputation.) I should also mention that it is possible to give a reasonably slick proo... | 6 | https://mathoverflow.net/users/6485 | 26759 | 17,516 |
https://mathoverflow.net/questions/26640 | 24 | Roughly speaking, I want to know whether one-relator groups only have 'obvious' free splittings.
>
> Consider a one-relator group $G=F/\langle\langle r\rangle\rangle$, where $F$ is a free group. Is it true that $F$ splits non-trivially as a free product $A \* B$ if and only if $r$ is contained in a proper free fact... | https://mathoverflow.net/users/1463 | Free splittings of one-relator groups | $\newcommand{\rank}{\operatorname{rank}}$I think [Grushko](http://en.wikipedia.org/wiki/Grushko_theorem) plus the [Freiheitssatz](http://en.wikipedia.org/wiki/Freiheitssatz) does the trick.
Suppose that $G=A \ast B$ is a one-relator group which splits as a free product non-trivially. By Grushko, $\rank(G)=\rank(A)+\r... | 18 | https://mathoverflow.net/users/1345 | 26767 | 17,524 |
https://mathoverflow.net/questions/21720 | 23 | Hi. Peter Roquette sent me an email asking for an example of a quadratic space in characteristic 2 having certain features. I have no idea on this, but maybe someone reading this does.
He would like an example of a field $K$ of characteristic 2 with the following two properties:
(1) The quaternion algebras over $K... | https://mathoverflow.net/users/3272 | Wanted: Quadratic Space in Characteristic 2 as a Counterexample to a Theorem of Arf | I forwarded this question to Detlev Hoffmann, who says that such examples exist. Specifically, you can produce such an example where there is, say, an anisotropic form of dimension 8 using characteristic 2 analogues of the techniques in Merkurjev's 1992 article "Simple algebras and quadratic forms". He says details can... | 17 | https://mathoverflow.net/users/6486 | 26772 | 17,529 |
https://mathoverflow.net/questions/26715 | 7 | $\underline{Background}$ :
Suppose $\tau$ is a preadditive category and $R$ a ring. Then one may form a new preadditive category $\tau \otimes R$ in the following way:
$\tau \otimes R$ has the same objects as $\tau$ and for objects $A$ and $B$, set $\tau \otimes R(A,B) := \tau(A,B)\otimes\_{\mathbb{Z}} R$. Composit... | https://mathoverflow.net/users/6474 | Is the tensorproduct of a triangulated category with a ring again triangulated? | I would imagine it is false in general that given a triangulated category $T$ the category $T\otimes R$ is also triangulated.
The following is a concrete counterexample. Consider $D^b(\mathbb{Z})$ and let $T = D^b(\mathbb{Z})\otimes \mathbb{Z}[x]$. In order for $T$ to be triangulated the morphism $\mathbb{Z} \stackre... | 8 | https://mathoverflow.net/users/310 | 26774 | 17,531 |
https://mathoverflow.net/questions/26776 | 32 | The total space of cotangent bundle of any manifold $M$ is a symplectic manifold.
Is it true/false/unknown that for any $M$, $T^\*M$ has Kähler structure?
Please support your claim with reference or counterexample.
| https://mathoverflow.net/users/5259 | Kähler structure on cotangent bundle? | This is true! I assume $M$ compact.
**Method 1.** Real algebraic geometry. Cf. [Fukaya, Seidel, and Smith - Exact Lagrangian submanifolds in simply-connected cotangent bundles](http://arxiv.org/abs/math/0701783). By a version of the Nash–Tognoli embedding theorem, one can realise $M$ as an real affine algebraic varie... | 41 | https://mathoverflow.net/users/2356 | 26781 | 17,536 |
https://mathoverflow.net/questions/26790 | 3 | Suppose we have a finite $\sigma$ -field $S$, of which $A$ and $B$ are member sets. Since $S$ is closed under union and complementation [by definition], it follows that $(A' \cup B')' = (A \cap B)' \in S$. From closure under complementation, we have that $A \cap B \in S$, implying that
$S$ is closed under intersections... | https://mathoverflow.net/users/6495 | Finite Topology vs sigma Field | Yes, it is also a topology on its union, the largest member of $S$. Since $S$ is finite, the arbitrary union rquirement amounts to finite union, which you have.
In fact,$S$ is a Boolean algebra, and since it is finite, it is isomorphic to a powerset algebra---the power set of the atoms of $S$ (the minimal non-empty ... | 6 | https://mathoverflow.net/users/1946 | 26792 | 17,541 |
https://mathoverflow.net/questions/26783 | 6 | Where can I find a comprehensive treatment of this important result at the level of a very advanced undergraduate/beginning graduate student? What works develop the relevant material in a cohesive and readable way? Assume only a familiarity with basic homological algebra and ring theory on the part of the reader in ass... | https://mathoverflow.net/users/6131 | Best exposition of the Proof of the Hilbert Syzygy Theorem by Eilenberg-Cartan | The place I learned it from is Chapter 19 of Eisenbud's *Commutative Algebra*. Most of the proofs in that section do not use material from previous chapters if I recall correctly. The route (which I think is what you are looking for) is to construct the Koszul complex of the residue field of a regular (graded) local ri... | 7 | https://mathoverflow.net/users/321 | 26798 | 17,542 |
https://mathoverflow.net/questions/26797 | 15 | Let $K$ be a finite extension of $\mathbb{Q}\_p$ and $E$ an elliptic curve over $K$ with good ordinary reduction.
The p-adic Tate module $T\_p(E)$ is (after tensoring with $\mathbb{Q}\_p$) a 2-dimensional $\mathbb{Q}\_p$-representation of $\mathop{\mathrm{Gal}}(\bar{K}/K)$.
It is reducible: the kernel of reduction ... | https://mathoverflow.net/users/1046 | Does the p-adic Tate module of an elliptic curve with ordinary reduction decompose? | Serre has shown that there exists a complementary subspace invariant under the Lie algebra $\mathfrak{g}$ *if and only if* E has complex multiplication. Otherwise the image of Galois is open in the Borel subgroup of $\operatorname{GL}\_2(\mathbb{Q}\_p)$. I learnt this from the paper by Coates and Howson ("Euler charact... | 17 | https://mathoverflow.net/users/2481 | 26800 | 17,543 |
https://mathoverflow.net/questions/26788 | 6 | Suppose a deck of cards consists of $a\_1+a\_2+\cdots+a\_k$ cards of $k$ types, where there are $a\_i$ indistinguishable cards of each type. How many shuffles does it take, on average, to randomize the deck? For example, $a\_i=4$ and $k=13$ gives a standard deck of playing cards; $a\_i=4$ for $1\le i\le9$, $i\_{10}=16$... | https://mathoverflow.net/users/6043 | Shuffling decks of cards where not all cards are distinguishable | A place to start is the recent preprint:
[Riffle shuffles of a deck with repeated cards](https://arxiv.org/abs/0905.4698)
Sami Assaf, Persi Diaconis, K. Soundararajan.
They also have another preprint about reducing this to a "rule of thumb", and cite some earlier work of Conger and Viswanath, available [here](http:... | 7 | https://mathoverflow.net/users/1102 | 26808 | 17,546 |
https://mathoverflow.net/questions/26810 | 6 | Are there examples of subshifts (that is, closed shift-invariant subsets of the full shift {$1...n$}${}^{\mathbb{Z}}$) on which the shift is topologically mixing, which admit a shift-invariant probability measure of full support, but no invariant ergodic measure of full support ?
I guess the answer is no, but I can't... | https://mathoverflow.net/users/6129 | topologically mixing subshifts without ergodic measures | A construction of a topologically transitive mixing subshift with a fully supported invariant measure, but no fully supported ergodic measure, is given by Benjamin Weiss in the article "Topological transitivity and ergodic measures", *Mathematical Systems Theory* 1971. It gives a direct combinatorial construction, with... | 8 | https://mathoverflow.net/users/1840 | 26813 | 17,547 |
https://mathoverflow.net/questions/26823 | 1 | Trying to solve for the area enclosed by $x^4+y^4=1$. A friend posed this question to me today, but I have no clue what to do to solve this. Keep in mind, we don't even know if there is a straightforward solution. I think he just likes thinking up problems out of thin air.
Anyway, the question becomes more general, ... | https://mathoverflow.net/users/5562 | Area enclosed by x^4 + y^4 = 1 | I always prefer not to skip $dx$:
$$
I\_n=\int\_0^1(1-x^n)^{1/n}dx.
$$
After the change of variable $t=x^n$, the integral becomes the beta integral,
$$
I\_n=\frac1n\int\_0^1(1-t)^{1/n}t^{1/n-1}dt
=\frac1n\frac{\Gamma(1+1/n)\Gamma(1/n)}{\Gamma(1+2/n)}
=\frac1n\frac{\Gamma(1/n)^2\cdot 1/n}{\Gamma(2/n)\cdot 2/n}
\to1 \qua... | 7 | https://mathoverflow.net/users/4953 | 26825 | 17,551 |
https://mathoverflow.net/questions/26815 | 12 | What is the idea behind deformation to the normal cone and what are examples of its applications?
| https://mathoverflow.net/users/nan | deformation to the normal cone | Here's a place to see the normal cone side-by-side with other familiar constructions, that I learned from Fulton's "Intersection Theory". Here $X \subset Y$.
Start with the space $Y \times {\mathbb P}^1$, thought of as a trivial family over ${\mathbb P}^1$. Blow up the subscheme $X \times \infty$. Now we still have a... | 20 | https://mathoverflow.net/users/391 | 26827 | 17,552 |
https://mathoverflow.net/questions/26814 | 1 | I know that the fibre of $A\_{g,n}$ over $\mathbf{F}\_p$ is quasi-projective (of what dimension?). Can one exhibit some smooth projective subvarieties of high dimension in it? What are references for its geometry?
| https://mathoverflow.net/users/nan | projective subvarieties of the moduli space of abelian varieties | The dimension is $g(g+1)/2$. The supersingular locus gives a large projective subvariety but I don't recall whether it is smooth or not. For references, look up the many papers of F. Oort.
| 5 | https://mathoverflow.net/users/2290 | 26835 | 17,556 |
https://mathoverflow.net/questions/10048 | 8 | Does anyone know if there is something that can be said (ideally under at most very mild hypotheses) in group cohomology (let's even restrict to degree 1) that is similar to Serre's twisting, but in the case where the thing you want to twist by is not inner? I'm not expecting the statement to hold word-for-word, just t... | https://mathoverflow.net/users/2047 | substitute for Serre's twisting when the "twisting" is outer | The short answer is that there is not much to say about the relationship between $H^1(G, B)$ and a twist $H^1(G, B\_c)$ where $c$ is a cocycle taking values in $Aut(B)$. (I am going to write $B\_c$ for the twist instead of Serre's notation $\_cB$ for the sake of easy typesetting.) You can get a good feel for what is po... | 6 | https://mathoverflow.net/users/6486 | 26836 | 17,557 |
https://mathoverflow.net/questions/26838 | 6 | Let $p$ be an odd prime. An old theorem of Jacobi asserts that $p$ has exactly $8(p+1)$ representations as a sum of four squares of integers (solutions counted with order and sign). What is the most effective way to enumerate these solutions computationally? Can it be done in time $p^{1+\varepsilon}$, or even in time $... | https://mathoverflow.net/users/1464 | Enumerating representations of an integer as a sum of squares | Form the set $S$ of all squares less than $p$. This has $O(\sqrt{p})$ elements, and writing them down takes $O(\sqrt{p} \log p)$ time. (You don't have to implement fast multiplication to do this; just compute the list of squares by successively adding odd numbers.)
Let $T$ be the set of all integers expressible as th... | 8 | https://mathoverflow.net/users/297 | 26841 | 17,560 |
https://mathoverflow.net/questions/26832 | 60 | This is an elementary question (coming from an undergraduate student) about algebraic numbers, to which I don't have a complete answer.
Let $a$ and $b$ be algebraic numbers, with respective degrees $m$ and $n$. Suppose $m$ and $n$ are coprime. Does the degree of $a+b$ always equal $mn$?
I know that the answer is "y... | https://mathoverflow.net/users/6506 | Degree of sum of algebraic numbers | The following answer was communicated to me by Keith Conrad:
See:
>
> M. Isaacs, Degree of sums in a separable field extension, Proc. AMS 25
> (1970), 638--641.
>
>
>
>
> [http://alpha.math.uga.edu/~pete/Isaacs70.pdf](http://alpha.math.uga.edu/%7Epete/Isaacs70.pdf)
>
>
>
Isaacs shows: when $K$ has cha... | 41 | https://mathoverflow.net/users/1149 | 26859 | 17,572 |
https://mathoverflow.net/questions/26857 | 11 | $\DeclareMathOperator\Rep{Rep}\DeclareMathOperator\GL{GL}\DeclareMathOperator\SW{SW}$Let $S\_k$ be the symmetric group. Let $F$ be an algebraically closed field. Let $\Rep(S\_k)$ be the category of representations of $S\_k$ over $F$.
Let $\Rep(\GL\_n(F))$ be the category of algebraic representations of $\GL\_n(F)$.
We ... | https://mathoverflow.net/users/5082 | Schur-Weyl duality in positive characteristic | The questions here have certainly been explored (though not definitively) in many recent papers or preprints on arXiv. Look for example at the arXiv paper by Stephen Doty [Link](https://arxiv.org/abs/math/0610591), as well as many others by Steve and/or his collaborators. Most of the arXiv papers have subject listing R... | 5 | https://mathoverflow.net/users/4231 | 26862 | 17,573 |
https://mathoverflow.net/questions/26861 | 17 | Is it possible to construct (without using Axoim of Choice) a totally ordered set S with cardinality larger than $\mathbb{R}$?
**Motivation:** A total ordering is often called a “linear ordering”. I have heard the following explanation: “If you have a total ordering on a set S, you can plot the set on the real line s... | https://mathoverflow.net/users/2097 | Explicit ordering on set with larger cardinality than R | Yes. By [Hartogs' theorem](http://en.wikipedia.org/wiki/Hartogs_number), there is an ordinal that has no injection into $R$. The minimal such ordinal is the smallest well-ordered cardinal not injecting into $R$. It is naturally well-ordered by the usual order on ordinals. None of this needs AC.
One can think very co... | 20 | https://mathoverflow.net/users/1946 | 26864 | 17,575 |
https://mathoverflow.net/questions/26860 | 8 | It might be a stupid question.
How to glue perverse sheaves? I am considering the following example. Flag variety of $sl\_2$, which is $P^1$. Consider category of perverse sheaves on $P^1$. Denoted by $Perv(P^1)$. The big cells of flag variety of $sl\_2$ give the affine cover for it. In this case, they should be two ... | https://mathoverflow.net/users/1851 | Gluing perverse sheaves? | Beilinson's *How to glue perverse sheaves* explains how one can glue perverse sheaves on a variety from perverse sheaves on a closed subvariety and its open complement (assuming that the closed subvariety is the set of zeros of a regular function on the big variety).
Gluing perverse sheaves from an open covering is a... | 7 | https://mathoverflow.net/users/2106 | 26872 | 17,581 |
https://mathoverflow.net/questions/26606 | 12 | Let $g \geq 2$, and consider the moduli space $\bar M\_{g,n}$ of stable *n*-pointed curves of genus *g*. There is a natural forgetful map to $\bar M\_g$, which forgets the markings and contracts any resulting unstable component. I am thinking about what the fibers of this map look like. Consider first the *n*-fold fibe... | https://mathoverflow.net/users/1310 | The fibers of M_{g,n} \to M_g and the Fulton-MacPherson compactification | You don't mean that the forgetful map contracts any rational component, only those touching less than three nodes. Also, the n-fold fibered power is not singular where several markings coincide at a smooth point. After all, a fibered product of smooth morphisms is smooth. It will, however, disagree with $\overline{M}\_... | 14 | https://mathoverflow.net/users/6522 | 26873 | 17,582 |
https://mathoverflow.net/questions/26874 | 3 | Given a finite group G, its complexified ring of finite-dimensional complex representations is isomorphic to its algebra of class functions, by the trace map $\mathrm{Tr}\_\rho: g \mapsto \mathrm{Tr}(\rho(g))$. These class functions can in turn can be shown to correspond to the elements in the center of the group algeb... | https://mathoverflow.net/users/799 | Ring homomorphism from the representation ring into group algebra? | Off the top of my head: the representation ring is really indexed by the set of irreps (well, OK, a set of representatives from equivalence classes thereof) and is indeed commutative and semisimple (at least over the complex numbers) so isomorphic to an appropriate direct product of copies of the ground field.
The ce... | 6 | https://mathoverflow.net/users/763 | 26882 | 17,586 |
https://mathoverflow.net/questions/26155 | 10 | What is the necessary condition on a ring that guarantees the number of minimal non-zero ideals to be finite? Neither Noetherian or Artinian condition seems sufficient, and the ring being semisimple seems too strong.
| https://mathoverflow.net/users/5292 | Finite number of minimal ideals | (Inspired by Graham's comment) For simplicity I will consider the case $(R,m)$ is a Noetherian, local ring.
A non-zero minimal ideal better be principal. Also, if $(x)$ is such ideal, then for any $y\in m$, the ideal $(xy)$ has to be $0$, so $mx=0$.
Let $I= \{x\in R|mx=0\}$, the *socle* of $R$. Since $Im=0$, $I$ ... | 10 | https://mathoverflow.net/users/2083 | 26886 | 17,589 |
https://mathoverflow.net/questions/26892 | 42 | Most of us are at least somewhat curious about what's going on in areas of mathematics outside our own area of research. If a significant breakthrough is made, and can be stated in language that we understand, then we would enjoy hearing about it. Now, for truly fantastic breakthroughs like Fermat's Last Theorem or the... | https://mathoverflow.net/users/3106 | How do you find out the latest news in fields other than your own? | It's always hard to follow what's happening, especially in fields other than your own. I don't have any silver bullets, just a few time tested things that require serious work.
1. Talk to colleagues. If you are in a research department, always go to a colloquium. MR can be used both before and after the talk to tie i... | 18 | https://mathoverflow.net/users/5740 | 26896 | 17,595 |
https://mathoverflow.net/questions/26895 | 3 | Consider the following equation for $X(t)$:
$$X(t)=e^{-bt}X(0)+\sigma\int\_{0}^{b}e^{-b(t-s)}dW(t) \, ,$$
where $0 < b, \sigma\in\mathbb{R} $, $X(0)$ is the initial distribution of $X(t)$, independent of the Brownian motion $W(t)$. I want so show that $dX(t)= -bX(t)dt+\sigma dW(t)$, but I am getting stuck on compu... | https://mathoverflow.net/users/5136 | how to find derivative of a stochastic process? | If you interpret the stochastic integral in the Ito-sense (often used in finance) you'll have to use Ito's lemma to evaluate it:
See e.g. here: [Ito's lemma](http://en.wikipedia.org/wiki/Ito_lemma)
Alternatively you could interpret it in the Stratonovich-sense (often used in physics):
See e.g. here: [Stratonovi... | 9 | https://mathoverflow.net/users/1047 | 26902 | 17,597 |
https://mathoverflow.net/questions/26871 | 17 | This question is inspired by [Relation between Hecke Operator and Hecke Algebra](https://mathoverflow.net/questions/19684/relation-between-hecke-operator-and-hecke-algebra)
I remember having heard of yet another way of looking at Hecke operators acting on the spaces of modular forms for classical congruence subgroups... | https://mathoverflow.net/users/2349 | Hecke operators acting as correspondences? | The answer to your last question is yes, modulo my own misunderstandings. In Scholl's "Motives for modular forms", §4 ([DigiZeitschreiften](http://www.digizeitschriften.de/dms/img/?PID=GDZPPN00210749X)), the Hecke operators are defined in this way, and I think the equivalence of these definitions is implied by the diag... | 13 | https://mathoverflow.net/users/1310 | 26905 | 17,599 |
https://mathoverflow.net/questions/26912 | 3 | I wonder whether it is impossible to write the nth [Motzkin number](http://www.research.att.com/~njas/sequences/A001006) as a sum of a fixed number of, say, hypergeometric terms. To illustrate what I mean: $n!+(2n)!$ is not a hypergeometric term, but it is written as a sum of two hypergeometric terms.
I'd also apprec... | https://mathoverflow.net/users/3032 | "Closed" form for Motzkin and related numbers | Chapter 8 of Petkovsek--Wilf--Zeilberger $A=B$ starts as follows:
"If you want to evaluate a given sum in closed form, so far the tools that have been
described in this book have enabled you to find a recurrence relation with polynomial
coefficients that your sum satisfies. If that recurrence is of order 1 then you a... | 5 | https://mathoverflow.net/users/4953 | 26918 | 17,603 |
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