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https://mathoverflow.net/questions/28391 | 3 | I came across this following way of defining connection and curvature which is not so obviously equivalent to the definitions as familiar in Riemannian Geometry books like say by Jost.
If $E$ is a complex vector bundle over a manifold $M$ then one defines the space of vector valued $p$-differential forms on them as $... | https://mathoverflow.net/users/2678 | Definition of the curvature tensor | You need to write $\omega X$ as $\omega \otimes X$.
If you prove that $R: \Gamma(E) \to \Omega^2(M,E)$ is tensorial - i.e. $R(X,Y)(fS) = fR(X,Y)S$ where $S\in\Gamma(E)$, $f\in\mathcal{C}^\infty(M)$ and $X,Y\in TM$, then it follows that this mapping is indeed an element of ${\Omega} ^2(M,End(TM))$, because then the v... | 2 | https://mathoverflow.net/users/6818 | 28393 | 18,553 |
https://mathoverflow.net/questions/28386 | 14 | **The question** Consider a topological space $X$ and a family of sheaves (of abelian groups, say) $\; \mathcal F\_i \;(i\in I)$ on $X$. Is it true that
$$H^\*(X,\prod \limits\_{i \in I} \mathcal F\_i)=\prod \limits\_{i \in I} H^\*(X,\mathcal F\_i) \;?$$
According to Godement's and to Bredon's monographs this is corre... | https://mathoverflow.net/users/450 | The cohomology of a product of sheaves and a plea. | The answer to the first question is almost always no, see *Roos, Jan-Erik(S-STOC)
Derived functors of inverse limits revisited. (English summary)
J. London Math. Soc. (2) 73 (2006), no. 1, 65--83.* .
**Addendum**: The crucial point is that infinite products are not exact. The most precise counterexample statement is ... | 8 | https://mathoverflow.net/users/4008 | 28398 | 18,557 |
https://mathoverflow.net/questions/28357 | 8 | **Motivation and background** This question is motivated by the problem of classifying the (two-sided) closed ideals of the Banach algebra $\mathcal{B}(L\_\infty)$ of all (bounded, linear) operators on $L\_\infty$ $(=L\_\infty[0,1])$. As far as I am aware, the only known nontrivial ideals in $\mathcal{B}(L\_\infty)$ ar... | https://mathoverflow.net/users/848 | Factoring operators $L_\infty \longrightarrow L_2$ as the composition of $n$ strictly singular operators, $n\in \mathbb{N}$ | I think this is an outline of a proof, Phil. It is enough to factor $I\_{\infty,2}$ as the product of strictly singular operators, where $I\_{\infty,2}$ is the identity from $L\_\infty(\mu)$ to $L\_2(\mu)$ with $\mu$ a probability (since every operator from $L\_\infty$ to $L\_2$ is $2$-summing). I guess it can be assum... | 6 | https://mathoverflow.net/users/2554 | 28405 | 18,562 |
https://mathoverflow.net/questions/28389 | 3 | A-The addition of the Grothendieck Universe Axiom (for every set x, there exists a set y that is a universe and contains x as member element) to ZFC (ZFC+GU) is considered as giving an almost good solution permitting the insertion of Category theory inside Set theory. It is known that we obtain an equivalent theory (ZF... | https://mathoverflow.net/users/30395 | About Grothendieck Universe and Tarski's A and A' Axioms | The answer to question 1 is affirmative for GU and IN but
negative for TA. That is, the proper class formulation of
TA is not equivalent to TA, unless both are inconsistent.
Asserting that every ordinal is below an inaccessible
cardinal is clearly equivalent in ZF to asserting that the
inaccessible cardinals form a p... | 6 | https://mathoverflow.net/users/1946 | 28406 | 18,563 |
https://mathoverflow.net/questions/28379 | 4 | From time to time I find myself wishing to calculate basic statistics on words in the English language. For example, today I found myself wanting a graph of the number of English words vs. their length.
Admittedly, such queries usually arise for me in the context of conversational/recreational purposes, but with the ... | https://mathoverflow.net/users/3623 | Good quality data/packages for statistical/structure analysis of words in the English language | I am very sure this kind ofn things is done in the free (free as in freedom, not as in beer!)
statistical language R. .. See www.r-project.org, and then repeat your question on
the r-help mailing list, where you are sure to get informed answers about how to do it in R.
That it can be done tere, and is done there, is ... | 1 | https://mathoverflow.net/users/6494 | 28408 | 18,565 |
https://mathoverflow.net/questions/28394 | 3 | Suppose we have a topological group $G$, then the multiplication map $\mu$ and the diagonal map $\Delta$ provide the cohomology $H^\ast(G;R)$ (with Pontryagin coproduct and cup coproduct) and homology $H\_\ast(G;R)$ (with Pontryagin product and diagonal coproduct) of $G$ with the structure of a Hopf algebra (assuming v... | https://mathoverflow.net/users/798 | When are the homology and cohomology Hopf algebras of topological groups equal? | The mod $2$ cohomology of $\mathrm{SO}\_n$ is not an exterior algebra as soon
as $n\geq3$ (there is a unique non-zero element in degree $1$ whose square is
non-zero, think of the case $\mathrm{SO}\_{3}$ which is $3$-dimensional real
projective space) while the homology ring is an exterior algebra.
Another example is ... | 7 | https://mathoverflow.net/users/4008 | 28413 | 18,570 |
https://mathoverflow.net/questions/28418 | 4 | The title says it all. Despite heavy googling I have not been able to find anything. What I am interested in, is theory (maybe modelling), not for the moment finite difference methods as approximations to partial differential equations! Books, papers, webpages, ....
| https://mathoverflow.net/users/6494 | Reference/Introduction to partial difference(NOT differential!) equations | [Partial Difference Equations](http://books.google.com/books?id=1klnDGelHGEC) by Sui Sun Cheng? A good start (in the context of reading the classics) is the paper by [Courant-Friedrichs-Lewy](http://www.amath.washington.edu/~narc/win08/papers/courant-friedrichs-lewy.pdf). Also useful is a list with [more books](http://... | 4 | https://mathoverflow.net/users/3993 | 28426 | 18,577 |
https://mathoverflow.net/questions/28421 | 22 | It is well known that if $M, \Omega$ is a symplectic manifold then the Poisson bracket gives $C^\infty(M)$ the structure of a Lie algebra. The only way I have seen this proven is via a calculation in canonical coordinates, which I found rather unsatisfying. So I decided to try to prove it just by playing around with di... | https://mathoverflow.net/users/4362 | The Jacobi Identity for the Poisson Bracket | The Jacobi identity for the Poisson bracket does indeed follow from the fact that $d\Omega =0$.
I claim that (twice) the Jacobi identity for functions $f,g,h$ is precisely
$$d\Omega(X\_f,X\_g,X\_h) = 0.$$
To see this, simply expand $d\Omega$.
You will find six terms of two kinds:
* three terms of the form
$$X\_... | 27 | https://mathoverflow.net/users/394 | 28431 | 18,581 |
https://mathoverflow.net/questions/28415 | 18 | Let $X, Y$ be normed vector spaces, where $X$ is infinite dimensional. Does there exist a linear map $T : X \rightarrow Y$ and a subset $D$ of $X$ such that $D$ is dense in $X$, $T$ is bounded in $D$ (i.e. $\sup \_{x\in D, x \neq 0} \frac{\|Tx\|}{\|x\|}<\infty $), but $T$ is not bounded in $X$?
| https://mathoverflow.net/users/4928 | Unbounded operator bounded in a dense subset | Matthew's answer reminded me of a fact that makes this easy: if $X$ is a normed space (say, over $\mathbb{R}$) and $f : X \to \mathbb{R}$ is a linear functional, then its kernel $\ker f$ is either closed or dense in $X$, depending on whether or not $f$ is continuous (i.e. bounded). The proof is trivial: $\ker f$ is a s... | 28 | https://mathoverflow.net/users/4832 | 28434 | 18,584 |
https://mathoverflow.net/questions/28435 | 11 | I am writing a small thing in which I am required to use manually formatted \bibitem entries rather than a BibTeX file. This takes a lot of time, and I probably get some of the formatting wrong. Is there a way of producing such entries automatically, for example from MathSciNet or from the BibTeX items created by MathS... | https://mathoverflow.net/users/349 | Automatically extract a bibitem (not BibTeX!) from MathSciNet? | Hi Andreas. When you run BibTeX on a .bib file then it produces a .bbl file which more-or-less contains your bibliography in \bibitem format. So I recommend exporting everything BibTex style from MathSciNet, creating a .bib file, running BibTeX, and then making any remaining changes to the resulting .bbl file (which yo... | 10 | https://mathoverflow.net/users/5830 | 28439 | 18,586 |
https://mathoverflow.net/questions/28428 | 25 | I recently learned of the result by Carleson and Hunt (1968) which states that if $f \in L^p$ for $p > 1$, then the Fourier series of $f$ converges to $f$ pointwise-a.e. Also, [Wikipedia](http://en.wikipedia.org/wiki/Convergence_of_fourier_series#Norm_convergence) informs me that if $f \in L^p$ for $1 < p < \infty$, th... | https://mathoverflow.net/users/2318 | Convergence of Fourier Series of $L^1$ Functions | The answer to your first question is no. There is an $L^1$ function with Fourier series not converging in measure.
In the Kolmogorov example of an $L^1$ function $f$ with a.e. divergent Fourier series, there is in fact a set of positive measure $E$ and a subsequence $n\_k$ such that for all $x$ in $E$, the absolute v... | 17 | https://mathoverflow.net/users/6129 | 28445 | 18,591 |
https://mathoverflow.net/questions/28422 | 13 | Let $W$ be a Coxeter group and let $P\_W(q) = \sum\_{w \in W} q^{\ell(w)}$ be its Poincare series. When $W$ is the Weyl group of a simple algebraic group $G$ (hence $W$ is finite), $P\_W(q)$ is the generating function describing the cells in the Bruhat decomposition of the flag variety $G/B$, $B$ a Borel.
What happe... | https://mathoverflow.net/users/290 | Does the Poincare series of a Coxeter group always describe a "flag variety"? | I think that Shrawan Kumar's book "Kac-Moody groups, their flag varieties, and representation theory" will contain the flag varieties (which are really ind-varieties in the non-finite case) that you are looking for.
A crystallographic Coxeter group is one of the form $\langle s\_1,\ldots,s\_n| s\_i^2=(s\_i s\_j)^{m\_... | 6 | https://mathoverflow.net/users/425 | 28447 | 18,592 |
https://mathoverflow.net/questions/28446 | 2 | Let $S$ be a subset of $\mathbb{R}^n$. I would like to call $S$
1. a Lipschitz(1) hypersurface if for every $x\in S$ there is a hyperplane $H$ so that the orthogonal projection onto $H$ is a bi-Lipschitz map from a neighbourhood of $x$ in $S$ onto an open subset of $H$, and
2. a Lipschitz(2) hypersurface if for every... | https://mathoverflow.net/users/802 | Are these two notions of Lipschitz hypersurface equivalent? | Counterexample in $\mathbb R^3$: Let $C$ be the cube $max(|x|,|y|,|z|)\le 1$. Let $S$ be the intersection of $C$ with $z=0$. Choose a piecewise linear (PL) homeomorphism from the boundary of $C$ to itself such that the boundary of $S$ goes to a very zigzaggy set. Extend to a PL (therefore Lipschitz) homeomorphism from ... | 7 | https://mathoverflow.net/users/6666 | 28449 | 18,594 |
https://mathoverflow.net/questions/28438 | 20 | Mathematics is not typically considered (by mathematicians) to be a solo sport; on the contrary, some amount of mathematical interaction with others is often deemed crucial. Courses are the student's main source of mathematical interaction. Even a slow course, or a course which covers material which one already knows t... | https://mathoverflow.net/users/6779 | Mathematics and autodidactism | I think the crucial distinction here is not between books and people (after all, books are written by people!) or between physical contact and electronic contact, but between *non-interaction* and *interaction*. In my view, the crucial benefit that interaction provides is *the ability to have your questions answered an... | 18 | https://mathoverflow.net/users/3106 | 28454 | 18,598 |
https://mathoverflow.net/questions/10666 | 20 | My question is about [nonstandard analysis](http://en.wikipedia.org/wiki/Non-standard_analysis), and the diverse possibilities for the choice of the nonstandard model R\*. Although one hears talk of *the* nonstandard reals R\*, there are of course many non-isomorphic possibilities for R\*. My question is, what kind of ... | https://mathoverflow.net/users/1946 | Isomorphism types or structure theory for nonstandard analysis | Under a not unreasonable assumption about cardinal arithmetic, namely $2^{<c}=c$ (which follows from the continuum hypothesis, or Martin's Axiom, or the cardinal characteristic equation t=c), the number of non-isomorphic possibilities for \*R of cardinality c is exactly 2^c. To see this, the first step is to deduce, fr... | 9 | https://mathoverflow.net/users/6794 | 28457 | 18,601 |
https://mathoverflow.net/questions/28463 | 2 | The [Wikipedia article on numerical differentiation](http://en.wikipedia.org/wiki/Numerical_differentiation#Practical_considerations) mentions the formula
$$
h=\sqrt \epsilon \times x
$$
where $\epsilon$ is the machine epsilon (approx. $2.2\times 10^{-16}$ for 64-bit IEEE 754 doubles), to calculate the optimum "sma... | https://mathoverflow.net/users/6861 | Optimum small number for numerical differentiation | If you click through to the reference given for the Wikipedia piece, you'll find an answer. The formula given there is to take $h$ to be roughly $\sqrt{\epsilon\_f}x\_c$, where $\epsilon\_f$ isn't necessarily "machine epsilon," but more to the point, where $x\_c$ isn't necessarily $x$.
| 4 | https://mathoverflow.net/users/3684 | 28465 | 18,606 |
https://mathoverflow.net/questions/28469 | 14 | Let $L\diagup K$ be a Galois extension of fields satisfying $\left[L:K\right] < \infty$. Let $B$ be a finite-dimensional (as a $K$-vector space) $K$-algebra. Then, the Galois group $G$ of $L\diagup K$ acts on the $L$-algebra $L\otimes\_K B$ (although not by $L$-linear homomorphisms), thus also on its unit group $\left(... | https://mathoverflow.net/users/2530 | Hilbert 90 for algebras | It's actually easier to go the other way around. Finite dimensional modules over
an algebra $A$ fulfils the Krull-Remak-Schmidt theorem of being isomorphic to a
direct sum of indecomposable modules with the indecomposable factors unique up
to isomorphism. If now $L\bigotimes\_KU$ and $L\bigotimes\_KV$ are isomorphic as... | 17 | https://mathoverflow.net/users/4008 | 28474 | 18,612 |
https://mathoverflow.net/questions/28481 | 2 | Let $K$ be an algebraic function field of one variable. Then we can define its genus. On the other hand, it can also be seen as a scheme, so we can define the arithmetic and geometric genus. Could anyone please tell me the relation between these definitions?
| https://mathoverflow.net/users/3849 | on the genus of a function field | The definitions coincide, with some caveats: basically for a curve, there is a single notion of genus, which applies equally to smooth curves over algebraically closed fields, and to their function fields; and also over the complex numbers to the associated Riemann surface as two-dimensional manifold. See <http://en.wi... | 1 | https://mathoverflow.net/users/6153 | 28482 | 18,617 |
https://mathoverflow.net/questions/28459 | 9 | Background
----------
I need to solve polynomials in multiple variables using [Horner's scheme](http://en.wikipedia.org/wiki/Horner_scheme) in Fortran90/95. The main reason for doing this is the increased efficiency and accuracy that occurs when using Horner's scheme to evaluate polynomials.
I currently have an imp... | https://mathoverflow.net/users/6860 | How to implement Horner’s scheme for multivariate polynomials? | The paper you cite, "On the multivariate Horner scheme" (Pena, Sauer) has an explicit algorithm specified on p.3. The remaining challenge is to penetrate the notation and conventions in the paper
laid out in the first three pages far enough to turn their algorithm presentation into code.
It also seems that this paper... | 7 | https://mathoverflow.net/users/6094 | 28490 | 18,620 |
https://mathoverflow.net/questions/28485 | 20 | I recently came across an interesting phenomenon which confused me slightly, concerning integral points on varieties.
For example, consider $X = \mathbb{A}\_{\mathbb{Z}}^{n+1} \setminus \{0\}$, affine $n$-space over $\mathbb{Z}$ with the origin removed. Naively, one would guess that $X(\mathbb{Z})$ is the set of inte... | https://mathoverflow.net/users/5101 | Integral points on varieties | Your belief is correct. A $\mathbb{Z}$-point has to reduce to an $\mathbb{F}\_p$-point for all $p$, which kills examples with gcd > 1.
If you want to make this precise, try writing down an explicit description of X by patching affine pieces. All the essential ideas are already there in $\mathbb{A}\_{\mathbb{Z}}^1 \b... | 14 | https://mathoverflow.net/users/2481 | 28491 | 18,621 |
https://mathoverflow.net/questions/28453 | 9 | I understand that a generic $G$-polynomial $f(t\_1,...,t\_n)[X]$ over field $k$ has Galois group $G$ over $k(t\_1,...,t\_n)$. And basically any $G$ extension of $k$ should be generated by a realization of $f$.(even a bit stronger but that is not the point here).
Now as much as I understand, our motivation for hunting... | https://mathoverflow.net/users/6776 | Why do generic polynomials work in reality? | Adding unto Boyarsky's answer: Stephen Cohen has given [quantative bounds](http://www.ams.org/mathscinet-getitem?mr=628276) for how often generic polynomials work. If I've skimmed his paper correctly, when the coefficients are integers chosen from the interval $[-N, N]$, the probability that the Galois group comes out ... | 6 | https://mathoverflow.net/users/297 | 28494 | 18,623 |
https://mathoverflow.net/questions/28437 | 2 | Hi,
I have this math modeling problem that I need help with. If I have 3 data sources, each being updated at different frequencies, what would be the best way to rank them so the less frequent sources don't get pushed down too quickly? In other words, I want the less likely source to be "weighed down" in its spot. Als... | https://mathoverflow.net/users/6855 | Ranking sources at variable(random) frequencies | You can model the arrivals as a poisson distribution with different arrival rates λi for each one of your sources.
$f(n\_i;\lambda\_i) = \dfrac{\lambda\_i^{n\_i} e^{-\lambda\_i}} { n\_i!}$
You can then assume that the arrival rates are random draws from a gamma distribution which would let you pool information ac... | 1 | https://mathoverflow.net/users/4660 | 28498 | 18,626 |
https://mathoverflow.net/questions/28486 | 7 | In any presheaf topos, there exists an object called *Lawvere's segment*, which can be described as the presheaf $L:A^{op}\to Set$ such that for each object $a\in A$, $L(a)=\{x\hookrightarrow\ h\_a: x\in Ob([A^{op},Set])\}$.
That is, $L$ assigns to each object $a\in Ob(A)$ the set of monomorphisms with target $h\_a(... | https://mathoverflow.net/users/1353 | An explicit description of Lawvere's segment in the category of simplicial sets | I've never heard this called 'Lawvere's segment' before, but your $L$ is the subobject classifier in the presheaf topos $[A^{\mathrm{op}},\mathrm{Set}]$. In presheaf toposes generally, the subobject classifier $\Omega$ is the presheaf that sends an object $a$ to the set of all sieves on $a$, i.e. the set of subobjects ... | 5 | https://mathoverflow.net/users/4262 | 28499 | 18,627 |
https://mathoverflow.net/questions/27721 | 3 | Hi,
We have a real, non-singular and symmetric matrix M of size n by n, with diagonal elements 0's. Its eigenvalues and eigenvectors are computed.
Now we wish to change its diagonal elements arbitrarily to maximize the multiplicy of an eigenvalue ( unnecessary to be the same value with old one).
For example, let... | https://mathoverflow.net/users/6679 | Maximize the multiplicity of an eigenvalue | Let $W = M + D$, where $M$ is the original $n \times n$ matrix and $D$ is the added diagonal matrix that we want to determine.
$W$ is symmetric, thus diagonalizable by an adjoint action of the orthogonal group.
Larger multiplicities in the eigenvalues of $W$ imply smaller dimensions of the adjoint orbits.
For examp... | 1 | https://mathoverflow.net/users/1059 | 28500 | 18,628 |
https://mathoverflow.net/questions/28523 | 2 | Let $f$ be a Hilbert polynomial, and $X := Hilb\_h(P^d\_{F\_p})$ a Hilbert scheme defined over $F\_p$. Then there is an absolute Frobenius map $F: X \to X$. I'm even interested in the case $f \equiv 1$, so $X = P^d$.
Which of the following is a sensical question?
1. What is the family over $X$ induced by pulling ba... | https://mathoverflow.net/users/391 | What is the family derived from the absolute Frobenius on the Hilbert scheme? | Indeed since the map on the base is not an isomorphism (apart from silly cases such as when $X$ is empty: the Frobenius endomporphism of a locally finite type $\mathbb{F}\_p$-scheme is an isomorphism if and only if the scheme is etale), the pullback cannot be universal. Functorially, $F$ carries a family of closed subs... | 7 | https://mathoverflow.net/users/6773 | 28529 | 18,641 |
https://mathoverflow.net/questions/28530 | 2 | Let $A$ and $B$ be two $4\times 4$ matrices. Using Newton's identities, one can prove that if
$$\det(A) = \det(B)\quad \text{and}\quad \mathrm{tr}(A^i) = \mathrm{tr}(B^i)$$ for $i=1,2,3$, then $A$ and $B$ have the same characteristic polynomial, thus the same eigenvalues.
I'm interested in pairs of matrices $A$ and $... | https://mathoverflow.net/users/1162 | On matrices that almost have the same eigenvalues | Such matrices will have a characteristic polynomial $z^4+a\_3z^3+a\_2z^2+a\_1z+a\_0$ with the same $a\_3$, $a\_2$, $a\_0$ but distinct $a\_1$. You can generate a plenty of diagonal such matrices by picking roots of such two polynomials.
I cannot vouch that they were not studied but I am pretty certain that nothing gro... | 6 | https://mathoverflow.net/users/5301 | 28531 | 18,642 |
https://mathoverflow.net/questions/28525 | 3 | Construction
------------
Suppose I have a density matrix $\rho$ which is proportional to a projector $P$ formed by tensoring together $N$ small projectors $P^{(i)}$ of rank 2:
$P^{(i)} = |a\rangle\_i\langle a| + |b\rangle\_i\langle b|$
$P=P^{(1)}\otimes \cdots \otimes P^{(N)} = \bigotimes\_{i=1}^N P^{(i)}$
$\r... | https://mathoverflow.net/users/5789 | What is the entropy of a density matrix which is the sum of two unitarily equivalent projectors? | First observation, you can unitarily rotate the first term without changing the spectrum so you can just consider a single unitary with out loss of generality. Call the rotated term $Q$.
Next, it helps to know about the canonical form for two projectors. Given two projectors P and Q in general position, you can alway... | 2 | https://mathoverflow.net/users/1171 | 28537 | 18,647 |
https://mathoverflow.net/questions/28524 | 0 | Let $X$ be a complex normal projective variety, let $|L|$ be a non empty linear series on $X$ and let $b(|L|)$ be its base ideal.
Suppose $f:X'\rightarrow X$ is a log resolution of the ideal $b(|L|)$.
Is $f$ a log resolution of the linear series $|L|$ (even if $X$ is not smooth)?
If it is do you have a proof or a r... | https://mathoverflow.net/users/6430 | Log resolutions of linear series | I don't think so. Suppose for example that $X$ is smooth, and the base locus of $|L|$ is set-theoretically a divisor with normal crossing, but it has an embedded component. In this case $X$ itself will be a log-resolution of the base locus, but not of the linear system $|L|$.
| 1 | https://mathoverflow.net/users/4790 | 28546 | 18,652 |
https://mathoverflow.net/questions/23427 | 32 | Just yesterday I heard of the notion of a fundamental group of a topos, so I looked it up on the [nLab](http://ncatlab.org/nlab/show/fundamental+group+of+a+topos), where the following nice definition is given:
If $T$ is a Grothendieck topos arising as category of sheaves on a site $X$, then there is the notion of loc... | https://mathoverflow.net/users/259 | Fundamental groups of topoi | The profinite fundamental group of $X\_{fppf}$ as you define it is again the etale fundamental group of X. More precisely, the functor (of points)
$f : X\_{et} \to \mathrm{Sh}\_{fppf}(X)$
is fully faithful and has essential image the locally finite constant sheaves (image clearly contained there, as finite etale ma... | 33 | https://mathoverflow.net/users/3931 | 28555 | 18,660 |
https://mathoverflow.net/questions/28532 | 24 | Suppose I want to understand classical mechanics.
Why should I be interested in arbitrary poisson manifolds and not just in symplectic ones?
What are examples of systems best described by non symplectic poisson manifolds?
| https://mathoverflow.net/users/2837 | Classical mechanics motivation for poisson manifolds? | For many reasons and purposes, it is the Poisson bracket, not the symplectic form, that plays a primary role.
* Equations of motion and, more generally, the evolution of observables have an easy form:
$$ \frac{\partial f}{\partial t}=\{H,f\}.$$
* Conserved quantities form a Poisson subalgebra:
$$\{H,F\}=\{H,G\}=0 \im... | 30 | https://mathoverflow.net/users/5740 | 28562 | 18,662 |
https://mathoverflow.net/questions/28569 | 27 | Let $G$ be a complex connected Lie group, $B$ a Borel subgroup and $W$ the Weyl group. The Bruhat decomposition allows us to write $G$ as a union $\bigcup\_{w \in W} BwB$ of cells given by double cosets.
One way I have seen to obtain cell decompositions of manifolds is using Morse theory. Is there a way to prove the ... | https://mathoverflow.net/users/798 | Is there a Morse theory proof of the Bruhat decomposition? | Here's a partial answer. What I'm about to say is taken from Section 2.4 of Chriss and Ginzburg's *Representation Theory and Complex Geometry*. The references I use will be from this book. There aren't complete proofs there, but there are references to complete proofs.
The Bruhat decomposition on $G$ (we'll assume $G... | 33 | https://mathoverflow.net/users/321 | 28575 | 18,673 |
https://mathoverflow.net/questions/28560 | 9 | Let $X$ be the set of $k\times k$ matrix with entries in $\mathbb{C}$, and let $M\in X$. The group $GL(k,\mathbb{C})$ acts on $X$ by conjugation, and according to the Jordan decomposition theorem (see e.g [wikipedia](http://en.wikipedia.org/wiki/Jordan_form)) somewhere in the orbit containing $M$ is a block diagonal ma... | https://mathoverflow.net/users/5124 | Jordan Form Over a Polynomial Ring | The short answer is "no". It is not too difficult to construct \*invariants, but the *canonical forms* are hard.**a** What follows is not a full answer, but a useful way to think about the question.
The problem of classifying $k\times k$ matrices over a commutative ring $R$ up to conjugacy is equivalent to the probl... | 13 | https://mathoverflow.net/users/5740 | 28588 | 18,683 |
https://mathoverflow.net/questions/28526 | 42 | Suppose that you graduate with a good PhD in mathematics, but don't necessarily want to go into academia, with the post-doc years that this entails. Are there any other options for continuing to do "real math" professionally?
For example, how about working at the NSA? I don't know much of what is done there -- is it ... | https://mathoverflow.net/users/3028 | "Industry"/Government jobs for mathematicians | I have worked in academia, at the research center of a telecommunications company (Tellabs), and at two different [FFRDCs](http://en.wikipedia.org/wiki/List_of_federally_funded_research_and_development_centers) (MIT Lincoln Laboratory and IDA). At all of the non-academic jobs, I have done "real math," published papers,... | 37 | https://mathoverflow.net/users/3106 | 28589 | 18,684 |
https://mathoverflow.net/questions/28553 | 16 | On the way to defining Cartier divisors on a scheme $X$, one sheafifies a presheaf **base-presheaf** of rings $\mathcal{K}'(U)=Frac(\mathcal{O}(U))$ **on open affines $U$** to get a sheaf $\mathcal{K}$ of "meromorphic functions".1
(**ETA**: See [Georges Elencwajg's answer](https://mathoverflow.net/questions/28553/ext... | https://mathoverflow.net/users/84526 | Extra principal Cartier divisors on non-Noetherian rings? (answered: no!) | In the setup in the question, it should really say "we could have *invertible* meromorphic functions on Spec($A$) that don't come Frac($A)^{\times}$", since those are what give rise to "extra principal Cartier divisors". This is what I will prove cannot happen. The argument is a correction on an earlier attempt which h... | 21 | https://mathoverflow.net/users/3927 | 28591 | 18,685 |
https://mathoverflow.net/questions/28590 | 11 | I derived this definition by searching for a representation of a family of sets. I am quite sure that someone should have thought to this before, because it seems to be quite straightforward given a family of sets. However, I did not find anything suitable on google or on wikipedia.
Let a family of sets, say $A\_1, \... | https://mathoverflow.net/users/6882 | I am searching for the name of a partition (if it already exists) | Your building blocks are known as the *atoms* in the [Boolean algebra](http://en.wikipedia.org/wiki/Boolean_algebras) or [field of sets](http://en.wikipedia.org/wiki/Field_of_sets) generated by the $A\_i$. Each building block will consist of points that have the same pattern of answers for membership in the various $A\... | 19 | https://mathoverflow.net/users/1946 | 28592 | 18,686 |
https://mathoverflow.net/questions/28595 | 6 | My question is a doubt I had in the last point to the first answer to this MO question - ["Algebraic" topologies like the Zariski topology?](https://mathoverflow.net/questions/14314/algebraic-topologies-like-the-zariski-topology)
Can one associate a Riemann surface to any arbitrary field extension? The statement ther... | https://mathoverflow.net/users/2720 | Does there exist a Riemann surface corresponding to every field extension? Any other hypothesis needed? | Zariski introduced an abstract notion of Riemann surface associated to, for example, a finitely generated field extension $K/k$. It's a topological space whose points are equivalence classes of valuations of $K$ that are trivial on $k$, or equivalently valuation rings satisfying $k\subset R\_v\subset K$. If $A$ is a fi... | 12 | https://mathoverflow.net/users/6666 | 28600 | 18,692 |
https://mathoverflow.net/questions/24234 | 2 | For personal research, I'm doing some analysis on collected data and trying to develop relationships between two variables where the data is collected through a data logger. I'm hypothesising that a = alpha \* b for any point in my data logger.
Anyways I plotted the 5,000+ x,y points below, and the only relationship ... | https://mathoverflow.net/users/152 | Statistical Data Analysis | To find the relationship between the data try this tool - to all of my knowledge this is the best one available (at least I am very excited about it)
<http://ccsl.mae.cornell.edu/eureqa>
| 2 | https://mathoverflow.net/users/1047 | 28603 | 18,695 |
https://mathoverflow.net/questions/28615 | 4 | Given a regular Tetrahedron *A* (i.e. each edge of *A* has same length), is it possible to split *A* into several smaller regular tetrahedra of equal size? I.e. smaller tetrahedra should completely fill volume of *A*, and they should not overlap.
This can be done in 2D with a triangle and square, and it can be done i... | https://mathoverflow.net/users/6883 | Tetrahedron splitting/subdivision | The answer is: No. There is a somehwat rambling discussion [here](http://answers.google.com/answers/threadview/id/497054.html). Let $B$ be a smaller tetrahedron that is jammed into the apex of $A$. It fills the solid angle there completely.
Let $e$ be a base edge of $B$. Then one cannot fill the neighborhood of $e$ by ... | 6 | https://mathoverflow.net/users/6094 | 28620 | 18,707 |
https://mathoverflow.net/questions/28622 | 8 | The Zoll surfaces have the property that all of their geodesics are closed.
If one futher stipulates that all geodesics are also *simple*, i.e., non-self-intersecting,
does this leave only the sphere?
Apologies for the simplicity of this question, but I am not finding an answer in the literature,
and I suspect many j... | https://mathoverflow.net/users/6094 | Surfaces all of whose geodesics are both closed and simple | From Guillemin's "The Radon transform on Zoll surfaces", it follows that there are deformations of $S^2$ which keep all geodesics closed AND simple.
| 12 | https://mathoverflow.net/users/1441 | 28627 | 18,712 |
https://mathoverflow.net/questions/28647 | 54 | Is it possible to partition $\mathbb R^3$ into unit circles?
| https://mathoverflow.net/users/3375 | Is it possible to partition $\mathbb R^3$ into unit circles? | The construction is based on a well ordering of $R^3$ into the least ordinal of cardinality continuum. Let $\phi$ be that ordinal and let $R^3=\{p\_\alpha:\alpha<\phi\}$ be an enumeration of the points of space. We define a unit circle $C\_\alpha$ containing $p\_\alpha$ by transfinite recursion on $\alpha$, for some $\... | 69 | https://mathoverflow.net/users/6647 | 28650 | 18,727 |
https://mathoverflow.net/questions/28649 | 5 | Let be $G=(V,E)$, where $V=\{1,\ldots,n\}$ and $E=\{\{i,j\}\subset V;|i-j|\leq k\}$ and $k<n$.
For which values of $k\geq 2$, can we count explicitly the number of Hamiltonian paths in $G$ ?
| https://mathoverflow.net/users/2386 | How many Hamiltonians Paths there are in almost regular graph ? | S. Kitaev defines Path schemes $P(n,M)$ as graphs with vertex set $\{1,2,\dots,n\}$ and edges $(i,j)$ iff $|i-j|\in M$. Hamiltonian graphs on path schemes were mentioned in "On uniquely k-determined permutations" by S. Avgustinovich and S. Kitaev. The formula is not simple even in the case where $M=\{1,2\}$ ([here](htt... | 5 | https://mathoverflow.net/users/2384 | 28654 | 18,730 |
https://mathoverflow.net/questions/28656 | 23 | Every student of set theory knows that the early axiomatization of the theory
had to deal with spectacular paradoxes such as Russel's, Burali-Forti's etc.
This is why the (self-contradictory) unlimited abstraction axiom ($\lbrace x | \phi(x) \rbrace$
is a set for any formula $\phi$) was replaced by the limited abstrac... | https://mathoverflow.net/users/2389 | Intuitive and/or philosophical explanation for set theory paradoxes | George Boolos has a number of very readable (to the non-expert like me) essays on this subject. Try "The Iterative Conception of Set" in [Logic, Logic and Logic](http://books.google.com/books?id=2BvlvetSrlgC&lpg=PP1&ots=sWb27wzVgM&dq=boolos%20logic%20logic%20harvard&pg=PP1#v=onepage&q=boolos%20logic%20logic%20harvard&f... | 5 | https://mathoverflow.net/users/1233 | 28661 | 18,733 |
https://mathoverflow.net/questions/28673 | 0 | Here's something that I'd like to use in my thesis.. but Im feeling too lazy to write a proof of it, I feel pretty sure this is correct though. I have a feeling that this can be found in a book on category theory. So maybe someone can point me to a reference (I have only used Adamék, Herrlich and Strecker so far).
Co... | https://mathoverflow.net/users/1245 | Need a reference for cones and limits that does this... | Yes -- filtered/directed colimits commute with finite limits. See Mac Lane, *Categories for the Working Mathematician*, theorem IX.2.1.
Edit: Oh, I thought you meant colimits, but it seems you meant limits. But limits always commute with each other (CWM IX.2).
| 3 | https://mathoverflow.net/users/4262 | 28675 | 18,741 |
https://mathoverflow.net/questions/28683 | 6 | The word problem (from wikipedia).
>
> Given a semi-Thue system T: = (Σ,R)
> and two words , can u be transformed
> into v by applying rules from R?
>
>
>
This problem is undecidable, but with a certain restriction, it is decidable.
The Restriction:
All the rules in R are of the form A->B where A and B are... | https://mathoverflow.net/users/6886 | Computational complexity of the word problem for semi-Thue systems with certain restrictions | The problem is at least NP-hard. Indeed, it is at least PSPACE-hard.
The reason the original semi-Thue rewrite system is undecidable is that
it reduces the halting problem. Given any Turing machine
program $e$ and input $x$, one sets up a rewrite system acting
on strings that code information about the Turing
computa... | 10 | https://mathoverflow.net/users/1946 | 28697 | 18,758 |
https://mathoverflow.net/questions/28462 | 20 | Question: Why do so few $n\equiv 3 \bmod 8$ have an odd number of representations in the form $$n=x\_0^2 + x\_1^2 + \dots + x\_{10}^2$$ with $x\_i \geq 0$?
Note that $x\_i\geq 0$ spoils the symmetry enough that this is not the usual "number of representations as a sum of squares" question. The question may be ill-pos... | https://mathoverflow.net/users/935 | Why are there usually an even number of representations as a sum of 11 squares | Throughout $N>0,$ and $N \equiv 3 \pmod 8.$ Let $I$ be the number of ordered triples $(a,d,e) \;\mbox{with} \; a,d,e \geq 0,$ such that
$$a^2+2 d^2+8 e^2=N.$$ I'll use a result of Gauss on sums of 3 squares to show that if there are 3 or more primes whose exponent in the prime factorization of $N$ is odd, then $I$ is e... | 14 | https://mathoverflow.net/users/6214 | 28711 | 18,770 |
https://mathoverflow.net/questions/28669 | 15 | I have a real-valued function $f$ on the unit disk $D$ that is fairly well behaved (real-analytic everywhere) and would like to find the integral $\int\_D f(x,y)dxdy$ numerically. After much searching, the best methods I've been able to find have been the simple quadrature rules in Abramowitz and Stegun that sample $f$... | https://mathoverflow.net/users/1233 | Numerical integration over 2D disk | See the [Encyclopedia of Cubature Formulas](http://nines.cs.kuleuven.be/research/ecf/). The site is password protected, but the maintainer will give a password to anyone who asks.
| 6 | https://mathoverflow.net/users/136 | 28713 | 18,771 |
https://mathoverflow.net/questions/28717 | 25 | Has any work been done on Singmaster's conjecture since Singmaster's work?
The conjecture says there is a finite upper bound on how many times a number other than 1 can occur as a binomial coefficient.
Wikipedia's article on it, written mostly by me, says that
* It is known that infinitely many numbers appear exa... | https://mathoverflow.net/users/6316 | Singmaster's conjecture | There is an upper bound of $O\left(\frac{(\log n)(\log \log \log n)}{(\log \log n)^3}\right)$ due to Daniel Kane: see "[Improved bounds on the number of ways of expressing *t* as a binomial coefficient](http://www.emis.de/journals/INTEGERS/papers/h53/h53.pdf)," Integers 7 (2007), #A53 for details.
| 26 | https://mathoverflow.net/users/428 | 28718 | 18,775 |
https://mathoverflow.net/questions/28593 | 2 | Let $A$ be a complex torus of (complex) dimension 2 and $X$ the associated Kummer variety $A/\sigma$, where $\sigma(x)=-x$. I would like to compute the cohomology of $X$ with $\mathbb{Z}$ coefficients. My initial instinct was to use Mayer-Vietoris, but the exact sequence involves the cohomology of the quadratic cone mi... | https://mathoverflow.net/users/6310 | Betti Cohomology of singular Kummer Surface | I missed that the question concerned the singular Kummer surface (which I think
historically was what was what was called the Kummer surface but our current fixation on
non-singularity has changed that) so one needs a few more steps than Barth,
Peters, van de Ven: Compact complex surfaces (which will be my reference be... | 5 | https://mathoverflow.net/users/4008 | 28724 | 18,780 |
https://mathoverflow.net/questions/28707 | 3 | If a set is equipped with a dense, complete order then the corresponding topological space is connected - and hence, so are its continuous images, even in unordered spaces. What happens if we remove the completeness condition on the order, and consider a general densely ordered space X?
I know that f is continuous fr... | https://mathoverflow.net/users/4336 | What, if anything, can be said about continuous images of densely ordered spaces? | Let me observe that Noah's excellent answer generalizes to solve the full case.
**Theorem.** Every topological space is a continuous image of a dense linear order.
Proof. Let $\kappa$ be any ordinal number. Let $P$ be the linear order $\mathbb{Q}\times\kappa$, under the lexical order, which is obtained by replacin... | 8 | https://mathoverflow.net/users/1946 | 28751 | 18,798 |
https://mathoverflow.net/questions/28757 | 3 | Is there standard notation for
(1) exterior algebras
(2) free graded commutative algebras
(3) divided polynomial algebras ?
I've seen (and used) $\Lambda$, $\Gamma$, $\Delta$ etc. used for some or all of these things, and I have no idea if there is a consensus about which notation goes with which algebra.
| https://mathoverflow.net/users/3634 | Notation for algebras | It's pretty standard to use $\bigwedge(V)$ or $\Lambda(V)$ for the exterior
algebra on a vector space $V$ and $\bigwedge^k(V)$ or $\Lambda^k(V)$
for the $k$-th graded part. For symmetric algebras $S(V)$ or $\mathrm{Sym}(V)$
etc. are frequent notations with again $S^k(V)$ or $\mathrm{Sym}^k(V)$
for the graded parts. I w... | 5 | https://mathoverflow.net/users/4213 | 28759 | 18,803 |
https://mathoverflow.net/questions/28743 | 5 | After getting stuck with the
[previous positivity](https://mathoverflow.net/questions/28374)
(it probably sounds too [complex](https://mathoverflow.net/questions/28612/do-names-given-to-math-concepts-have-a-role-in-common-mistakes-by-students/28614#28614)),
I would like to give a version of the problem which is of most... | https://mathoverflow.net/users/4953 | A plausible positivity | The sum $\sum a\_n/n$ can be negative. Below I construct a finite sequence; one can always add a negligibly small tail to get infinitely many non-zeroes.
Begin with $a\_1=1$ and $a\_2=-1$.
This gives $A\_2=0$ and the partial sum of the main series is $1-1/2=1/2$.
Then, repeat 100 times the following procedure:
Pick... | 8 | https://mathoverflow.net/users/4354 | 28760 | 18,804 |
https://mathoverflow.net/questions/28766 | 2 | Would anybody be able to share a Mathematica/Matlab/other script for calculating Onsager's exact solution for the magnetisation of the 2d Ising model? I would be most grateful of one in order to test my MC simulations of the system.
| https://mathoverflow.net/users/6915 | Mathematica/Matlab/other for calculating Onsager's exact solution to the 2d Ising model | Not that it's directly relevant, but I have code for the generator matrix of a 1D Glauber-Ising model that could probably be reworked into 2D...
---
```
function y = glauber1d(symb,n,varargin);
% produces the generator matrix etc for a 1D Glauber-Ising model of n spins
% call as either glauber1d(1,n) for a less... | 4 | https://mathoverflow.net/users/1847 | 28768 | 18,808 |
https://mathoverflow.net/questions/28779 | 1 | Hi,
I would like to do this:
```
fit <- test( measured_values, fitted_values )
```
Where:
* the return value from the `test` function is: **0 < fit < 1**.
* **measured\_values** are the observed data.
* **fitted\_values** are the data for the curve produced by GAM for the **measured\_values**.
What test can ... | https://mathoverflow.net/users/5908 | Goodness-of-fit test for Generalized Additive Model | I meant only what is usually meant and you're being completely cryptic.
A wikipedia article titled "generalized additive model" says you've got
$$
g(\operatorname{E}(Y))=\beta\_0 + f\_1(x\_1) + f\_2(x\_2)+ \cdots + f\_m(x\_m)
$$
and then says "The functions $f\_i(x\_i)$ may be fit using parametric or non-parametric m... | 2 | https://mathoverflow.net/users/6316 | 28786 | 18,820 |
https://mathoverflow.net/questions/28788 | 147 | A while back I saw posted on someone's office door a statement attributed to some famous person, saying that it is an instance of the callousness of youth to think that a theorem is trivial because its proof is trivial.
I don't remember who said that, and the person whose door it was posted on didn't remember either.... | https://mathoverflow.net/users/6316 | Nontrivial theorems with trivial proofs | Bertrand Russell proved that the general set-formation principle known as the Comprehension Principle, which asserts that for any property $\varphi$ one may form the set $\lbrace\ x \mid \varphi(x)\ \rbrace$ of all objects having that property, is simply inconsistent.
This theorem, also known as the [Russell Paradox]... | 145 | https://mathoverflow.net/users/1946 | 28791 | 18,824 |
https://mathoverflow.net/questions/28784 | 8 | Let $A$ be an algebra over an algebraically closed field $k.$ Recall that if $A$ is
a finitely generated module over its center, and if its center is a finitely generated
algebra over $k,$ then by the Schur's lemma all simple $A$-modules are finite dimensional
over $k.$
Motivated by the above, I would like an exampl... | https://mathoverflow.net/users/6277 | Example of an algebra finite over a commutative subalgebra with infinite dimensional simple modules | Doc, this is a stinker. Your condition (2) forces your algebra to be finitely generated PI, and every little hare knows that simple modules over such algebras are finite-dimensional. See 13.4.9 and 13.10.3 of McConnell-Robson...
| 5 | https://mathoverflow.net/users/5301 | 28816 | 18,842 |
https://mathoverflow.net/questions/28610 | 18 | Suppose we have a ($n-1$ dimensional) unit sphere centered at the origin: $$ \sum\_{i=1}^{n}{x\_i}^2 = 1$$
Given some some $d \in [0,1]$, what is the probability that a randomly selected point on the sphere, $ (x\_1,x\_2,x\_3,...,x\_n)$, has coordinates such that $$|x\_i| \leq d$$ for all $i$?
This is equivalent to f... | https://mathoverflow.net/users/5768 | Probability of a point on a unit sphere lying within a cube | Denote the median of $\max\_{i=1,\dots,n}|x\_i|$ on the sphere by $M\_n$. It is known that the ratio between $M\_n$ and $\sqrt{\log n/n}$ is universally bounded and bounded away from zero. If you take $d=M\_n$ then the quantity you are looking for is exactly $1/2$. It is also known that the $\infty$-norm ($\max\_{i=1,\... | 16 | https://mathoverflow.net/users/6921 | 28818 | 18,843 |
https://mathoverflow.net/questions/28736 | 8 | Let G and H be locally compact groups, and let $\theta:G\rightarrow H$ be a continuous group homomorphism. This induces a \*-homomorphism $\pi:C^b(H) \rightarrow C^b(G)$ between the spaces of bounded continuous functions on H and G.
If $\theta$ is an injection with closed range, then as locally compact groups are nor... | https://mathoverflow.net/users/406 | Group homomorphisms and maps between function spaces | The answer is yes, at least if the group $G$ is metrizable $\iff$ $G$ is Hausdorff and has countable basis of neighborhoods of the identity element $e$. This follows from the following general statement.
>
> **Proposition.** Let $G$ and $G'$ be topological groups, with $G$ locally compact and metrizable and $f:G\to... | 2 | https://mathoverflow.net/users/5740 | 28820 | 18,845 |
https://mathoverflow.net/questions/28800 | 2 | It seems that the ratio of those authors allowing a field to be a Dedekind domain to those who do not is almost 50 - 50. Why such a bewildering lack of consensus for such an elementary notion?
| https://mathoverflow.net/users/5292 | 0 dimensional Dedekind domain? | Historically Emmy Noether's paper introducing the concept "Dedekind domain" certainly included fields (see e.g. Kleiner's book on the history of abstract algebra, which gives axioms). She was characterising the scope of unique factorisation into prime ideals. Now, a question that might actually be answered is "what hap... | 5 | https://mathoverflow.net/users/6153 | 28840 | 18,858 |
https://mathoverflow.net/questions/28843 | 7 | Suppose I know what the category of free algebras for a particular monad look like. Can I then describe what the category of Eilenberg–Moore algebras look like? E.g. Suppose that I have a good handle on a category $D$ and that I have two functors $F$ and $G$, with $G:D \to C$ and $F$ left-adjoint to $G$ and $F$ essenti... | https://mathoverflow.net/users/4528 | Eilenberg–Moore algebras in terms of Kleisli ones | One nice result is Street's theorem 14 in *The formal theory of monads*, generalized in *Elementary cosmoi*, which says that $C^T$ is isomorphic to the full subcategory of $[(C\_T)^{\mathrm{op}}, \mathrm{Set}]$ containing those presheaves that become representable when precomposed with the inclusion $C \to C\_T$. That ... | 11 | https://mathoverflow.net/users/4262 | 28858 | 18,868 |
https://mathoverflow.net/questions/28826 | 57 | While reading the answer to another Mathoverflow question, which mentioned the Poisson summation formula, I felt a question of my own coming on. This is something I've wanted to know for a long time. In fact, I've even asked people, who have probably given me perfectly good answers, but somehow their answers have never... | https://mathoverflow.net/users/1459 | How does one use the Poisson summation formula? | The existing answers list some important situations where Poisson Summation plays a role, the application to proving the functional equation of $\theta$ and hence of $\zeta$ being my personal favourite. My best answer to Tim's question as he actually asked it might be: why not have it in mind to try using it whenever y... | 33 | https://mathoverflow.net/users/5575 | 28867 | 18,874 |
https://mathoverflow.net/questions/28850 | 1 | It seems an easy problem but I couldn't prove it.
Let $M$ be a manifold with boundary and $N\subset \mathrm{int}(M)$ is a strong deformation retract of $M$.
Then I wonder whether $M-N$ is homotopic equivalent to the boundary of $M$ or not.
| https://mathoverflow.net/users/6569 | Is the complement of a strong deformation retract of a manifold M homotopic equivalent with the boundary of M? | Take a compact contactible manifold $C$ whose boundary is not a homotopy sphere sphere (those are no so easy to construct, but it has been done long ago). Then removing a point (or equivalently a small ball) from the interior of $C$ gives a manifold that is homotopy equivalent (by excison in homology) to the boundary o... | 6 | https://mathoverflow.net/users/1573 | 28870 | 18,876 |
https://mathoverflow.net/questions/28865 | 6 | In Paul Halmos' Measure Theory book, section 53, he defines a content on a locally compact Hausdorff space to be a set function, $\lambda$ that is additive, subadditive, monotone, and $0\le\lambda(C)<\infty$ for all $C$ compact. The "Borel sets" he considers(section 52) in this book are the smallest $\sigma$-ring gener... | https://mathoverflow.net/users/2048 | Haar Measure Existence/A problem with Borel sets and regularity. | The situation is indeed a delicate one and one needs to carefully check the conventions before transferring a result from one context to another. The situation is summarized in Royden's *Real Analysis*, though with just a few examples.
For a given space $X$, the main players are:
* $\mathcal{B}a$ — the σ-algebra ge... | 4 | https://mathoverflow.net/users/2000 | 28874 | 18,879 |
https://mathoverflow.net/questions/28856 | 3 | What is the motivic cohomology $H^{p,q}(\mathbf{P}^n,\mathbf{Z})$ of projective space? By the projective bundle formula, one has
$H^{p,q}(\mathbf{P}^n,\mathbf{Z})$ =
$\oplus\_{i=0}^n\mathrm{Hom}\_\mathbf{DM}(\mathbf{Z}(i)[2i],\mathbf{Z}(q)[p])$
=
$\oplus\_{i=0}^n\mathrm{Hom}\_\mathbf{DM}(\mathbf{Z}, \mathbf{Z}(q-i)... | https://mathoverflow.net/users/nan | The motivic cohomology of projective space | Motivic cohomology is an absolute invariant not a geometric one. The projective bundle formula is purely geometric. It reduces the computation of the motivic cohomology of the projective space to that of the base:
$$
H^{p,q}(\mathbb{P}^n\_k) = \bigoplus\_{i=0}^n H^{p-2i,q-i}(Spec(k))
$$
In general $H^{\bullet,\bullet... | 10 | https://mathoverflow.net/users/1985 | 28876 | 18,881 |
https://mathoverflow.net/questions/28869 | 7 | Kenneth Kunen in his “The Foundations of Mathematics” writes:
1. ‘Set theory is the study of models of ZFC’ (p. 7)
2. ‘Set theory is the theory of everything’ (p. 14)
With (1) Kunen is pointing to a change in the intended use of the axioms of ZFC: ‘there are two different uses of the word “axioms”: as “*statements ... | https://mathoverflow.net/users/6466 | How to think like a set (or a model) theorist. | I highly recommend reading Andrej Bauer's [excellent answer](https://mathoverflow.net/questions/23060/set-theory-and-model-theory/23077#23077) to an earlier question along the same lines. To summarize the situation in a few words: the fact that set theory is anterior to model theory does not preclude the model theoreti... | 6 | https://mathoverflow.net/users/2000 | 28881 | 18,883 |
https://mathoverflow.net/questions/28878 | 14 | What is Borel-de Siebenthal theory?
| https://mathoverflow.net/users/6772 | What is Borel-de Siebenthal theory? | I'm not sure the term "theory" is appropriate here, but the joint paper by Borel and de Siebenthal has had considerable influence in Lie theory over the years: MR0032659 (11,326d)
Borel, A.; De Siebenthal, J.,
Les sous-groupes fermés de rang maximum des groupes de Lie clos.
Comment. Math. Helv. 23, (1949). 200--221. (... | 9 | https://mathoverflow.net/users/4231 | 28884 | 18,884 |
https://mathoverflow.net/questions/24552 | 46 | Brian Conrad indicated a while ago that many of the results proven in AG using universes can be proven without them by being very careful ([link](https://mathoverflow.net/questions/15897/in-what-topology-dm-stacks-are-stacks/15910#15910)). I'm wondering if there are any results in AG that actually depend on the existen... | https://mathoverflow.net/users/1353 | What interesting/nontrivial results in Algebraic geometry require the existence of universes? | My belief is that no result in algebraic geometry that does
not explicitly engage the universe concept will fully
require the use of universes. Indeed, I shall advance an
argument that no such results actually need anything beyond
ZFC, and indeed, that they need much less than this. (But please note, I answer as a set ... | 52 | https://mathoverflow.net/users/1946 | 28913 | 18,901 |
https://mathoverflow.net/questions/27960 | 10 | A set $M$ is called *amenable* if it is transitive and satisfies the following conditions:
1. For all $x,y\in M$, $\{x,y\}\in M$
2. For all $x\in M$, $\bigcup x \in M$
3. $\omega \in M$
4. For all $x,y \in M$, $x\times y \in M$
5. ($\Sigma\_0$ comprehension) Whenever $\Phi$ is a $\Sigma\_0$ formula of one free variab... | https://mathoverflow.net/users/6649 | When does replacement (accidentally) hold in amenable sets? | I think the following is a counterexample to your specific question. Let AH be the set of those $x$ such that (1) each element of $TC\{x\}$ has cardinality at most $\aleph\_\omega$ and (2) all but finitely many elements of $TC\{x\}$have cardinality strictly smaller than $\aleph\_\omega$. (By $TC\{x\}$, I mean the trans... | 7 | https://mathoverflow.net/users/6794 | 28918 | 18,903 |
https://mathoverflow.net/questions/28871 | 7 | Question:
---------
---
Let $D:A\to (X\downarrow C)$ be a $\kappa$-good $S$-tree rooted at $X$ for a collection of morphisms $S$ in $C$, where $\kappa$ is a fixed uncountable regular cardinal. Then according to the proof of Lemma A.1.5.8 of *Higher Topos Theory* by Lurie, for any $\kappa$-small downward-closed $B... | https://mathoverflow.net/users/1353 | K-good trees and K-compactness of colimits over K-small downwards-closed subposets (500 point bounty if answered by Midnight EST)) | Does this work?
To prove $D(\alpha)$ is $\kappa$-compact for all $\alpha$ in $A$, assume otherwise, that there exists some counterexample. Then, by the fact $A$ is well-ordered, there is a minimal counterexample (i.e., there is a minimal element $\alpha$ in the set of $\gamma \in A$ such that $D(\gamma)$ is not $\ka... | 5 | https://mathoverflow.net/users/2926 | 28922 | 18,906 |
https://mathoverflow.net/questions/28892 | 16 | I was searching on MathSciNet recently for a certain paper by two mathematicians. As I often do, I just typed in the names of the two authors, figuring that would give me a short enough list. My strategy was rather dramatically unsuccessful in this case: the two mathematicians I listed have written 80 papers together!
... | https://mathoverflow.net/users/1149 | Which pair of mathematicians has the most joint papers? | We get 135 matches for "Author=(Jimbo, Michio and Miwa, Tetsuji)" in mathscinet.
| 21 | https://mathoverflow.net/users/36665 | 28933 | 18,912 |
https://mathoverflow.net/questions/28945 | 16 | There are infinite graphs which contain all finite graphs as induced subgraphs, e.g. the Rado graph or the coprimeness graph on the naturals.
>
> Are there infinite groups which
> contain all finite groups as
> subgroups?
>
>
>
| https://mathoverflow.net/users/2672 | Infinite groups which contain all finite groups as subgroups | Yes, plenty. The group only has to contain all finite permutation groups. Perhaps the most
straightforward example would be the permutations of a countable set. That is bijections
which fix all but a finite set.
| 66 | https://mathoverflow.net/users/3992 | 28946 | 18,917 |
https://mathoverflow.net/questions/28941 | 3 | Can we find finite metabelian (ie with derived length 2) groups with arbitrarily many distinct degrees of irreducible complex characters?
If we cannot, can we somehow find a bound of the form $|cd(G)|\leq f(dl(G))$ for some "interesting" function $f$ (linear would be very cool for instance).
The motivation is that ... | https://mathoverflow.net/users/4614 | Are there finite metabelian groups with arbitrarily many character degrees? | Sure, you just take direct products of metabelian groups with different character degrees: cd(G×H) = cd(G)×cd(H) and (G×H)′ = G′×H′.
I suggest taking G(p) = AGL(1,p) = Hol(p) to be the normalizer of a Sylow p-subgroup in the symmetric group of degree p, for each prime p, but there are lots of examples. For instance:
... | 3 | https://mathoverflow.net/users/3710 | 28950 | 18,918 |
https://mathoverflow.net/questions/28747 | 4 | Two short questions:
* Is there any work classifying the lattice of subcategories of an arbitrary (sufficiently small) category $\mathcal{C}$, similar to the way that the set of subsets of set $\mathcal{S}$ is isomorphic to the functions $\mathcal{S}\to\mathbf{2}$, where $\mathbf{2}$ is a two point set?
* Is there st... | https://mathoverflow.net/users/2620 | Lattice of subcategories: subobject classifier in Cat | In his comment on Finn Lawler's response, Mike Shulman points out that Cat has no subobject classifier in the 2-categorical sense. It also fails to have a subobject classifier in the 1-categorical sense. To see this, note that if Cat had a subobject classifier, then it would be a topos, as it is cartesian closed. Howev... | 6 | https://mathoverflow.net/users/6485 | 28960 | 18,926 |
https://mathoverflow.net/questions/28947 | 76 | The recent question about the most prolific collaboration interested me. How about this question in the opposite direction, then: can anyone beat, amongst contemporary mathematicians, the example of Christopher Hooley, who has written 91 papers and has yet to coauthor a single one (at least if one discounts an obituary... | https://mathoverflow.net/users/5575 | Least collaborative mathematician | Lucien Godeaux wrote more than 600 papers and not one of them is a joint paper. He cowrote a textbook in projective geometry. Mathscinet records only 15 citations to all these papers! But there is something called Godeaux surfaces which is mentioned in the literature. This is about the weirdest example I know.
<http:... | 80 | https://mathoverflow.net/users/2290 | 28973 | 18,934 |
https://mathoverflow.net/questions/28832 | 6 | Dear all.
Let
$$
f(x) = \sum\_{k \in \mathbb{Z}} \hat{f}(k) \exp(2\pi \mathrm{i} kx)
$$
be a function given by usual fourier series.
Since my original question hasn't got any answer yet, and I came across another related question, I am just adding it. Denote by $T\_n$ the set of all trigonometric polynomials of ... | https://mathoverflow.net/users/3983 | Approximation by analytic functions | The answer to the modified question is given by [Jackson-type theorems](http://en.wikipedia.org/wiki/Jackson%27s_inequality).
The classic book by N.I. Akhiezer which is quoted in the Wikipedia article contains a number of specialised results on optimal approximation by trigonometric polynomials.
A typical optimal ... | 2 | https://mathoverflow.net/users/5371 | 28975 | 18,935 |
https://mathoverflow.net/questions/28974 | 7 | Let $K$ be a number field and $\mathfrak{p}$ be a place of good reduction. It is easy to see that the reduction map on prime-to-$p$ torsion $A(K)[p'] \hookrightarrow A\_{\mathfrak{p}}(\kappa(\mathfrak{p}))$ is injective.
But if $p > e(\mathfrak{p}/p) + 1$, the reduction map is even injective on $p$-torsion. This can ... | https://mathoverflow.net/users/6960 | $p$-torsion in the Mordell-Weil group of Abelian varieties injecting in reduction | One has the finite flat group scheme $\mathbb Z/p$ over $\mathcal O\_{K\_{\mathfrak p}}$
(I write $K\_{\mathfrak p}$ for the $\mathfrak p$-adic completion of $K$, and
$\mathcal O\_{K\_{\mathfrak p}}$ for its integer ring),
as well as
the finite flat group scheme $A[p]$. Giving a $p$-torsion point over $K\_{\mathfrak ... | 7 | https://mathoverflow.net/users/2874 | 28981 | 18,939 |
https://mathoverflow.net/questions/28967 | 16 | While doing some research on polytopes I came to the following question. Maybe it's already somewhere but anyway I'll post it here.
Let $X\subset \mathbb{R}^3$ be such that, for every plane $P$, $P\cap X$ is simply connected. Is $X$ convex?
I'm not sure, but I think maybe it's necesary to assume some well behaving ... | https://mathoverflow.net/users/4619 | A characterization of convexity | How about some tomography? This should work if $X$ is open. Assume $X\subset \mathbb R^3$ is nonempty and for every plane $H$ the intersection $H\cap X$ is either contractible or empty. (Note that an open subset of the plane is contractible if it is ((nonempty,) connected, and) simply connected.)
Claim 1: $X$ is cont... | 8 | https://mathoverflow.net/users/6666 | 28987 | 18,942 |
https://mathoverflow.net/questions/28744 | 5 | In their book "Theory of sets" Bourbaki suggested a general theory of isomorphism.
(See also <http://www.tau.ac.il/~corry/publications/articles/pdf/bourbaki-structures.pdf> )
The example of an untransportable relation (i.e. formula) in the book involves 2 principal base sets.
Are there examples of untrasportable ... | https://mathoverflow.net/users/5761 | Bourbaki theory of isomorphism, examples of untransportable formulas | An example of untrasportable sentence, when there is only one principal base set X, may be the following one:
All elements of the set X are finite sets,
Because, by definition, the truth value of a transportable sentence must be preserved under all bijections from the set X. Obviously, there exists a bijection fro... | 3 | https://mathoverflow.net/users/5761 | 28989 | 18,944 |
https://mathoverflow.net/questions/28986 | 2 | Okay, we know that
$$ \frac{sin(x)}{x} = \prod\_{n=1}^{\infty} \Big(1-\frac{x^2}{n^2\cdot\pi^2}\Big) $$ .
Is there some known (trigonometric(?)) function that is equal to the following infinite product?
$$ \prod\_{n=1}^{\infty} \Big(1-\frac{x}{n\cdot\pi}\Big) $$
I'd be happy as well if someone could provide m... | https://mathoverflow.net/users/93724 | Closed form of divergent infinite product? | It's a **divergent** infinite product. You might as well ask for the sum of
$$\sum\_{n=1}^\infty\frac{x}{n\pi}.$$
You can "cure" the divergence by multipliying each term by a suitable factor, so
for instance
$$f(x)=\prod\_{n=1}^\infty e^{x/n\pi}\left(1-\frac{x}{n\pi}\right)$$
does converge (as the $n$-th term is like $... | 7 | https://mathoverflow.net/users/4213 | 28990 | 18,945 |
https://mathoverflow.net/questions/28980 | 3 | I need a reference for the proof that the complex orthogonal group
$SO\_{2n+1}($ℂ$) = \{A\in SL\_{2n+1}($ℂ$): A^TA = Id\}$ is simple in a group theoretical sense (if it is true).
How about the simplicity of $SO\_{2n+1}(K)$ in general (i.e. $K$ an arbitrary infinite field)?
It there any criterion? It seems that if $K$... | https://mathoverflow.net/users/6923 | Simplicity of (complex) orthogonal groups | The structure of classical groups goes back a long way and has been treated in a number of books, but in varying generality (arbitrary fields, various commutative rings, etc.). One older source in French is J.A. Dieudonne's concise Springer Ergebnisse volume *La geometrie des groupes classiques* (1963). A probably more... | 4 | https://mathoverflow.net/users/4231 | 28994 | 18,948 |
https://mathoverflow.net/questions/28999 | 43 | [This recent MO
question](https://mathoverflow.net/questions/28945/infinite-groups-which-contain-all-finite-groups-as-subgroups),
answered now several times over, inquired whether an
infinite group can contain every finite group as a
subgroup. The answer is yes by a variety of means.
So let us raise the stakes: Is th... | https://mathoverflow.net/users/1946 | Is there a universal countable group? (a countable group containing every countable group as a subgroup) | There isn't a countable group which contains a copy of every countable group
as a subgroup. This follows from the fact that there are uncountably many
2-generator groups up to isomorphism.
The first example of such a family was discovered by B.H. Neumann. A clear
account of his construction can be found in de la Harp... | 56 | https://mathoverflow.net/users/4706 | 29001 | 18,951 |
https://mathoverflow.net/questions/29007 | 4 | This is not a math question as much as a process question. For the first time in my (very short) career, I find myself doing one of those messy calculations, where each 'line' of the calculation can spread over a page or three. Essentially all of the calculation is trivial if I'm willing to write some reasonable inequa... | https://mathoverflow.net/users/6966 | Medium-Sized Calculations and Organization | Mathematica can be very useful for this kind of thing. If you're good enough at it you can force it to go through calculations pretty much step-by-step if you need it too. You can also export its output to LaTeX, which is very nice and saved me a lot of work on various physics homeworks!
For more complicated things,... | 2 | https://mathoverflow.net/users/3329 | 29013 | 18,959 |
https://mathoverflow.net/questions/28997 | 51 | For many standard, well-understood theorems the proofs have been streamlined to the point where you just need to understand the proof once and you remember the general idea forever. At this point I have learned three different proofs of the Birkhoff ergodic theorem on three separate occasions and yet I still could prob... | https://mathoverflow.net/users/4362 | Does anyone know an intuitive proof of the Birkhoff ergodic theorem? | I don't know whether this helps or whether you've already seen this before, but this made a lot more intuitive sense to me than the combinatorial approach in Halmos's book.
The key point in the proof is to prove the maximal ergodic theorem. This states that if $M\_T$ is the maximal operator $M\_T f= \sup\_{n >0} \fra... | 25 | https://mathoverflow.net/users/344 | 29014 | 18,960 |
https://mathoverflow.net/questions/28879 | 7 | A von Dyck group is a group with presentation $< a,b | a^m=b^n=(ab)^p=1 >$ with m,n,p natural numbers. Is it known which of these groups are solvable and which are not? Is there a reference for this? Thanks.
| https://mathoverflow.net/users/3804 | von dyck groups and solvability | You might try Generators and Relations for Discrete Groups by Coxeter and Moser.
Specifically for 1/m + 1/n + 1/p = 1 there are only 3 cases up to permutation, (2,3,6), (2,4,4) and (3,3,3). Map a and b to an appropriate root of unity to get a homomorphism onto C\_6, C\_4, or C\_3, respectively. The kernel of the map... | 5 | https://mathoverflow.net/users/6787 | 29032 | 18,974 |
https://mathoverflow.net/questions/28992 | 14 | I am just reading about Iwasawa theory about Coates and Sujatha's book on Iwasawa Theory. I was wondering that since Iwasawa thought about the whole theory from the analogy of curves over finite fields, so what should be the analog of the module $U\_\infty$/$C\_\infty$ in the curve case (if there is any) where $U\_\inf... | https://mathoverflow.net/users/2081 | A question about Iwasawa Theory | There is a very close analogy but to unravel it requires some work.
So take $X$ a smooth curve over $\mathbb F\_{\ell}$ (more generally you could take $X$ a scheme over $\mathbb F\_{\ell}$) and let $\mathscr F$ be a smooth sheaf of $\mathbb Q\_{p}$-vector spaces on $X$ (you could be much more general in your choice o... | 9 | https://mathoverflow.net/users/2284 | 29059 | 18,990 |
https://mathoverflow.net/questions/29033 | -1 | Define $$x\_{k+1}(t)=\frac{3x^4\_k(t)+6(1-t)x\_k^2(t)-(1-t)^2}{8x\_k^3(t)},$$
with $x\_0(t)=1$. It is not difficult to see $x\_k(t)$ converges to $\sqrt{1-t}$, whose (Maclaurin expansion) has negative coefficients unless the first one. Let
$$x\_k(t)=\sum\limits\_{i =0}^{\infty}c\_{k,i}t^{i}, \mid t\mid<1$$
Is it ... | https://mathoverflow.net/users/6858 | A classical analysis problem | This is not the first time I am fighting with positivity (nonnegativity).
But this problem looks not natural enough for a standard technique, and
it seems to me that the resulting sequence $x\_k(t)$ is always between
$\sqrt{1-t}$ and $1$, in the sense that the expansions of $x\_k(t)-\sqrt{1-t}$
and $1-x\_k(t)$ have non... | 3 | https://mathoverflow.net/users/4953 | 29075 | 18,997 |
https://mathoverflow.net/questions/29073 | 5 | Let $G$ be a linear algebraic group over an algebraically closed field $k$, and $T$ a maximal torus of $G$.
Suppose we have two cocharacter $\mu, \mu' : \mathbb{G}\_m \to T$, which are conjugate under $G$ i.e. there exists $g \in G$ such that $\mu'(z) = g\mu(z)g^{-1}$.
**Question.** Can we always choose $g \in G$ s... | https://mathoverflow.net/users/1046 | Conjugate cocharacters in a maximal torus | Yes. That $\mu'(z) = g \mu(z) g^{-1}$ means that $g T g^{-1}$ centralizes the image of $\mu'$. Thus, $T$ and $g T g^{-1}$ are maximal tori in the centralizer
of the image of $\mu'$, so there exists $h$ in this centralizer such that $g T g^{-1} = h T h^{-1}$. Now replace $g$ by $h^{-1} g$. It doesn't seem that you need ... | 7 | https://mathoverflow.net/users/6982 | 29081 | 19,000 |
https://mathoverflow.net/questions/29074 | 9 | The [separation axioms](http://ncatlab.org/nlab/show/separation+axioms) have exploded a little since the original list of four! Amongst them can be found "completely regular" spaces and "perfectly normal" spaces. The former is well-known: a point can be separated from a disjoint closed subset by a continuous real-value... | https://mathoverflow.net/users/45 | Is there a notion of a "perfectly regular" topological space? | An answer is: completely regular plus countable pseudocharacter, the latter means that points are $G\_\delta$-sets. In completely regular spaces a point is a $G\_\delta$-set iff it is the zero-set of a continuous function, the proof is just like that for arbitrary closed $G\_\delta$-sets in normal spaces: if $\lbrace x... | 8 | https://mathoverflow.net/users/5903 | 29093 | 19,004 |
https://mathoverflow.net/questions/29088 | 11 | Martin Gardner kept voluminous correspondence with amateur and professional mathematicians worldwide throughout his career. His files are a treasure trove of information about all areas of recreational mathematics. Does anybody here know whether any portion of those files will be made available to the public for resear... | https://mathoverflow.net/users/3106 | What is happening to Martin Gardner's files? | According to a simple Google search, the papers were donated to Stanford.
<http://www.oac.cdlib.org/data/13030/6s/kt6s20356s/files/kt6s20356s.pdf>
| 18 | https://mathoverflow.net/users/454 | 29094 | 19,005 |
https://mathoverflow.net/questions/29102 | 7 | In the book "A = B" by Petkovesk, Wilf, and Zeilberger, [(downloadable here)](http://www.math.upenn.edu/~wilf/Downld.html), the authors provide several algorithmic methods for finding closed forms or recurrences for sums involving e.g. binomial coefficients. Even more exciting, their methods provide seemingly short cer... | https://mathoverflow.net/users/6950 | How long are the certificates produced by the Zeilberger and WZ methods for solving combinatorial sums (A=B)? | Already for the case of Gosper summation (single-variable), it is known that things can get exponentially larger than the input, because the 'answer' fundamentally depends on the *dispersion* of the input term. You will find much more comprehensive answers in the papers of [Sergei Abramov](http://www.ccas.ru/sabramov/p... | 6 | https://mathoverflow.net/users/3993 | 29106 | 19,008 |
https://mathoverflow.net/questions/29115 | 4 | I just read a proof and, after struggling some time with a mental leap, I think that it uses tacitly the following:
Let $\kappa$ be a regular cardinal, $\theta > \kappa$ a regular cardinal too then:
$ S \subset \kappa$ is stationary if and only if
$\forall \mathcal{A} = (H(\theta), \in, <,..) \exists M \prec \mathca... | https://mathoverflow.net/users/4753 | A characterization of stationarity? | (I first wanted to give an answer, but I was not quick enough. I then wanted to add a small comment and found out after 20 minutes that I had insufficient reputation.)
The comment was regarding 2) of oktan's original query: having $H(\theta)$
in the structure is overkill: it suffices to have $( \kappa, <, \in, C)$. (... | 10 | https://mathoverflow.net/users/6942 | 29128 | 19,019 |
https://mathoverflow.net/questions/29123 | 1 | Can a harmonic function defined on the upper half-plain (or any domain which is unbounded) be extended to the point at infinity. If so, under what condition. What happens to the mean value property then ? Do we still get a integral representation of some sort. Please suggest a reference.
Thank you.
| https://mathoverflow.net/users/6766 | Extension of harmonic function at infinity | One has this type of representations for Herglotz functions under certain growth conditions.
Herglotz means here that the function maps the upper half plane into the upper half plane.
This can for example be found in the spectral theory book by Teschl ( <http://www.mat.univie.ac.at/~gerald/ftp/book-schroe/index.html... | 1 | https://mathoverflow.net/users/3983 | 29132 | 19,023 |
https://mathoverflow.net/questions/29104 | 26 | As a person who has been spending significant time to learn mathematics, I have to admit that I sometimes find the fact uncovered by Godel very upsetting: we never can know that our axiom system is consistent. The consistency of ZFC can only be proved in a larger system, whose consistency is unknown.
That means proof... | https://mathoverflow.net/users/1229 | Why are proofs so valuable, although we do not know that our axiom system is consistent? | If you like, you can view proofs of a statement in some formal system (e.g. ZFC) as a certificate that a counterexample cannot be found without demonstrating the inconsistency of ZFC, which would be a major mathematical event, and probably one of far greater significance than whether one's given statement was true or f... | 68 | https://mathoverflow.net/users/766 | 29133 | 19,024 |
https://mathoverflow.net/questions/29138 | 4 | If $E$ is a complex vector bundle over a manifold $M$ then one defines the space of vector valued $p$-differential forms on them as $\Omega^p(M,E) = \Gamma ( \wedge ^p (T^\*M) \otimes E) $
The connection be defined as the map , $\nabla: \Gamma(E) \rightarrow \Omega^1(M,E)$ satisfying $\nabla(fX) = df\otimes X+f\nabla... | https://mathoverflow.net/users/2678 | Proving the basic identity which implies the Chern-Weil theorem | (1) No, a connection is not a section of $\Omega^1(M,\mathrm{End}(E))$: a section would act tensorially and not satisfy the Leibniz rule. The connection is $\mathbb{C}$ linear and not $C^\infty(M,\mathbb{C})$ linear.
(2) Since you have a vector bundle over some manifold, by definition there is some complex vector sp... | 2 | https://mathoverflow.net/users/3948 | 29158 | 19,040 |
https://mathoverflow.net/questions/27805 | 31 | Given a solution $S$ of the Navier-Stokes equations, is there a way to make mathematically precise a statement like: "$S$ is [turbulent](https://en.wikipedia.org/wiki/Turbulence) in the spacetime region $U$"?
And if such a definition exists, are there any known exact solutions of Navier-Stokes exhibiting turbulence?
... | https://mathoverflow.net/users/745 | Is there a mathematically precise definition of turbulence for solutions of Navier-Stokes? | There is probably no universally accepted mathematical definition of turbulence. (By the way, is there a physical one?) Moreover, the prevailing definitions seem to be highly volatile and time-dependent themselves.
A few notable examples.
* In the Ptolemaic [Landau–Hopf theory](https://en.wikipedia.org/wiki/Landau%... | 19 | https://mathoverflow.net/users/5371 | 29159 | 19,041 |
https://mathoverflow.net/questions/29136 | 7 | The theta function of a lattice is defined to be
$$ \vartheta\_\Lambda = \sum\_{v\in\Lambda} q^{{\Vert v\Vert}^2}$$
which yields as a coefficient of *qk* the number of vectors of norm-squared *k*.
On the other hand, the Jacobi theta function is given by
$$ \vartheta(u,q) = \sum\_{n=-\infty}^\infty u^{2n}q^{n^2}$$
and... | https://mathoverflow.net/users/1703 | Is there any literature on multivariable theta functions? | There are three ways to view theta functions
1. as classical homomorphic functions in
vector z and/or period matrix T
2. as matrix coefficients of a representation of the
Heisenberg and/or Metaplectic grp
3. as sections of Line bundles on the Abelian variety
and/or moduli space of the abelian variety
Ram Murty's [T... | 7 | https://mathoverflow.net/users/5372 | 29165 | 19,045 |
https://mathoverflow.net/questions/29149 | 5 | Given a regular tessellation, i.e. either a platonic solid (a tessellation of the sphere), the tessellation of the euclidean plane by squares or by regular hexagons, or a regular tessellation of the hyperbolic plane.
One can consider its isometry group $G$. It acts on the set of all faces $F$. I want to define a symm... | https://mathoverflow.net/users/3969 | Symmetric colorings of regular tessellations | The answer is yes. Moreover, for every two different faces $A$ and $B$ there is a symmetric coloring assigning different colors to $A$ and $B$.
The isometry group $G$ is residually finite, hence here is a normal finite index subgroup $H$ of $G$ that contains no elements (except the identity) sending $A$ to itself or ... | 5 | https://mathoverflow.net/users/4354 | 29168 | 19,046 |
https://mathoverflow.net/questions/29160 | 1 | **Background**
This can be generalised, but let me be fairly concrete. Let $X$ be a simply-connected riemannian manifold and let $G$ denote the Lie group of isometries, assumed nontrivial. Let $F < G$ be a finite subgroup acting freely and consider the smooth quotient $X/F$ with the induced riemannian structure.
Th... | https://mathoverflow.net/users/394 | A question about iterated quotients in riemannian geometry | The group $D$ is the preimage of $E$ in $N(F)$, so it is as you expect. The finiteness hypothesis can be weakened, which is important for many applications. Things become clearer if one thinks categorically in terms of the universal properties.
Say an arbitrary group $G$ acts freely and properly discontinuously and ... | 5 | https://mathoverflow.net/users/6773 | 29171 | 19,048 |
https://mathoverflow.net/questions/29118 | 26 | I am trying to prove $\sum\limits\_{j=0}^{k-1}(-1)^{j+1}(k-j)^{2k-2} \binom{2k+1}{j} \ge 0$. This inequality has been verified by computer for $k\le40$.
Some clues that might work (kindly provided by Doron Zeilberger) are as follows:
1. Let $Ef(x):=f(x-1)$, let $P\_k(E):=\sum\_{j=0}^{k-1}(-1)^{(j+1)}\binom{2k+1}{j}... | https://mathoverflow.net/users/6989 | Is the sum $\sum\limits_{j=0}^{k-1}(-1)^{j+1}(k-j)^{2k-2} \binom{2k+1}{j} \ge 0?$ | Your expression is the difference of two central Eulerian numbers ,
$$A(k):=\sum\_{j=0}^{k-1}(-1)^{j+1}(k-j)^{2k-2}{2k+1 \choose j}=\left \langle {2k-2\atop k-2} \right \rangle-\left \langle {2k-2\atop k-3} \right \rangle$$
as you can easily deduce from their closed formula. The positivity of $A(k)$ is just due to... | 41 | https://mathoverflow.net/users/6101 | 29179 | 19,051 |
https://mathoverflow.net/questions/29166 | 7 | I am reading about the L-functions of elliptic curves and I was thinking about the root number as the product of local root numbers. So my question is how to think about the local root numbers geometrically or arithmetically. I have also read that even though the functional equation is conjectural (in different cases) ... | https://mathoverflow.net/users/2081 | Local root number | Yes, if $X$ is a variety over an extension $K$ of $\mathbb Q\_p$, then the $\ell$-adic cohomology spaces
$H^i(X,\mathbb Q\_{\ell})$ are $\ell$-adic representations of $G\_{K}$,
which give rise to Weil--Deligne representations. (See Tate's Corvallis article,
for example.) The resulting Weil--Deligne representation is co... | 12 | https://mathoverflow.net/users/2874 | 29181 | 19,052 |
https://mathoverflow.net/questions/29161 | 4 | If we want to define a sheaf F on a topological space X and we have a basis B for the topology of X, what we can do is to define objects and restrictions for guys in B, check that they satisfy the "B-sheaf axioms" and then use
Theorem 1: the B-sheaf extends uninquely to the whole of X.
I was wondering if there's a ... | https://mathoverflow.net/users/3701 | Does the concept of a basis for a topology on a category exist? | Let $S$ be your Grothendieck site. What you want is a subcategory $j:B \hookrightarrow S$ such that the Grothendieck topology of $S$ restricts to $B$ in the sense that every covering sieve of $b \in B$ can be refined by one coming from a family $(U\_i \to b)\_i$ with each $U\_i \in B$, AND such that $j^\*:Sh(B) \to Sh(... | 5 | https://mathoverflow.net/users/4528 | 29182 | 19,053 |
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