parent_url stringlengths 37 41 | parent_score stringlengths 1 3 | parent_body stringlengths 19 30.2k | parent_user stringlengths 32 37 | parent_title stringlengths 15 248 | body stringlengths 8 29.9k | score stringlengths 1 3 | user stringlengths 32 37 | answer_id stringlengths 2 6 | __index_level_0__ int64 1 182k |
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https://mathoverflow.net/questions/122096 | 2 | I am looking at words $\alpha\_1 \ldots \alpha\_n$, where $\alpha\_j \in \{ 1, \ldots, j \}$. Thinking of $\alpha\_j$ as a height, these words can be interpreted as left-to-right paths on the positive square lattice which stay under the diagonal. They can also be thought as a Lehmer code for permutations. Obviously, th... | https://mathoverflow.net/users/13923 | Statistics on Lehmer codes | The papers <http://www4.ncsu.edu/~savage/PAPERS/ESofLHPandEPforIS.pdf> and <http://www4.ncsu.edu/~savage/PAPERS/rational.pdf> contain some information about statistics on generalizations of Lehmer codes called **$s$**-*inversion sequences*.
| 3 | https://mathoverflow.net/users/2807 | 122129 | 68,632 |
https://mathoverflow.net/questions/122124 | 2 | I posted this on mathstack and it is not a homework problem, I am doing a modeling problem and this ODE comes up. I don't know whether it could be considered to be a research problem yet. Please help me to close it if it is not placed at the right place.
I have a problem with finding the closed form solution of the f... | https://mathoverflow.net/users/31522 | closed form solution of an ODEs | This is a second degree linear ODE with rational function coefficients, and this problem has been completely solved algorithmically by Kovacic in the mid-eighties:
Kovacic, Jerald J. "An algorithm for solving second order linear homogeneous differential equations." Journal of Symbolic Computation 2, no. 1 (1986): 3-4... | 7 | https://mathoverflow.net/users/11142 | 122132 | 68,635 |
https://mathoverflow.net/questions/122092 | 5 | This is a reference-request about a very simple statement.
The Riemann hypothesis is well-known to be equivalent to
$$(1)\ \ \ \pi(x) = \mathrm{Li}(x)+O(x^{1/2} \log x)$$
and to
$$(2)\ \ \theta(x)=x+O(x^{1/2} \log^2 x).$$
Here as usual, $\pi(x) = \sum\_{p \leq x} 1$ and $\theta(x) = \sum\_{p \leq x} \log p$.
>
> ... | https://mathoverflow.net/users/9317 | Equivalence of two well-known forms of (RH): reference-request | The argument that (2) implies (1) is given as equation (13.5) in Montgomery and Vaughan's **Multiplicative Number Theory I. Classical Theory**. A similar partial summation argument (probably two lines long instead of one) will establish that (1) implies (2).
| 7 | https://mathoverflow.net/users/5091 | 122138 | 68,639 |
https://mathoverflow.net/questions/122148 | 3 | Recently I was reading about integral equations and I am a beginner in it. There was a constant reference to the non-availability of methods to find the exact solutions and hence lot of approximation methods were referred.
I wanted to know that how effective these methods are while resolving problems and is it really ... | https://mathoverflow.net/users/22242 | approximation methods in integral equations | [Kendall Atkinson](http://www.math.uiowa.edu/~atkinson/) has done a big work in numerical solutions of integral equations. As with almost all other kinds of equations, the explicit solution of the integral equation is rarely known.
On the other hand, you shall ask yourself what is meant by the explicit solution. Usu... | 2 | https://mathoverflow.net/users/11768 | 122150 | 68,646 |
https://mathoverflow.net/questions/122119 | 11 | Are there any obstructions known which prevent an even dimensional orientable manifold from
being symplectic? I am a novice in this area so I unfortunately I cannot make the question more precise. What I have in mind is raw and is as follows:from the point of handle body theory you can put a symplectic structure on eac... | https://mathoverflow.net/users/31475 | what prevents a manifold to be symplectic? | Robert's comment is spot on. Describing symplectic manifolds constructively via handle attachment is particularly tricky because you're dealing with a geometric structure. Typically if you figure out a new way to build symplectic manifolds via surgery, it's worth writing a paper about. The most basic ways to build symp... | 11 | https://mathoverflow.net/users/31530 | 122152 | 68,647 |
https://mathoverflow.net/questions/122153 | 1 | Is there any characterization for the left primitive rings which are left duo ? Clearly division rings satisfy this property.
| https://mathoverflow.net/users/nan | A question on primitive rings | The class of rings in the question is exactly the class of Division rings. Here is another proof based on the characterization of primitive rings.
**Proof.** Since you have asked about primitive rings I can assume that you are familiar with the **Jacobson's Density Theorem**. Therefore $R$ can be regarded as a dense... | 2 | https://mathoverflow.net/users/nan | 122161 | 68,651 |
https://mathoverflow.net/questions/122147 | 5 | How is the Diamagnetic inequality born? Why is it call this name?
Diamagnetic inequality： $\big|\nabla|u|(x)\big|\leq \big|(\nabla+iA)u(x)\big|$.
| https://mathoverflow.net/users/31527 | why is it called the diamagnetic inequality? |
>
> Why is it called this name?
>
>
>
Diamagnetism is a quantum effect in which material under an applied external magnetic field generate their own "opposite" magnetic field. The most prominent example happens in [superconductors](http://en.wikipedia.org/wiki/Meissner_effect).
A consequence of this is that m... | 9 | https://mathoverflow.net/users/3948 | 122165 | 68,652 |
https://mathoverflow.net/questions/122121 | 8 | Let $(M,g)$ be a compact Riemannian manifold, no boundary.
I would like to know when is it possible to endow the tangent bundle $TM$ with an euclidean Riemannian metric.
Do you know of a good reference?
| https://mathoverflow.net/users/31521 | Metric on tangent bundle | I'll answer the particular question that Cecilia ultimately asked, after she elaborated her question in response to our comments.
I believe that she was asking this: Given a Riemannian $n$-manifold $(M^n,g)$, when does there exist a metric $\hat g$ on $TM$ such that each point in $M$ lies in some coordinate chart $x... | 15 | https://mathoverflow.net/users/13972 | 122166 | 68,653 |
https://mathoverflow.net/questions/122162 | 5 | Let $R$ be a henselian ring. Is $R[[x]]$ also a henselian ring?
| https://mathoverflow.net/users/5274 | Formal power series over a henselian ring | Yes, it is.
More generally, the following result holds.
>
> **Proposition.** If $R$ is henselian at the maximal ideal $\mathfrak{m}$, then $R[[x\_1, \ldots, x\_n]]$ is henselian at the maximal ideal lying over $\mathfrak{m}$.
>
>
>
A reference is the paper by N. Sankharan [*A Theorem on Henselian Rings*](http... | 11 | https://mathoverflow.net/users/7460 | 122170 | 68,656 |
https://mathoverflow.net/questions/122114 | 1 | Let $B$ be a Banach space. Let $\{Y\_{n}\}$ be a sequence of $B$ valued random variables.
Assume
1. $P(\{Y\_{n}\} \mbox{is bounded}) = 1$,
2. fo every $\epsilon>0$, there exists a finite dimensional subspace $F$ such that $P(\limsup\_{n} q\_{F}(Y\_{n})\leq \epsilon) = 1$.
Then show that $P(\{Y\_{n}\} \mbox{ is r... | https://mathoverflow.net/users/nan | Almost sure convergence Banach Space valued Random Variable | For $\omega\in\Omega$, define $S(\omega):=\(Y\_n(\omega),n\in\mathbb N\)$. We have to show that $P(\omega\mid S(\omega)\mbox{ is relatively compact})=1$.
Considering $\varepsilon=1/j$, we can see that there is $\Omega'$ of probability $1$ such that for each $\omega\in \Omega'$,
1. $\sup\_n\lVert Y\_n(\omega)\rVer... | 0 | https://mathoverflow.net/users/17118 | 122172 | 68,657 |
https://mathoverflow.net/questions/121517 | 10 | This question arises from an application to graphical models in probability theory, but I have abstracted that part out so only algebra remains. Let $\mathbb{R}$ denote standard field of real numbers and $\mathcal{R} = (\mathbb{R}\cup\{\infty\},\min,+)$ the tropical semiring. Let $\mathbb{Z}\_n$ denote the cyclic group... | https://mathoverflow.net/users/5963 | Efficient computation of "discrete infimal convolution" | After a bit more google searching, I found the 2006 paper [Necklaces, Convolutions, and X+Y](http://compgeom.cs.uiuc.edu/~jeffe/pubs/pdf/necklace.pdf), which addresses this problem. The nine authors give an $O(n\sqrt{n})$ algorithm in the nonuniform linear decision tree model (I'm having little trouble pinning down the... | 2 | https://mathoverflow.net/users/5963 | 122178 | 68,659 |
https://mathoverflow.net/questions/34909 | 9 | The reduced $C^\*$-algebra of a non-abelian free group $G$ has a unique trace. Hence, there is no chance to separate conjugacy classes of group elements using traces on $C^\star\_{red} G$. On the other side, for the group ${\mathbb Z}$, separation is clearly possible.
>
> **Question:** Let $G$ be an amenable group.... | https://mathoverflow.net/users/8176 | Conjugacy classes and reduced group $C^*$-algebra of an amenable group | There is an easy example, namely $SL\_3(F)$ where $F$ is the algebraic closure of some finite field. This group does not admit non-trivial characters (a result of Kirillov) and is locally finite, hence amenable. This gives a negative answer to the first question.
Sorry, I deleted my previous answer since it contained... | 5 | https://mathoverflow.net/users/8176 | 122181 | 68,661 |
https://mathoverflow.net/questions/122159 | 1 | Let us have a look on the proof of Theorem 2 in [P. G. Spain, [Boolean algebras of projections](http://journals.cambridge.org/action/displayAbstract?fromPage=online&aid=3073512), *Proceedings of the Edinburgh Mathematical Society* (Series 2) **19**, 03, March 1975, 287-289]
The author claims in the proof of Theorem 2... | https://mathoverflow.net/users/31512 | Is this result of Spain correct? | I haven't looked at the paper but your counterexample is mistaken. The basis projections generate not $B(l^p)$ but the algebra of multiplication operators, which is isometrically isomorphic to $l^\infty$ and hence is a von Neumann algebra.
| 4 | https://mathoverflow.net/users/23141 | 122185 | 68,664 |
https://mathoverflow.net/questions/122186 | 3 | Hello all,
Assume NxN matrix A of complex values. I want to calculate the sum of all elements of its inverse.
The problem is that calculating the inverse is computationally expensive and since I am looking only for the sum of its elements, I thought there might be something smarter to do.
Note: the real part of A... | https://mathoverflow.net/users/31543 | Sum of elements of inverse matrix | The sum of the elements of a matrix $M$ is $e^T M e$, where $e$ is the vector of all ones.
So, instead of computing the inverse, you should solve the system $Ax=e$ and then compute $e^Tx$. This might look like a simple trick, but solving linear systems is faster than computing inverses in basically all settings.
Of... | 11 | https://mathoverflow.net/users/1898 | 122188 | 68,665 |
https://mathoverflow.net/questions/122164 | 9 | I'm writing a paper on classical Morse Theory and I'm interested in applying Morse functions to the computation of homology groups of a compact manifold $M$. The standard way in which this is done is by using Theorem 3.5 in [Milnor's textbook](https://books.google.it/books/about/Morse_Theory.html?id=A9QZZ3S_QxwC&redir_... | https://mathoverflow.net/users/47757 | A description of cellular boundary maps in terms of a Morse function | Under certain conditions, (Morse-Smale being one, but not sufficient) the stratification by unstable manifolds of a Morse flow on a compact manifolds gives a cellular decomposition; see the paper *[On the Space of Trajectories of a Generic Vector Field](https://arxiv.org/abs/1101.0778)* by Burghelea & co, and the paper... | 8 | https://mathoverflow.net/users/20302 | 122192 | 68,666 |
https://mathoverflow.net/questions/122163 | 1 | Hello,
I'm wondering if there exists general results insuring exhaustion by compact sets of a given topological space ?
nicolas
| https://mathoverflow.net/users/31535 | General criteria for exhaustion by compact sets | David White's result may be generalized to necessary and sufficient conditions for when a locally compact space is $\sigma$-compact. A locally compact space is $\sigma$-compact if and only if it is Lindelof. In particular, since every second countable space is Lindelof, every locally compact and second countable space ... | 2 | https://mathoverflow.net/users/22277 | 122194 | 68,667 |
https://mathoverflow.net/questions/122175 | 9 | Can anything be said about connectedness of a smooth manifold M from some property of Diff(M) in an analogous way Like C(X) has no idempotents iff X is connected.
| https://mathoverflow.net/users/30081 | Diff(M) and connectedness | Let $M$ be a compact oriented manifold. The following hold if and only if $M$ is connected.
1) $\text{Diff}\_0(M)$ is simple.
This was proven by Thurston if $M$ is connected; see
MR1445290 (98h:22024)
Banyaga, Augustin(1-PAS)
The structure of classical diffeomorphism groups. (English summary)
Mathematics and i... | 14 | https://mathoverflow.net/users/317 | 122203 | 68,670 |
https://mathoverflow.net/questions/122180 | 5 | Let $\mathfrak{g}\_1\subset \mathfrak{g}\_2$ be a Lie algebra embedding. Assume both are semisimple. For instance take the standard diagonal embedding $\mathfrak{sl}(2, \mathbb{C})\subset \mathfrak{sl}(3, \mathbb{C})$. This lifts to an embedding $U(\mathfrak{sl}(2, \mathbb{C}))\subset U(\mathfrak{sl}(3, \mathbb{C}))$ o... | https://mathoverflow.net/users/31261 | Lie algebra embeddings and the center of their enveloping algrabras | Take a look at "Shifted Schur Functions"
Andrei Okounkov, Grigori Olshanski
<http://arxiv.org/abs/q-alg/9605042>
Section 10: "Coherence property of quantum
immanants and shifted Schur polynomials"
In particular formulas 10.4, 10.5 - they discuss "averaging operators"
Z(U(gl(n)) -> ZU(gl(N)) , n < N
and later pr... | 4 | https://mathoverflow.net/users/10446 | 122212 | 68,677 |
https://mathoverflow.net/questions/122217 | 6 | For a discrete group G, if $M$ is a direct/inverse limit of $M\_i$, is $H^i(G, M)$ the direct/inverse limit of the $H^i(G, M\_i)$? Of course, cohomology commutes with finite direct sums, but how about general limits/colimits over finite categories? Thank you!
| https://mathoverflow.net/users/26081 | Which limits does group cohomology commute with? | Ken Brown shows in
* *Homological criteria for finiteness*, Comment. Math. Helv. 50 (1975), 129–135, doi:[10.1007/BF02565740](https://doi.org/10.1007/BF02565740), ([free author version](https://pi.math.cornell.edu/%7Ekbrown/scan/1975.0050.pdf))
that group cohomology for a group G commutes with direct limits iff G i... | 7 | https://mathoverflow.net/users/15934 | 122220 | 68,680 |
https://mathoverflow.net/questions/122219 | 3 | Let $M$ be a Riemannian manifold of dimension $n$. Let $N\subset M$ be a subset with smooth boundary $\Sigma=\partial N$. If one assume the second fundamental form $II$ with respect to inner normal direction of $\Sigma$ is nonnegative. (here we use the convention that the second fundamental form of round sphere in $\ma... | https://mathoverflow.net/users/30176 | Positivity of second fundamental form implies global convexity? | From the comment of Willie Wong , the answer is NO.
Yet simpler example is $\mathbb{S}^1\times\mathbb{R}$, where you have locally convex discs which are not globally convex.
If you want "yes" as an answer, you have to assume bit more. For example simply connectedness plus sectional curvature $\le 0$ (It was proved ... | 7 | https://mathoverflow.net/users/1441 | 122226 | 68,682 |
https://mathoverflow.net/questions/122215 | 0 | For all $s>0$ define for $\epsilon\in(0,1)$ the function:
\begin{equation}
g(\epsilon)=\sum\_{k=0}^{\infty}(1+k)^s(\sqrt{1-\epsilon})^k.
\end{equation}
Prove that $\exists C>0$ and $\phi(s)$ such that:
\begin{equation}
g(\epsilon)\leq C \epsilon^{\phi(s)}.
\end{equation}
| https://mathoverflow.net/users/31551 | An interpolation inequality. | $$
g(\epsilon)=\sum\_{k\ge 1} k^se^{\frac {k-1}2\ln(1-\epsilon)}
\lesssim \int\_0^{+\infty} x^s e^{-a\epsilon x} dx=
\int\_0^{+\infty}x^s e^{-ax}dx\epsilon^{-s-1}
$$
where $a$ is a fixed constant. So
$$
C=\int\_0^{+\infty}x^s e^{-ax}dx,\quad \phi(s)=-s-1.
$$
| 0 | https://mathoverflow.net/users/21907 | 122229 | 68,683 |
https://mathoverflow.net/questions/122182 | 10 | I'm currently trying to relate two descriptions of the curvature and torsion of a connection and am running into some confusion.
I know that an affine connection $A$ on an $n$-dimensional manifold $M$ can be split into two parts $A = \omega + e$, where $\omega$ takes values in the Lie algebra of rotations $\mathfrak{... | https://mathoverflow.net/users/14601 | Relating curvature and torsion of a connection to those of a curve | This is not really an answer, but I think it might help you look at things a little bit differently. It is not at all clear that Cartan chose the word 'torsion' to describe the 'translation' component of the curvature because it was related to the torsion of a curve in flat space or had anything to do with developing m... | 13 | https://mathoverflow.net/users/13972 | 122235 | 68,686 |
https://mathoverflow.net/questions/122135 | 0 | Let $f:X\dashrightarrow Y$ be a birational map of smooth projective varieties, i.e., there exist open subsets $U\_1, \subset X$ and $U\_2 \subset Y$ such that $f|\_{U\_1} : U\_1 \rightarrow U\_2$ is an isomorphism.
There is an homomorphism induced by $f$ given by
$$ f\_\* : Div(X) \rightarrow Div(Y) , D\mapsto \ov... | https://mathoverflow.net/users/31524 | Zero-cohomology of birational varieties | So, the latest formulation is much better, but there are still some problems with this setup. You can push-forward cycles via morphisms, but rational maps are trickier.
First of all if the inverse of $f$ contracts a divisor, then $f\_\*$ does not (necessarily) respect linear equivalence. Here is an example. Let $X$ ... | 3 | https://mathoverflow.net/users/10076 | 122240 | 68,688 |
https://mathoverflow.net/questions/122202 | 1 | Let $\widehat{f}(\xi)$ be Fourier transform of $f$ given by
\begin{align}
\widehat{f}(\xi)=\int\_{\mathbb{R}^n} e^{-ix\cdot\xi}f(x)dx.
\end{align}
Suppose that $\widehat{f}(\xi)$ is nonnegative and locally integrable function, easily seems (by inverse Fourier transform) that
\begin{align}
\Vert f\Vert\_{L^{\infty}} ... | https://mathoverflow.net/users/nan | nonnegative Fourier Transform | If $\hat f$ is nonnegative, then (up to a factor),
$$f(0)=\int \hat f=\Vert \hat f \Vert\_1 = \Vert f \Vert\_\infty.$$
| 10 | https://mathoverflow.net/users/12120 | 122252 | 68,696 |
https://mathoverflow.net/questions/122254 | 5 | I have seen some similar questions to this one on here recently, so I hope this isn't redundant. Basically, suppose I have two cofiber sequences of spectra (or perhaps just work in some general homotopy category or something) $X\overset{f}\to Y\to Cf$ and $X'\overset{g}\to Y'\to Cg$. I'd like to look at an induced filt... | https://mathoverflow.net/users/11546 | Filtration on Smash Product of Cofibers | This is an excellent question that I have thought a lot about. I'd rather answer it in a more general context that was motivated by what I knew to be true in the stable homotopy category. The reference is "The additivity of traces in triangulated categories", #99 on my web site. The essential point there is to formulat... | 8 | https://mathoverflow.net/users/14447 | 122258 | 68,698 |
https://mathoverflow.net/questions/122259 | 12 | Cantor's diagonalization construction, on a certain view, furnishes functions
$$d\_X:{\rm Injections}(X,P(X))\rightarrow P(X)$$
that satisfy $\forall X\forall i\ \ d\_X(i)\not\in i(X)$
In ZF, can one prove the existence of such functions with the added requirement
that $d\_X(i)$ actually depends only on the image ... | https://mathoverflow.net/users/10909 | Cantor's diagonal argument and ZF | If there is such a function then there is an injection from $\omega\_1$ to $2^{\omega}$.
(Set $\: X = \omega \:$, $\:$ send the finite ordinals to the corresponding singletons,
then extend to $\omega\_1$ with [transfinite recursion](https://planetmath.org/TransfiniteRecursion).)
If there is such a function an... | 14 | https://mathoverflow.net/users/nan | 122261 | 68,699 |
https://mathoverflow.net/questions/122238 | 7 | Suppose $L$ is a nilpotent finite-dimensional Lie algebra over $\mathbb{Q}$ of class $c$. We can define an associated graded Lie algebra to $L$ that, as a vector space, is:
$$\bigoplus\_{i=1}^c \gamma\_i(L)/\gamma\_{i+1}(L)$$
where $\gamma\_i$ denotes the $i^{th}$ member of the lower central series, and where the L... | https://mathoverflow.net/users/3040 | Malcev Lie algebra and associated graded Lie algebra | You can find an example on pages 151-152 of Y.Shalom, "Harmonic analysis, cohomology, and the large-scale geometry of amenable groups", Acta Math, vol. 192 (2004) 119-185.
In this example, the corresponding nilpotent groups are not even quasi-isometric to each other.
| 4 | https://mathoverflow.net/users/21684 | 122267 | 68,701 |
https://mathoverflow.net/questions/122127 | 1 | Let $H$ be a subgroup of a [lattice-ordered group](https://www.encyclopediaofmath.org/index.php/Lattice-ordered_group) $G$. Suppose that $H$ with the induced order is a lattice (but a priori not a sublattice), so that $H$ is a lattice-ordered group too. For $a, b\in H$, let $ c=\inf\_H(a, b) \in H$ and let $d= \inf\_G(... | https://mathoverflow.net/users/30267 | Subgroup of lattice-ordered group | No. A counterexample (essentially from Bourbaki's *Algèbre* VI.1 Exercice 12 a)) is the following.
We furnish $\mathbb{Z}$ with its usual structure of ordered group and consider the product of ordered groups $G=\mathbb{Z}^3$. This is a lattice, and for $(x,y,z),(u,v,w)\in G$ we have $$\textstyle\sup\_G((x,y,z),(u,v,w... | 2 | https://mathoverflow.net/users/11025 | 122269 | 68,702 |
https://mathoverflow.net/questions/122237 | 18 | It is a well-known fact that for any compact topological space $\rm S$, the set its points is in bijection with the set of maximal ideals of $\mathcal C(\rm S)$, the algebra of continuous functions $\rm S \to \mathbb R$.
Now consider the circle $\rm S^1$ with its usual $\sigma$-algebra. Note $\mathcal M$ the algebra ... | https://mathoverflow.net/users/27840 | Maximal ideals of the algebra of measurable functions | We shall give a complete characterization of all the maximal ideals in $\mathcal{M}/\mathcal{N}$ in terms of ultrafilters. For simplicity, we shall classify all maximal ideals in $\mathcal{M}$ since the maximal ideals in $\mathcal{M}/\mathcal{N}$ are in a one-to-one correspondence with all maximal ideals in $\mathcal{M... | 13 | https://mathoverflow.net/users/22277 | 122272 | 68,705 |
https://mathoverflow.net/questions/122243 | 1 | I am interested in solving **binomial systems** of the form
$$
\begin{cases}
a\_1 x\_1^{d\_{11}} x\_2^{d\_{12}} \cdots x\_n^{d\_{1n}} +
b\_1 x\_1^{d\_{11}} x\_2^{d\_{12}} \cdots x\_n^{d\_{1n}} &= 0 \\\\
\vdots &\vdots \\\\
a\_m x\_1^{d\_{m1}} x\_2^{d\_{m2}} \cdots x\_n^{d\_{mn}} +
b\_m x\_1^{d\_{m1}} x\_2^{d\_{m2... | https://mathoverflow.net/users/21522 | Recommendations for binomial system solver | Let $I$ be the ideal generated by your binomial equations after clearing denominators.
As a general rule with binomial equations, disregard coefficients in the first run (see here for why and how: <http://arxiv.org/abs/1107.4699> ). I would first solve the system where all coefficients are set to be one, which means ... | 3 | https://mathoverflow.net/users/5495 | 122275 | 68,706 |
https://mathoverflow.net/questions/122277 | 3 | Let $\mathcal{L}=L\_1\cup \ldots \cup L\_k$ be an arrangement of lines in $\mathbb{P}^2$ with $k>2$ and $t\_k=0$ ($t\_r$ is defined to be the number of $r$-fold points of the arrangement for $r\in \mathbb{N}$). For any natural number $n>1$ we may associate with $\mathcal{L}$ the following function field $$F\_{\mathcal{... | https://mathoverflow.net/users/24132 | Algebraic surface of a line arrangement | Regarding 1: If $X$ is a variety with function field $K$, $L$ a finite separable extension of $K$, we can construct a normal variety $Y$, "normalization of $X$ in $L$", with function field $L$ and a finite map $Y\to X$. This is done as follows: for any affine open $X\_i = Spec(R)$ in $X$, take $Y\_i = Spec(S)$ where $S... | 2 | https://mathoverflow.net/users/3847 | 122280 | 68,709 |
https://mathoverflow.net/questions/122288 | 1 | Just as the title explains, is the Sorgenfrey Line monotonically monolithic [(see the definition)](https://math.stackexchange.com/questions/297948/what-is-the-relation-between-kappa-monolithic-and-monotonically-monolithic)?
| https://mathoverflow.net/users/18465 | Is the Sorgenfrey Line monotonically monolithic? | No, the Sorgenfrey line $\mathbb{R}\_l$ is not even $\aleph\_0$-monolithic.
Note that any network $\mathcal{N}$ on $\mathbb{R}\_l$ must have at least $2^{\aleph\_0}$ elements, since for each $x \in \mathbb{R}\_l$ there is an $N\_x \in \mathcal{N}$ such that $x \in N\_x \subseteq [x,x+1)$. On the other hand $\mathbb{... | 3 | https://mathoverflow.net/users/17836 | 122294 | 68,714 |
https://mathoverflow.net/questions/122227 | 4 | The following got no answer on mathstackexchange. I believe it not to be hard, but maybe it is a little specialized?
All varieties will be over $\mathbb{C}$ and projective unless stated otherwise.
In Beauville - complex algebraic surfaces, the following is described: Let $S$ be a smooth surface and $p \in S$ a poin... | https://mathoverflow.net/users/29657 | Divisor class group on blowup of nodal surface | This is a partial answer to the part concerning Weil divisors.
I prefer to use the terminology of cycles of codimension $1$ instead of Weil divisors. The group of Weil divisors modulo rational equivalence is the Chow group $A^1$.
Let $f : \tilde{S}\to S$ be any proper birational morphism of integral normal algebr... | 4 | https://mathoverflow.net/users/3485 | 122296 | 68,715 |
https://mathoverflow.net/questions/120768 | 5 | Define $Stab^{+}(\Omega )$={ $\phi \in GL^{+}(V)$ : $\phi^{\*}\Omega=\Omega$ }.
we say three-form $\Omega\in\wedge^{3}V^{\*}$ is non-degenerate , if $i\_X\Omega\neq 0$ for all $X\in V$-{0}
Let $V\cong \mathbb{R}^{6}$ and $\Omega\in\wedge^{3}V^{\*}$ be non-degenerate(by sense of Hitchin) . Then we know $Stab^{+}(\Om... | https://mathoverflow.net/users/nan | Recognizing the stablizer of a degenerate three forms in six dimension | First of all, it is not true that there are only two types of nondegenerate $3$-forms in $\Lambda^3(V^\ast)$ when $\dim\_\mathbb{R}V=6$. There are actually $3$ such nondegenerate orbits. Here is the complete list of $\mathrm{GL}(V)$-orbit types in this space: Let $e^1,\ldots, e^6$ be a basis of $V$. Then the following ... | 15 | https://mathoverflow.net/users/13972 | 122317 | 68,725 |
https://mathoverflow.net/questions/122308 | 6 | When a Riemannian manifold is of Hessian Type (i.e., a Riemannian manifold which its metric is Hessian)
| https://mathoverflow.net/users/nan | When a Riemannian manifold is of Hessian Typ | First, the definition: A Riemannian $n$-manifold $(M^n,g)$ is *of Hessian type* if there exist $(n{+}1)$ functions $x^1,\ldots,x^n, u$ on $M$ such that $dx^1\wedge\cdots\wedge dx^n\not=0$ and such that
$$
g = \frac{\partial^2u}{\partial x^i\partial x^j} dx^idx^j
$$
(of course, the summation convention is in force for t... | 14 | https://mathoverflow.net/users/13972 | 122319 | 68,726 |
https://mathoverflow.net/questions/122207 | 5 | I'm looking for a reference for the following fact.
Suppose $A$ is a finitely generated associative commutative unital algebra over an algebraically closed field of characteristic zero. Let $S^n(A)$ be the $n$-th symmetric power of $A$. That is, $S^n(A)$ is the subalgebra of $A^{\otimes n}$ fixed by the natural actio... | https://mathoverflow.net/users/29738 | A nice generating set for the symmetric power of an algebra | Since your algebra is finitely generated, you only really need this result in the case when $A=\mathbb{C}[t\_1,\ldots,t\_n]$.
It seems that this result first appeared in
F. Junker, *Über symmetrische Funktionen von mehreren Reihen von Veränderlichen*, Math. Ann. 43 (1893), 225-270
(see [here](http://gdz.sub.un... | 4 | https://mathoverflow.net/users/1306 | 122320 | 68,727 |
https://mathoverflow.net/questions/121272 | 2 | Suppose $U\subset\mathbb{Z}\_{2k}$ with $|U|=k$. Let $U^c$ denote the complement of $U$.
Let $v\in \mathbb{Z}\_{2k}^{\times}$. How much is it known about $U+vU^c$?
For example: When $U+v\complement U = \mathbb{Z}\_{2k}$?
Yet another example: if $U$ is aperiodic (i. e., its isotropy group is trivial), Kemperman's ... | https://mathoverflow.net/users/31285 | About a sumset in $\mathbb{Z}_{2k}$ | The majority of facts I wanted to know about this situation are contained in Theorem 7.5 (p. 75) of David J. Grynkiewicz's Ph. D. thesis "Sumsets, Zero-Sums and Extremal Combinatorics".
Thank you very much to those who offered their remarks.
| 3 | https://mathoverflow.net/users/31285 | 122330 | 68,734 |
https://mathoverflow.net/questions/122311 | 4 | We can define projective structure on a manifold in two ways.
First we can define it as a maximal atlas of charts from the open subsets of manifold to the projective space, such that transitions maps are locally the elements of the projective general linear group.
Second, we can define it as a torsion-free projectively... | https://mathoverflow.net/users/25609 | Projectively equivalent connections | I assume you are asking why two definitions of the projective structures, one given in terms of atlas, and another given as the existence of projectively flat connection, coincide.
If two connections are projectively equivalent, then their geodesics (considered as unparameterized curves) coincide, and vice versa. If... | 8 | https://mathoverflow.net/users/14515 | 122334 | 68,736 |
https://mathoverflow.net/questions/122125 | 74 | Suppose most mathematical research papers were freely accessible online.
Suppose a well-organized platform existed where responsible users could write comments on any paper (linking to its doi, Arxiv number, or other electronic identifier from which it could be retrieved freely), or even ``mark it up'' (pointing to ... | https://mathoverflow.net/users/31518 | Math Annotate Platform? | I think such a thing would provide immense value. In particular I can think of instances when the following sorts of comments would have saved me a great deal of time:
(1) No need to read pages XX-XXX, here is a one paragraph argument.
(2) This result has since been strengthened, see ...
(3) The following claims... | 36 | https://mathoverflow.net/users/4707 | 122342 | 68,740 |
https://mathoverflow.net/questions/122282 | 4 | A matrix of $k$ rows and $n$ columns is filled with the numbers $1,2,\ldots,k$ such that the following conditions are satisfied:
1. Every column contain all the numbers form 1 to $k$ ***without repetition***.
2. The variance of the elements of each row is calculated. The matrix is filled in such a way that the total ... | https://mathoverflow.net/users/23388 | Most inconsistent ranking | $\newcommand{\E}{\mathrm{E}}$
$\newcommand{\Var}{\mathrm{Var}}$
$\newcommand{\Cov}{\mathrm{Cov}}$
Extending Kevin's answer let me show that
$$S\_{k,n} \leq \frac{k^3-k}{12}.$$
It is easy to see that $$\Var(X) \leq \E\[(X-c)^2\]$$
for any random variable $X$ and any constant $c$.
Let $a\_\{i,j\}$ denote the ent... | 4 | https://mathoverflow.net/users/8733 | 122347 | 68,742 |
https://mathoverflow.net/questions/122322 | 6 | **Background**
Let $U\_q(sl(2))$ be the quantum group associated with $sl(2)$ i.e. the associative algebra with 1 over $Q(q)$ generated by $x^+,x^-,K,K^{-1}$ with relations
$$KK^{-1}=K^{-1}K=1$$
$$Kx^+K^{-1}=q^2x^+,Kx^-K^{-1}=q^{-2}x^-$$
$$x^+x^{-}-x^{-}x^+=\frac{K-K^{-1}}{q-q^{-1}}$$
Here $q$ is indefinite, in part... | https://mathoverflow.net/users/31556 | Question about unusual highest weight modules for $U_q(sl(2))$ | When talking about modules for such a quantized enveloping algebra, you have to be explicit about whether the module is finite dimensional or not. (For affine or other infinite dimensional Lie algebras of interest, the parallel question is whether modules are integrable or not.) In Jantzen's introductory AMS text *Lect... | 4 | https://mathoverflow.net/users/4231 | 122348 | 68,743 |
https://mathoverflow.net/questions/122041 | 1 | [Andrew Boucher's General Arithmetic (GA2)](http://www.andrewboucher.com/papers/ga.pdf) is a weak sub-theory of second order Peano Axioms (PA2). GA has second order induction and a single successor axiom:
$$\forall x \forall y \forall z\bigr((Sx=y \land Sx=z)\to(y=z)\bigl)$$
Boucher proves multiplication is commuta... | https://mathoverflow.net/users/26766 | Why Does Induction Prove Multiplication is Commutative? | This is an answer to the edited questions Russell has added. Joel David Hamkins' reply in comments to the question is completely correct, but I'll take advantage of the greater space here.
Let (R,0,1,+,\*) be a ring. Define
Sx = x + 1 and
B = {x | $\forall P(P0 \land \forall y\forall z(Py \land Sy,z \to Pz) \to... | 6 | https://mathoverflow.net/users/20716 | 122349 | 68,744 |
https://mathoverflow.net/questions/122298 | 4 | Regarding the published version of "Motivic cohomology groups are isomorphic to higher Chow groups in any characterstic" (IMRN) available at:
<http://imrn.oxfordjournals.org/content/2002/7/351.full.pdf>
I would appreciate any elucidation on the following two points:
**(1)** a short assertion is made on the second... | https://mathoverflow.net/users/4235 | Voevodsky's proof in any characteristic (for motivic and Chow) | Voevodsky's statement (1) is plainly false if $k$ is not of finite transcendence degree over $k\_0$. For then the fraction field of any $O\_{X,x}$ is of infinite transcendence degree over $k\_0$ whereas the fraction field of $O\_{X\_0,x\_0}$ always has finite transcendence degree over $k\_0$.
But anyway here's a more... | 5 | https://mathoverflow.net/users/20233 | 122357 | 68,746 |
https://mathoverflow.net/questions/122361 | 7 | I am mainly interested in varieties over an algebraic closed field $k$ (or $\mathbb{C}$). The classification of complex surface is established in the last century and known as [Enriques–Kodaira classification](http://en.wikipedia.org/wiki/Enriques%E2%80%93Kodaira_classification). In higher dimension, people realized th... | https://mathoverflow.net/users/31593 | Is there an Enriques–Kodaira-like classification of Fano threefolds? | Yes, Fano threefolds have been completely classified and one has explicit projective models of them. See V.A. Iskovskih, Yu. G. Prokhorov: Algebraic Geometry V: Fano varieties. Encyclopaedia of Math. Sciences 47, Springer-Verlag, Berlin 1999. See also [Andreas Ott's thesis](http://people.mpim-bonn.mpg.de/andreaso/files... | 8 | https://mathoverflow.net/users/3996 | 122365 | 68,748 |
https://mathoverflow.net/questions/122352 | 4 | Let $G$ be a compact, connected, simply-connected Lie group with centre $Z(G)$, and consider the Lie group $G/Z(G)$. I believe that for $G$ a classical group, the Lie group $G/Z(G)$ is sometimes called a projective classical group. What is known about the integral cohomology $H^\*(G/Z(G);\mathbb{Z})$? I am particularly... | https://mathoverflow.net/users/25358 | Cohomology of Projective Classical Lie Groups | Ancient history! I don't remember exactly what is in it, but I think the paper by Paul Baum and William Browder "The cohomology of quotients of classical groups" Topology 3 1965 305–336,
considers the cohomology of these groups. Of course, even back then, calculations were made in mod $p$ cohomology first, relying on t... | 5 | https://mathoverflow.net/users/14447 | 122368 | 68,751 |
https://mathoverflow.net/questions/108965 | 8 | Recently, it was conjectured by the [paper](http://arxiv.org/abs/1204.4709) of Aganagic and Vafa that the $Q$-deformed $A$-polynomials can be identified with the augmentation polynomials of the knot contact homology. The $Q$-deformed $A$-polynomial of a knot $K$ can be obtained by finding the difference equation of min... | https://mathoverflow.net/users/17644 | What is knot contact homology? | Check out these lecture notes (and the references listed therein) <http://www.renyi.hu/~cast2012/ng-cast.pdf>
1) Visualize elements of $ST^{\*}\mathbb{R}^{3}$ as unit length vectors -- not necessarily based at the origin -- in $\mathbb{R}^{3}$. The projection $\pi$ onto $\mathbb{R}^{3}$ send each vector to its basepo... | 5 | https://mathoverflow.net/users/31530 | 122369 | 68,752 |
https://mathoverflow.net/questions/122354 | 17 | I've been working through some notes on differential cohomology for the past few months. I feel like I have a pretty decent grasp on the concepts and its construction, at least for differential extensions of ordinary cohomology, like Deligne cohomology.
My friend recently asked me about some of the uses of different... | https://mathoverflow.net/users/10546 | Results from Differential Cohomology | The paper
Chern, Shiing Shen; Simons, James
*Characteristic forms and geometric invariants.*
Ann. of Math. (2) 99 (1974), 48–69
seems to be seminal in the subject, and gives a flavour of how differential cohomology came about. It suggests that Chern and Simons discovered the theory "by accident", when looking for ... | 7 | https://mathoverflow.net/users/8103 | 122380 | 68,756 |
https://mathoverflow.net/questions/122382 | 0 | Let $R$ denote the set of all real numbers. $B$ is any Bernstein set of $R$.
>
> Bernstein Set: A subset of the real line that meets every uncountable closed subset of the real line but that contains none of them. It's from wiki.
>
>
>
We topologize $R$ now: the set $B$ is discrete and its complement has the ... | https://mathoverflow.net/users/18465 | How to see such space is Lindelof? | Note that the open subsets of (what I will denote by) $\mathbb{R}\_B$ are of the form $U \cup A$ where $U \subseteq \mathbb{R}$ is open in the usual topology, and $A \subseteq B$ is arbitrary.
Suppose that $\{ U\_i \cup A\_i : i \in I \}$ is an open cover of $\mathbb{R}\\_B$. Note that there is a countable $I\_0 \su... | 3 | https://mathoverflow.net/users/13653 | 122383 | 68,758 |
https://mathoverflow.net/questions/122390 | 2 | Let $f\_0(T),f\_1(T) \in \mathbb R [T]$ be polynomials. Let $F(T,X):=X^2+f\_1(T)X+f\_0(T)$. Then the set $M(t):=\{x \in \mathbb R \mid F(t,x) \le 0\}$ is bounded. Its volume $V(t)$ is $\sqrt{\max\{0,f\_1(T)^2-4f\_0(T)\}}$. In particular this is piecewise continously differentiable and monotonous, and the number of piec... | https://mathoverflow.net/users/31600 | Volumes of families of semialgebraic sets | A theorem by Jean-Philippe Rolin and Jean-Marie Lion asserts a similar property in the analytic category. See *[Intégration des fonctions sous-analytiques et volumes des sous-ensembles sous-analytiques](https://eudml.org/doc/75301)*, Annales de l'institut Fourier (1998) **48** (3), p.755-767.
I quote their abstract ... | 2 | https://mathoverflow.net/users/10696 | 122392 | 68,763 |
https://mathoverflow.net/questions/99122 | 9 | [Fekete's (subadditive) lemma](http://en.wikipedia.org/wiki/Subadditivity#Properties) takes its name from a 1923 paper by the Hungarian mathematician Michael Fekete [1]. A historical overview and references to (a couple of) generalizations and applications of the result are found in Steele's book on probability and com... | https://mathoverflow.net/users/16537 | Generalizations and relative applications of Fekete's subadditive lemma | Since you mentioned Kingman's subadditive ergodic theorem, you may find interesting the following *semi-uniform subadditive ergodic theorem*:
Let $T \colon X \to X$ be a continuous map of a compact metric space $X$.
If $f\_n \colon X \to [-\infty,+\infty)$ is a subadditive sequence ($f\_{n+m} \le f\_n + f\_m \circ T^... | 5 | https://mathoverflow.net/users/1516 | 122404 | 68,769 |
https://mathoverflow.net/questions/122394 | 5 | The group-algebra of an abelian group is commutative, so we can consider the spectrum of this algebra. Are there any information about the abelian group that we can obtain from such considerations? That is to say, could we study abelian groups by considering the spectrum and the scheme of its group-algebra?
Since I ... | https://mathoverflow.net/users/11059 | Spectrum and scheme of the commutative group-algebra of an abelian group. | The spectrum of the group algebra of a commutative group is called a diagonalizable group scheme. This is defined in [SGA 3](http://www.math.jussieu.fr/~polo/SGA3/) Exposé VIII Section 1. Several geometric characterizations of group-theoretic properties are given in Proposition 2.1. A lot more is written in later secti... | 11 | https://mathoverflow.net/users/121 | 122414 | 68,773 |
https://mathoverflow.net/questions/122406 | 3 | The goal of this question is to recall a certain mathematical fact -not in my field- that I was once briefly told and that I have fogotten, and also to collect similar results.
The fact, I think, was about the undecidability (read: independence from ZFC or maybe ZF axioms) of a certain seemingly very "natural" sente... | https://mathoverflow.net/users/4721 | Undecidability and holomorphic functions (Reference request) | Very likely the fact that you are trying to remember is the interpolation problem solved by Erdős.
<http://www.renyi.hu/~p_erdos/1964-04.pdf>
| 6 | https://mathoverflow.net/users/23141 | 122426 | 68,777 |
https://mathoverflow.net/questions/122424 | 6 | Let $l$, $p$ be primes. Is it true that the functor of taking invariants under pro-$p$-group $P$ of finite-dimensional $\mathbb Q\_l$-vector spaces ($l\neq p$) is an exact functor?
Thanks!
NOTE 1: I am not assuming that the action is discrete.
NOTE 2: I am assuming the action of the group on the vector space is c... | https://mathoverflow.net/users/36285 | Taking invariants under pro-p-group is exact? | Yes, I think taking invariants is exact in your context.
**Proof:** Let be $G$ a pro-p group and $0 \to A \to B \to C\to 0$ a short exact sequence (s.e.s) of finite dim. $\mathbb{Q}\_l$-vector spaces on which $G$ acts continuously and $\mathbb{Q}\_l$-linearly. There is a long exact sequence (l.e.s.) [N, 2.3.2]
$$0... | 6 | https://mathoverflow.net/users/10194 | 122433 | 68,779 |
https://mathoverflow.net/questions/122411 | 11 | Let us consider the Arithmetic Mean -- Geometric Mean inequality for nonnegative real numbers:
$$ GM := (a\_1 a\_2 \ldots a\_n)^{1/n} \le \frac{1}{n} \left( a\_1 + a\_2 + \ldots + a\_n \right) =: AM. $$
It is known that the converse inequality ($\ge$) holds if and only if all the $a\_i$'s are the same.
Therefor... | https://mathoverflow.net/users/8460 | Almost-converses to the AM-GM inequality | Power mean inequality can give many bounds for the difference between AM and GM. Most simple is
$$AM - GM \leq \max\_i a\_i - \min\_i a\_i.$$
Another bound is
$$AM - GM \leq AM - HM = \frac{a\_1+\dots+a\_n}{n} - \frac{n}{1/a\_1 + \dots + 1/a\_n}$$
etc.
See <http://en.wikipedia.org/wiki/Generalized_mean#Generalized_... | 18 | https://mathoverflow.net/users/7076 | 122444 | 68,782 |
https://mathoverflow.net/questions/122442 | 1 | The functional equations for [Dedekind zeta functions](http://en.wikipedia.org/wiki/Dedekind_zeta_function) (zeta functions attached to rings of integers in algebraic number fields) come from functional equations of theta functions like $\sum\_{n \in \mathbb{Z}} q^{n^2}$, where $q = e^{2 \pi i z}$, which in turn come f... | https://mathoverflow.net/users/683 | Functional equations of zeta functions over global fields | Have a look at Bump Automorphic representations. The adelic picture is given there as does Tate's Thesis. I think Bump works in the global field context there. The main difference is that the adelic norm map has discrete image in the function field case, but that doesn't harm much.
Although Tate's proof works as well... | 2 | https://mathoverflow.net/users/10400 | 122448 | 68,784 |
https://mathoverflow.net/questions/122429 | 10 | I'm looking for a reference for a certain pair of monads on $Cat$. One problem is that I don't know the modern way of thinking about some basic things, so excuse me if my presentation is naive.
First some notation. Let $C$ and $D$ be small categories and let $F\colon C\to Cat\;$ and $G\colon D\to Cat\;$ be functors. ... | https://mathoverflow.net/users/2811 | Reference for my monads? | What you are describing is an example of Max Kelly's notion of **club**, closely connected with the concept of operad. The original references date back to the 70's; one reference is
* G.M.Kelly. On clubs and doctrines. In Category Seminar, Sydney 1972/1973. Springer LNM 420, pp. 181-257 (1974).
Actually, a club i... | 9 | https://mathoverflow.net/users/2926 | 122451 | 68,785 |
https://mathoverflow.net/questions/122466 | 8 | An open 3-manifold is *tame* if it is homeomorphic to the interior of a compact manifold. Is there a (known) example of an open 3-manifold that is not tame, has finitely generated fundamental group and universal cover homeomorphic to $\mathbb R^3$?
| https://mathoverflow.net/users/1573 | Non-tame 3-manifolds covered by the Euclidean space | Yes, see the following paper of Freedman and Gabai for lots of examples:
Freedman and Gabai, Covering a nontaming knot by the unlink. Algebr. Geom. Topol. 7 (2007), 1561–1578.
| 10 | https://mathoverflow.net/users/1335 | 122469 | 68,789 |
https://mathoverflow.net/questions/122468 | 5 | Gowers' dichotomy theorem asserts that every Banach space either contains an HI subspace or a subspace having an unconditional basis. There are examples of HI spaces without quotients having unconditional bases (was Argyros the first who proved that?). This strange phenomenon tempts me to ask
whether every reflexive... | https://mathoverflow.net/users/31622 | Quotients with unconditional bases | Every quotient of the original Gowers-Maurey space is HI; Ferenczi proved this. Argyros-Felouzis produced examples of reflexive spaces that are not HI (e.g. contain some $\ell\_p$) and have HI duals. Any of these spaces give an example of a space such that no quotient has an unconditional basis. I think it's safe to sa... | 5 | https://mathoverflow.net/users/15388 | 122479 | 68,793 |
https://mathoverflow.net/questions/122412 | 1 | From a bisimplicial space $T$, one can consider the simplicial spaces $\underline p \mapsto T\_{pp} $, $\underline p \mapsto |\underline q \mapsto T\_{pq} |$, and $\underline q \mapsto |\underline p \mapsto T\_{pq} |$, where $| \cdot |$ denotes geometric realisation. In a lemma (used in proving Theorems A and B, in 'Hi... | https://mathoverflow.net/users/19313 | Simplicial sets from bisimplicial sets, and their realisations. | In the term "canonical presentation", "presentation" just means that the diagram you wrote is a coequalizer diagram *presenting* $T$ as being obtained from "simpler" bisimplicial spaces (so, yes, "cokernel of a pair" means "coequalizer"). This use of the term presentation comes from algebra, where presenting, say, a gr... | 3 | https://mathoverflow.net/users/644 | 122482 | 68,795 |
https://mathoverflow.net/questions/122231 | 1 | I am looking for a closed-form formula for the following sum:
$\displaystyle \sum\_{k=0}^{N}{\frac{\sin^{2}(\frac{k\pi}{N})}{a \cdot \sin^{2}(\frac{k\pi}{N})+1}}=\sum\_{k=0}^{N}{\frac{1}{a+\csc^{2}(\frac{k\pi}{N})}}$.
Is such a formula known?
| https://mathoverflow.net/users/22051 | Trigonometric identity needed for sums involving secants | Two other references to similar sums are
Bruce C. Berndt and Boon Pin Yeap, [Explicit evaluations and reciprocity theorems for finite trigonometric sums](http://www.math.uiuc.edu/~berndt/articles/trigsums.pdf), Advances in Applied Mathematics
Volume 29, Issue 3, October 2002, Pages 358--385
and
Ira Gessel, [Gen... | 5 | https://mathoverflow.net/users/10744 | 122484 | 68,797 |
https://mathoverflow.net/questions/122438 | 6 | Let M be a compact Riemannian surface of genus$\geq 2$.
Can M have a globally defined Killing field ?
Can M have a Killing field defined on M-(finite set of points)?
| https://mathoverflow.net/users/31610 | Compact surface with genus$\geq 2$ with Killing field | Of course, there's always the Killing field $X\equiv0$. :)
Seriously, here's a different proof and an argument that addresses the 'suppose one leaves out a finite number of points' question:
Taking the orientation double cover if necessary, we can assume that $M$ is connected and orientable with $g\ge 2$. The hypot... | 6 | https://mathoverflow.net/users/13972 | 122485 | 68,798 |
https://mathoverflow.net/questions/122476 | 4 | In Calculus we teach that if the $a\_n$ are positive and decreasing with limit equal to zero, then the alternating series $\sum\_n (-1)^na\_n$ converges. One can in general not leave out the assumption that the $a\_n$ (eventually) decrease, as the example $a\_{2n}:=1/n$ and $a\_{2n+1}:=1/2^n$ shows. However, most examp... | https://mathoverflow.net/users/22873 | Alternating series test for non-decreasing terms | $$f(x) = \frac{2(\cos \pi x/2)^2}{x} + \frac{(\sin \pi x/2)^2}{2^{(x-1)/2}}$$ is certainly a function in the class you describe, but $f(2n) = 1/n$ and $f(2n+1)=1/2^n$, which breaks the alternating series test as you've remarked.
| 9 | https://mathoverflow.net/users/5091 | 122487 | 68,799 |
https://mathoverflow.net/questions/122462 | 4 | The following is a result of Woodin (the proof is found after Theorem 5 of "Generalizations of the Kunen Inconsistency" by J.D.Hamkins, G.Kirmayer and N.L.Perlmutter):
(Woodin) Let $V[G]$ be a set-forcing extension of $V$. Then there is no non-trivial elementary embedding $$j:V[G]\prec V.$$
Corollary 6 gives the the... | https://mathoverflow.net/users/5697 | Set forcing and ultrapowers | The paper you refer to is available here: [Generalizations of the
Kunen inconsistency](http://jdh.hamkins.org/generalizationsofkuneninconsistency/).
In your question, it seems that things got switched around a
little when you went from the theorem to the corollary, since in
the paper Corollary 6 asserts that if $j:V\... | 4 | https://mathoverflow.net/users/1946 | 122490 | 68,802 |
https://mathoverflow.net/questions/122504 | 4 | Suppose that $X$ is a Banach space (or more generally, Frechet space) such that $X$ is the closure of the span of a compact (*in the original topology*) subset $K$. Do we know anything "nice" about $X$, from this information alone?
If $X$ is the span of $K$, without needing take a closure, then a Baire category argum... | https://mathoverflow.net/users/16107 | Compactly generated Banach spaces | I think not. Suppose X has a countable subset S whose span is dense in X. Enumerate S as a sequence, then by rescaling you can assume the sequence converges to zero in norm. This gives a countable relatively compact set whose linear span is dense in X.
| 12 | https://mathoverflow.net/users/763 | 122506 | 68,812 |
https://mathoverflow.net/questions/122473 | 2 | There have been questions lately about almost equitable partitions in graphs, for example [this one](https://mathoverflow.net/questions/79841/eigenvectors-and-partitions-of-graphs) which provides the definition.) Every equitable partition is almost equitable. The converse is true for regular graphs but not in general. ... | https://mathoverflow.net/users/8008 | What are some interesting almost equitable partitions which are not equitable? | Well, my favourite example: There has been some research on the Laplacians on infinite graphs lately, say [here](http://arxiv.org/abs/1201.0118) and [here](http://arxiv.org/abs/1105.0395). The authors deduce a few decompositions for the Laplacians on (usually infinite) trees and "perturbed trees" under certain symmetry... | 1 | https://mathoverflow.net/users/26039 | 122508 | 68,813 |
https://mathoverflow.net/questions/122509 | 5 | I have two questions on complex geometry.
First one is that why the existence of almost complex structure on tangent bundle on real 2n-dimensional manifold is a topological question?
Wikipedia describes it as a topological question. I think that mean there is some homology or cohomology group associated to topologi... | https://mathoverflow.net/users/29334 | Two questions on complex geometry | (1) Why is the existence of an almost-complex structure a topological question? Suppose $M$ is a $2n$-manifold. The tangent bundle is classified by some map $M\to BGL\_{2n}(\mathbb{R})$; $M$ admits an almost-complex structure if and only if this map admits a lift to $BGL\_{n}(\mathbb{C})$ (that is, an almost-complex st... | 11 | https://mathoverflow.net/users/6950 | 122514 | 68,815 |
https://mathoverflow.net/questions/93772 | 10 | Consider a set $A = \{ a\_1,a\_2,\ldots, a\_n \} $ of vectors in $\mathbb{R}^d$, which lie in a common affine hyperplane. Two convex polytopes may be obtained from $A$, namely the convex hull of the vectors in $A$, $conv(A)$ and the zonotope generated by vectors in $A$, $Z(A)$.
Both polytopes can be viewed as the proje... | https://mathoverflow.net/users/15054 | Triangulations of polytopes and tilings of zonotopes | Triangulations of polytopes are "more fundamental" than cubical tilings of zonotopes. By the [Cayley trick](http://www.ams.org/mathscinet-getitem?mr=1763304), every cubical tiling of a zonotope can be seen as a triangulation of the Cayley lifting of the segments defining it. The latter is equal to their Lawrence liftin... | 5 | https://mathoverflow.net/users/15054 | 122515 | 68,816 |
https://mathoverflow.net/questions/122522 | -1 | I have a question in seeing this
$$U(n)=\frac{U(n)}{U(n-1)} \* \frac{U(n-1)}{U(n-2)}\*\cdots \*\frac{U(2)}{U(1)}\*U(1)$$
So, group U(n) is written as product of quotient spaces.
Is quotient space, for example $\frac{U(n)}{U(n-1)}$ , as topological space the same as
quotient gropup i.e. set of cosets?
How to prove... | https://mathoverflow.net/users/31636 | Group or manifold ? | I'm not sure what you mean by the star product but yes, you see a quotient G/H as the set of cosets. There are topological requirements on H though, for instance being closed etc... at least for G/H to be a differentiable manifold, I'm not sure about topological manifold.
Finally, the idea to prove the sphere stateme... | 0 | https://mathoverflow.net/users/19545 | 122524 | 68,822 |
https://mathoverflow.net/questions/121892 | 1 | Are f.g. projective modules (of constant rank) free over the ring $A$ which is the total quotient ring of a reduced non-Noetherian commutative ring. Note that dimension of $A$ need not be $0$.
| https://mathoverflow.net/users/31461 | Are f.g. projective modules free over total quotient ring of a reduced non-noetherian commutative ring | I've posted an answer [here](https://math.stackexchange.com/questions/296109/are-finitely-generated-projective-modules-free-over-the-total-ring-of-fractions).
| 2 | https://mathoverflow.net/users/23950 | 122526 | 68,823 |
https://mathoverflow.net/questions/122477 | 5 | Let $G$ be a connected, simply-connected, complex, semisimple Lie group with Lie algebra $\frak{g}$. Suppose that $X\in\frak{g}$ is a nilpotent element (i.e. that $ad\_X:\frak{g}\rightarrow\frak{g}$ is a nilpotent endomorphism), and let $C\_G(X)\subseteq G$ denote its stabilizer with respect to the adjoint representati... | https://mathoverflow.net/users/25358 | Stabilizers for nilpotent adjoint orbits of semisimple groups | To supplement what Francois Ziegler [says](https://mathoverflow.net/a/122520), I'd point out that the structure of semisimple complex Lie groups has been developed piecemeal over a century or so. The basic results on nilpotent elements in the Lie algebra are by now fairly old, but refinements continue to be made. The "... | 3 | https://mathoverflow.net/users/4231 | 122529 | 68,825 |
https://mathoverflow.net/questions/114429 | 14 | I apologize in advance if this question is terribly naive. I've just recently learned a bit of rigid analytic geometry with the hopes of understanding some basic facts about eigenvarieties.
In the literature (e.g. in Coleman-Mazur) I've seen people talk about the $\mathbf{C}\_p$-points of a rigid analytic space over... | https://mathoverflow.net/users/4351 | How does one make sense of the $\mathbf{C}_p$-points of a rigid analytic space over $\mathbf{Q}_p$? | I think in Coleman-Mazur, it is just to be taken in the sense of $\mathbb C\_p$-points of $X\_{\mathbb C\_p}$. This is compatible with the more abstract definition you propose. Indeed, as $\mathrm{Sp}(\mathbb C\_p)$ is just one point, it is enough to work with an affinoide space $X$ associated to an affinoide algebra $... | 4 | https://mathoverflow.net/users/3485 | 122531 | 68,826 |
https://mathoverflow.net/questions/122532 | 3 | For any monoid $M$, we can naturally construct a semiring $S$ as follows:
1. Let the additive monoid of $S$ be the free commutative monoid on $M$
2. Let the multiplicative monoid of $S$ be $M$
Then, if you make multiplication distribute over addition, you get a semiring.
This has an extremely simple interpretatio... | https://mathoverflow.net/users/24611 | Semiring naturally associated to any monoid? | It is at least sometimes called a "monoid semiring" by analogy with "[group ring](http://en.wikipedia.org/wiki/Group_ring)". As such it would be notated $S = \mathbb{N\_0}[M]$ (or $\mathbb{N}[M]$ depending how you define things).
By the way, the ring $\mathbb{Z}[M]$ you define in #1 is a commutative ring, but not a f... | 8 | https://mathoverflow.net/users/5963 | 122533 | 68,827 |
https://mathoverflow.net/questions/122536 | 0 | Let us work over the complex numbers for simplicity. Consider a curve $C$ presented as a cyclic cover of some lower genus curve $C'$. When $C'$ has genus $0$, we can write $C$ as the normalization of the projective curve given by
$$
y^n= \Pi (x- \alpha\_i)
$$
where $\alpha\_i \in \mathbb{C}$ are the branch points.
Qu... | https://mathoverflow.net/users/23434 | When is a cyclic cover hyperelliptic? | Consider an automorphism of a hyperelliptic curve. It is sufficient to ask what the quotient by this automorphism can be. Because there is a unique hyperelliptic projection to $\mathbb P^1$, it is canonical, hence the automorphism factors through to an automorphism of $\mathbb P^1$. As a finite order automorphism of $\... | 4 | https://mathoverflow.net/users/18060 | 122538 | 68,828 |
https://mathoverflow.net/questions/122537 | 11 | I am studying orbifolds with view towards Chen-Ruan cohomology. I have been struggling with inertia orbifolds but have no intuition about them at this point. I would appreciate your motivating me by answering the following questions:
1. What is the intuition behind inertia orbifolds? How should one think of them?
2. ... | https://mathoverflow.net/users/31553 | What is the intuition behind the inertia orbifold (or stack)? | I am not sure what kind of answer you are looking for. But if you have a stack $X$, then the inertia stack $IX$ is basically the gadget parametrizing pairs $(x,\sigma)$ where $x$ is a point of $X$ and $\sigma$ is in the isotropy group at $x$ (an automorphism of $x$). Informally, the locus where you have automorphism gr... | 24 | https://mathoverflow.net/users/1310 | 122543 | 68,831 |
https://mathoverflow.net/questions/122547 | 1 | Let $\imath :T^{n}\rightarrow X$ is a special Lagrangian n-Torus so that $\imath(T^{n})=L$ and all small special Lagrangian deformations of $L$ are flat then why $L$ has Tubular neighbourhood which is fibred by special Lagrangian Tori?
| https://mathoverflow.net/users/nan | special Lagrangian n-Torus has Tubular neighbourhood? | NB: As the OP pointed out, the first version of my answer was incomplete, since it didn't address the 'tubular neighborhood' part of the claim. Here's a better (but still not complete) version of an answer
This is a *partly* consequence of Maclean's theorem. Since the $n$-torus itself is flat, a basis of the harmonic... | 6 | https://mathoverflow.net/users/13972 | 122549 | 68,834 |
https://mathoverflow.net/questions/122552 | 6 | Does anybody know a proof (or a reference to proof) of the following combinatorial question: Let $a,b,c,\ldots=1,\ldots,n$ enumerate elements of a set of size $n$, e.g. vertices of an $n-1$ symplex. Let $C\_{ab}$ be a cocycle, i.e. an object satisfying $C\_{ab}=-C\_{ba}$ and $$C\_{ab}+C\_{bc}+C\_{ca}=0.$$
This can be ... | https://mathoverflow.net/users/31641 | A simple (?) combinatorial question | Choose an arbitrary number $D\_n$ and define $D\_i = C\_{in}+D\_n$ for $i \neq n$. So $C\_{ij} = D\_i - D\_j$.
The identity we want to prove is
$$\sum\_{a=1}^n (-1)^{a+1} \prod\_{\begin{matrix} 1 \leq b < c \leq n \\ b,c \neq a\end{matrix}} (D\_b-D\_c) = 0.$$
The product is the Vandermonde determinant $\det (D\_j^k... | 13 | https://mathoverflow.net/users/297 | 122555 | 68,836 |
https://mathoverflow.net/questions/121926 | 1 | For an uncountable collection of uncountable sets of real-valued random variables (i.e. measurable with respect to a $\sigma$-algebra) $\{S\_i\}\_{i\in I}$, with
$\inf \left(\bigcap\_{i\in I}S\_i \right)= \sup\{\inf S\_i|{i\in I}\}$
I want to to show
$\mathrm{ess}\inf \left(\bigcap\_{i\in I}S\_i \right)= \mathrm{... | https://mathoverflow.net/users/11011 | From point-wise to essential supremum of a set of real-valued measurable functions | Okay. Give $[0,1]$ Lebesgue measure. For each $t \in [0,1]$ let $S\_t = \{1\_{\{t\}}\} \cup \{a\cdot 1\_{[0,1]}: a \geq 1\}$.
Then $\bigcap S\_t = \{a\cdot 1\_{[0,1]}: a \geq 1\}$ and its inf is the function $1\_{[0,1]}$. For each $t$ the inf of $S\_t$ is the function $1\_{\{t\}}$, and the sup of these is also the fu... | 2 | https://mathoverflow.net/users/23141 | 122561 | 68,838 |
https://mathoverflow.net/questions/122562 | 0 | Suppose I have a (2,4) complete intersection 3-fold $X\subset\mathbb P\_{\mathbb C}^5$ with 118 nodal ($A\_1$) singularities (if it simplifies things then assume it's the intersection of the Grassmanian $G(2,4)$ with a quartic). It is known (for example see Theorem 4.3 and Corollary 4.4 in Dimca's "Singularities and To... | https://mathoverflow.net/users/13139 | Effect of crepant resolution on the torsion in homology of a complete intersection 3-fold | If I am not mistaken, the mapping $\tilde X\to X$ blows down a disjoint union of 2-dimensional spheres, so it follows from the homology sequence of pair that it induces isomorphisms of both $H\_1$'s and $H\_4$'s (with integer coefficients).
| 2 | https://mathoverflow.net/users/29992 | 122564 | 68,840 |
https://mathoverflow.net/questions/122557 | 20 | I would like to know if there are compact (n-1)-manifolds $N$ that are not spheres but such that there is a manifold with boundary $M$ which satisfies the following two properties:
* $\partial M\cong N$
* $M-\partial M\cong \mathbb{R}^n$
I am primarily interested in this question in the category of smooth manifolds... | https://mathoverflow.net/users/10707 | What manifolds are boundaries of euclidian spaces ? | $N$ has to be a homotopy-sphere. So as long as it's dimension isn't $4$, there's a proof that it has to be the standard $S^{n-1}$.
These arguments appear in the Kosinski book on smooth manifolds. The basic idea goes like this.
1) $N$ is simply connected. There's the inclusion map $N \to M$, but if $p \in int(M)$ ... | 22 | https://mathoverflow.net/users/1465 | 122568 | 68,842 |
https://mathoverflow.net/questions/122566 | 3 | Hi
Let $n$ be an odd natural number (sufficiently large), and $ 1\leq t,k < n $. Let $A= \{ a\_{1},a\_{2},\ldots,a\_{t} \} $, $B= \{ b\_{1},b\_ {2},\ldots,b\_{n-t} \} $, $C= \{ c\_{1},c\_{2} \ldots,c\_{k} \} $ and $D= \{ d\_{1},d\_{2},\ldots,d\_{n-k} \} $ be subsets of $ \{ 1,2,\ldots,n \} $ such that $A\cup B=C\cup ... | https://mathoverflow.net/users/8725 | Generators of symmetric group | Counterexample: Everything in $\langle(123)(456789),(123456)(789)\rangle$ acts cyclically on the set partition $\lbrace1,4,7\rbrace \lbrace2,5,8\rbrace \lbrace 3, 6, 9 \rbrace$. This type of counterexample works for any odd composite $n$.
Thanks to Michael Zieve for pointing out that the following is incorrect:
~~... | 6 | https://mathoverflow.net/users/2954 | 122574 | 68,847 |
https://mathoverflow.net/questions/122596 | 9 | A Tannakian category is defined to be a rigid monoidal abelian category over a field with an exact tensor functor to the category of vector spaces. (It is always equal to the category of representations of some pro-algebraic group.)
Is there such a thing as a rigid monoidal abelian category over a field without an ex... | https://mathoverflow.net/users/18060 | Rigid monoidal abelian category without an exact tensor functor to Vect | Let $C$ be a [fusion category](http://ncatlab.org/nlab/show/fusion+category) with simple objects $X\_i$. If $X$ is an object, then $X \otimes X\_i = \bigoplus N\_{ij} X\_j$ for some matrix $N\_{ij}$ of non-negative integers. The largest non-negative real eigenvalue of this matrix is called the [Frobenius-Perron dimensi... | 10 | https://mathoverflow.net/users/290 | 122598 | 68,856 |
https://mathoverflow.net/questions/86230 | 12 | Given a field $k$ of characteristic $0$. For every $k$-bialgebra $A$, let $\mathrm{Prim} A$ denote the $k$-vector subspace of $A$ consisting of all primitive elements of $A$.
What conditions can we put on two $k$-bialgebras $A$ and $B$ to ensure that $\mathrm{Prim}\left(A\otimes B\right) = k\otimes \left(\mathrm{Prim... | https://mathoverflow.net/users/2530 | Primitive elements of a tensor product of bialgebras | Well first when we restrict to the case when $A, B$ are filtered (see Bourbaki for example), in this case $\log\_{\*}$ always converges at $Id$ (as $Id=e+I\_+$, $e$ being the unit for the convolution, it suffices to remark that $I\_+^{\*N}(h)=0$ for $N=N(h)$ large enough).
Now, in the general case, you can adapt the... | 10 | https://mathoverflow.net/users/25256 | 122606 | 68,861 |
https://mathoverflow.net/questions/122581 | 2 | if f is in $H^{1}$ the Hardy space with integrable norm, and if f is not identically zero, then the zeroes of f (certainly countable in number) satisfy the Blaschke condition.Can any "Blaschke condition" be defined if hardy space is considered on the upper-half plane instead of unit disk. Curious to know if this carrie... | https://mathoverflow.net/users/30081 | Blaschke condition on upper half plane | Blaschke condition in the upper half-plane is $$\sum\left|\Im\frac{1}{z\_k}\right|<\infty.$$
For the proof follow the advice given above, that is prove it yourself, or look in a book.
In addition to the books mentioned above, I recommend Koosis Hardy spaces, or Levin,
Distribution of values of entire functions, or de B... | 3 | https://mathoverflow.net/users/25510 | 122611 | 68,862 |
https://mathoverflow.net/questions/122610 | 8 | I have $A^{1/2} B A^{1/2} \preceq I$ for two PSD matrices $A$ and $B$, and I'd like to know if that implies $\|AB\|\_2 \leq 1.$
The argument I was using to show this is that for any two square matrices $A$ and $B,$ it is always the case that $\|AB\|\_2 = \|BA\|\_2.$ I thought I read that this equality does hold in a... | https://mathoverflow.net/users/2586 | When is spectral norm of AB equal to that of BA? | I suppose that your norm is either the Schur-Frobenius $\|F\|=({\rm Tr}FF^T)^{1/2}$ or that subordinated to the Euclidan norm $\|F\|=\rho(FF^T)$. In either case, the equality $\|AB\|=\|BA\|$ follows from the equality of the spectra of $ABBA$ and $BAAB$ (mind that $A$ and $B$ are symmetric). This is true because both ma... | 14 | https://mathoverflow.net/users/8799 | 122612 | 68,863 |
https://mathoverflow.net/questions/121353 | 3 | It is known, that maximal matchings (i.e. matchings with the maximal number of edges) and optimal matchings (i.e. matchings for which the sum of edge weights is optimal) can be calculated in polynomial time.
I would like to know whether the same is true for optimal matchings with a given fixed number of edges.
The ... | https://mathoverflow.net/users/31310 | complexity of finding optimal matchings of given fixed size | With a little trick you can use the algorithm for finding an optimal matching to solve this.
Let the graph $G$ have $n$ vertices and you want to find the optimal matching with size $k$. Lets suppose that there is a matching of size $k$.
The first remark is that if you add the same constant to all edge weights then ... | 3 | https://mathoverflow.net/users/24634 | 122621 | 68,867 |
https://mathoverflow.net/questions/122619 | 1 | Getting nowhere with maple using its triangularize and groebner decompositions for even moderate size systems with any symbolic factors. Any suggestions on how better to approach this would be appreciated. To put a shape on this query I append a sample problem case below, u,v,w,x,y,z are variables, a,b,c,d constants.
... | https://mathoverflow.net/users/31664 | Reducing system of polynomials with symbolic factors | **Added more details:** From specializing $a,b,c,d$ to random integers it seems that your system is $0$-dimensional, and $z$ is the root of a degree $16$ polynomial $f\_{a,b,c,d}(z)$. You might compute these degree $16$ polynomials for many integer tuples $a,b,c,d$, and try to guess/interpolate the dependencies of the ... | 4 | https://mathoverflow.net/users/18739 | 122625 | 68,868 |
https://mathoverflow.net/questions/122632 | 1 | I want to implement the algorithm of graph partitioning of sparse directed graph. In this algorithm after computing the transition matrix ,we should compute the stationary distribution of the random walk. I don't know how to compute the stationary distribution of random walk in such sparse directed graph.Is there any s... | https://mathoverflow.net/users/38345 | Stationary distribution for directed graph |
>
> In this algorithm after computing the transition matrix,
> we should compute the stationary distribution
>
>
>
The stationary distribution is proportional to one of the left or right eigenvectors of the transition matrix, and the corresponding eigenvalue is 1. You can compute this using matlab or other soft... | 2 | https://mathoverflow.net/users/27564 | 122638 | 68,874 |
https://mathoverflow.net/questions/122643 | 7 | The classifying space $B\mathcal{C}$ of a small category $\mathcal{C}$ is by definition the geometric realization of the nerve of $\mathcal{C}$. Now let $\mathcal{C}\_1$ and $\mathcal{C}\_2$ be two small categories. Any functor $F : \mathcal{C}\_1 \rightarrow \mathcal{C}\_2 $ induces a continuous map $BF: B\mathcal{C}\... | https://mathoverflow.net/users/31670 | realization of maps between classifying spaces of categories | Here's a counterexample. Let $M$ be a discrete monoid which is not a group. Consider the
map
$$
M = \hom (\Bbb N, M) \to \text{maps}\_{\*}(B\Bbb N , BM) = \Omega BM
$$
($\Omega BM =$ the based loops of $BM$ which is the same thing as
Segal's group completion of $M$).
This map is not a $\pi\_0$ surjection since $M$ is... | 4 | https://mathoverflow.net/users/8032 | 122651 | 68,880 |
https://mathoverflow.net/questions/122601 | 3 | Let $X \in R^n$ be a random vector such that
$$P(|X\_i| > \epsilon) > e^{-\epsilon^2}$$
What is a tight bound on
$$P(\sum\_{i=1}^n |X\_i| > \epsilon)$$
and on
$$P(\max\_{1\le i\le n} |X\_i| > \epsilon)?$$
The $X\_i$ can be arbitrarily dependent. The best I can get for both bounds is $n e^{-\epsilon^2}$ (u... | https://mathoverflow.net/users/9118 | concentration inequality for averages of dependent random variables | Without further assumptions you can't do better than the union bound (which should be $n e^{-\epsilon^2}$ as you've written things). If $X\_i$ are identically distributed and the events $(|X\_i| > \epsilon\_0)$ are disjoint then you get equality in the union bound for the maximum whenever $\epsilon \ge \epsilon\_0$. If... | 8 | https://mathoverflow.net/users/1044 | 122652 | 68,881 |
https://mathoverflow.net/questions/122658 | 5 | In several set theories, among which Quine's NF is one of the best known and most extensively
investigated, the existence of a Universal set V can be proved. V is the set of all sets-indeed
all "proper classes", including V itself, are elements of V. Although it might be considered
"meaningless" to ask what the "cardin... | https://mathoverflow.net/users/4423 | A question about Universal sets. | I believe that not much is known about NF itself and large cardinals - keep in mind that NF disproves the axiom of choice (Specker, 1953), and many large cardinal notions are not robustly defined in the absence of choice, in the sense that definitions equivalent over ZFC become non-equivalent without AC.
The system N... | 7 | https://mathoverflow.net/users/8133 | 122659 | 68,883 |
https://mathoverflow.net/questions/122656 | 4 | I read somewhere that Maltcev proved that for any finitely generated torsion free Nilpotent group $G$ there are canonical generators, i.e.
$g\_{1},\ldots,g\_{k}$ such that any $g \in G$ can be written uniquely as:
$g\_{1}^{a\_{1}}g\_{2}^{a\_{2}} \cdots g\_{k}^{a\_{k}}$, where the $a$-s are in $Z$.
I could not find the ... | https://mathoverflow.net/users/31679 | On Canonical generators of torsion free nilpotent group | An alternative latinization of the name is Malcev (or Mal'cev). It is less fashionable now, but most of the literature uses that spelling.
1. No. The Heisenberg group $[a,b]=c$, $c$ central is a 3-dimensional algebraic group with has canonical expression $a^nb^mc^p$, but quartic growth. Since $b^na^n=a^nb^nc^{-n^2}$,... | 3 | https://mathoverflow.net/users/4639 | 122669 | 68,887 |
https://mathoverflow.net/questions/122588 | 5 | Let $Card$ be the class of infinite cardinals and $p\colon Card^2\to Card$ be given by $(\kappa,\lambda)\mapsto\kappa^\lambda$.
Assuming GCH it is known that $p(\kappa,\lambda)$ is either $\kappa$ (if $\lambda < cof(\kappa)$) or $\kappa^+$ (if $cof(\kappa)≤\lambda ≤\kappa$) or $\lambda^+$ (if $\kappa < \lambda$).
... | https://mathoverflow.net/users/30363 | do behavior of gimel or GCH determine all infinte products of cardinals? | I guess this is the final result:
For $(\kappa(i)\mid i<\delta)$ nondecreasing sequence of infinite cardinals we have for $\delta=\lambda\_0\alpha\_0+\ldots+\lambda\_l\alpha\_l$ (every ordinal can be written this way) and $\delta\_k:=\lambda\_0\alpha\_0+\ldots+\lambda\_{k}\alpha\_{k}$ with cardinals $\lambda\_0>\ldo... | 1 | https://mathoverflow.net/users/30363 | 122676 | 68,893 |
https://mathoverflow.net/questions/122672 | 2 | How do one prove (or disprove) that $\Omega S^{2}$ and $\Omega(S^{2} \vee S^{2})$ are non commutative Hopf spaces?
I thought this is a question for math.stackexchange, but not many people even viewed that question. Seems like it should be easy but right now I have no idea how to do it. Any hint?
Definition: If $X$... | https://mathoverflow.net/users/19186 | Non-commutativity of certain Hopf spaces | Let us run the Serre SS for homology with $\mathbb{Z}/2$ coefficients for the fibration
$\Omega(S^{2} \vee S^{2}) \to \* \to S^{2} \vee S^{2}$
Let $x,y \in H\_{2}(S^{2} \vee S^{2})$ be the generators. Since these guys has to die, there exists $x\_{1}, y\_{1} \in H\_{1}(\Omega(S^{2} \vee S^{2}))$ which are hit by $... | 0 | https://mathoverflow.net/users/19186 | 122679 | 68,895 |
https://mathoverflow.net/questions/122548 | 4 | There is a standard notation $\mathrm{ZF}[n]$ for Zermelo Fraenkel set theory with the power set axiom restricted to saying the set of natural numbers has $n$ successive power sets $\beth\_0\dots\beth\_n$.
Is there a similarly standard notation for the extension of $\mathrm{ZF}[n]$ by an axiom saying every set has a... | https://mathoverflow.net/users/38783 | Notation for upperbound power sets. | In modern set theory $H(\kappa)$ denotes the collection of sets $X$ such that $X$ is hereditarily of cardinality strictly less than $\kappa$.
>
>
> >
> > So, the statement you are asking for can be conveniently expressed as $V=H(\kappa)$, where $\kappa=\beth\_{n}^+$.
> >
> >
> >
>
>
>
| 3 | https://mathoverflow.net/users/9269 | 122685 | 68,898 |
https://mathoverflow.net/questions/122629 | 9 | Consider a function whose Fourier transform is supported on a half-ray:
$$
A(t)=\int\_0^\infty \omega(E) e^{-iEt}d E,
$$
where I can suppose $\omega(E)\geq 0$ and any suitable regularity conditions on $\omega(E)$ in the limit $E\rightarrow\infty$. I am interested in results constraining the rate of decay of $A(t)$ in t... | https://mathoverflow.net/users/20729 | The Paley-Wiener theorem and exponential decay. | The idea behind this theorem is actually simple.
For the Fourier transform to make sense (with usual understanding of the integral) the function
$\omega$ must be summable, that is in $L^1(0,+\infty)$. This immediately implies that the Fourier integral converges not only on the real line but also in the lower half-pl... | 18 | https://mathoverflow.net/users/25510 | 122695 | 68,904 |
https://mathoverflow.net/questions/122687 | 0 | Is there a dual concept for the p-primary component of an abelian group? Please name some books/papers where it is studied.
| https://mathoverflow.net/users/20947 | Dual concept for the p-primary component | EDIT: Note that in what follows I interpret "dual" as "dual in the sense of Pontryagin-Van Kampen's duality". If you had something different in mind I think you should have specified...
Well, first of all you should start finding a dual notion for the torsion part. Indeed, a given Abelian discrete Abelian group $G$ i... | 1 | https://mathoverflow.net/users/24891 | 122712 | 68,907 |
https://mathoverflow.net/questions/122703 | 5 | I have a question on "Torelli theorems" in algebraic geometry. "Torelli theorems" say about how the period map of a given family of varieties behave. If I understand correctly, global Torelli theorems are the most strong ones and basically classify the varieties you have, that is, the period map is isomorphism. Local T... | https://mathoverflow.net/users/31553 | What do generic Torelli theorems claim? | I would formulate it as follows: global Torelli says that the period map is injective/an immersion (but not necessarily an isomorphism!!), local Torelli says that the period map has injective differential, and generic Torelli says that the period map is generically injective (injective on a Zariski open dense set).
| 4 | https://mathoverflow.net/users/1310 | 122715 | 68,910 |
https://mathoverflow.net/questions/122689 | 9 | This is a question about pathologies.
Let $X/\mathbb{C}$ be an irreducible projective variety smooth over $\mathbb{C}$. Then, the singular cohomology groups $H^i(X, \mathbb{C})$ have a hodge decompositon, and hodge theory tells us that the betti numbers (and hodge numbers) are not completely random. Eg, the odd betti... | https://mathoverflow.net/users/25854 | Betti numbers of Proper nonprojective varieties | I think, that many basic restrictions, that you have for complex projective varieties still hold for proper smooth complex varieties. Let me show that $b\_2>0$, and $b\_{2n-2}>0$.
Suppose that $X^{2n}$ is a proper smooth complex variety. Then there is a birational morphism $\phi: Y^{2n}\to X^{2n}$ from a *projective ... | 10 | https://mathoverflow.net/users/943 | 122718 | 68,911 |
https://mathoverflow.net/questions/122704 | 43 | Why do all measure theory textbooks present the concept of push-forward measure, but never the concept of pull-back measure? Doesn't the latter exist?
It's true that the naive treatment of such a concept would sometimes lead to contradictions. For instance, let $p:\mathbb{R}^2\rightarrow\mathbb{R}$ be given by $p(x,y... | https://mathoverflow.net/users/54780 | Pullback measures | To define pullbacks of measures we need some additional data,
because otherwise one would be able
to obtain a canonical measure on an arbitrary measurable space M
by pulling back the canonical measure on the point along the unique map M→pt.
One natural choice for such additional data is
a choice of measure on each fi... | 55 | https://mathoverflow.net/users/402 | 122719 | 68,912 |
https://mathoverflow.net/questions/122721 | 4 | I have a proof in ZFC, using AC and the axiom of foundation, that given any proper class A, every set can be injected into A. I wonder if we could have a proof of this that does not use Foundation.
Gérard Lang
| https://mathoverflow.net/users/30395 | Does ZFC without Foundation prove that given any proper class A, every set can be injected into A ? | No. This principle is known as **The Injection Principle**
See in Jech **Axiom of Choice**, Chapter 9, Problems 3 and 4 both give us a models of ZF+Atoms (so foundation fails) *with choice* in which the injection principle fails.
| 5 | https://mathoverflow.net/users/7206 | 122726 | 68,915 |
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