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/229612 | 23 | Let us agree on the following: a "homology theory" means a functor $h\_\*$ from the category of pointed CW complexes to the category of graded abelian groups, together with natural isomorphisms $h\_{\*+1}(\Sigma X)\cong h\_\*(X)$, such that the functor $h\_\*$ is homotopy invariant, sends a cofiber sequence to an exact... | https://mathoverflow.net/users/6249 | A homology theory which satisfies Milnor's additivity axiom but not the direct limit axiom? | I think it's a great question because not everybody is aware of this. A consequence of Brown representability theorem (Adams's version, which is highly non-trivial and depends on homotopy groups of spheres being countable) is that any spectrum $X$ fits into an exact triangle
\[\coprod\_{j\in J}Z\_j\longrightarrow \copr... | 14 | https://mathoverflow.net/users/12166 | 229637 | 106,958 |
https://mathoverflow.net/questions/229626 | 5 | Azumaya originally defined an Azumaya algebra (which he called a proper maximally central algebra) to be an algebra A which is a free module of finite rank over its centre Z such that the natural map
$$A\otimes\_Z A^{\mathrm{op}}\to \mathrm{End}\_Z(A)$$
is an isomorphism. More modern definitions (e.g., Knus, Quadratic ... | https://mathoverflow.net/users/85918 | An example of an Azumaya algebra that isn't free over its centre | It suffices to find a vector bundle $E$ on an affine variety $V$ such that the vector bundle $\mathcal{E}nd(E)$ is nontrivial: then this vector bundle gives a non-free, projective Azumaya algebra over $\mathcal{O}(V)$. This will be the case if some Chern class of $\mathcal{E}nd(E)$, say $c\_2$, is nontrivial in the Cho... | 7 | https://mathoverflow.net/users/40297 | 229644 | 106,961 |
https://mathoverflow.net/questions/229568 | 6 | We know that the irreducible representations $R\_i$ of a group $G$ can give rise to a fusion ring: $R\_i\otimes R\_j = \oplus\_k N^{ij}\_k R\_k$.
I wonder if the following statement is true or not:
If $G$ is a simple finite group, then its fusion ring is different from the fusion rings of any other groups.
| https://mathoverflow.net/users/17787 | Uniqueness of the fusion ring for simple finite group | The fusion ring, as a ring with basis, contains the same information as the character table. So your question, phrased in language more familiar to finite group theorists, is:
>
> Is a finite simple group determined by its character table among all finite groups?
>
>
>
Here is how to see the above claim. The ... | 8 | https://mathoverflow.net/users/290 | 229648 | 106,962 |
https://mathoverflow.net/questions/229523 | 7 | This question is somewhat related to a [previous one](https://mathoverflow.net/questions/100981/ultrainfinitism-or-a-step-beyond-the-transfinite), where I asked for new forms of infinite beyond the cardinal hierarchy.
Using forcing techniques, at least the ones I know of, one starts from a ground model and then enla... | https://mathoverflow.net/users/15293 | Taller models of ZFC |
>
> Regarding Question 1:
>
>
>
It is certainly possible (i.e., consistent with ZFC) that there is only one transitive model of ZFC. For example, if there are transitive models $M$ and $N$ of ZFC such that $o(M)<o(N)$ (where $o(M)$ is the ordinal height of $M$), then there are ordinals $\alpha < \beta$ such tha... | 13 | https://mathoverflow.net/users/9269 | 229649 | 106,963 |
https://mathoverflow.net/questions/229646 | 7 | Let $c\_1,\dots, c\_d$ be the first $d$ moments of the standard normal distribution. Does the point $(c\_1,\dots, c\_d)$ lie in the convex hull of the set $\{(t,t^2,\dots,t^d)\colon t\in[-b,b]\}$, for a sufficiently large $b$?
Here is another more involved question: Is there a way we can tell how $b$ scales with $d$... | https://mathoverflow.net/users/82358 | Gaussian and the convex hull of moment curves | Let $P\_d$ be the set of all probability measures $\mu$ on $\mathbb R$ whose first $d$ moments $c\_1,\dots, c\_d$ are the same as those of the standard normal distribution $\gamma$.
By Theorem 3.1 in **[[1](http://www.jstor.org/stable/3689944)]**
(and the sentence following it there), there is an extreme measure $\nu... | 4 | https://mathoverflow.net/users/36721 | 229656 | 106,966 |
https://mathoverflow.net/questions/228767 | 3 | In the study of adjoint orbits in a complex semi-simple lie algebra, there is a well known object known as a "sheet". These are the irreducible components of the union of orbits of the same dimension. I will refer to a nice paper of [De-Graaf-Elashvili](http://arxiv.org/abs/0905.2743) for more concrete definitions and ... | https://mathoverflow.net/users/26208 | Affine analog of the theory of sheets | Levi subalgebras appear in the classification of sheets because they are the centralizers of semisimple elements of the Lie algebra. So given *any* element of ${\mathfrak g}$, we can associate a pair $({\mathfrak l},{\mathcal O}\_{\mathfrak l})$ where ${\mathfrak l}$ is the centralizer of the semisimple part, and ${\ma... | 3 | https://mathoverflow.net/users/26635 | 229661 | 106,968 |
https://mathoverflow.net/questions/229653 | 6 | Dictionary between algebra and geometry is somewhat one of the main concepts in modern mathematics. So commutative $C^\*$ algebras are one-to-one with locally compact Hausdorff spaces.
So it is natural to be curious how one can see properties of the manifold from algebra of functions.
In the realm of algebraic functio... | https://mathoverflow.net/users/10446 | Can $C^*$-algebra of continuous functions on $R^n$ ($S^n$) be characterized alternatively? | Yes, it can be defined as the univeral commutative $C^\*$-algebra with unit, generated by $n+1$ self adjoint elements $x\_1,...,x\_{n+1}$ subject to the relation $x\_1^2+...+x\_{n+1}^2=1$. Here universal mean the following: $A$ with generators $(a\_j)\_j$ and relations $(r\_k)\_k$ is called universal if whenever there ... | 10 | https://mathoverflow.net/users/24078 | 229670 | 106,973 |
https://mathoverflow.net/questions/229667 | 5 | According to the countable generator theorem, every ergodic invertible measure-preserving transformation has a generating partition.
What are the generating partitions of the dyadic odometer ? I don't find the answer in textbooks. Is there a "canonical" one among them ?
| https://mathoverflow.net/users/21339 | What are the generating partitions of the odometer? | Quite a nice one is the two-set partition $A,A^c$, where $A$ is the set of points with an even number of terminal 0's:
$$
A = \bigcup\_{k \geq 0} A\_k
$$
where $A\_k = \bigl\{(x\_1, x\_2, \ldots) \mid x\_1=\ldots=x\_{2k}=0 \text{ and } x\_{2k+1}=1 \bigr\}$.
Or, if you work with the odometer acting on the space $[0,1[$... | 4 | https://mathoverflow.net/users/11054 | 229674 | 106,975 |
https://mathoverflow.net/questions/229658 | 2 | Bourgain and Demeter's proof of the $L^2$ decoupling conjecture decouples $\|f\|\_{L^p}$ into an $L^2$ sum of $\|f\_\theta\|\_{L^p}$, where $\hat f$ is supported on a curved hypersurface $S$, where $\theta$ ranges over $\delta^{1/2} \times \cdots \times \delta^{1/2} \times \delta$-sized slab neighborhoods of $S$, and w... | https://mathoverflow.net/users/37103 | Decoupling in mixed norm spaces | For hypersurfaces there is a decent chance that the methods can be adapted to handle $L^q\_t L^p\_x$ type norms, but not $L^p\_x L^q\_t$ type norms. This is because the parabolic rescalings used in the Bourgain-Demeter argument (see Proposition 4.1 of <http://arxiv.org/pdf/1403.5335.pdf> ) react well with the former ty... | 4 | https://mathoverflow.net/users/766 | 229703 | 106,981 |
https://mathoverflow.net/questions/229696 | 5 | Given a graph $G$, the *total graph of $G$*, denoted $T(G)$, is the graph with vertex set $V(G) \cup E(G)$, where $a$ and $b$ are adjacent in $T(G)$ if and only if they are adjacent or incident in $G$.
>
> Is there any characterization of properties of two graphs $G$ and $H$ such that $T(G)$ is isomorphic to $T(H)... | https://mathoverflow.net/users/82845 | Characterization of non-isomorphic graphs but isomorphic total graphs? | It is not difficult to see that both $G$ and $L(G)$ are disjoint induced subgraphs of the graph $T(G)$. For your question, in the below paper:
[1] M. Behzad and H. Radjavi, The total group of a graph, Proc. Amer. Math. Soc.
19 (1968), 158-163. MR 36 #1358.
It is proved that, the graph $G$ is isomorphic to the graph... | 9 | https://mathoverflow.net/users/19885 | 229722 | 106,985 |
https://mathoverflow.net/questions/229600 | 8 | This is a question related to the statistical model behind independent component analysis (ICA).
We assume that $Z \sim N(0,1)$. Our goal is to construct a random variable $X$ that follows a Gaussian mixture distribution and satisfies
$$
E[X^j] = E[Z^j]
$$
for $j \leq k-1$, while
$$
E[X^k] - E[Z^k] = \epsilon.
$$... | https://mathoverflow.net/users/82358 | Moment matching: construction of a mixture of Gaussian distribution with lower moments identical to Gaussian | If non-zero means are allowed, the answer is yes (and it is based on quite different considerations, and so, is presented separately from the previous answer).
Indeed, let $Z$ be a standard normal random variable (r.v.).
Let $P\_k$ denote the set of all probability measures $\nu$ on $\mathbb R$ such that $\int\_{\mat... | 5 | https://mathoverflow.net/users/36721 | 229723 | 106,986 |
https://mathoverflow.net/questions/229726 | 0 | $\newcommand{\mbf}{\mathbf}$
Hi all,
I've been thinking about the following question for a while now, and got a little stuck trying to solve it. Hopefully, someone here might be able to help.
For starters, let $K$ be some field, and led $G$ be the group of $K$-points of the group $\mbf{G}=\mathrm{Sp}\_{2n}$, for so... | https://mathoverflow.net/users/14443 | Do tori in a symplectic group always have invariant maximal isotropic subspaces? | When $K$ is finite or $p$-adic the answer to your question is negative. Indeed there exist maximal tori which are anisotropic. Those tori are not included in any proper parabolic subgroup, so they cannnot stabilize a non trivial totally isotropic subspace.
| 6 | https://mathoverflow.net/users/4767 | 229727 | 106,988 |
https://mathoverflow.net/questions/229708 | 5 | Let $X$ be a set and let $\cal A$ be a non-empty subset of $P(X)$ with the property that whenever $A\_1 \subseteq A\_2 \subseteq \cdots $ is an increasing chain of elements of $\cal A$ then $\cup\_i A\_i \in \cal A$. Let
$f : \cal A\longrightarrow \mathbb{R}$ be increasing and bounded from above. Is it true that there... | https://mathoverflow.net/users/85969 | A weak kind of fixed point | If not, then we can find an increasing and continuous sequence $(A\_\alpha: \alpha < \omega\_1)$ of elements of $\mathcal{A}$ such that $\alpha < \beta \implies f(A\_\alpha) < f(A\_\beta)$. This is a contradiction.
---
**Edit**. Let me add more details. By contradiction, the following holds:
**$(\*)$**: For any... | 6 | https://mathoverflow.net/users/11115 | 229741 | 106,990 |
https://mathoverflow.net/questions/221353 | 5 | Are there some references about the center of quantum affine algebras? I searched on google and only find [the paper](http://arxiv.org/pdf/q-alg/9412007v1.pdf). In particular, what is the center of $U\_q(\widehat{\mathfrak{sl}\_2})$. Thank you very much.
| https://mathoverflow.net/users/11877 | Center of quantum affine algebras | For affine (quantum or not) algebras the center appears only for special value of element "c" - the so-called "critical level".
The second order central element related to "Sugavara construction" is well-known for a long time.
Somewhat nice explicit formulas for the center of affine (not quantum) algebras for type A\_... | 2 | https://mathoverflow.net/users/10446 | 229750 | 106,994 |
https://mathoverflow.net/questions/229744 | -3 | Let $G$ be a group with the property $G=G\_e\dot{\cup} G\_o$ with
$G\_oG\_o\subseteq G\_e\leq G$.
($\dot{\cup}$ denotes disjoint union, $\leq$ is subgroup notation, and $G\_o^{-1}=\{x^{-1}: x\in G\_o\}$)
We observe that
(1) $G\_o^{-1}=G\_o$, $G\_oG\_o=G\_e$, $G\_oG\_e=G\_eG\_o=G\_o$;
(2) $|G:G\_e|=2$, $|G|=2|... | https://mathoverflow.net/users/40520 | Even-odd partitioned groups! | Group $G$ has subsets $G\_e$ and $G\_o$ as mentioned above, iff $G$ has a normal subgroup of index 2.
If $G\_e$ is a normal subgroup of index 2, then $G\_e$ and $G\_o=xG\_e$, $x\notin G\_e$, satisfied the condition of the problem.
Now, suppose such subsets $G\_e$ and $G\_o$ exist. Since $G\_e$ is a subgroup, and $G... | 3 | https://mathoverflow.net/users/84700 | 229752 | 106,996 |
https://mathoverflow.net/questions/227774 | 3 | The [Hadamard circulant conjecture](https://mathoverflow.net/questions/55524/status-of-the-hadamard-circulant-conjecture) states that there do not exist circulant Hadamard matrices with more than $4$ columns.
An $n$ by $n$ Hadamard matrix where the entries are chosen from $\{-1,1\}$ has determinant $\pm n^{n/2}$.
... | https://mathoverflow.net/users/45564 | Bounds for maximum determinant of circulant matrices | I have not seen this question discussed in the literature either, but I will propose an answer here.
To begin with, I think that since $n^{n/2}$ is the maximum possible absolute value that the determinant of *any* $\{-1,1\}$-matrix can have (Hadamard upper bound), the constant $c$ in the question (if it exists) must ... | 1 | https://mathoverflow.net/users/84637 | 229754 | 106,998 |
https://mathoverflow.net/questions/229384 | 20 | I am interested in a problem closely related to a problem stated by Jan Mycielski in his paper *Can One Solve Equations in Group?* (The American Mathematical Monthly, 1977, <http://www.jstor.org/stable/2321255>).
Let $p\_1, q\_1,...., p\_m, q\_m$ be fixed integers (both positive or negative). I am interested in the i... | https://mathoverflow.net/users/13015 | Solving equations in SO(3) : an open problem by Jan Mycielski | Let me collect a number of known results:
i) $\alpha$ can be arbitrarily small, see my paper
Andreas Thom, Convergent sequences in discrete groups, Canad. Math. Bull. 56 (2013), no. 2, 424–433.
ii) There is some interest in estimating how small $\alpha$ can be in terms of the word length, this has been studied ... | 14 | https://mathoverflow.net/users/8176 | 229762 | 107,002 |
https://mathoverflow.net/questions/229525 | 8 | Let $\omega$ be a word in the free group on generators $x\_1,x\_2,\ldots,x\_n,g\_1,g\_2,\ldots,g\_k$, where $n>0$ and $k\geq 0$. For any finite group $G$ and elements $g\_1,g\_2,\ldots g\_k$ in $G$ we obtain an equation:
$$\omega(x\_1,x\_2,\ldots,x\_n,g\_1,g\_2,\ldots,g\_k)=e$$
Let $S=S\_{\omega,g\_1,g\_2,\ldots,g\_k}$... | https://mathoverflow.net/users/16678 | Finite groups: equations with many solutions | In the case $k=0$ this is a conjecture of Dixon. See <http://people.math.carleton.ca/~jdixon/Prgrpth.pdf>, Section 1.2. I don't know of any reason it should become obviously false for $k>0$, so I guess it's open.
| 1 | https://mathoverflow.net/users/20598 | 229764 | 107,003 |
https://mathoverflow.net/questions/229729 | -1 | Let $A\in M\_n(\mathbb C)$ be arbitrary. I'm interested to know How should $A^{\alpha}$ be defined for real $\alpha\in [0,\infty)$? When $A$ is nonsingular, we can define $A^{\alpha}=\exp(\alpha \log(A))$ for $\alpha \in [0,1]$ and $A^{\alpha}=A^{[\alpha]}.A^{\alpha-[\alpha]}$ ([.] denote the intiger part) for $\alpha ... | https://mathoverflow.net/users/83050 | How should $A^α$ be defined for real $α ∈ [0,∞)$ and $A\in M_n(\mathbb C)$? | Let me suggest a solution since I don't think that any of the above comments address the essentials. Firstly, there is no need to use the logarithm to attack this problem. One can, of course, but I think that its use just muddies the water. There is a standard path to address this kind of problem---the Jordan canonical... | 0 | https://mathoverflow.net/users/81335 | 229772 | 107,008 |
https://mathoverflow.net/questions/229746 | -1 | I would like to know whether Langlands' functoriality conjecture implies that the Selberg class coincides with the class of automorphic L-functions and, if so, whether this class is closed under Rankin-Selberg convolution or not.
Many thanks in advance.
| https://mathoverflow.net/users/13625 | Consequences of Langlands functoriality conjecture | The Langlands functoriality conjecture implies that automorphic $L$-functions belong to the Selberg class, but not the other way (i.e. the other direction is not known to follow from this conjecture). Regarding to Rankin-Selberg convolutions, I don't think that this operation has been defined precisely for the Selberg ... | 4 | https://mathoverflow.net/users/11919 | 229774 | 107,009 |
https://mathoverflow.net/questions/229775 | 3 | Let $k$ be an algebraically closed field, and $X$ a connected smooth projective curve over $X$. Let $F$ be the function field of $k$. Let $G$ be an algebraic group over $k$ (assume that it is smooth, or reductive, if needed). Can you explain why $H^1 (Gal(F^{sep}/F) , G(F^{sep}))$ is zero, or provide a simple reference... | https://mathoverflow.net/users/2095 | Galois cohomology of a non-abelian group over a function field | You need to assume that $G$ is connected or else that is false. Assuming that $G$ is connected, this follows immediately from Steinberg's Theorem, i.e., the proof of Serre's Conjecture I. There is a copy of Steinberg's paper as an appendix of Serre's book on Galois Cohomology. Serre explains in the book how to deduce y... | 7 | https://mathoverflow.net/users/13265 | 229776 | 107,010 |
https://mathoverflow.net/questions/229785 | 10 | If $(M, \omega)$ is a symplectic manifold, is it possible to embed (or injectively immerse) it *symplectically* into a sufficiently large $(\Bbb R ^{2N}, \Omega)$, (with the usual symplectic structure)?
The only thing that I have found is a theorem by Gromov which is conceptually very nice, but which seems to be very... | https://mathoverflow.net/users/54780 | Is there a Nash-type theorem for symplectic manifolds? | Not as stated, there isn't, at least for closed manifolds. For the form $\Omega$ is exact, but on a closed manifold a symplectic form cannot be exact, since a power of it is a volume form. This would apply to immersions as well as embeddings. In particular, this explains the hypothesis of being exact in the Gromov theo... | 11 | https://mathoverflow.net/users/3460 | 229787 | 107,014 |
https://mathoverflow.net/questions/229651 | 3 | For a commutative ring $R$, consider $R[[t]]$-modules $$t^k R[[t]]^n \subset M \subset t^{-k} R[[t]]^n \subset R((t))^n.$$ It is known that if $t^{-k} R[[t]]^n / M$ is finitely generated projective $R$-module, then $M$ is finitely generated projective $R[[t]]$-module.
One line of proof, as found in <http://web.stanfo... | https://mathoverflow.net/users/2095 | A technical question about affine grassmanian | I am being told it is better not to leave answers in comments. So:
I think you can find more details in Martin Kreidl's thesis, available at [uni-due.de/~hx0051/Dissertation.pdf](http://uni-due.de/~hx0051/Dissertation.pdf).
| 2 | https://mathoverflow.net/users/17988 | 229791 | 107,015 |
https://mathoverflow.net/questions/229763 | 2 | I have seen in at least two different places ([here](https://epub.ub.uni-muenchen.de/3843/1/3843.pdf), p. 183; and [here](http://www.mathematik.uni-muenchen.de/~aehlig/EST/rathjen2.pdf), last slide) the *Tait calculus* defined the following way.
Here $\Gamma$ denotes a set of formulas $\{A\_1, \ldots, A\_k\}$, which ... | https://mathoverflow.net/users/27742 | A derivation in Tait calculus | There's no *rule* that lets you get from a deduction of $\Gamma$ to a deduction of $\Gamma,\Delta$. However it's an easy lemma that, given a deduction of $\Gamma$, there is also a deduction of $\Gamma,\Delta$: go by induction on the deduction, adding $\Delta$ to every intermediate rule as well.
You might ask why it's... | 6 | https://mathoverflow.net/users/8991 | 229796 | 107,018 |
https://mathoverflow.net/questions/229710 | 5 | Let $k$ be algebraically closed with characteristic $0$. For a scheme $X$, let $FEt(X)$ be the category of finite etale covers of $X$. What can be said about $FEt(X \times \mathbb{A}^1)$ and the functor $- \times\_k \mathbb{A}^1: FEt(X) \rightarrow FEt(X \times \mathbb{A}^1)$?
It seems to me that the functor induced... | https://mathoverflow.net/users/24706 | $\mathbb{A}^1$-invariance of categories of Finite Etale Covers | I'm a bit surprised that this was never answered - if I'd known I'd posted something more helpful than a snippy question. Yes, $FEt(X) \to FEt(X \times \mathbb A^1)$ is an equivalence of categories in characteristic zero. This assertion is in fact *equivalent* to $\pi\_1^{et}(X) \to \pi\_1^{et}(X \times \mathbb A^1)$ b... | 6 | https://mathoverflow.net/users/1310 | 229797 | 107,019 |
https://mathoverflow.net/questions/229827 | 21 | What is the correct definition of an ind-scheme?
I ask this because there are (at least) two definitions in the literature, and they really differ.
Definition 1. An ind-scheme is a directed colimit of schemes inside the category of presheaves on the Zariski site of affines schemes, where the transition maps are clo... | https://mathoverflow.net/users/2841 | Definition of ind-schemes | There is in fact no difference between the two definitions if you take your site to be the category of affine schemes – while it is true that the forgetful functor from sheaves to presheaves does not preserve colimits in general, for the purposes of this definition, only filtered colimits matter, and those are preserve... | 14 | https://mathoverflow.net/users/11640 | 229836 | 107,025 |
https://mathoverflow.net/questions/229830 | 11 | Montgomery's pair correlation conjecture states that the distribution of the pair correlations of the zeroes of the Riemann zeta function (normalized to have average spacing 1) is given by the function
$$
g(u) = 1 - \left(\frac{\sin(\pi u)}{\pi u}\right)^2 + \delta(u).
$$
See <https://en.wikipedia.org/wiki/Pair_correla... | https://mathoverflow.net/users/46852 | Riemann zeta function: pair correlations vs. neighbor spacings | This next-nearest-neighbor distribution of the Riemann zero's is addressed in [Mehta's book on random-matrix theory.](https://books.google.nl/books/about/Random_Matrices.html?id=Kp3Nx03_gMwC&redir_esc=y) It is well reproduced by that of the Gaussian Unitary Ensemble (GUE), compare black curve and black data points:
!... | 10 | https://mathoverflow.net/users/11260 | 229837 | 107,026 |
https://mathoverflow.net/questions/229833 | 12 | Let $\beta\mathbb N$ is the set of ultrafilters on $\mathbb N$ and $\mathscr F\in\beta\mathbb N$. Assume that $l\_{\mathscr F}\in\big(\ell^\infty(\mathbb N)\big)^{\!\*}$ is the functional which assigns to a bounded sequence $\{a\_n\}\_{n\in\mathbb N}\subset\mathbb C$ its limit, with respect to the ultrafilter $\mathscr... | https://mathoverflow.net/users/43681 | Description of $\big(\ell^\infty(\mathbb N)\big)^{\!*}$ via ultrafilters | Identify $\mathbb{N}$ with $\mathbb{Q}\cap[0,1]$ via a bijection, and consider the subspace $C([0,1])\subset\ell^\infty(\mathbb{N})$ of sequences which extend to a continuous function on $[0,1]$. When we restrict each $l\_{\mathscr{F}}$ to $C([0,1])$, we get evaluation at some point of $[0,1]$ (namely, the limit of $\m... | 20 | https://mathoverflow.net/users/75 | 229840 | 107,027 |
https://mathoverflow.net/questions/225644 | 37 | Yesterday Bourgain, Demeter and Guth [released a preprint](http://arxiv.org/abs/1512.01565) proving (up to endpoints) the so-called main conjecture of the Vinogradov's Mean Value Theorem for all degrees. This had previously been only known for degree 3 by work of Wooley. See [this survey](http://arxiv.org/abs/1404.3508... | https://mathoverflow.net/users/630 | The Bourgain-Demeter-Guth breakthrough and the Riemann zeta function? | With respect to the recent breakthrough, Bourgain states in this [preprint](http://arxiv.org/pdf/1601.08173v1.pdf):
>
> Concerning applications to the zeta-function, our work as it stands does
> not lead to further progress. The reason for this is that we did not explore
> the effect of large $k$ (possibly depend... | 11 | https://mathoverflow.net/users/12176 | 229842 | 107,028 |
https://mathoverflow.net/questions/229596 | 0 | I am trying to work through a paper *Instability in Parallel Flows Revisited* by Friedlander and Howard, and there are a couple steps in the beginning that I do not understand. I apologize in advance for being long-winded, but I want to be candid.
Consider a steady two-dimensional parallel-shear incompressible flow
$... | https://mathoverflow.net/users/82505 | Linearized stream function | First compare your derivation to the following. Consider the Navier-Stokes equation with the plane symmetric ansatz, which reduces to $v = (v\_1,v\_2)$ solving
$$ \partial\_t v + (v\cdot \nabla)v + \nabla p = \epsilon \triangle v $$
(We ignore the external force for now, since we are only concerned with the linearized... | 1 | https://mathoverflow.net/users/3948 | 229853 | 107,032 |
https://mathoverflow.net/questions/229856 | 2 | I post this question in [MSE](https://math.stackexchange.com/questions/1629349/the-inverse-of-laplacian-for-different-orders) couple of days before and get no response. So I repost it here for better luck. Thank you!
---
Let $u,v\in C\_c^\infty(\Omega)$ and $\Omega\subset \mathbb R^N$ is open bounded with smooth ... | https://mathoverflow.net/users/62560 | The inverse of Laplacian operator for different orders | **Disclaimer**
The below argument applies directly to the previous version of the question. For the current version the basic argument still hold, using that
$$ \| \nabla u\|\_2 = \langle u, -\Delta u\rangle $$
and so behaves nicely for eigenfunctions of the Laplacian.
To make the argument also work for $C^\infty... | 0 | https://mathoverflow.net/users/3948 | 229861 | 107,035 |
https://mathoverflow.net/questions/229868 | 23 | Previosly my impression on this subject was that hyperelliptic cryptography systems (as well as other possible cryptosystems based on abelian varieties of dimension $>1$) have no advantages over elliptic curve ones and are more difficult to implement. Yet Google search has demonstrated me that people are working on imp... | https://mathoverflow.net/users/2191 | Is hyperelliptic cryptography "practical"? | I am not aware of anybody seriously considering hyperelliptic curves for actual real-world usage, beyond toys, and I would be rather surprised to hear differently from anyone.
As you say, hyperelliptic provide comparatively few (if any!) advantages over elliptic curves but have the huge disadvantage that virtually n... | 18 | https://mathoverflow.net/users/8338 | 229873 | 107,039 |
https://mathoverflow.net/questions/229882 | 2 | I learned **basic** results (regarding weak convergence) about Banach-space valued functions of a single real variable when learning PDE. (See e.g. Appendix E in Evans's *Partial Differential Equations*) I can only find a treatment about this topic in Yosida's *Functional Analysis*, which is quite a classical old book.... | https://mathoverflow.net/users/nan | reference request: simple facts about vector-valued $L^p$ spaces | Too long for a comment...
If I am not mistaken, the last (=6th) edition of Yosida's Functional Analysis is from 1995, so it seems you want something not older than that. Here is a list:
1. Marsden, Ratiu, Abraham - Manifolds, Tensor Analysis, and Applications: lecture notes from 2007 available online, it has a full... | 3 | https://mathoverflow.net/users/1849 | 229886 | 107,044 |
https://mathoverflow.net/questions/229829 | 0 | Let $k$ be an algebraically closed field, $X$ a connected smooth curve over $k$, $G$ a connected reductive group over $k$, and $B \subset G$ a Borel subgroup.
Given a $G$-torsor $E$ on $X$ in the etale topology, I would like to understand why it has a reduction to $B$, in the simple case above (usually one finds some... | https://mathoverflow.net/users/2095 | Existence of $B$-reduction of a $G$-torsor on a curve | It is not too difficult to show that $B\backslash E$ is a scheme -- see e.g. <http://arxiv.org/pdf/1308.3078.pdf>, Prop. 4.4.
| 2 | https://mathoverflow.net/users/6772 | 229894 | 107,047 |
https://mathoverflow.net/questions/229884 | 6 | What is the definition of $E\_\infty$-operad in the category of chain complexes over $\mathbb{Z}/p\mathbb{Z}$? J. Smith in <http://arxiv.org/abs/math/0004003> define it for complexes over $\mathbb{Z}$ (Definition 2.24). Sorry but I'm confused with how does the version in
$\mathbb{Z}/p\mathbb{Z}$ must looks like. Thanks... | https://mathoverflow.net/users/86066 | Definition of E-infinity operad | Operads $\mathcal{C}$ can be defined in any symmetric monoidal category, and then
$E\_{\infty}$ operads are specified in accordance with the (or a) notion of equivalence
relevant to that category. In any category, I prefer to insist that they be $\Sigma$-free,
in the sense that the symmetric group $\Sigma\_n$ acts fr... | 11 | https://mathoverflow.net/users/14447 | 229895 | 107,048 |
https://mathoverflow.net/questions/229892 | 4 | I just stumbled on a set of prime-generating polynomials of the form $$9 n^2-3 H n+H (H+1)/4$$ (where $H$ is a Heegner number $>11$), which generate the same number of distinct primes as their more familiar counterparts. There is some overlap in the primes generated by $n^2 - n + 41$ and $9 n^2-489 n+6683$ for example,... | https://mathoverflow.net/users/45057 | Unfamiliar prime-generating polynomials related to Heegner numbers | According to [Wikipedia](https://en.wikipedia.org/wiki/Heegner_number),
Rabinowitz showed that $n^2 + n + p$ is prime for $n=0, \ldots, p-2$ iff $p = (1+H)/4$, $H$ a Heegner number. If you substitute $n = 3 m - (H+1)/2$, you get your polynomial $9 m^2 - 3 H m + H(H+1)/4$.
| 5 | https://mathoverflow.net/users/13650 | 229901 | 107,051 |
https://mathoverflow.net/questions/229867 | 7 | The nLab says the following about closed monoidal functor categories:
>
> Let $C$ be a complete closed monoidal category and $I$ any small category. Then the functor category $[I, C]$ is closed monoidal with the pointwise tensor product, $(F \otimes G)(x) = F(x) \otimes G(x)$.
>
>
>
Now I wonder what the right... | https://mathoverflow.net/users/25527 | What is the right adjoint of the tensor product in a closed monoidal functor category? | The formula you give is correct. However, I prefer to use a power/cotensor rather than an indexed product, for while they are equivalent for ordinary categories, only the former gives the correct formula for the general enriched case. So, if $\mathscr{A}$ is a $\mathscr{V}$-category and $\mathscr{X}$ is a closed monoid... | 8 | https://mathoverflow.net/users/57405 | 229913 | 107,053 |
https://mathoverflow.net/questions/229915 | 9 | Let $B$ be the symmetric monoidal category of finite sets and bijections with disjoint union. Let $C$ be a symmetric monoidal category. Is there a standard name for a lax monoidal functor $F:B \to C$? In other words, we are considering a sequence $F(n)$ of $S\_n$-representations in $C$ (i.e. a species in $C$) together ... | https://mathoverflow.net/users/1310 | Does this notion related to species/operads/FI-modules have a name? | Depending on whether you want it to agree with the symmetric structure or only with monoidal structure, this would be usually referred to, respectively, as twisted commutative algebras or twisted associative algebras. See, for example, <http://arxiv.org/pdf/0710.3392.pdf>, and a more classical reference <http://www.sci... | 8 | https://mathoverflow.net/users/1306 | 229919 | 107,054 |
https://mathoverflow.net/questions/229504 | 2 | When reading the books or papers on p-adic Hodge theory, non trivial example of p-adic representation seems to be only the example of Tate curves. To be sure, I had read the very readable introduction by [L.Berger](http://perso.ens-lyon.fr/laurent.berger/articles/article05.pdf) and there are some calculations for ellip... | https://mathoverflow.net/users/85711 | Examples of p-adic representations | Not sure what you mean by "coming from given equations". If you're looking for concrete examples, you can start with abelian representations of $G\_K = Gal(K^{alg}/K)$ where $K$ is a finite extension of $Q\_p$. Choose a Lubin-Tate character $\chi\_K : G\_K \to O\_K^\times$ and see what kinds of representations you can ... | 3 | https://mathoverflow.net/users/5743 | 229925 | 107,056 |
https://mathoverflow.net/questions/229920 | 3 | Let $\mathcal{C}$ and $\mathcal{D}$ be triangulated categories, and suppose that there exists an additive equivalence $F: \mathcal{C} \to \mathcal{D}$. Suppose further that $\mathcal{C} = \text{add}(C\_1 \oplus \dots \oplus C\_n)$ and $\mathcal{D} = \text{add}(D\_1 \oplus \dots \oplus D\_n)$, where the $C\_i$ and $D\_i... | https://mathoverflow.net/users/78095 | Is this additive equivalence a triangulated equivalence? | No, just take any category equipped with a shift functor satisfying you condition on indecomposables and which admits more than one triangulated structure, and $F$ the identity functor. If you want an explicit example, consider the category of finitely gnerated modules over the dual numbers $k[\epsilon]/(\epsilon^2)$, ... | 4 | https://mathoverflow.net/users/12166 | 229934 | 107,058 |
https://mathoverflow.net/questions/229953 | 14 | Martin-Löf Extensional Type theory differs from its intensional counterpart in that it contains the so-called *reflection rule* that says that if $p : x = y$, then actually $x \equiv y$ (i.e. $x$ and $y$ are definitionally or judgementally equivalent). Its known that this causes strong normalization to fail and type-ch... | https://mathoverflow.net/users/84804 | Why the reflection rule trivializes higher paths in Martin-Löf Extensional Type theory? | The point is that the reflection rule makes $p = \mathsf{refl}\_x$ a well-formed expression. This turns out to be incredibly dangerous: now we can prove it by induction on equality.
More precisely:
1. In a context where $x$ and $y$ are variables of the same type $T$ and $p$ is a variable of type $x = y$, we have $x... | 16 | https://mathoverflow.net/users/11640 | 229957 | 107,068 |
https://mathoverflow.net/questions/229907 | 4 | If points $A$, $B$, $C$ form a triangle in euclidean space and $D$ is another point in the plane of the triangle, the problem is to show that :
$\frac{AB}{DA + DB} + \frac{BC}{DB + DC} \ge \frac{AC}{DA + DC}$
[Figure here](https://i.stack.imgur.com/IWKZj.png)
I verified that it holds for more than several millio... | https://mathoverflow.net/users/86078 | Inequality from a point in plane to a triangle OR Inequality on a quadrilateral | I would like to streamline the proof a bit.
If $AC=0$ then the inequality in question is obvious. So, assume that $AC\ne0$. Multiplying both sides of the inequality by $(AD + BD) (AD + CD) (BD + CD)$, rewrite it as
$$
L:=(AD + CD) (AD\ BC + AB\ BD + BC\ BD + AB\ CD)
-AC (AD + BD) (BD + CD)\ge0.
$$
Clearly, $L$ i... | 3 | https://mathoverflow.net/users/36721 | 229967 | 107,070 |
https://mathoverflow.net/questions/229959 | 18 | Coming from a number-theoretic background, I certainly care about $L$-functions and in particular automorphic ones. For automorphic forms on $SL\_2(\mathbb{Z}) \backslash SL\_2(\mathbb{R})$, $L$-function can be interpreted as the Mellin transform of Fourier expansion at the cusp.
If we look at automorphic forms on $S... | https://mathoverflow.net/users/31814 | Do we care about multiple zeta functions? | It may help clarify things to work out a specific example, although the OP may know this. In case $n=3$, the double Dirichlet series evaluates as
$$ \sum\_{m,n=1}^{\infty} \frac{A\_F(m,n)}{m^{w} n^s} = \frac{L(\overline{F},w) L(F, s)}{\zeta(s+w)}.$$
Here $\overline{F}$ is the contragredient of $F$.
This is known as Bum... | 11 | https://mathoverflow.net/users/2627 | 229969 | 107,071 |
https://mathoverflow.net/questions/229172 | 4 | Is there a Killing vector field on a complete Riemannian manifold $M$ with finite volume that satisfies the condition
$$\displaystyle\liminf\_{r\rightarrow +\infty} \displaystyle\frac{1}{r} \displaystyle\int\_{ B(2r)/B(r) } |X| d\nu\_g > 0, $$
where $B(r)$ denotes the geodesic ball of radius $r$ and centre $p$, whe... | https://mathoverflow.net/users/85681 | Is there an example of a Killing vector field on a complete Riemannian manifold with finite volume? | Consider a warped product of the hyperbolic plane $H$ with a circle $S^1$,
where the length of the circle is rescaled by a function $f\colon H\to(0,1]$
that is radially symmetric around $o\in H$. Write $f(x)=f(r)$ where $d(o,x)=r$ by abuse of notation. A sphere at distance $r$ in $H$
has volume $2\pi\sinh r$, so $f$ ne... | 5 | https://mathoverflow.net/users/70808 | 229975 | 107,074 |
https://mathoverflow.net/questions/229970 | 6 | Let $L/K$ be a finite Galois extension, write $G:= Gal(L/K)$. Denote by $R = Res(\mathbb{G}\_m)$ the Weil restriction of $\mathbb{G}\_m$, from $L$ to $K$. I want to show that its first Galois cohomology vanishes: $H^1(G, R(L)) = 0$.
**Question:** Is there a simple way to do this, without calculation (something formal... | https://mathoverflow.net/users/2095 | First Galois cohomology of Weil restriction of $\mathbb{G}_m$ | There is a general argument that is slightly more elementary than what you wrote. By standard properties of Weil restrictions, we have $R(L) = \prod\_{\sigma} \mathbb{G}\_m(L)$, where the product is taken over the different $K$-linear embeddings of $L$ into some algebraic closure $\overline{K}$ and where $G$ acts on th... | 5 | https://mathoverflow.net/users/17907 | 229984 | 107,076 |
https://mathoverflow.net/questions/229989 | 5 | For points $a,b\in \mathbb{R}^n\setminus \{0\}$ denote $$d(a,b)=\frac{\|a-b\|}{\|a\|+\|b\|}.$$
[This](https://mathoverflow.net/questions/229907/inequality-from-a-point-in-plane-to-a-triangle-or-inequality-on-a-quadrilateral) question by Ritesh Ahuja (positive answered by Iosif Pinelis) says that $d$ is a metric. My que... | https://mathoverflow.net/users/4312 | What is this distance about? | The first distance is a special case of the so-called [p-q relative metric](https://books.google.com/books?id=q_7FBAAAQBAJ&lpg=PA362&ots=QY9TbHWREw&dq=p%20q%20relative%20metric&pg=PA362#v=onepage&q=p%20q%20relative%20metric&f=false), which is in general defined on any Ptolemaic space as
\begin{equation\*}
\frac{\|x-y\... | 5 | https://mathoverflow.net/users/8430 | 229991 | 107,079 |
https://mathoverflow.net/questions/228876 | 4 | Birch has a conjecture about which automorphic forms on $PGL(2)$ are the lifts from nonsplit $O(3)$. Temporarily ignore global issues, and focus on the local nonarchimedian picture. The automorphic representations of $PGL(2)$ are representations of $GL(2)$ with trivial central character, and Jacquet-Langlands describes... | https://mathoverflow.net/users/6084 | Birch's conjecture from Representation Theory | I haven't looked at Birch's conjecture, but yes, Jacquet and Langlands characterize the image of of their transfer. Here is what it is in terms of representation theory. Fix a quaternion division algebra $D$ over a number field $F$. Let $S$ be the set of places at which $D/F$ is ramified. Then a cuspidal automorphic re... | 3 | https://mathoverflow.net/users/6518 | 230000 | 107,083 |
https://mathoverflow.net/questions/217951 | 13 | Are the following mutually consistent (relative to large cardinals)?
(1) There are no $\omega\_2$-Aronszajn trees.
(2) There is an $\omega\_1$-Kurepa tree.
In the models I know of the tree property at $\omega\_2$, it also holds that there are no *weak* Kurepa trees on $\omega\_1$ (also called Canadian trees).
| https://mathoverflow.net/users/11145 | tree properties on $\omega_1$ and $\omega_2$ | I wrote a short note with the consistency proof, which can be found at <http://www.math.cmu.edu/users/jcumming/papers/kurepa/kurepa.pdf>. It is pretty rough, please tell me if there are problems.
| 13 | https://mathoverflow.net/users/86131 | 230004 | 107,086 |
https://mathoverflow.net/questions/230007 | 2 | I have seen the expectation of a random vector expressed as the solution to the optimization problem:
\begin{equation}
\mathbb{E}[X]=argmin\_{v \in \mathbb{R}^n}\mathbb{E}[\|X-v\|\_{l^2}^2](:= \int\_{\Omega} \|X(\omega)-v\|^2 dP(\omega)).
\end{equation}
---
My question is... can we express the covariance matrix o... | https://mathoverflow.net/users/36886 | Covariance matrix as optimization problem solution? | I believe you can do something like this:
Let $X$ be zero mean. Then the covariance matrix is $ \text{cov}(X) = \mathbb{E}[XX^T]$. Hence, by the same argument:
\begin{align\*}
\text{cov}(X) = \text{arg} \min\_{\Sigma} \; \mathbb{E} \| XX^T - \Sigma\|\_F^2
\end{align\*}
If $X$ not zero mean, let $X'$ be an independent ... | 3 | https://mathoverflow.net/users/36687 | 230008 | 107,088 |
https://mathoverflow.net/questions/229479 | 7 | In my research, I've come twice now towards a skew field $K$ that satisfies the following:
> $$\text{for all non-central element $a$, the map }\quad x\mapsto ax-xa\quad\text{ is onto.}$$
I am hoping such a field does not exist but am unable to prove it.
**Does this condition ring a bell** to some of you ?
**An... | https://mathoverflow.net/users/18583 | Existence of a skew field with surjective inner derivations | In *The range of derivations on a skew field and the equation ax-xb=c*, Journal of the Indian Math. Soc. 37 (1973), 61--69, P.M. Cohn shows the following :
> **Theorem (Cohn 1973).** Given a field $K$ with centre $k$, there is an extension field $L$ (still with $k$ as centre) such that the equation $$ax-xa=c$$ has a... | 3 | https://mathoverflow.net/users/18583 | 230011 | 107,089 |
https://mathoverflow.net/questions/230019 | 0 | Let $A,B$ be complete Boolean algebras and $\varphi,\psi:A\rightarrow B$ be
maps preserving $0,1$, and arbitrary joins and meets. Let $C$ be the equalizer of these two; so $C=\left\{a\in A:\varphi(a)=\psi(a)\right\}$. It can be easily checked that $C$ itself is a complete Boolean algebra.
**Question:** Let $a\in A$... | https://mathoverflow.net/users/78655 | About equalizer of Boolean algebras | The condition $\varphi(a)\notin\mathfrak{r}\iff \psi(a)\notin\mathfrak{r}$ is equivalent to $\varphi(a)=\psi(a)$, since an element of a Boolean algebra is determined by the prime ideals that contain it (if $a\neq b$, then there is a prime ideal containing $1+a+b$, which then must contain exactly one of $a$ and $b$). So... | 2 | https://mathoverflow.net/users/75 | 230020 | 107,092 |
https://mathoverflow.net/questions/230015 | 2 | Given a prime number $p$, can you give me concrete examples of fields $\mathbf F$ of characteristic $p$ and quaternion algebras $\mathbb H(\mathbf F)$ over $\mathbf F$ such that $\mathbb H(\mathbf F)$ is a (non-commutative) division ring ?
| https://mathoverflow.net/users/18583 | Quaternion algebra in characteristic $p$ | If $F$ is a finite field, the unique central division $F$-algebra is $F$ itself. This is Wedderburn's theorem :
<https://en.wikipedia.org/wiki/Wedderburn%27s_little_theorem>
In particular there is no central quaternion $F$-algebra. In contrast there are central division $F$-algebras of arbitrary (square) dimension ... | 4 | https://mathoverflow.net/users/4767 | 230027 | 107,096 |
https://mathoverflow.net/questions/230036 | 2 | Given a category $C$, is there a name for the following category:
* $\mathrm{Obj}(D) = \left\{ (x, y, f) \middle| x, y \in \mathrm{Obj}(C), f \in C(x, y) \right\}$
* $D((x, y, f), (x', y', f')) = \left\{ (g, h) \middle| g \in C(x', x), h \in C(y, y'), h \circ f \circ g = f' \right\}$
This is, if you want, the final... | https://mathoverflow.net/users/66017 | The source-side-opposite of the arrow category | As Zhen Lin suggests, what you are describing is called the "twisted arrow category" $C\_\#$ in Mac Lane's *Categories for the Working Mathematician* (see exercise IX.6.3, p.227). It goes by other names as well: for example "category of factorizations". A quick way of defining $C\_\#$ is as the category of elements of ... | 7 | https://mathoverflow.net/users/1015 | 230043 | 107,101 |
https://mathoverflow.net/questions/229565 | 4 | Let $p<1$ be a constant. Consider two sets $A,B$, each with $n$ vertices. For each pair $(a,b)\in A\times B$, the edge between $a$ and $b$ appears with probability $p$, independently of the remaining edges. From this [question](https://mathoverflow.net/questions/222769/matching-with-probabilistic-edges) (which referenc... | https://mathoverflow.net/users/79906 | Probability bound for perfect matching | Let $q=1-p$ be a probability that there is no edge between two given vertices. For any subsets $A'\subset A$, $B'\subset B$, $|B'|+|A'|=n+1$, denote by $X(A',B')$ the following event: there are no edges between $A'$ and $B'$. The absence of a perfect matching is a union of these events by Hall theorem.
Thus the follow... | 2 | https://mathoverflow.net/users/4312 | 230047 | 107,104 |
https://mathoverflow.net/questions/229981 | 1 | My question comes as a natural follow-up of the previous one which concerned [symplectic manifolds](https://mathoverflow.net/questions/229785/is-there-a-nash-type-theorem-for-symplectic-manifolds): if $(M, P)$ is a Poisson manifold, what embedding theorems are there into some target space (I am looking mostly at embedd... | https://mathoverflow.net/users/54780 | Nash-type theorems for Poisson manifolds | Not in this form. Poisson maps are rank decreasing. If you have a Poisson map $f$ from $(M,\pi\_1)$ to $(M,\pi\_2)$ then ${\mathrm rank}\,\pi\_1(x)\ge {\mathrm rank}\,\pi\_2(f(x))$. Therefore if the target manifold is symplectic then $f$ should be a submersion.
Even $\mathbb R^2\hookrightarrow \mathbb R^4$ with respe... | 1 | https://mathoverflow.net/users/6032 | 230052 | 107,107 |
https://mathoverflow.net/questions/230054 | 0 | Question
--------
We now define the following "ugly" function:
$$ A\_c(s,r,n,m) =
\begin{cases}
1 & \text{ if only $sr+nm=2c$ } \\ 0 & \text{otherwise}
\end{cases}
$$
How does the "ugly" function asymptotically behave?
$$ \sum\_{n=1}^c \sum\_{m=1}^c \sum\_{r=1}^c \sum\_{s=1}^c A\_c(s,r,n,m) \sim (?)$$
Does ... | https://mathoverflow.net/users/78872 | Asymptotics of "ugly" function elucidate Goldbach's conjecture? | Your sum can be rewritten as
$$ D(N)=\sum\_{n=1}^{N-1}d(n)d(N-n),$$
where $N=2c$, and $d(m)$ is the number of divisors of $m$. This is a so-called "binary additive divisor sum", and it has been studied widely. In particular, Ingham (1927) showed that
$$ D(N) = (1+o(1))\frac{6}{\pi^2}\sigma\_1(N)(\log N)^2,$$
where $\si... | 9 | https://mathoverflow.net/users/11919 | 230057 | 107,110 |
https://mathoverflow.net/questions/159372 | 8 | **Motivation:** This question is related to my study of hyperbolic Coxeter polytopes. In general, if one put some restrictions on the type of their dihedral angles (say, all dihedral angles are equal to $\pi/2$), an the type of the polytope (all its vertices are proper, or all are ideal, or of mixed type), then one obt... | https://mathoverflow.net/users/39331 | Classifying two-faces of four-polytopes | I dont know the anser to the specific question. It seems that for the study of hyperbolic Coxeter polytopes even if using some properties of general simple 4-polytope one needs to use the very restricted nature of these Coxeter polytopes. Overall the question of prescribing the sizes of 2-faces of simple (and general) ... | 3 | https://mathoverflow.net/users/1532 | 230059 | 107,111 |
https://mathoverflow.net/questions/229899 | 5 | My question is inspired by the following observation:
**Claim:** It is *not* possible to choose $n$ subsets of the universe $[n]$, each of size $\Omega(n)$, such that for each subset $S$ and each element $s \in S$, there is another subset $S'$ such that $S \cap S' = \{s\}$.
**Proof:** Suppose, towards a contradicti... | https://mathoverflow.net/users/25121 | How often can subsets of a universe intersect exactly once? | It seems that for every $k\geq 2$, there exists a desired cpllection of $6k$ subsets of $[3k]$ with size $k$ each.
Let $a\_1$, $a\_2$, $a\_3$ be three rays with a common origin $O$, and let $s\_1,\dots,s\_k$ be $k$ different circles centered at $O$. Denote $x\_{ij}=a\_i\cap s\_j$; we may identify $[3k]$ with the set... | 3 | https://mathoverflow.net/users/17581 | 230062 | 107,113 |
https://mathoverflow.net/questions/230009 | 1 | This is a follow-up question on [this](https://mathoverflow.net/questions/229185/collection-of-projection-operators-in-finite-dimension-and-algebraic-techinques/229295#229295). Let $A$ be a von Neumann algebra and $P$ be its projection lattice.
For $p,s,q \in P$, let us define $ p \perp q \mid s \iff ps^\perp q = 0$... | https://mathoverflow.net/users/36687 | A relation among projections of a von Neumann algebra | I think it is false - let $r=1$ and take any distinct rank $1$ projections $p,q,s\in M\_2$ so $p\vee q=p\vee s=q\vee s=1$.
| 2 | https://mathoverflow.net/users/38085 | 230073 | 107,118 |
https://mathoverflow.net/questions/230082 | 8 | I am trying to get some insights for the combinatorial argument of Pitts (in his PhD thesis 'Existence and regularity of minimal surfaces in Riemannian manifolds', Princeton University Press, 1981) to prove the existence of suitable class of varifolds in the so-called Almgren-Pitts theory. Pitts quotes the reference 'T... | https://mathoverflow.net/users/56191 | Almgren's mimeographed lectures notes on varifolds | Here is the story behind these notes, and a redirect to [On the First Variation of a Varifold](https://spencerthughes.files.wordpress.com/2012/06/allard-on-the-first-variation-of-a-varifold.pdf), W.K. Allard (1972).
a quote from: [Selected Works of Frederick J. Alm... | 10 | https://mathoverflow.net/users/11260 | 230085 | 107,120 |
https://mathoverflow.net/questions/230068 | 2 | Is the following limit result correct: $$\lim\limits\_{x \to 0^{+}}\sum\limits\_{n=0}^{\infty} \binom{z}{n} e^{-xn\log n} = 2^{z}$$ where, $z \in \mathbb{C}$, and the notation $\displaystyle \binom{z}{n} = \frac{1}{n!}\prod\limits\_{k=0}^{n-1}(z-k)$
I verified the result for $z = -1$ case and the limit is indeed $\df... | https://mathoverflow.net/users/62680 | Strange limit problem involving $\binom{z}{n} e^{-xn\log n}$ with $z \in \mathbb{C}$ | If $\Re z>0$, the series $\sum\_n \binom{z}n$ converges absolutely (by Raabe test, for example), thus we may replace $x$ to 0 and just need to check that this sum equals $2^z$. This follows from another relaxation: $\sum \binom{z}{n}t^n=(1+t)^z$ for $0<t<1$ and we may let $t$ tend to 1.
Next, consider the case $-1<\R... | 5 | https://mathoverflow.net/users/4312 | 230088 | 107,122 |
https://mathoverflow.net/questions/230087 | 1 | Let $S$ be the space of all loops in $\mathbb{R}^2$, i.e. all continuous mappings $\gamma:[0,1]\to \mathbb{R}^2$ such that $\gamma(0)=\gamma(1)$. Is there any canonical interpretation of $S$ as a Banach space? If so, is its dual space easy to define?
| https://mathoverflow.net/users/86166 | The space of loops as a Banach space | The space of loops in $\mathbb R^2$ is the same as $C(\mathbb T)$, the continuous functions of the circle into the complex numbers. This is a C$^\*$-algebra and thus a Banach space.
It is well known that the dual of $C(\mathbb T)$ is $M(\mathbb T)$, the space of all Radon measures of the circle.
| 2 | https://mathoverflow.net/users/76593 | 230093 | 107,124 |
https://mathoverflow.net/questions/230089 | 1 | Given an undirected connected graph on $n$ nodes, let $S$ be the subspace of vectors $x \in \mathbb{R}^n$ which satisfy $$\sum\_{j \in N(i)} x\_j = 0,$$ for all $i=1, \ldots, n$. Here $N(i)$ is the set of neighbors of node $i$; the graph may have self-loops so that it is possible that $i \in N(i)$. I am interested in u... | https://mathoverflow.net/users/86167 | Vectors which average to zero over any graph neighborhood | Note that what you defined is simply the dimension of the nullspace of the [adjacency matrix](https://www.wikiwand.com/en/Adjacency_matrix) of $G$. This is usually called the *nullity* $\eta(G)$ of $G$. Apparently, this parameter is significant in Chemistry. If $\eta(G)>0$ for a molecular graph, then the corresponding ... | 5 | https://mathoverflow.net/users/2233 | 230094 | 107,125 |
https://mathoverflow.net/questions/221717 | 10 | Lojasiewicz's theorem asserts that if $F: \mathbb{R}^n\to \mathbb{R}$ is a real-analytic function in a neighborhood of its critical point $0$, then there exist constants $\theta\in (0,1/2]$, $\gamma\ge2$ and $\delta>0$ such that
$$(1)~~~\operatorname{dist}(x,V)^{\gamma}\leq \|\nabla F(x)\|$$
$$(2)~~~|F(x)-F(0)|^{1-\the... | https://mathoverflow.net/users/nan | Optimal exponent in the Lojasiewicz-Simon gradient inequality | For Inequality (2) in Simon's Theorem 3, a discussion of when the optimal exponent, $\theta=1/2$, is attained can be found (along with many references) in my paper [Lojasieicz-Simon gradient inequalities for analytic and Morse-Bott functionals on Banach spaces and applications to harmonic maps](http://arxiv.org/abs/151... | 7 | https://mathoverflow.net/users/86172 | 230097 | 107,127 |
https://mathoverflow.net/questions/229998 | 6 | Looking into the Wigner-Weyl transformation mapping Hilbert space operators to functions on phase-space, I've run up against the need for a symplectic convolution
$$[F\star G](x,p) = \int \!dy\,dk\, F(y,k)G(x-y,p-k)e^{i (xk-yp)}$$
or more compactly
$$[F\star G](\alpha) = \int \!d\beta\, F(\beta)G(\alpha-\beta)e^{... | https://mathoverflow.net/users/5789 | Is there an alternate name for the symplectic convolution? | This operation (generalised slightly by replacing $e^{i (xk-yp)}$ by $e^{i\lambda (xk-yp)}$ for a parameter $\lambda$) is known as "twisted convolution" in the harmonic analysis literature, see e.g. Chapter XII.3.3 of Stein's "[Harmonic analysis](http://www.ams.org/mathscinet-getitem?mr=1232192)". (Side note: the citat... | 5 | https://mathoverflow.net/users/766 | 230098 | 107,128 |
https://mathoverflow.net/questions/230069 | 8 | I'll write $\mathbf{Z}^n$ for the integral quadratic form $x\_1^2 + \cdots + x\_n^2$. For which values of $n$ is $\mathbf{Z}^n$ unique in its genus, i.e. isolated in Kneser's graph? In particular can someone verify or shoot down the following guesses:
1. For $n \leq 8$, any lattice in the same genus as $\mathbf{Z}^n$... | https://mathoverflow.net/users/1048 | What's in the genus of the cubic lattice? | [*edited* mostly to add information about $n > 9$]
**David Treumann**'s guess is correct: ${\bf Z}^n$ is unique in its genus
*iff* $n \leq 8$, and for $n = 9$ the genus consists of only ${\bf Z}^9$ and
${\bf Z} \oplus E\_8$.
The comments indicate two ways to prove this,
using $p$-neighbors (as implemented in MAGMA)... | 8 | https://mathoverflow.net/users/14830 | 230102 | 107,130 |
https://mathoverflow.net/questions/230115 | 4 | Let $R$ be an integral domain. Let $x,y\in R\setminus\{0\}$ be distinct. Let $\hat R$ be the $x$-adic completion of $R$ (the ring of all sequences $(r\_n+Rx^n)\_{n\ge0}$ where for $n\ge0$, $r\_n\in R$ and $r\_n+Rx^{n}=r\_{n+1}+Rx^n$). Denote the image of $r\in R$ under the canonical ring-homomorphism by $\hat r$. Local... | https://mathoverflow.net/users/51389 | Completeness of Localizations of Completions of Commutative Rings | No, $\hat{R}\_y$ need not be $\hat{x}$-adically complete. The polynomial ring $R=k[x,y]$ is a counterexample. The $x$-adic completion of $R$ is identified with $k[y][[x]]$, the ring of power series in $x$ whose coefficients are polynomials in $y$. If we invert $y$, we get a ring whose elements are power series in $x$ w... | 4 | https://mathoverflow.net/users/5263 | 230123 | 107,138 |
https://mathoverflow.net/questions/230113 | 9 | A few years ago a [nice paper](http://arxiv.org/abs/1211.1036) surveyed the differences in quality between papers submitted to arXiv and those submitted to arXiv's rough cousin, viXra. However, that paper was about generic contributions to natural sciences, whereas [this post](https://www.quora.com/Are-there-any-seriou... | https://mathoverflow.net/users/26039 | Hard maths on viXra? | From a quick search I found [Adjugates of Diophantine Quadruples](http://vixra.org/abs/0907.0024) by Philip Gibbs, which was originally posted on the viXra and has since appeared in [INTEGERS](http://www.degruyter.com/view/j/integ.2010.10.issue-2/integ.2010.015/integ.2010.015.xml). I do not think it meets the quota of ... | 14 | https://mathoverflow.net/users/2233 | 230124 | 107,139 |
https://mathoverflow.net/questions/230126 | 1 | I am currently using a non-systematic, pseudo-random method for finding prime-generating polynomials, based on the [Bateman-Horn method](https://dl.dropboxusercontent.com/u/5188175/BatemanHorn.pdf) for finding likely candidates, and then narrowing down. I have been so far, only searching for quadratics, searching up to... | https://mathoverflow.net/users/45057 | Methods for searching for prime generating polynomials | Another approach is to maximize to constant $C$ in
[Bateman–Horn conjecture](https://en.wikipedia.org/wiki/Bateman%E2%80%93Horn_conjecture)
The constant is:
$$C = \prod\_p \frac{1-N(p)/p}{(1-1/p)}$$
For linear $f(x)$, you can make it unbounded rational.
Let $n\#$ denote primorial, the product of the first $n$
p... | 1 | https://mathoverflow.net/users/12481 | 230135 | 107,142 |
https://mathoverflow.net/questions/230142 | 0 | Consider the direct limit of an indexed family $\{a\_n\}\_{n\in \omega}$:
$\require{AMScd}$
\begin{CD}
a\_0 @>>> \ldots @>>> a\_n @>>> a\_{n+1} @>>> \ldots\\
@Vi\_0VV \ldots @. \ldots @Vi\_nVV @Vi\_{n+1}VV \ldots \\
a @= \ldots @= a @= a @= \ldots
\end{CD}
Let $f:a \to b$ be an arrow from $a$ to $b$.
We can map $f... | https://mathoverflow.net/users/7507 | Mapping an arrow from the direct limit of a diagram to the family of arrows from the diagram | The composition of the cone over $a$ with $f$.
In case you are asking how to call the passage from $f : (\sum\_i A\_i) \to B$ to $(f\_i : A\_i \to B)\_i$, that would be "precomposition with the injections $A\_j \to \sum\_i A\_i$" but you will not like the answer because you are looking for an "established" term. Appa... | 2 | https://mathoverflow.net/users/1176 | 230144 | 107,145 |
https://mathoverflow.net/questions/202254 | 3 | Let $X$ be a variety over $\mathbb{C}$ with finite quotient singularities, i.e. every point has a Zariski-open neighbourhood isomorphic to $U/H$ where $U$ is a smooth variety and $H$ is a finite group acting on it.
Now assume we have a connected reductive group $G$ acting on $X$ with finite stabilizers, let $\mathcal... | https://mathoverflow.net/users/69630 | GIT quotient of variety with finite quotient singularities | Since I asked the question, I have come to realize that in my application, I know more about $X$ and the action of $G$. For this application, $X$ has finite quotient singularities as it is the coarse moduli space of a smooth DM stack and the action of $G$ comes from an action of the group on this stack in the sense [Ro... | 0 | https://mathoverflow.net/users/69630 | 230152 | 107,147 |
https://mathoverflow.net/questions/226186 | 9 | Let $f$ be a real valued continuous algebraic function on $\mathbb R^n$. Suppose the zero set of $f$ is bounded, i.e., if $|x|$ is large enough, $f(x)\neq 0$. Is there any estimate of the sort $|f(x)|\ge c(1+|x|)^{-N}$ for some $c>0$ and $n\in\mathbb N$? A prototypical example of such an inequality is $\sqrt{1+x^2}-x\g... | https://mathoverflow.net/users/37103 | Decay of real continuous algebraic functions at infinity | Your function $f$ is, in particular, a continuous semi-algebraic function on $\mathbf R^n$, i.e., its graph is a semi-algebraic subset of $\mathbf R^{n+1}$. Such functions are known to have sub-polynomial growth. More precisely (cf. Bochnak, Coste, Roy: Real algebraic geometry, Proposition 2.6.2, p. 43):
Let $S\subse... | 2 | https://mathoverflow.net/users/85592 | 230158 | 107,148 |
https://mathoverflow.net/questions/230154 | 5 | **Question.** Is it true that to check that a model category is right proper, it suffices to check the property for weak equivalences with fibrant codomain ? (if the domain is also fibrant, the pullback is always a weak equivalence). Or is there a close statement that I can't remember (browsing nLab did not help me) ?
... | https://mathoverflow.net/users/24563 | How to simplify the proof of right-properness? | To complete the argument you need to apply K. Brown's Lemma. Call your model category $\mathcal{M}$, then the map $Z \to Y$ induces a pullback functor $\mathcal{M} \downarrow Y \to \mathcal{M} \downarrow Z$ and the lemma implies that it preserves weak equivalences between fibrations over $Y$. If you define $V \to Y$ as... | 5 | https://mathoverflow.net/users/12547 | 230167 | 107,152 |
https://mathoverflow.net/questions/230153 | 3 | This question is motivated by [Frankl's union-closet sets conjecture](https://en.wikipedia.org/wiki/Union-closed_sets_conjecture).
Let $n\in\mathbb{N}$ and set $[n] = \{0,1,\ldots,n\}$. We say that a family ${\cal A} \subseteq {\cal P}([n])$ is *union-closed* if $\emptyset\notin{\cal A}$ and $A,B\in {\cal A}$ implie... | https://mathoverflow.net/users/8628 | Discrepancy of elements in minimal members of a union-closed set | Surely yes.
Firstly, we may achieve $\max\{w(j)\colon j\in M\}=|\mathcal A|$ for free. Just take any union-closed family of subsets on $[n-1]$, and add $n$ to every set.
So it suffices to make $\min\{w(j)\colon j\in M\}$ small compared with $\mathcal A$. To make this, take $2^{[n-2]}\setminus\{\varnothing\}$ and ad... | 4 | https://mathoverflow.net/users/17581 | 230178 | 107,157 |
https://mathoverflow.net/questions/229875 | 12 | Let $C\_\mu$ be the size of the conjugacy class in $S\_n$ of permutations whose cycletype is the partition $\mu\vdash n$. Let $\chi$ be the characters of the irreducible representations of $S\_n$.
Let $\omega\vdash m$ and let $\theta\vdash(n+m)$. I am interested in the sum
$$ \frac{1}{n!}\sum\_{\mu\vdash n} C\_\mu\c... | https://mathoverflow.net/users/83671 | A sum over characters of the symmetric group | Since $\chi\_\theta(\mu\cup\omega)=\langle s\_\theta,p\_\mu p\_\omega\rangle$, your sum is given by
$$ \frac{1}{n!}\sum\_{\mu\vdash n}
C\_\mu\chi\_\lambda(\mu)\langle s\_\theta,p\_\mu p\_\omega\rangle =
\left\langle s\_\theta,p\_\omega\cdot \frac{1}{n!}\sum\_{\mu\vdash
n} C\_\mu \chi\_\lambda(\mu)p\_\mu\right\rangl... | 12 | https://mathoverflow.net/users/2807 | 230182 | 107,158 |
https://mathoverflow.net/questions/230129 | 0 | In the proof of the existence of weak solutions to the NSE (*[Navier-Stokes Equations](http://www.ams.org/mathscinet/search/publdoc.html?arg3=&co4=AND&co5=AND&co6=AND&co7=AND&dr=all&pg4=AUCN&pg5=TI&pg6=PC&pg7=ALLF&pg8=ET&review_format=html&s4=Foias&s5=Navier-Stokes%20Equations&s6=&s7=&s8=Books&vfpref=html&yearRangeFirs... | https://mathoverflow.net/users/nan | In the proof of the existence of weak solutions to the NSE | *To provide some context*, first let $\Omega$ be a bounded domain in $\mathbb{R}^3$ with smooth boundary.
The popular way of constructing weak solutions is by Galerkin approximation with the eigenfunctions of the Stokes operator (due to Hopf in 1951, or see Ladyzhenskaya's [book](http://rads.stackoverflow.com/amzn/cl... | 4 | https://mathoverflow.net/users/nan | 230184 | 107,160 |
https://mathoverflow.net/questions/230143 | -1 | Let $A$ be a well-quasi-ordered infnite set. Does there exist an order-preserving bijection $f:A\to A^\*$, where $A^\*$ is the free monoid over $A$ under the subword ordering? Would this subword ordering make $A^\*$ well-quasi-ordered as well (reference to Corollary $1.7$ of [2]).
References:
1. <http://www.math.ha... | https://mathoverflow.net/users/82691 | Order-Perserving Bijection $f:A\to A^*$? | Unless I have misunderstood, the answer seems to be no.
Let $A=\omega+1$ with the usual order, or indeed, any well-quasi-ordering with an object having infinitely many predecessors. Such an object would get mapped to an element of $A^\*$, which we can think of as a finite term over $A$, and every such term has only ... | 1 | https://mathoverflow.net/users/1946 | 230193 | 107,161 |
https://mathoverflow.net/questions/230139 | 4 | Consider the ribbon category of finite-dimensional representations of $\mathcal{U}\_q(\mathfrak{sl}(2))$, with twist $\theta$. If $V$ is the vector representation, then $\theta\_V$ is multiplication by $q^{-3}$. As described e.g. in Snyder-Tingley <http://arxiv.org/abs/0810.0084>, one could use an alternate twist $\the... | https://mathoverflow.net/users/8041 | Can the ribbon category of f.d. reps of $\mathcal{U}_q(\mathfrak{sl}(2))$ be modified so the twist is trivial on the vector representation? | There is a category built from the HOMFLYPT skein relation in the same way that $\mathcal U\_q(\mathfrak sl(2))$ is built from the Kauffman skein relation. It is a version of $\mathcal U\_q(\mathfrak gl(t))$, where $t$ is a variable which I will use to parameterize the value of the unknot, and where I don't have the de... | 3 | https://mathoverflow.net/users/78 | 230194 | 107,162 |
https://mathoverflow.net/questions/230162 | 13 | A student of mine keeps coming again and again and telling *"I've found a formula $n\mapsto f(n)$ giving all primes"* or sometimes *"infinitely many primes"*, where $f$ is a classical function (I mean made of exponentials and polynomials, just like Mersenne primes). Then follows a long discussion, me saying *"nice try,... | https://mathoverflow.net/users/39552 | Elementary prime-generating sequences | Do a web search for Diophantine Representation of prime numbers. You will find an article of that title containing a polynomial of degree 25 in 26 variables whose range intersected with the positive integers is exactly the prime numbers.
There are general results for polynomials of one variable which extend to severa... | 7 | https://mathoverflow.net/users/3402 | 230196 | 107,163 |
https://mathoverflow.net/questions/230127 | 15 | Prove, if possible in an elementary way, that $\sum\_{n=1}^{\infty}\frac{1}{p\_n(p\_{n+1}-p\_n)}$ converges/diverges, where $p\_n$ denotes the $n^{\textrm{th}}$ prime.
| https://mathoverflow.net/users/85402 | Does the sum $\sum_{n=1}^{\infty}\frac{1}{p_n(p_{n+1}-p_n)}$ converge? | By popular demand, I am converting my comment above to an answer. Yes, the series converges. In the paper
Erdös, Paul(H-AOS); Nathanson, Melvyn B.(1-CUNY7) On the sum of the reciprocals of the differences between consecutive primes. Number theory (New York, 1991–1995), 97–101, Springer, New York, 1996
the authors s... | 23 | https://mathoverflow.net/users/16510 | 230201 | 107,166 |
https://mathoverflow.net/questions/229890 | 10 | Suppose that $\Gamma$ is a finitely generated nonsolvable subgroup of $GL(n, R)$. Is it in the literature that $\Gamma$ has a nonsolvable finite quotient? I know how to prove it (the hardest ingredient is due to Nori), but would prefer to give a reference instead of a proof.
| https://mathoverflow.net/users/21684 | Nonsolvable finite quotients of matrix groups | It is a straightforward consequence of the combination of two old results of Malcev and most likely Malcev was aware of this consequence.
1) The first is the well-known Malcev's residual finiteness result (every finitely generated linear group over, say, a field, is residually finite), or rather the stronger result t... | 7 | https://mathoverflow.net/users/14094 | 230207 | 107,168 |
https://mathoverflow.net/questions/230209 | 2 | Let $C$ be the Cantor set as a compact Abelian topological group, isomorphic to countable product of $\mathbb{Z}/2\mathbb{Z}$.
Its normalized Haar measure is denoted by $\mu$.
>
> Is there a positive continuous map $f:C \to \mathbb{Q}$ which is not a locally constant map but satisfy $\int\_{C} f^{2}d\mu=1 $?
>
>... | https://mathoverflow.net/users/36688 | Rational valued functions on the Cantor set with $\int_{C} f^{3}d\mu=1 $ | No. For example, let $S\_n=\{x\in C\colon x\_1=x\_2=\ldots=x\_{n-1}=0; x\_n=1\}$.
Now inductively choose a sequence of rationals $q\_n$ such that $\sum\_{n=1}^N q\_n^2\mu(S\_n)\in (1-2\cdot 3^{-N},1-3^{-N})$. You can check you have $q\_n=\Theta(2^n/3^n)$, so that 1) $f$ takes countably many values; 2) $f(x)\to 0$ as $x... | 3 | https://mathoverflow.net/users/11054 | 230215 | 107,171 |
https://mathoverflow.net/questions/230145 | 2 | Is there any sufficient condition in terms of moments under which
$$ \sum\_{n=1}^{\infty} X\_n$$ diverges a.s.?Here $X\_n$ are not independent
I am given that $\sum\_n E[X\_n]$ diverges. Actually, I am interested in the following special case:
Let $\{F\_n\}$ be a filtration, and $\{A\_n\}$ be a sequence of events ... | https://mathoverflow.net/users/77923 | Divergence of general random series and a special case | Here is one simple sufficient condition:
$\sum \mathsf{E} X\_n$ diverges, and $\sum (\mathsf{Var} X\_n)^{1/2} < \infty$.
Indeed, the variance condition ensures that $\sum (X\_n - \mathsf{E} X\_n)$ converges in $L^2$, so if $\sum \mathsf{E} X\_n$ diverges, so does $\sum X\_n$ (say, in probability).
| 3 | https://mathoverflow.net/users/22758 | 230222 | 107,173 |
https://mathoverflow.net/questions/230224 | 13 | The splitting principle is as follows.
>
> Given a vector bundle $E \to X$ with $X$ compact Hausdorff, there is a compact Hausdorff space $F(E)$ and a map $p: F(E) \to X$ such that the induced map $p^\*: K^\*(X) \to K^\*(F(E))$ is injective and $p^\*(E)$ splits as the sum of line bundles.
>
>
>
My question is,... | https://mathoverflow.net/users/86235 | Intuition/idea behind a proof of the splitting principle? | In the topological language you are using, $F(E)$ is the space of "orthogonal splittings". That is to say, $p^{-1}(x)$ is the space of all ways to write the fiber $E\_x$ as an orthogonal sum of one dimensional spaces. Since it is the "space of splittings", there is a tautological splitting over it. "$\square$"
---
... | 8 | https://mathoverflow.net/users/297 | 230229 | 107,176 |
https://mathoverflow.net/questions/230243 | 2 | **Definition:**
A graph $G$ is chordal if every induced cycle in $G$ has length 3, and is co-chordal if the complement graph $G^c$ is chordal.The co-chordal
cover number, denoted $cochord (G)$, is the minimum number of co-chordal subgraphs required to cover the edges of $G$.
**Definition:**
Induced matching in a gr... | https://mathoverflow.net/users/85929 | Induced matching of cycle | **Yes.** Let $C\_n$ be the cycle on $n$ vertices. The size of a largest induced matching in $C\_n$ is exactly $\lfloor \frac{n}{3} \rfloor$, since we can take at most every third edge of $C\_n$. On the other hand, since $K\_n$ minus a $3$-edge path is chordal, we have $cochord(C\_n) \leq \lceil \frac{n}{3} \rceil$. Thu... | 2 | https://mathoverflow.net/users/2233 | 230245 | 107,182 |
https://mathoverflow.net/questions/230070 | 3 | Given two real $BV$ functions $u$ and $v$ in an open interval $(a,b)$ consider the set
$A=\{x: \text{both } u \text{ and } v \text{ are continuous at } x \text{ and } u(x)=v(x)\}$
is it true that $|Du|(A)=|D(v)|(A)$?
At first glance it seems absurd that something like that holds but I find it hard to come up with... | https://mathoverflow.net/users/86159 | Is the variation of two BV functions the same in the set in which they coincide? | Yes, this works. Let me write $\mu$ and $\nu$ for the (signed) measures induced by $u$ and $v$, respectively. I then claim that $\mu|\_A=\nu|\_A$, and this will then imply that the total variations agree also.
To prove my claim, it suffices to discuss $\mu(U), \nu(U)$ for open $U\supseteq B$, where $B$ is a Borel sub... | 0 | https://mathoverflow.net/users/48839 | 230250 | 107,185 |
https://mathoverflow.net/questions/230260 | 2 | Consider the statement
>
>
> >
> > For any infinite set $X$ there is an injection $\varphi$ from $(X\times\{0\}) \cup (X\times\{1\})$ into $X$.
> >
> >
> >
>
>
>
Does this imply the ${\sf AC}$?
| https://mathoverflow.net/users/8628 | Does $|(X\times\{0\}) \cup (X\times\{1\})| \leq |X|$ for $X$ infinite imply ${\sf AC}$? | It's way over my head, but it seems that this question was answered in the negative by Gershon Sageev, An independence result concerning the axiom of choice, Ann. Math. Logic 8 (1975), 1-184.
| 4 | https://mathoverflow.net/users/43266 | 230267 | 107,190 |
https://mathoverflow.net/questions/230270 | 2 | This question is motivated by [Frankl's union-closet sets conjecture](https://en.wikipedia.org/wiki/Union-closed_sets_conjecture).
Let $X$ be a non-empty set. We say that a family ${\cal A} \subseteq {\cal P}(X)$ is *union-closed* if $\emptyset\notin{\cal A}$ and $A,B\in {\cal A}$ implies $A\cup B\in{\cal A}$.
For ... | https://mathoverflow.net/users/8628 | Frankl's union-closed sets conjecture for infinite families | **Counterexample.** Identify the infinite cardinal $\kappa$ with its initial ordinal, and let $\mathcal A$ be the set of all final segments of $\kappa.$
| 9 | https://mathoverflow.net/users/43266 | 230272 | 107,191 |
https://mathoverflow.net/questions/230271 | 2 | This is a follow-up question to [Does $|(X\times\{0\}) \cup (X\times\{1\})| \leq |X|$ for $X$ infinite imply ${\sf AC}$?](https://mathoverflow.net/questions/230260/does-x-times-0-cup-x-times-1-leq-x-for-x-infinite-imply-s)
Consider the statements
>
>
> >
> > $(\text{S}1)$ For any infinite set $X$ there is an in... | https://mathoverflow.net/users/8628 | Some very weak statements on choice | $S1$ is equivalent to "every infinite set is Dedekind-infinite", so $\sf ZF<ZF+(S1)$, because it is consistent that infinite Dedekind-finite sets exist.
$\sf ZF+(S1)<ZF+(S2)$ follows using results from [this answer](https://math.stackexchange.com/questions/393196/the-relationship-of-frak-mm-m-to-ac), which implies th... | 4 | https://mathoverflow.net/users/30186 | 230273 | 107,192 |
https://mathoverflow.net/questions/230281 | 1 | It is known that each non-decreasing continuous function $\phi$ induces a $\sigma$-additive measure $d\phi$ such that $\int\_0^1 f(x) d\phi(x)$ exists for every bounded real-valued Baire function $f$. This follows because every Borel set is measurable with respect to $d\phi(x)$, and for every Baire subset $A \subseteq ... | https://mathoverflow.net/users/12976 | A question about Borel sets on the unit interval | Unless I'm missing something, your measures $d\phi$ are precisely the atomless finite Borel measures (equivalently, Baire measures) on $[0,1]$. Then your condition on $A$ is that it is [universally measurable](https://en.wikipedia.org/wiki/Universally_measurable_set). In that case, the answer to your question is No: th... | 5 | https://mathoverflow.net/users/4832 | 230286 | 107,195 |
https://mathoverflow.net/questions/230258 | 19 | Let $S$ be a scheme (although I am more than happy to have $S=\text{Spec}(k)$ for a field $k$) and $\mathsf{AlgSp}/S$ the category of algebraic spaces over $S$.
Does there exist an embedding $\varphi:\mathsf{AlgSp}/S\hookrightarrow \mathsf{LRS}$ (where $\mathsf{LRS}$ is the category of locally ringed spaces) such th... | https://mathoverflow.net/users/38867 | Algebraic spaces as locally ringed spaces | Probably you want your functor $\varphi$ to restrict to the "identity functor" on the full subcategory of algebraic spaces that are schemes (which doesn't seem to be a purely formal consequence of your hypotheses). And to be "reasonable" you likely want such a hypothetical functor $\varphi$ to carry open immersions to ... | 18 | https://mathoverflow.net/users/81332 | 230295 | 107,200 |
https://mathoverflow.net/questions/230103 | 12 | Let $G$ be a semisimple algebraic group. (I'm already interested in the case $G=SL\_2$.)
Let $\mathcal C$ be a semisimple rigid monoidal abelian category endowed with pair of exact tensor functors $Rep\_G \to \mathcal C \to Vect$ that whose composition is the forgetful functor $Rep\_g \to Vect$. Assume every object o... | https://mathoverflow.net/users/18060 | Is there a finite test for isomorphisms of rigid monoidal abelian categories? | In the situation you describe here, the category $\mathcal{C}$ will automatically be symmetric, and the functors will automatically be symmetric functors. The reason is the following: If you have two objects $X$ and $Y$ of $\mathcal{C}$ which are direct summands of $F(A)$ and $F(B)$ respectively (where $F:Rep-G\rightar... | 4 | https://mathoverflow.net/users/41644 | 230301 | 107,204 |
https://mathoverflow.net/questions/230299 | 4 | I am looking for a proof that:
$$\prod\_\limits{m=0}^\infty \dfrac{1}{(1-x^{5m+1})}=\sum\_\limits{i=0}^\infty \dfrac{x^i}{\prod\_\limits{j=1}^i (1-x^{5j})}$$
The left hand side expands into:
$$\dfrac{1}{(1-x)(1-x^6)(1-x^{11})\dots}$$
which gives the number of partitions into parts $1\mod5$.
The right hand sid... | https://mathoverflow.net/users/70355 | mod 5 partition identity proof | More generally, there is an equality of power series in two variables
$$
\prod\_\limits{m=0}^\infty \dfrac{1}{(1-t x^{5m+1})}=\sum\_\limits{i=0}^\infty \dfrac{t^i x^i}{\prod\_\limits{j=1}^i (1-x^{5j})}.
$$
The coefficient of $t^i x^n$ on the left hand side is the number of partitions of $\frac{n-i}{5}$ into at most $i$... | 12 | https://mathoverflow.net/users/5263 | 230306 | 107,205 |
https://mathoverflow.net/questions/230279 | 8 | Is there a graph with maximum degree three that has 3 degree two vertices that must get the same (resp. different) color in every 3-coloring of the graph?
I'm interested in *any* similar results as well.
ps. Note that if we change the maximum degree to four, then it is easy to construct such graphs. Just take a lon... | https://mathoverflow.net/users/955 | How many uniquely colored degree two vertices in 3-coloring of subcubic graph? | No such graph exists (that is, you cannot have a subcubic graph with three degree-$2$ vertices all forced to the same color). Suppose that such a graph exists; we may assume the graph is connected. The vertices of degree $2$ forced to the same color must be pairwise nonadjacent (otherwise, your graph has no $3$-colorin... | 8 | https://mathoverflow.net/users/6322 | 230329 | 107,214 |
https://mathoverflow.net/questions/229869 | 2 | Assume $X$ is a smooth projective curve of genus $g\geq 3$ over $\mathbb{C}$ and let $M$ be the (singular) moduli space of semistable rank two vector bundles with trivial determinant on $X$. Then there is a desingularization of $M$ due to a more general construction of Kirwan.
As far as I understand the idea, one use... | https://mathoverflow.net/users/70593 | What finite groups are stabilizers in Kirwan's desingularization construction? | I think the non-trivial ones are $\mathbb{Z}/2$ and $\mathbb{Z}/2 \oplus \mathbb{Z}/2$. This is proved on page 24 of O'Grady's paper : <http://arxiv.org/pdf/alg-geom/9708009.pdf>
Vorsicht : O'Grady deals in that paper with the moduli space of semis-stable sheaves (with respect to a generic polarization) of rank $2$ w... | 1 | https://mathoverflow.net/users/37214 | 230351 | 107,222 |
https://mathoverflow.net/questions/230342 | 4 | I'd like some introductory references for deformation theory in algebraic geometry. I'm interested in survey articles too but I primarily want references which give all the definitions and go through the basics carefully and also give some idea of the link between deformation theory and intersection theory and/or K-the... | https://mathoverflow.net/users/48273 | Reference request for an introduction to deformation theory in algebraic geometry | You could try Sernesi's book (deformations of algebraic schemes).
| 5 | https://mathoverflow.net/users/13503 | 230353 | 107,223 |
https://mathoverflow.net/questions/230332 | 2 | Let $S\_k$ be the space of weight 12 cusp forms of $\Gamma\_0(p)$, ($p$ prime), then Sage tells that $\dim S\_k^{\text{new}}=\dim S\_k-2$. Thus the old forms spans a 2-dimensional subspace. One of the obvious eigenbase vectors is $\Delta(z)=q\prod\_{n=1}^\infty(1-q^n)^{24}$. What does the other eigenbase vector look li... | https://mathoverflow.net/users/31134 | Weight 12 cusp forms for $\Gamma_0(p)$ | Per suggestion of GH from MO in the comments, I'm turning my comment into an answer.
In general, the space $S\_{k}(\Gamma\_0(N))$ decomposes as $S\_{k}^{\text{new}}(\Gamma\_0(N)) \oplus S\_{k}^{\text{old}}(\Gamma\_0(N))$. Here the space of oldforms is spanned by the forms $f(dz)$, where $f$ is a newform of weight $k$... | 11 | https://mathoverflow.net/users/30726 | 230354 | 107,224 |
https://mathoverflow.net/questions/229714 | 40 |
>
> Let $(a\_{n})\_{n \ge 1}$ be a sequence of integers such that for all $n \ge 2$:
>
>
> $0\le a\_{n-1}+\frac{1-\sqrt{5}}{2}a\_{n}+a\_{n+1} <1$.
>
>
> Prove that the sequence $(a\_{n})$ is periodic.
>
>
>
This question was asked at the Miklos Schweitzer Competition 2005, [problem 2](http://www.math.u-szeg... | https://mathoverflow.net/users/70464 | The sequence $a_{n+1}=\left\lceil \frac{-1+\sqrt{5}}{2}a_{n}-a_{n-1} \right\rceil$ is periodic | Define the Fibonacci numbers by $f\_0 = 0$, $f\_1 = 1$, $f\_{n+1} = f\_n + f\_{n-1}$, and the golden ratio by $\phi = \frac{\sqrt{5}+1}{2}$. Then it is easy to check that $f\_n = \frac{\phi^n - (\tfrac{-1}{\phi})^n}{\sqrt{5}}$ and $(\phi-1)f\_n = f\_{n-1}-(\tfrac{-1}{\phi})^n$.
We will show that if five terms in a ro... | 14 | https://mathoverflow.net/users/2363 | 230364 | 107,225 |
https://mathoverflow.net/questions/230315 | 5 | I recently read a footnote in [Chern's article](http://www.ams.org/notices/199609/chern.pdf) stating that a non-Riemmanian Finsler manifold does not possess normal coordinates.
As I'm still new to non-Riemmanian Finsler geometry I don't see why this is the case. Can someone provide me with a simple example in conjun... | https://mathoverflow.net/users/36886 | No normal coordinates on general Finsler manifolds | I think it's important to keep two things separate here:
First, if $(M,F)$ is a smooth Finsler manifold (which means that $F^2:TM\to [0,\infty)$ is smooth and strongly convex away from the zero section of $TM$), then the unit sphere bundle $\Sigma\subset TM$ (aka the *tangent indicatrix*) is a smooth hypersurface in... | 14 | https://mathoverflow.net/users/13972 | 230369 | 107,227 |
https://mathoverflow.net/questions/230255 | 6 | Let $(V,\omega)$ be a $2n$-dimensional linear symplectic space, and $(\mathbb{P}V,\theta\_\omega)$ the corresponding $(2n-1)$-dimensional contact manifold.
Given a smooth $(n-1)$-dimensional smooth projective variety $X\subset \mathbb{P}V$, I can define its **orthogonal**
$$
X^\perp:=\{ \pi\in \mathbb{P}V^\*\mid\pi=v... | https://mathoverflow.net/users/22606 | Symplectic orthogonality and projective duality: how do they work together? | **Corrected (Partial) Answer**
Given a smooth variety $X\subset \mathbb{P}V$ that is the projectivization of a smooth punctured cone $\hat X\subset V\setminus\{0\}$, the space $X\_\bullet$ that the OP defines is a variety in $\mathbb{P}V$ described as
$$
X\_\bullet
= \bigl\{[v]\in\mathbb{P}V\ \bigl|\ \exists w\in \h... | 4 | https://mathoverflow.net/users/13972 | 230376 | 107,228 |
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