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https://mathoverflow.net/questions/283259 | 11 | Given a Riemannian manifold, I have a notion of volume for each of my chains, so it makes sense to ask for a representative of a homology class with the smallest volume. Are there conditions for when such a representative exists?
I know it is not always possible: for example, the generator of $H^1$ of the punctured ... | https://mathoverflow.net/users/57044 | Smallest volume representatives of homology | Finding volume-minimizing representatives of homology classes is one of the major applications of [geometric measure theory](https://en.wikipedia.org/wiki/Geometric_measure_theory).
It's a theorem of [Federer and Fleming](https://www.jstor.org/stable/1970227) that any nontrivial integral $k$-homology class of a smoo... | 15 | https://mathoverflow.net/users/353 | 283265 | 125,297 |
https://mathoverflow.net/questions/283251 | 5 | Consider a connected (we define connected components by defining the set of vertices where every vertex has one neighbour) sublattice $V$ of the square lattice $V \subset\mathbb{Z}^2.$
On this we define the discrete Laplacian as $T:\ell^2(V) \rightarrow \ell^2(V)$ by
$$ (Tf)(x)=\sum\_{y \text{ neighbour of } x}(f(y... | https://mathoverflow.net/users/115840 | The spectrum of the discrete Laplacian | Yes, I'm sure this is possible, and I think there will be many ways to do it, but I don't have a fully worked out rigorous argument right now. But let me throw out some ideas. First of all, a 1D Laplacian with a random potential (aka *Anderson model*)
$$
(Hf)(n) = f(n+1)+f(n-1)+q(n)f(n)
$$
on $\ell^2(\mathbb Z)$ will h... | 3 | https://mathoverflow.net/users/48839 | 283267 | 125,298 |
https://mathoverflow.net/questions/283233 | 10 | I asked this [question](https://math.stackexchange.com/q/2463353/660) on Mathematics Stack Exchange, but got no answer.
I don't understand why the definition of a sheaf (Definition 17.3.1 (ii)) given in the book
[KS] **[Categories and Sheaves](https://doi.org/10.1007/3-540-27950-4)** by Kashiwara and Schapira
is ... | https://mathoverflow.net/users/461 | Equivalence of the definitions of a sheaf in SGA4 and in "Categories and Sheaves" | $\DeclareMathOperator\im{im}$Actually I couldn't quite figure out how to do a Ken Brown sort of argument, but here's an argument that works:
Let $L$ denote the *usual* sheafification functor, a la Grothendieck and Verdier etc. Then I claim it's enough to show $L$ sends local epimorphisms to epimorphisms and local mon... | 10 | https://mathoverflow.net/users/6936 | 283271 | 125,301 |
https://mathoverflow.net/questions/283240 | 5 | Let $V$ be a finite-dimensional complex vector space with a linear action of a complex reductive group $G$. Suppose that $\omega\_0$ and $\omega\_1$ are two $G$-invariant complex symplectic bilinear forms on $V$. Is there always a $G$-equivariant isomorphism $\varphi:V\to V$ such that $\varphi^\*\omega\_1=\omega\_0$?
... | https://mathoverflow.net/users/115839 | Equivalence of $G$-invariant symplectic forms | It is true that any two $G$-invariant symplectic forms on a representation are equivalent. A proof of this fact is provided in F. Knop: "Classification of multiplicity free symplectic representations", J. Algebra 301 (2006) 531–553. See Thm 2.1 (b). Since the question is so natural, I suspect that thismust have been ob... | 7 | https://mathoverflow.net/users/89948 | 283280 | 125,303 |
https://mathoverflow.net/questions/283284 | 14 | Let $A$ and $B$ be two $n \times n$ matrices with entries in $\mathbb Z\_p$, the $p$-adic integers. Is it true that $A$ and $B$ are conjugate iff they're conjugate over $\mathbb Q\_p$ and over $\mathbb F\_p$?
| https://mathoverflow.net/users/16914 | Similar matrices over $\mathbb Z_p$ | A counterexample is
$$
A=\left[\begin{array}{cc}0&2\\8&0\end{array}\right],\hspace{5mm}B=\left[\begin{array}{cc}0&4\\4&0\end{array}\right]\in M\_2(\mathbb{Z}\_2).
$$
The matrices are conjugate in $\mathbb{Q}\_2$ because they have the same eigenvalues $\pm 4$, and they are conjugate in $\mathbb{F}\_2$ because both are $... | 23 | https://mathoverflow.net/users/5263 | 283286 | 125,305 |
https://mathoverflow.net/questions/283277 | 2 | Let $(S,\circ)$ be a semigroup with identical involution. Of course, we know that products of positive definite functions on $S$ are again positive definite. I'm interested in the other direction, meaning:
Suppose $V\colon S \to\mathbb{R}$ is a positive definite function, which factorises as $V=V\_{1}\cdot V\_{2}$ (... | https://mathoverflow.net/users/85194 | Factorisation of positive definite functions | The answer is no, $V\_{1}$ is not positive definite in general. E.g., let $(S,\circ)=(\mathbb R,+)$. Let $V(x)=1$ and $V\_2(x)=(e^x+e^{-x})/2$ for $x\in\mathbb R$. Then $V$ and $V\_2$ are positive definite (with respect to the identical involution).
However, $V\_1=V/V\_2=1/\cosh=\text{sech}$ is not positive definite.... | 4 | https://mathoverflow.net/users/36721 | 283296 | 125,310 |
https://mathoverflow.net/questions/283214 | 0 | Let $\theta\_\alpha$ be the smallest ordinal such that there is no first-order formula $\phi$ such that:
$$\exists\beta\_0,\beta\_1...\beta\_n\in\alpha\forall x(\phi(x,\theta\_{\beta\_0},\theta\_{\beta\_1}...\theta\_{\beta\_n})\Leftrightarrow x=\theta\_\alpha)$$
Unless $\alpha=0$, in which case $\theta\_0$ is the s... | https://mathoverflow.net/users/111429 | The smallest initial ordinal which is not defined using first-order formulas with parameters of smaller ordinals? | Under reasonable assumptions (to avoid such things as pointwise-definable models), there is indeed a $\kappa$ with $\theta\_\kappa=\kappa$ and $cf(\kappa)=\omega$.
To see this, let $M(\alpha)$ be the supremum of the ordinals definable with parameters $<\alpha$. *Note that $M(\alpha)$ is not necessarily defined in an ... | 2 | https://mathoverflow.net/users/8133 | 283303 | 125,311 |
https://mathoverflow.net/questions/283272 | 14 | By doing some calculations on the generating function of matching polynomials of cycles I made the following interesting observation:
* For all positive integers $n>1$ and $k <n $, the number of matchings of size $k $ in $C\_{2n} $ is equal to the number of matchings of the same size in the disjoint union of two $C\_... | https://mathoverflow.net/users/51663 | Number of matchings of even cycles | There is in fact a topological(?) proof of this statement and the following generalization, essentially due to [Péter Csikvári](http://math.mit.edu/~csikvari/Schrijver_Gurvits_LMC_V6) (Section 4, Lemma 4.2).
We define a double cover (or ''2-lift'') $H$ of a graph $G$ as follows:
consider $G$ as a topological space w... | 8 | https://mathoverflow.net/users/81295 | 283318 | 125,316 |
https://mathoverflow.net/questions/283310 | 5 | For $n\in\mathbb{N}^{+}$, let $c\_{n}$ denote the number of simple non-isomorphic cycle matroids of graphs on $n$ vertices. That is, let
$$A(n)=\{M(G)\;;\;G\text{ is a graph on }n\text{ vertices}\},$$
and let $B(n)$ be a largest subset of $A(n)$ such that no two elements of $B(n)$ are isomorphic (as matroids). Then... | https://mathoverflow.net/users/70629 | What upper bounds are known on the number of non-isomorphic cycle matroids? | It seems now that the question really boins down to the number of non-isomorphic graphs on $n$ vertices. Denote this number by $f(n)$. Clearly,
$$
2^{n\choose 2}\geq f(n)\geq \frac{2^{n\choose 2}}{n!}=\frac{2^{n\choose 2}}{n^{O(n)}},
$$
these two numbers being sufficiently close on the logarithic scale. (Moreover,
on... | 4 | https://mathoverflow.net/users/17581 | 283333 | 125,322 |
https://mathoverflow.net/questions/283331 | 5 | Consider the set of irrationals $\mathbb{I} \cap (0,1)$. Define the map $f$ that takes
$x= \sum\_{n\ge 1} \frac{a\_n}{3^n}$ to $\sum\_{n\ge 1} \frac{ \phi(a\_n)}{3^n}$, where $\phi$ is the permutation of $\{0,1,2\}$, $0\mapsto 1 \mapsto 2\mapsto 0$. Is it possible that both $x$ and $f(x)$ are algebraic for some $x$ ... | https://mathoverflow.net/users/47264 | Changing values of digits of an algebraic irrational number | Let $x\_j = \sum\_{n \geq 1} \mathrm{1}\_{a\_n=j} 3^{-n}$. Then
$$
x\_0 + x\_1 + x\_2 = \frac{1}{2} \\
x\_1 + 2x\_2 = x \\
x\_0 + 2x\_1 = f(x).
$$
If $x$ and $f(x)$ are algebraic then the equations above would yield that $x\_j$ is algebraic for each $j$. It is conjectured and generally believed that any irrational alg... | 9 | https://mathoverflow.net/users/21724 | 283334 | 125,323 |
https://mathoverflow.net/questions/283343 | 10 | Let $\Phi$ be a finite crytallographic root system. Let $\Phi^+$ be the positive roots and $\alpha\_1$, ..., $\alpha\_n$ be the simple roots. For $\beta = \sum c\_i \alpha\_i$ in $\Phi^+$, we define $h(\beta) = \sum c\_i$. For $\beta = \sum c\_i \alpha\_i$ and $\gamma = \sum d\_i \alpha\_i$, we define $\beta \preceq \g... | https://mathoverflow.net/users/297 | Why is the root poset is graded by height? | Let $(\cdot,\cdot)$ be a positve definite Weyl group invariant product on $\mathbb{R}\Phi$. Let $\beta$ and $\gamma$ be positive roots with $\beta\leq \gamma$ and $h(\gamma)-h(\beta)\geq 2$. Let $v=\gamma-\beta$.
Let $\alpha\_i$ be a simple root which occurs in $v$. Suppose for want of a contradiction that $\beta+\al... | 7 | https://mathoverflow.net/users/425 | 283350 | 125,326 |
https://mathoverflow.net/questions/283328 | 7 | Let $G$ be a simple graph such that $\omega(G)\leq\lfloor\frac{\Delta(G)+1}{2}\rfloor+1$ where $\Delta(G)$ is the maximal degree of $G$. Is it true that
\begin{equation}
\chi(G)\leq \lfloor\frac{\Delta(G)+1}{2}\rfloor+2?
\end{equation}
A similar problem can be found [here](http://onlinelibrary.wiley.com/doi/10.1002/(SI... | https://mathoverflow.net/users/90655 | Upper bound for chromatic number of graphs with $\omega(G)\leq\lfloor\frac{\Delta(G)+1}{2}\rfloor+1$ | As far as I know, your conjecture is an open problem, because Reed's conjecture is still open for $\omega=2$. For $\omega=2$, your conjecture is essentially equivalent to Reed's conjecture. That is, up to rounding, both conjectures assert that every triangle-free graph has chromatic number at most $\Delta/2+2$.
As m... | 6 | https://mathoverflow.net/users/2233 | 283353 | 125,327 |
https://mathoverflow.net/questions/283352 | 3 | Suppose I created the following random number generator.
A trusted person choose a irrational number. That can easily defined and computed by a computer. Like square root of a prime.
Every time the random number generator is called it calculate the first digit, then the second digit and it goes on.
On a given mom... | https://mathoverflow.net/users/49488 | Example of irrational number with a pattern in digits | $0.1234567891011121314151617181920212223242526272829303132333435363738394041424344454647484950\ldots$
| 7 | https://mathoverflow.net/users/11260 | 283359 | 125,329 |
https://mathoverflow.net/questions/283337 | 7 | My question is:
>
> If $p\colon X \to B$, $q\colon Y \to B$ are proper submersions, is there a characterization of $H\_\*(X \times\_B Y)$ in terms of $H\_\*(X)$, $H\_\*(Y)$, $H\_\*(B)$ that is simpler than the Eilenberg-Moore spectral sequence?
>
>
>
Ideally, what I would like is a chain complex $A$ built from... | https://mathoverflow.net/users/20391 | Eilenberg-Zilber-type theorem for good fiber products? | There are fibrations
\begin{align\*}
F \to X & \to B \\
G \to Y & \to B \\
F\times G \to X\times\_B Y & \to B \\
F\times G \to X\times Y & \to B\times B \\
F \to X\times\_BY &\to Y \\
G \to X\times\_BY &\to X
\end{align\*}
and various comparison maps between them. You are probably better off reasoning indirectly ... | 6 | https://mathoverflow.net/users/10366 | 283377 | 125,335 |
https://mathoverflow.net/questions/283382 | 4 | Let $X$ be a complete metric space. Are all [$\tau$-additive](http://en.wikipedia.org/wiki/%CE%A4-additivity) Borel measures on $X$ tight?
In Bogachev's "Measure Theory", vol. 2, in the proof of Theorem 8.9.4 (end of page 213) it says:
>
> Note that if $X$ is a complete metric space, then $\mathcal{M\_\tau}(X)=\m... | https://mathoverflow.net/users/30366 | $\tau$-additive measures on a complete metric space are tight | The equality $\mathcal M\_\tau(X)=\mathcal M\_t(X)$ for a complete metric space $X$ follows from three facts:
1) For any finitely additive measure $\mu$ on $X$ its support $supp(\mu)$ (i.e., the set of points $x\in X$ whose any neighborhood $O\_x$ has positive measure $\mu(O\_x)$) has countable cellularity and hence ... | 6 | https://mathoverflow.net/users/61536 | 283396 | 125,342 |
https://mathoverflow.net/questions/283418 | 1 | Let $f(x,y) \in \mathbb C\{x,y\}$ be a holomorphic function-germ at zero. Let $f\_x, f\_y$ denote its partial derivatives. What is the proof of the following statement?
>
> If the $\mathbb C$-algebra $\mathbb C\{x,y\}/(f\_x,f\_y)$ is finite dimensional, then $(f\_x,f\_y)$ is a regular sequence.
>
>
>
Saying th... | https://mathoverflow.net/users/111673 | Jacobian ideal regular sequence | Here's the "sophisticated" proof: if you have a sequence of elements $r\_1,\dots,r\_n$ in a commutative ring $R$ and the Krull dimension of $R/(r\_1,\dots,r\_n)$ is $\dim R - n$, then call the sequence a system of parameters. In a Cohen-Macaulay ring, every system of parameters is a regular sequence. Cohen-Macaulayness... | 1 | https://mathoverflow.net/users/321 | 283420 | 125,346 |
https://mathoverflow.net/questions/266718 | 2 | **Question:** I'd like to know if there is some reference or reasonable way to develop curve theory in a plane with degenerate metric $(\Bbb R^2, {\rm d}s^2 ={\rm d}x^2)$.
**Context:** In Lorentz-Minkowski space $\Bbb L^3 = (\Bbb R^3, \langle \cdot,\cdot\rangle\_L = {\rm d}x^2+{\rm d}y^2 - {\rm d}z^2)$ one can consid... | https://mathoverflow.net/users/54656 | Relation of pseudo-torsion with curvature in degenerate plane | I'll convert Robert's comment into an answer:
>
> The difficulty is that the group of symmetries of the 'metric' ${\rm d}x^2$ on $\Bbb R^2$ is the set of transformations of the form $$\phi(x,y) = (\pm x+c, f(x,y)),$$where $c$ is a constant and $f\_y(x,y)$ is nowhere vanishing. In particular, any curve of the form $... | 0 | https://mathoverflow.net/users/54656 | 283421 | 125,347 |
https://mathoverflow.net/questions/283417 | 3 | I am trying to read the paper [*Marginally Trapped Submanifolds in Space Forms with Arbitrary Signature*](https://arxiv.org/pdf/1309.3875.pdf) by Henri Anciaux, but I think that there is a mistake in Lemma $1$, in page $5$:
>
> The second fundamental form $h$ is collinear to $\nu$ if and only if the mean curvature ... | https://mathoverflow.net/users/54656 | Possible mistake in classification of marginally trapped submanifolds of $\Bbb R^{n+2}_{p+1}$ | The proof of the forward implication is perhaps more complicated than necessary, but it is true.
The proof of the reverse implication is wrong. In fact here's a counterexample. Consider the space $\mathbb{R}^{4}\_2$ with the quadruple null coordinates $(u,v,w,z)$ where the metric is $du\otimes dv + dv \otimes du + d... | 4 | https://mathoverflow.net/users/3948 | 283425 | 125,348 |
https://mathoverflow.net/questions/283111 | 6 | Let $\mathcal{E}$ be a rank two vector bundle on $\mathbb{P}^2$ fitting in the following exact sequence
$$0\rightarrow \mathcal{O}\_{\mathbb{P}^2}\rightarrow \mathcal{E}\rightarrow \mathcal{I}\_p(-1)\rightarrow 0$$
where $\mathcal{I}\_p$ is the ideal sheaf of a point $p\in\mathbb{P}^2$. Then $c\_1(\mathcal{E})=-1$ ... | https://mathoverflow.net/users/nan | Intersection numbers in $\mathbb{P}^1$-bundles | Here are the details.
Let $L\_p\cong\mathbb{P}^1$ be a line in $\mathbb{P}^2$ through $p$. Your exact sequence restricted to $L\_p$ yields the following exact sequence
$$0\rightarrow\mathcal{O}\_{\mathbb{P}^1}(1)\rightarrow\mathcal{E}\_{|L\_p}\rightarrow\mathcal{O}\_{\mathbb{P}^1}(-2)\rightarrow 0
$$
and hence $\math... | 1 | https://mathoverflow.net/users/14514 | 283428 | 125,349 |
https://mathoverflow.net/questions/283427 | 3 | Let $(X\_1, X\_2, \dots, X\_n)$ be i.i.d. ${\cal N}(0,1)$. We construct a random circulant matrix $M$:
$$M = \frac{1}{\sqrt n}\begin{pmatrix}X\_1 &X\_2 &X\_3 \dots &X\_n\\ X\_n &X\_1 & X\_2 \dots &X\_{n-1}\\ \vdots &\vdots &\vdots &\vdots\\X\_2 &X\_3 &X\_4 \dots &X\_1\end{pmatrix}.$$
My questions are the following:... | https://mathoverflow.net/users/20062 | Are random circulant matrices almost orthonormal? | The diagonal elements of $P=\frac{1}{N}MM^T$, like
$$P\_{11}=\frac{1}{N}\sum\_{i=1}^NX\_i^2,$$
satisfy $ \langle P\_{11}\rangle=1$ and $ \langle P\_{11}^2\rangle=1+2/N$ (variance decreases like $N^{-1}$).
On the other hand, off-diagonal elements like
$$P\_{12}=\frac{1}{N}\sum\_{i=1}^{N}X\_iX\_{i+1} $$
satisfy $ \lang... | 7 | https://mathoverflow.net/users/78061 | 283432 | 125,353 |
https://mathoverflow.net/questions/283434 | -2 | There are six neatly-stacked circles of radius r.
➊ Circle with centre at {−1,0}.
➌ Circle with centre at {+1,0}.
➋ Circle between ➊ and ➌ with centre at {x,0}, x ∈ ℝ, r−1 ≤ x−r, x+r ≤ 1−r.
➍ A circle is balanced atop ➊ and ➋, touching both.
➎ A circle is balanced atop ➋ and ➌, touching both.
➏ And a circle... | https://mathoverflow.net/users/36186 | Six stacked circles, not quite symmetrical | I don't know if this is really research-level math, but here is a quick answer. For each $i=1,\dotsc,6$, let $P\_i$ be the center of circle (i) and write $P\_i=(x\_i,y\_i)$. We have $P\_1=(-1,0)$, $P\_3=(1,0)$, and $P\_2=(x,0)$. Since Circle (4) is stacked on top of circles (1) and (2), $x\_4 = \frac{x\_1+x\_2}{2} = \f... | 0 | https://mathoverflow.net/users/88133 | 283441 | 125,356 |
https://mathoverflow.net/questions/283419 | 19 | Given a finite CW or simplicial decomposition of a space $X$ and a ring homomorphism $\varphi:\mathbb{Z}[\pi\_1(X)]\to F$ for a field $F$, if the $\varphi$-twisted homology is trivial, then the Reidemeister-Franz torsion $\tau^\varphi(X)\in F$ is an invariant of the twisted chain complex, well-defined up to multiplicat... | https://mathoverflow.net/users/43804 | Is there a geometric interpretation for Reidemeister torsion? | Yes, there is a very nice geometric interpretation over any manifold $X$ satisfying $\chi(X)=0$, for a version of Reidemeister torsion spelled out in Turaev's paper **"Euler structures, nonsingular vector fields, and torsions of Reidemeister type"**. Ian Agol's comment mentioned the Seiberg-Witten invariants, which equ... | 18 | https://mathoverflow.net/users/12310 | 283450 | 125,360 |
https://mathoverflow.net/questions/283439 | 2 | Suppose that $A$ is an bounded linear operator on a Hilbert space such that $\left\|A\right\| \leq 1$. Can we approximate $A$ by an operator $\tilde{A}$ such that $\tilde{A} = \sum\_{n=1}^N \alpha\_n R\_n$ where $\alpha\_j \in [-1, 1]$ and $R\_n$ are mutually orthogonal projections?
I asked this question on Math Stac... | https://mathoverflow.net/users/8435 | Approximation of an Operator | Assuming the $R\_n$ are *orthogonal* projections the answer is no, in general.
Since orthogonal projections on orthogonal subspaces commute, the sum $\tilde{A} = \sum\_n \alpha\_n R\_n$ commutes with its adjoint $\sum\_n \alpha\_n^\* R\_n$. Conversely, a bounded normal operator has an orthogonal spectral deposition.
... | 3 | https://mathoverflow.net/users/327 | 283452 | 125,361 |
https://mathoverflow.net/questions/283409 | 10 | If $A$ and $B$ are $C^\*$ algebras I will write $A \overline{\otimes} B$ for the maximal tensor product and $A \underline{\otimes} B$ for the minimal tensor product. If $G$ is a countable discrete group I will write $C^\*(G)$ for the full group $C^\*$ algebra of $G$. Let also $\mathbb{F}\_\infty$ be the free group on a... | https://mathoverflow.net/users/30721 | Strengthening of Connes' embedding conjecture | Your more general conjecture is not true. This property for a discrete group $G$ is equivalent to the fact that $G$ is WEP (its $C^{\ast}$-algebra has weak expectation property). In "Examples of hyperlinear groups without factorization property" Andreas Thom constructs a (hyperlinear) property (T) group that is not res... | 8 | https://mathoverflow.net/users/24953 | 283459 | 125,365 |
https://mathoverflow.net/questions/281948 | 8 | Let us call a series $\sum\_n x\_n$ in a Banach space "good" if there exists a permutation $\sigma:\mathbb N\to\mathbb N$ such that the rearranged series $\sum\_n x\_{\sigma(n)}$ converges.
Find a simple proof of the following theorem (which was proved by E.Steinitz in 1913 according to V.Kadets).
**Theorem.** A s... | https://mathoverflow.net/users/61536 | Is "weakly good" series in a finite-dimensional Banach space "good"? | A relatively short inductive proof of Steinitz Theorem can be founded in [this paper](https://arxiv.org/abs/1711.04136). Here we present a sketch of the proof, which is based on 3 lemmas whose proof is left to the reader. First we recall the formulation of the result.
**Theorem.** A series $\sum\_{n=1}^\infty x\_n$ i... | 3 | https://mathoverflow.net/users/61536 | 283461 | 125,367 |
https://mathoverflow.net/questions/278823 | 4 | Let $p$ be a prime. Let $\omega$ be a $ p $-th root of unity. We know that $ \chi\_\alpha (x) = \omega^{\alpha\cdot x} $ are the additive characters of $ \mathbb{Z}\_p $.
I have a question about bounding incomplete sums of these characters.
Let $ T $ be a subset of $ \mathbb{Z}\_p $ of size $ (p-1)/2 $. I want sta... | https://mathoverflow.net/users/113446 | About Averages of Incomplete Additive Character Sums | The trivial estimate here would be
$$ \mathbb E\,\Bigg|\sum\_{t\in T}\chi\_\alpha(t)\Bigg|
\le \Bigg( \mathbb E\,\Bigg| \sum\_{t\in T} \chi\_\alpha(t) \Bigg|^2 \Bigg )^{1/2} = \Bigg( \sum\_{t\_1,t\_2\in T} \frac1p\sum\_{\alpha\in\mathbb Z\_p} \omega^{\alpha\cdot(t\_1-t\_2)} \Bigg)^{1/2} = \sqrt{|T|} \le p^{1/2}. $$
D... | 2 | https://mathoverflow.net/users/9924 | 283466 | 125,369 |
https://mathoverflow.net/questions/282808 | 2 | Question
--------
**Is [Bregman divergence](https://en.wikipedia.org/wiki/Bregman_divergence) free of coordinates?**
Although it is invariant w.r.t. which local affine coordinate you take, is it possible to prove that it does not change w.r.t. an arbitrary change of coordinates?
(I am reading "Information Geometr... | https://mathoverflow.net/users/94075 | Is Bregman divergence independent of coordinates? | No. Away from a critical point, any function $\psi$ becomes linear in some coordinates. In such coordinates, the Bregman divergence $D\_{\psi}$ of that function $\psi$, as defined in your book (p. 14 equation (1.44)), vanishes. So the Bregman divergence of a function, as defined in your book, is not coordinate invarian... | 4 | https://mathoverflow.net/users/13268 | 283470 | 125,371 |
https://mathoverflow.net/questions/283467 | 15 | Recently I started working on a problem in Differential Geometry (where I'm not a specialist, so I apologize if this question turns out to have a trivial answer) and I had to consider the following situation.
Let $M \subset \mathbb{R}^n$ be a smooth submanifold of dimension $n-k$. By a *normal tube* of radius $\varep... | https://mathoverflow.net/users/7460 | Tubular neighborhoods of embedded manifolds | I suggest to look at the very clear proof of the tubular neighborhood theorem in
*Lee, John M.*, Introduction to smooth manifolds, Graduate Texts in Mathematics. 218. New York, NY: Springer. xvii, 628 p. (2002). [ZBL1030.53001](https://zbmath.org/?q=an:1030.53001)., pag 253, I think it's exactly what you need.
He ... | 14 | https://mathoverflow.net/users/13915 | 283477 | 125,375 |
https://mathoverflow.net/questions/283472 | 17 | Let $\mathbf B\_n$ be the braid group on $n$ strings.
>
> What is known about the cohomology of $\mathbf B\_n$ with coefficients in its integral group ring: $H^\*(\mathbf B\_n;\mathbb Z \mathbf B\_n)$?
>
>
>
| https://mathoverflow.net/users/23500 | Cohomology of braid groups with coefficients in the group ring | The braid groups $B\_n$ are Bieri-Eckmann duality groups of dimension $n-1$. It follows (either by definition or by a standard result depending on how you set things up) that $H^k(B\_n;\mathbb{Z}[B\_n])$ is $0$ for $k \neq n-1$ and is torsion-free for $k=n-1$.
Here is a brief description of how to see this. First, th... | 16 | https://mathoverflow.net/users/317 | 283478 | 125,376 |
https://mathoverflow.net/questions/283447 | 2 | I am studying an example of symplectic Lefschetz fibrations. As far as I know, given a Weinstein manifold $F$ and a collection $V\_1,\ldots,V\_k$ of exact framed Lagrangian spheres of $F$, there exists a unique up to deformation Lefschetz fibration $f:W\rightarrow\mathbb{C}$ whose regular fiber is $F$ and whose vanishi... | https://mathoverflow.net/users/115929 | Describing a Lefschetz fibration whose fiber is plumbing of $T^*S^n$ | Yes, it's possible for the specific case that you are looking at. For $k=1$, this is obvious. For $k\geq2$, such a Lefschetz fibration can be constructed by applying a standard construction to the standard Lefschetz fibration on $\mathbb{C}^{n+1}$, which is called stabilization. Concretely, a stabilization to a Lefsche... | 2 | https://mathoverflow.net/users/43423 | 283482 | 125,377 |
https://mathoverflow.net/questions/283481 | 7 | Let $f:X\rightarrow Y$ be a finite morphism between smooth projective varieties, and let $D$ be an effective nef but not ample divisor on $X$.
Consider the divisor $f\_{\*}D$ on $Y$. Is $f\_{\*}D$ nef but not ample as well ?
| https://mathoverflow.net/users/nan | Push-forward of nef divisors via finite morphisms | I suppose you want $f$ to be surjective, otherwise $f\_\*D$ is not defined. Then $f\_\*D$ is nef: for any curve $C\subset Y$, $\ (f\_\*D\cdot C)=(D\cdot f^\*C)\geq 0$. But it might very well be ample. Consider a smooth quadric $Q\subset \mathbb{P}^3$, and let $f:Q\rightarrow \mathbb{P}^2$ be the projection from a point... | 8 | https://mathoverflow.net/users/40297 | 283486 | 125,378 |
https://mathoverflow.net/questions/283444 | 8 | What is known about irreducible decomposition of tensor products of (infinite-dimensional) unitary irreducible representations of $SU(2,2)$ (or, more generally, simple groups of split rank greater than $1$)?
I am aware of some general and special results for split rank 1 groups, but I feel that the case of split rank... | https://mathoverflow.net/users/32985 | Tensor products of unitary irreducible representations of $SU(2,2)$ | People have mostly discussed tensoring of (continued) holomorphic discrete series and “ladder” representations — see all the references in Dvorsky ([2007](http://www.ams.org/mathscinet-getitem?mr=2296624)), maybe also Libine ([2017](http://www.ams.org/mathscinet-getitem?mr=3611096)).
For the principal series you ask ... | 1 | https://mathoverflow.net/users/19276 | 283487 | 125,379 |
https://mathoverflow.net/questions/283480 | -1 | In these [notes](http://ccirm.cedram.org/cedram-bin/article/CCIRM_2010__1_1_33_0.pdf) of Pezzini and the obvious question arises, is a sphere, a "spherical variety". I'm interested in $L^2(X)$ for the space:
$$ X = \{ x^2 + y^2 + z^2 = 1\} \subseteq \mathbb{R}^3 $$
In order to become a spherical variety I need a coupl... | https://mathoverflow.net/users/1358 | Is $x^2 + y^2 + z^2 = 1$ a spherical variety? | Yes, $S^2$ is $PSL\_2\mathbb{C}/B$ where $B$ is the Borel subgroup, so the Riemann sphere is spherical in that sense. Spherical varieties are called spherical because the Riemann sphere is one, with action of the rotation group: see here: [Why are they called Spherical Varieties?](https://mathoverflow.net/questions/532... | 2 | https://mathoverflow.net/users/13268 | 283491 | 125,380 |
https://mathoverflow.net/questions/273508 | 9 | While the fundamental group $\pi\_1$ preserves products, it is not true in general that an inverse limit of simply connected topological spaces is simply connected. I would like to know if similar things can happen with topological groups.
Let $G=\varprojlim\_{n}(G\_n,p\_{n+1,n}:G\_{n+1}\to G\_n)$ be the inverse limi... | https://mathoverflow.net/users/5801 | Must an inverse limit of simply connected groups be simply connected? | It seems that the inverse limit of simply connected Lie groups is simply connected. The argument uses the well-known fact that the second homotopy group of any Lie group is trivial (see e.g. [Homotopy groups of Lie groups](https://mathoverflow.net/questions/8957/homotopy-groups-of-lie-groups)).
Applying the long exac... | 5 | https://mathoverflow.net/users/61536 | 283493 | 125,381 |
https://mathoverflow.net/questions/283479 | 1 | Let $f: \mathbb Z\_{\ge 0} \to \mathbb C\_p$ be any function.
My understanding is that Mahler's theorem says
that $f$ extends to a continuous function
$f: \mathbb Z\_p \to \mathbb C\_p$ if and only if the
Mahler coefficients
\begin{align\*}
d\_0 &= f(0) \\
d\_1 &= f(1)-f(0) \\
d\_2 &= f(2)-2f(1)+f(0) \\
&\vdots \\
d\_n... | https://mathoverflow.net/users/70654 | Analytic p-adic functions from Mahler coefficients | For your first question: Theorem 54.4 (page 166) in Schikhof's "Ultrametric Calculus" says that $f$ is analytic if and only if
$$\lim\_{n\to\infty}\frac{d\_n}{n!}=0.$$
Here, analytic means that $f$ has a power series expansion which converges for all $|x|\le1$ (in $\mathbb{Q}\_p$, hence in $\mathbb{C}\_p$).
Actua... | 4 | https://mathoverflow.net/users/109085 | 283497 | 125,382 |
https://mathoverflow.net/questions/283357 | 7 | Let $R$ be a Riemann surface of genus $g\ge 2$ and $q$ an holomorphic quadratic differential on $R$. Together they determine a semi-translation structure: an atlas on $X$ such that its changes of charts are of the form $z\mapsto \pm z+c$ and a singular flat metric $|q|$ on $X$.
Suppose that $q$ has at least one singu... | https://mathoverflow.net/users/60675 | Length of simple closed curve in half-translation surface | Yes, this is true, and as far as I can tell the argument doesn't seem any different whether the surface is a pure translation surface or a semi-translation surface. The point is that the argument is purely local, and from this argument you can *deduce* from the given conditions on $c$ that the rotational holonomy aroun... | 1 | https://mathoverflow.net/users/20787 | 283501 | 125,384 |
https://mathoverflow.net/questions/283519 | 5 | Let $G = \langle V, E \rangle$ be an undirected, connected and weighted multigraph, with the weights given by a function $w: E \rightarrow N$. Consider any spanning tree $T$. Denote the edges of $T$ by $e\_1, e\_2, \ldots , e\_{|V|-1}$. Define $P\_T = \frac{\sum\_{i = 1}^{|V| - 1|} w(e\_i)}{|V| - 1}$, the arithmetic me... | https://mathoverflow.net/users/115989 | Spanning tree minimizing $F_T = \sum_{i = 1}^{|V| - 1|} (w(e_i) - P_T)^2$ | The problem is equivalent to finding $\min\_{e\_i, \lambda} \sum (w(e\_i) - \lambda)^2$, where $\lambda$ is a free real parameter subject to optimization as well as the edges of a spanning tree. Indeed, for a chosen set of edges $e\_i$ we have that $\sum (w(e\_i) - \lambda)^2$ is minimized when $\lambda$ is the mean of... | 5 | https://mathoverflow.net/users/106512 | 283523 | 125,389 |
https://mathoverflow.net/questions/283456 | 7 | Let $X$ be a noetherian local normal scheme, we may even assume that $X$ is complete if necessary.
Consider $X\times (\mathbb{A}^2-\{0\})$, is it true that the Picard group of this scheme vanishes?
| https://mathoverflow.net/users/80490 | Picard group of the product of a local normal scheme with $\mathbb{A}^2-\{0\}$ | My doubt in the first comment was incorrect. There are, indeed, many formal invertible sheaves without trivializations over $\text{Spf} \ k[[t]].$ Thus, for instance, over $R=\text{Spec}\ k[t]/\langle t^n\rangle$ there are invertible sheaves on $\mathbb{A}^2\_R\setminus\{0\}$ that are not isomorphic to the structure sh... | 5 | https://mathoverflow.net/users/13265 | 283527 | 125,390 |
https://mathoverflow.net/questions/283509 | 5 | Ordinary inductive types is initial algebras for free monads. However, HITs are not initial algebras for endofunctors but presented monads.
From nLab, [initial algebra of a presentable (infinity,1)-monad](https://ncatlab.org/nlab/show/initial+algebra+of+a+presentable+%28infinity%2C1%29-monad) and [blog comment](http... | https://mathoverflow.net/users/115253 | Models for Higher Inductive Types in Homotopy Type Theory | <https://arxiv.org/abs/1705.07088> . I've updated the nLab page.
There are also some slides available at my [web page](http://home.sandiego.edu/~shulman/papers/).
| 6 | https://mathoverflow.net/users/49 | 283532 | 125,392 |
https://mathoverflow.net/questions/283516 | 0 | $\omega\_1$ is generally referred to as an ordinal when it is rather more of a formula $\phi$ true for some sets in models of ZFC.
I started to wonder if $\omega\_1$ really *was* an ordinal, unlike $\beth\_1$ (which is a cardinal in every model but isn't always the same cardinal; it is more easily described as a form... | https://mathoverflow.net/users/115951 | On The Explicit-Ness of $\omega_1$ | CLAIM: A formula $\chi$ is $\phi$-intact in $T$ iff $T$ proves $\exists x: \phi(x)$ and moreover there is a model $M$ of $T$ satisfying $\lnot \exists x: \phi(x)\wedge\chi(x)$ (i.e., if $T$ does not prove the formula $\exists x: \phi(x)\wedge \chi(x)$).
Proof: It is clear that this property is necessary.
If $\phi... | 3 | https://mathoverflow.net/users/14915 | 283534 | 125,393 |
https://mathoverflow.net/questions/283530 | 1 | The [uniformity conjecture](https://mathoverflow.net/questions/223913/what-is-the-exact-statement-about-uniform-boundedness-of-rational-points-on-curv)
basically states that the number of rational points on a smooth curve of genus $g >1$ over number field is bounded.
If we drop smoothness, there is counterexample com... | https://mathoverflow.net/users/12481 | Can we drop smoothness in uniformity conjecture if we don't count singular points? | Yes. Consider the smooth projective model $\tilde{C}$ of your (geometrically irreducible) curve $C$ of (geometric) genus $g \ge 2$ over a number field $k$. Then the nonsingular points on $C$ are in bijection with a Zariski-open subset of $\tilde{C}$, and this bijection preserves $k$-rational points (because the canonic... | 4 | https://mathoverflow.net/users/21146 | 283535 | 125,394 |
https://mathoverflow.net/questions/283533 | 3 | Take $X$ a hypersurface in $\mathbb{C}P^{n+1}$ of degree $d$, denote $\Lambda\_X=H^n(X,\mathbb{Z})$. We know that $\Lambda\_X$ is a finitely generated Abelian group. I was wondering whether $\Lambda\_X$ is always free (for any $n$, $d$). Anyone know an answer or a refrence of this? Thanks a lot!
| https://mathoverflow.net/users/108424 | Freeness of the integral middle cohomology of a smooth hypersurface in $\mathbb{C}P^{n+1}$ | If $X$ is smooth, then by the Lefschetz hyperplane theorem, $H\_k(X; \mathbb{Z}) \cong H\_k(\mathbb{CP}^{n+1}; \mathbb{Z})$ for $k < n$. By the Universal Coefficient Theorem,
$$H^n(X; \mathbb{Z}) \cong \operatorname{Hom}(H\_n(X;\mathbb{Z}), \mathbb{Z})\oplus\operatorname{Ext}^1(H\_{n-1}(X; \mathbb{Z}), \mathbb{Z}).$... | 5 | https://mathoverflow.net/users/21564 | 283539 | 125,397 |
https://mathoverflow.net/questions/283020 | 5 | I would like to know if the following statement (or a more general version of it) is contained in some book or article:
**Statement.** Let $(U,g)$ be a complex manifold with a Kahler metric $g$ and let $x\in U$ be a point. Let $U\_x\subset U$ be a neighborhood of $x$ and $h$ be a Kahler metric on $U\_x$. Then there i... | https://mathoverflow.net/users/13441 | Varying a Kahler metric in a neighborhood of a point | This is not literally the answer that the OP wanted (a reference to the literature, which I am not aware of), but following the comments above let me write down the simple gluing argument.
Let $\widetilde{\max}$ be a regularized maximum function, and fix a small coordinate ball $B$ around $x$ (contained in $U\_x$), w... | 2 | https://mathoverflow.net/users/13168 | 283542 | 125,398 |
https://mathoverflow.net/questions/283529 | 11 | Let $X$ be a compact Riemann surface with boundary $\partial X$. Assume (for simplicity only) that $\partial X$ has a single connected component. Let us fix an orientation preserving diffeomorphism $\phi\colon S^1\tilde\to \partial X$ with the standard unit circle $S^1$ which is the boundary of the standard unit disk $... | https://mathoverflow.net/users/16183 | Gluing Riemann surfaces | The answer to the first question is "yes". This is called conformal gluing, and the proof is based on the following lemma due to Lavrentiev: Let $\phi$ be an increasing diffeomorphism of $[-1,1]$ onto itself. Then there is a simple curve
in the unit disk connecting $-1$ and $1$ breaking the disk into two domains
$D^+$ ... | 12 | https://mathoverflow.net/users/25510 | 283544 | 125,400 |
https://mathoverflow.net/questions/283518 | 4 | In page 9 of the introductory chaper of *Renormalization and Effective Field Theory* (the introductory chapter is available free [here](http://bookstore.ams.org/surv-170)), Kevin Costello defines a propagator $P$ for the Laplace operator $D$ in a Riemannian spacetime $M$.
In order to define a "length-scale version o... | https://mathoverflow.net/users/69531 | Spurious length-scale cutoff emerges in propagator defined in Costello's "Renormalization and EFT" | There is no space cutoff because it is $d(x,y)$ which depends on the two points. If you put a constant $c$ in your integral $\int\_c^\infty d\tau$ then yes you would have introduced a spurrious cutoff in position space.
If you are in $\mathbb{R}^d$ you can write the free massless propagator as
$$
(-\Delta)^{-1}(x,y)=\i... | 5 | https://mathoverflow.net/users/7410 | 283546 | 125,401 |
https://mathoverflow.net/questions/283295 | 2 | Let
* $(\Omega,\mathcal A,\operatorname P)$ be a probability space
* $T>0$
* $(\mathcal F\_t)\_{t\in[0,\:T]}$ be a filtration of $\mathcal A$
* $W$ be an $\mathcal F$-Brownian motion on $(\Omega,\mathcal A,\operatorname P)$
$\Phi:\Omega\times[0,T]\to\mathbb R$ is called **elementary $\mathcal F$-predictable** $:\Le... | https://mathoverflow.net/users/91890 | A dilemma about the definition of the stochastic integral $\int_a^b\Phi\:{\rm d}W$ | Conventionally, the stochastic integral $\Phi\cdot W$ is defined for $\Phi\in\mathcal I^2$ as a continuous martingale via the Ito isometry, and then $\int\_a^b \Phi\,dW$ is just *notation* for $(\Phi\cdot W)\_b-(\Phi\cdot W)\_a$. The subtlety you raise is basically our sloppy identification of $(\Phi\cdot W)\_{t\wedge ... | 2 | https://mathoverflow.net/users/42851 | 283550 | 125,402 |
https://mathoverflow.net/questions/283545 | 4 | Let $X$ be a compact Riemann surface with boundary. Let us shrink each connected component of the boundary into a point. We get a closed topological surface $Z$ with several marked points (which came from the components of the boundary of $X$) such that the complement $Z\_0$ of these points carries a complex structure.... | https://mathoverflow.net/users/16183 | Shrinking the boundary of a Riemann surface | No. Complex structure on $X$ "knows" whether an end is a hole or a puncture.
The conformal invariant responsible for this is called the extremal length of the family of closed curves surrounding the hole/puncture.
See Ahlfors, Lectures on quasiconformal mappings.
The simplest case is a sphere with a puncture (conform... | 4 | https://mathoverflow.net/users/25510 | 283552 | 125,404 |
https://mathoverflow.net/questions/283576 | 1 | Consider two standard normal random variable, X and Y.
They both have mean 0, and variance 1. But we don't know their dependency.
Is it possible for X+2Y to be nonsymmetric?
In another word, is it possible for P(X+2Y>0) = 1/2 to not hold.
I understand if they follow multivariate normal, the sum has to be norma... | https://mathoverflow.net/users/50800 | Is sum of dependent normal variables symmetric? | Yes - take $Y=-X$ if $X<0$ and $Y$ be "negative gaussian" independent of the value of $X$ if $X>0$. Then $X+2Y=-X>0$ for $X<0$ (i.e., $X+2Y$ is positive at least with probability 1/2); on the other hand, if $X>0$ and $Y<0$ they are conditionally independent, so that $X+2Y$ can also be positive with non-zero probability... | 3 | https://mathoverflow.net/users/8588 | 283578 | 125,413 |
https://mathoverflow.net/questions/283577 | -6 | I have known that $b\_2(CP^n\sharp CP^n)=2$, however I have no idear how to prove this fact ! I appreciate any help for this simple question! Thank you!
| https://mathoverflow.net/users/88727 | The Betti numbers of of $CP^n\sharp CP^n$ | This is one of those calculations which is easier with the general formalism of homology, rather than just Betti numbers. Using the [Mayer-Vietoris theorem](https://en.wikipedia.org/wiki/Mayer%E2%80%93Vietoris_sequence) a few times, one can show that if $M$ and $N$ are connected $n$-manifolds and $k\ne n-1,n$, then
$... | 3 | https://mathoverflow.net/users/97265 | 283579 | 125,414 |
https://mathoverflow.net/questions/283588 | 9 | It seems that as $n$ increases, the ratio $$\frac{\varphi(2^n-1)}{2^n-1},$$ where $\varphi$ denotes the Euler totient function,
takes on values reasonably often in the interval $(.3,.4)$.
Is there anything known about $$\lim \inf\_{n \rightarrow \infty}\frac{\varphi(2^n-1)}{2^n-1}?$$
| https://mathoverflow.net/users/17773 | How Composite can $2^n-1$ be, infinitely often? | Let $n \in \mathbf N^+$ and $a \in \mathbf N\_{\ge 2}$. Every prime $\le n+1$ that doesn't divide $a$, is a divisor of $a^{n!} - 1$ (by Fermat's little theorem). So we have $$\frac{\varphi(a^{n!} - 1)}{a^{n!} - 1} = \prod\_{p \,\mid\, a^{n!} - 1} \left(1 - \frac{1}{p}\right) \le \prod\_{a < p \le n+1} \left(1 - \frac{1... | 33 | https://mathoverflow.net/users/16537 | 283591 | 125,419 |
https://mathoverflow.net/questions/283589 | 3 | Suppose $(P,\leq)$ is a poset without maximal elements. For $X\subseteq P$ we set $X^u = \{p\in P: p \geq x \text{ for all } x\in X\}$ and call this the *set of upper bounds of* $X$. We say that $B\subseteq P$ is *unbounded* if $B^u = \emptyset$. Moreover we say $D\subseteq P$ is *dominating* if for all $p\in P$ there ... | https://mathoverflow.net/users/8628 | Unboundedness number and domination number of a poset $(P,\leq)$ | $\omega\times \omega\_1$ with the product (pointwise) order.
| 7 | https://mathoverflow.net/users/14915 | 283592 | 125,420 |
https://mathoverflow.net/questions/283614 | 2 | Is the following true for some prime $p$?
There exists some prime $\ell$ and some $n$ such that $PSL\_n(\mathbb{F}\_{\ell})$ contains nontrivial $p$-torsion, and moreover if $x \in PSL\_n(\mathbb{F}\_{\ell})$ has order $p$ and $0 < k < p$, then $x$ is conjugate to $x^k$.
| https://mathoverflow.net/users/116050 | Conjugacy of powers of elements in $PSL_n(\mathbb{F}_{\ell})$ | Yes. Take $\ell$ to be a primitive root mod $p$ (which exists by Dirichlet's theorem), $n=p-1$, then because $p$ divides $\ell^{p-1}-1$ there is a $p$-torsion element in $PSL\_n(\mathbb F\_\ell$. However, because $p$ is prime to $\ell$, all such elements are semisimple, hence conjugate to their $\ell^k$th powers for an... | 6 | https://mathoverflow.net/users/18060 | 283617 | 125,427 |
https://mathoverflow.net/questions/283607 | 1 | If $\mathbb{F}\_{q^n}$ is a finite field with $q^n$ elements ($q$ being a power of a prime $p$) we have the trace map $tr^n\_m:\mathbb{F}\_{q^n}\rightarrow \mathbb{F}\_{q^m}$ such that $x\mapsto x+F^m(x)+..+F^{n-m}(x)$, if $m\mid n$. So we can form the inverse limit of these maps, that will be
$$
\left\{(x\_n)\_{n\in\m... | https://mathoverflow.net/users/47300 | Limit of trace maps in finite fields | One answer to the question of "what abelian group" is "the inverse limit of a countable sequence of surjective maps of $\mathbb F\_p$-vector spaces, of increasing dimension". (We can use the factorial function to reduce the inverse system to a countable sequence.) It is easy to see that we can choose a basis such that ... | 6 | https://mathoverflow.net/users/18060 | 283620 | 125,428 |
https://mathoverflow.net/questions/283628 | 0 | I am trying to find whether the polynomial (monomial) functor $P : X \rightarrow X\times X $, i.e. $P(X) = X^2$, is monomorphic on objects, in other words, that if there exists an isomorphism $A\times A \overset {i} {\hookrightarrow} B \times B$, then there is also an isomorphism $A \overset {j} {\hookrightarrow} B$.
... | https://mathoverflow.net/users/29853 | Is "square" functor monomorphic on objects? | There is a [group](https://mathoverflow.net/questions/218113/a-is-isomorphic-to-a-oplus-mathbbz2-but-not-to-a-oplus-mathbbz/227443#227443) $A$ that is isomorphic to $A \times \mathbb Z \times \mathbb Z$ but not isomorphic to $A \times \mathbb Z$.
Taking $B = \mathbb A \times \mathbb Z$ produces a counterexample in th... | 4 | https://mathoverflow.net/users/18060 | 283631 | 125,431 |
https://mathoverflow.net/questions/283616 | 6 | I meant to assign to my class the following homework problem:
>
> If $u\in C^2((0,T)\times \Omega) \cap C^0([0,T]\times\bar{\Omega})$ where $\Omega$ is an open, bounded domain, is such that $\partial\_t u - \triangle u \leq - \epsilon < 0$ for some constant $\epsilon > 0$, then $u$ cannot have a local maximum on th... | https://mathoverflow.net/users/3948 | Maximum principle for heat equation, low regularity case | This version is still true: if $u$ had a local maximum at $(x,\,T)$, say with $u(x,\,T) = 0$, then $u \leq 0$ in a small parabolic cylinder centered at $(x,\,T)$. After rescaling we can assume that $u \leq 0$ in $\overline{B\_1} \times [0,\,1]$, with $u(0,\,1) = 0$.
Replacing $u$ with $u - \frac{\epsilon}{4n}t(|x|^2-... | 5 | https://mathoverflow.net/users/16659 | 283635 | 125,432 |
https://mathoverflow.net/questions/283269 | 2 | Let $M$ be a closed manifold with non-torsion $\pi\_2$, and $A$ a non-trivial free homotopy class of a map $f: S^2 \to M$.
Let $S$ be the set of (immersed) class $A$ surfaces in $(M,g)$ with mean curvature bounded from above by a fixed constant $C$. Is there a $C=C(A)>0$ sufficiently small, so that the $g$-area functio... | https://mathoverflow.net/users/16877 | Mean curvature upper bounds and area, or geodesic curvature upper bounds and length | The answer is No. Indeed it is no for spheres of all dimension assuming one can find an immersed class $A$ $S^n$ with mean curvature less than a given $C(A)$. The counterexamples can be generalized from the following example for $n=1$.
Suppose we have a closed surface $(M,g)$, and a free homotopy class $A$ geodesic ... | 0 | https://mathoverflow.net/users/16877 | 283638 | 125,434 |
https://mathoverflow.net/questions/283278 | 7 | Rado's conjecture (one of many equivalent formulations) states: any non-special tree has a non-special subtree of cardinality $\aleph\_1$.
"Special" means a tree can be decomposed into countably many antichains. Hence in particular, special trees have no cofinal branches of uncountable length.
I'm asking for a ref... | https://mathoverflow.net/users/23835 | Consistency of Rado's conjecture with not CH | Rado's conjecture holds in Mitchell's model (of course, start with a strongly compact instead of a weakly compact) granted the following: If $T$ if a non-special tree of height $\omega\_1$, then $T$ remains non-special after Cohen forcing $Add(\omega, \kappa)$ for any regular $\kappa$. It suffices to show the case wher... | 4 | https://mathoverflow.net/users/23835 | 283641 | 125,435 |
https://mathoverflow.net/questions/283639 | 0 | In my math essay of thesis I have defined the probability coupling as follows
$$\Pi(\mu,\nu)=\left\lbrace \pi \in \Omega \left\vert
\begin{matrix}
\pi(A\times\mathcal{Y})=\mu(A) \\
\pi(\mathcal{X} \times B)=\nu(B)
\end{matrix}\right. \right\rbrace, \quad
\begin{matrix}
A\subset \mathcal{X} \\
B\subset \mathcal{Y}
\... | https://mathoverflow.net/users/nan | How to characterize Radon Nikodym's derivative of a coupling with respect to any measure in the product space? | You have too much notation here - which is quite confusing. For instance, what is your "combined probability space" $\Omega$ (it seems to be $X\times Y$ as this is where your Radon-Nikodym derivatives are defined) and how its elements can be measures? As far as I understand, you consider the joint distributions for whi... | 1 | https://mathoverflow.net/users/8588 | 283643 | 125,437 |
https://mathoverflow.net/questions/283634 | 3 | Given $b\in \mathbb{R}\_{>1}$ is there $U\subseteq\mathbb{R}\_{\ge 0}$ such that $U+bU=\mathbb{R}\_{\ge 0}$ and $(U-U)\cap b(U-U)=\{0\}$ (or equivalently: $u+bv=u'+bv' \implies u=u', v=v'$)?
Here is an example of near miss: if $b=10$ define $U$ as the set of positive reals with $0$ digits in odd places. Then the deco... | https://mathoverflow.net/users/2480 | Are there unique additive decompositions of the reals? | Can't this be solved in the following usual manner?
Take the first ordinal $\phi$ of cardinality continuum, and let $\{p\_\alpha\colon \alpha<\phi\}$ be an enumeration of points in $\mathbb R\_{\geq 0}$, with $0$ being the minimal point. We construct the set $U$ by transfinite recursion on $\alpha$. Initially, we put... | 4 | https://mathoverflow.net/users/17581 | 283648 | 125,438 |
https://mathoverflow.net/questions/283608 | 2 | For a finite abelian group $G$ and a subset $S\subseteq G$ with $0\in S=-S$, let
$$ \alpha(S):=\max\{|A|\colon(A-A)\cap S=\varnothing\} $$
and
$$ \omega(S):=\max\{|A|\colon A-A\subseteq S\}. $$
These quantities are the independence number and the clique number, respectively, of the Cayley graph induced by $S$ on $... | https://mathoverflow.net/users/9924 | The product of the clique and independence numbers in the Cayley sum graph | If the set $S$ is arbitrary, then no strong variant of $(\*)$ holds unless $G$ has order $2$.
Indeed, consider a factorization of $G$, and suppose there is a factor $H \sim \mathbb{Z}\_n \neq \mathbb{Z}\_2$. Let $S$ be the set of all elements of $G$ such that their projection on $H$ is at least $n / 2$. But then $\al... | 1 | https://mathoverflow.net/users/106512 | 283651 | 125,440 |
https://mathoverflow.net/questions/283609 | 10 | Does there exist a smooth projective surface $S / \mathbb{C}$ equipped with a dominant morphism $\pi: S \to \mathbb{P}^1$ which has the following properties:
1. The fibre at infinity is "multiple", i.e. as divisors on $S$ one has $\pi^{-1}(\infty) = mD$ for some $m > 1$ and some divisor $D$.
2. All other fibres are i... | https://mathoverflow.net/users/5101 | A pencil with exactly one multiple fibre | The answer is *yes*.
In fact, it is possible to construct a rational elliptic fibration $f \colon S \to \mathbb{P}^1$ with exactly one multiple fibre of multiplicity $m \geq 2$, by starting from the blow-up of $\mathbb{P}^2$ at nine points that are the base locus of a pencil $\mathscr{P}$ of elliptic curves and then... | 5 | https://mathoverflow.net/users/7460 | 283661 | 125,443 |
https://mathoverflow.net/questions/283664 | 1 | Let
$$\theta(x)=\sum\_{p\leq x} \log p$$ be the Chebyshev function over primes $p$.
Computational evidence seems to suggest that $\theta(x) < x$ for every sufficiently large $x$.
But is it true ?
| https://mathoverflow.net/users/116072 | Inquiry on the Chebyshev $\theta$ function | No. Littlewood proved that $\theta(x) > x + c \sqrt{x} \log \log \log x$ holds for infinitely many integers $x$, for some $c > 0$. Cf the answer to [this Mathoverflow question](https://mathoverflow.net/questions/178635/lower-bounds-on-the-error-term-of-the-prime-number-theorem), noting that $\theta(x) = \psi(x) + O(\sq... | 7 | https://mathoverflow.net/users/21724 | 283665 | 125,444 |
https://mathoverflow.net/questions/283663 | 4 | Assume $f$ is an entire non-polynomial function of arbitrarily small exponential order ('zero'th order' if you're into calling it that). Is it possible that for all $n$ we have
$$|f^{\circ n}(z)| < M\_ne^{|z|}$$
Where $f^{\circ n}$ is the $n$-fold iterate of $f$. This seems a little wonky to actually happen but I'm... | https://mathoverflow.net/users/nan | Can iterates of a non-polynomial function be bounded by an exponential indefinitely? | It is sufficient to find a non-polynomial entire function $f$ such that $T(r) = \sup\_{|z| \leq r} |f(z)|$ satisfies $T^{\circ n}(r) = O\_n(e^r)$. Let $r \in \mathbb{R}\_+ \mapsto S(r) \in \mathbb{R}^\*\_+$ be a continuous function such that $\log S(r) / \log r$ is increasing and tends to infinity slower than any itera... | 2 | https://mathoverflow.net/users/21724 | 283669 | 125,446 |
https://mathoverflow.net/questions/220060 | 9 | I would like a reference/argument for the truth/falsity of the following statement:
The etale sheafification of the unramified Milnor-Witt K-theory (Nisnevich) sheaves are the (etale sheafification of?) unramified Milnor K-theory sheaves.
Thanks!
| https://mathoverflow.net/users/24706 | What is the etale sheafification of the (unramified) Milnor-Witt $K$-theory | We have a short exact sequence of sheaves of abelian groups (see e.g. Morel's book on $\mathbb{A}^1$-algebraic topology):
$$
0\to\mathbf{I}^{n+1}\to\mathbf{K}^{\rm MW}\_n\to \mathbf{K}^{\rm M}\_n\to 0,
$$
where $\mathbf{I}^{n+1}$ is the sheaf of $n+1$-th powers of the fundamental ideal in the Witt ring. Over a strictl... | 7 | https://mathoverflow.net/users/50846 | 283674 | 125,448 |
https://mathoverflow.net/questions/283662 | 5 | I came across the following function when reading the famous paper of Letac and Mora "Natural exponential families with cubic variance functions", i.e.,
$$f(x) = \frac{x^x e^{-x}}{\Gamma(x+2)}$$ for $x \ge 0$. Proposition 5.5 there showed via a transform of a Levy process that $f$ is a density on $(0,\infty)$ without t... | https://mathoverflow.net/users/14390 | Integral involving Gamma function: density of Kendall-Ressel family of distributions | According to Hankel's formula
$$
\frac{1}{\Gamma(z)}=\frac{i}{2\pi}\int\_C(-t)^{-z}e^{-t}dt,
$$
where $C$ is [Hankel contour](https://en.wikipedia.org/wiki/Hankel_contour). So
$$
\frac{x^x}{\Gamma(x+1)}=\frac{i}{2\pi}\int\_C(-t)^{-x-1}e^{-xt}dt,\quad x>0.
$$
Consider the integral
$$
I(b)=\int\_0^\infty \frac{x^xe^{-bx}... | 5 | https://mathoverflow.net/users/82588 | 283675 | 125,449 |
https://mathoverflow.net/questions/283659 | 3 | Suppose $G$ is a finite group with a split $BN$-pair, satisfying the commutator relations, and such that representatives in $N$ of elements of the Weyl group can be chosen in a nice way: if $W\ni w=s\_{i\_1}\cdots s\_{i\_r}$, then $N\ni\dot{w}=\dot{s}\_{i\_1}\cdots\dot{s}\_{i\_r}$. (I'm essentially assuming $G$ behaves... | https://mathoverflow.net/users/113597 | If $\chi\in\operatorname{Irr}(L_J)$ and $^\ast R^{L_J}_T(\chi)\neq 0$, does $(\mu,R^{L_I}_{L_J}(\chi))=0$ when $^\ast R^{L_I}_T(\mu)=0$? | By that adjointness we have that $(\mu,R^{L\_I}\_{L\_J}\chi)=({}^\*R^{L\_I}\_{L\_J}\mu,\chi)$. If this value is not zero, then it means that ${}^\*R^{L\_I}\_{L\_J}\mu$ contains a non-zero multiple of $\chi$, say it is $m\chi$. Then by the transitive property we see that ${}^\*R^{L\_I}\_{T}\mu$ contains $m({}^\*R^{L\_J}... | 1 | https://mathoverflow.net/users/56217 | 283680 | 125,452 |
https://mathoverflow.net/questions/283695 | 6 | Let $k$ be a field of characteristic $p$ and $G$ be a torsion free abelian group . Then the group ring $k[G]$ is an integral domain , let $k(G)$ denote its field of fractions . Then can we say anything about the transcendence degree of $k(G)$ over $F\_p$ in terms of $k$ and/or $G$ ? What about the same question for fie... | https://mathoverflow.net/users/nan | Transcendence degree of the fraction field of $k[G]$ for torsion free abelian group $G$ | It seems that the transcendence degree of $k(G)$ over $k$ should be the dimension of the $\mathbb Q$-vector space $G \otimes\_\mathbb Z \mathbb Q$. Indeed, passing from $G$ to $G \otimes \mathbb Q$ corresponds to adding roots of existing elements, so it does not alter the transcendence degree. Thus we are reduced to $\... | 7 | https://mathoverflow.net/users/82179 | 283698 | 125,459 |
https://mathoverflow.net/questions/283696 | 5 | In a large (possibly above $5000\times 5000$) matrix, the problem of finding all the eigenvalues and eigenvectors can be solved using iterative methods (Arnoldi, Lanczos etc.). However, there seems to be a convergence happening in these methods suggesting that one can quantify how many eigenvalues/eigenvectors are need... | https://mathoverflow.net/users/nan | In a large sparse matrix, how many eigenvalues/eigenvectors are “spurious”? | Antisymmetric matrices are normal, hence they can be diagonalized with an orthogonal matrix. So $\|A\|\_F^2=\sum |\lambda\_i|^2$. If you keep only the $k$ largest eigenvalues (ordered in modulus: $|\lambda\_1| \geq |\lambda\_2| \geq \dots \geq |\lambda\_k| \geq |\lambda\_{k+1}| \geq \dots \geq |\lambda\_n|)$ and replac... | 5 | https://mathoverflow.net/users/1898 | 283711 | 125,461 |
https://mathoverflow.net/questions/283716 | 22 | This question is inspired by some others on MathOverflow. Hecke operators are standardly defined by double cosets acting on automorphic forms, in an explicit way.
However, what bother me is that Hecke operators are also mentioned in the automorphic *representations* language. Those remain very mysterious to me: what... | https://mathoverflow.net/users/116092 | What is the matter with Hecke operators? | If $K$ is a local field and $\mathcal O$ is its ring of integers, we say an irreducible representation of $G(K)$ is unramified if it contains a vector invariant under $G(\mathcal O)$. It is known that such a representation naturally has a unique vector invariant under $G(\mathcal O)$. The Hecke eigenvalue of a double c... | 15 | https://mathoverflow.net/users/18060 | 283718 | 125,462 |
https://mathoverflow.net/questions/283700 | 7 | Let $f(n) = a\_n$ be a sequence taking values in $\mathbb C$ for $n=1,2,...$. Let $T\_m$ be the Hecke operators (of a fixed weight $k$) defined as usual in terms of the $a\_n$. That is:
$$T\_m(f)(n) = \sum\_{r>0, r|(m,n)}r^{k-1}a\_{mn/r^2}.$$
Suppose $f$ is such that it is an eigenvalue for all the $T\_m$, then is it... | https://mathoverflow.net/users/58001 | Is every eigenvector sequence for the Hecke operators a eigenform? | The answer is no. Your form $f$ would have to be of weight $k$, and of level $SL\_2(\mathbb Z)$, so should be in a finite dimensional vector space of dimension $d$. That would mean that the Hecke operators $T\_n$ acting on the space of all sequences would have at most $d$ different systems of eigenvalues.
But this is... | 7 | https://mathoverflow.net/users/9317 | 283725 | 125,466 |
https://mathoverflow.net/questions/259093 | 4 | In the very last page of Janelidze and Tholen's paper *Beyond Barr Exactness: Effective Descent Morphisms*, the authors relate the theory of fiber bundles (and covering spaces in particular) to descent and the lifting properties defining fibrations (in topology).
Below is an excerpt from the final two pages of the pa... | https://mathoverflow.net/users/69037 | Descent theory, fibrations, and bundles | Consider a bundle $\begin{smallmatrix}A\\ \downarrow\\ B \end{smallmatrix}$. Observe that a trivialization is furnished precisely by a space $F\to \bf 1$ whose pullback along $B\to \bf 1$ is isomorphic to $\begin{smallmatrix}A\\ \downarrow\\ B \end{smallmatrix}$. Indeed the trivialization is the isomorphism induced by ... | 1 | https://mathoverflow.net/users/69037 | 283731 | 125,471 |
https://mathoverflow.net/questions/282430 | 11 | Just wondering if anyone knows any references in the literature to bijections corresponding to the following simple generating function identities. Let $B(z)=\dfrac{1}{\sqrt{1-4z}}$ and $C(z)=\dfrac{1-\sqrt{1-4z}}{2z}$, the generating functions of the central binomial coefficients and Catalan numbers, respectively. I'm... | https://mathoverflow.net/users/113161 | Proofs of some combinatorial identities | Partial answer: Your first identity is
\begin{equation}
\sum\limits\_{k=0}^n \left(-1\right)^k \dbinom{2k}{k} \dbinom{2\left(n-k\right)}{n-k} = \left[n \text{ is even}\right] 2^n \dbinom{n}{n/2} ,
\end{equation}
where I am using the Iverson bracket notation. (That is, $[\mathcal{A}]$ denotes the truth value of a statem... | 7 | https://mathoverflow.net/users/2530 | 283736 | 125,472 |
https://mathoverflow.net/questions/283654 | 17 | Let $P$ be a convex polytope in $\mathbb{R}^3$ whose every vertex lies in the $\mathbb{Z}^3$ lattice.
**Question:** If $P$ contains exactly one lattice point in its interior, what is the maximum possible volume of $P$?
Notice that a convex lattice polytope with no interior lattice point can have arbitrarily large ... | https://mathoverflow.net/users/36904 | Volume of convex lattice polytopes with one interior lattice point | This is addressed in the 2013 paper (appeared in Advances in 2015) by [Averkov, Krumpelmann, Nill](https://arxiv.org/abs/1309.7967). The give a sharp bound for the volume of a lattice simplex with one interior lattice point (Theorem 2.2 in the paper), and an improved bound for a general lattice polyhedron with the same... | 9 | https://mathoverflow.net/users/11142 | 283740 | 125,475 |
https://mathoverflow.net/questions/283737 | 1 | Let $X$ be a smooth, projective variety and $Y \subset X$ a smooth, effective divisor. Consider now the natural map $i^\ast:H^2(X) \to H^2(Y)$. Then,
1. When is the image of $c\_1(\mathcal{O}\_X(Y)) \in H^2(X)$ under the map $i^\*$ non-zero?
2. If $Z$ is another smooth projective variety containing $Y$ as an effectiv... | https://mathoverflow.net/users/58203 | Self-intersection of divisors and Chern class | 1. Because homological equivalence and numerical equivalence coincide for divisors (up to torsion), we see that $i^\*(c\_1(\mathcal O\_X(Y))) = 0 \in H^2(Y,\mathbb Q)$ if and only if
$$\int\_Y i^\*c\_1(\mathcal O\_X(Y)) \cup \operatorname{cl}\_Y(C) = 0$$
for every curve $C \subseteq Y$ (where $\operatorname{cl}\_Y(C)$ ... | 2 | https://mathoverflow.net/users/82179 | 283742 | 125,476 |
https://mathoverflow.net/questions/283749 | 0 | Let $\kappa$ be an infinite ordinal, and suppose $E\subseteq {\cal P}(\kappa)$ with $|E| > \kappa$. Let us call $x\in \kappa$ *popular* if $$|\{e\in E: x\in e\}| > \kappa.$$
If $\text{Pop}(\kappa)$ is the collection of popular elements of $\kappa$, is it possible that $|\text{Pop}(\kappa)|<\kappa$?
| https://mathoverflow.net/users/8628 | "Popular" vertices in an infinite hypergraph | **Yes** if there exists a cardinal $\gamma$ with $|\gamma|<|\kappa|<|2^\gamma|$. Just take $E=2^\gamma$.
**No**, otherwise. Throw out all popular elements from the sets in $E$; some of these sets become identical, but each equivalence class has at most $|\kappa|$ elements, since $|2^{\mathop{\rm Pop}(\kappa)}|\leq|\k... | 3 | https://mathoverflow.net/users/17581 | 283750 | 125,477 |
https://mathoverflow.net/questions/283756 | 14 | Let $\varphi$ denote [Euler's totient function](https://en.wikipedia.org/wiki/Euler's_totient_function). It's [widely believed](https://en.wikipedia.org/wiki/Mersenne_prime) that $2^n-1$ is prime for infinitely many (prime) $n$, which, in turn, implies
$$\limsup\_{n \to \infty} \frac{\varphi(2^n-1)}{2^n-1} = 1.
$$
But ... | https://mathoverflow.net/users/16537 | The limit superior of $\varphi(2^n-1)/(2^n-1)$ | Let $n$ be a large prime. Then $2^n-1=p\_1\dots p\_k$ (possibly with equal factors), where $n\mid p\_i-1$ for all $i$. Therefore, $k\leq \log\_n(2^n-1)<\frac{n}{\log\_2n}$. Thus,
$$
\frac{\varphi(2^n-1)}{2^n-1}\geq\prod\_{i=1}^k\left(1-\frac1{p\_i}\right)
\geq \left(1-\frac1n\right)^{n/\log\_2n}\to1, \quad n\to\infty... | 30 | https://mathoverflow.net/users/17581 | 283759 | 125,478 |
https://mathoverflow.net/questions/283733 | 3 | Let $T$ be a measure preserving transformation on a measure space $(X, \mathscr{F}, m)$ with *infinite* measure $m$. Let $A \in \mathscr{F}$ be such that $X = \cup\_{k=0}^\infty T^{-k} A \pmod{m}$. Then first hitting time $\tau(x):= \inf\{k \ge 1: T^k x \in A\}$ is finite $m$-a.e. $x \in X$ and the first hitting mappin... | https://mathoverflow.net/users/116098 | Are induced transformations always measure-preserving on infinite measure spaces? | One basic thing: for $X=\mathbb Z$ with counting measure and $T:x\mapsto x+1$, $A$ satisfies the condition iff $\sup A=\infty$, i.e. $A=\{a\_1<...<a\_n<...\}$. Then $\varphi(a\_i)=a\_{i+1}$ and of course $\varphi\_A^{-1}(a\_1)=\emptyset$ so that $\varphi\_A$ is not counting measure preserving on $A$ if this means $|\va... | 2 | https://mathoverflow.net/users/75422 | 283762 | 125,479 |
https://mathoverflow.net/questions/283319 | 10 | It is well known that every smooth and every complex manifold equipped with a group action by a (compact) Lie group $G$ admits a *stratification by orbit types*.
I would like to know if there is a similar orbit type stratification in every one of the following cases:
1.) *smooth symplectic manifolds;*
2.) *holom... | https://mathoverflow.net/users/48531 | Orbit type stratification of a holomorphic symplectic manifold. | I have a partial answer to my own question that I posed 5 days ago. It turns out that at least in the case of a smooth symplectic manifold you can perform an orbit type stratification. Here is why:
Asume $X$ is a symplectic manifold equipped with a symplectic form $\omega$. Assume furthermore that a compact Lie group... | 1 | https://mathoverflow.net/users/48531 | 283763 | 125,480 |
https://mathoverflow.net/questions/283753 | 18 | **Question.** Does there exist a graph $G$ with $(\Delta(G),\chi(G),\omega(G))=(8,8,6)$?
**Remarks.**
Here,
* 'graph'='undirected simple graph'='irreflexive symmetric relation on a set'
* any number of vertices is permitted in the question (though, trivially, at least 8 vertices (and quite a bit more) vertices a... | https://mathoverflow.net/users/108556 | Does there exist a graph with maximum degree 8, chromatic number 8, clique number 6? | Yes, it exists. Take 5 triangles $T\_1,\dots,T\_5$ (all 15 vertices are distinct) and draw also all edges between $T\_i$ and $T\_{i+1}$, $i=1,2,3,4$, and between $T\_5$ and $T\_1$. All degrees are equal to 8, maximal clique is formed by two neighboring triangles, and $\chi=8$. Indeed, each color may appear at most twic... | 39 | https://mathoverflow.net/users/4312 | 283764 | 125,481 |
https://mathoverflow.net/questions/283640 | 1 | It is a well-known result by Woodin that the Hartogs number $h(\mathbb{R})$ (more commonly known as $\Theta$) is a Woodin cardinal (in HOD) assuming ZF + AD + DC. This is equivalent to $h(\mathcal{P}(\aleph\_0))$. However, I was wondering the simple question of this very generalization; specifically the following quest... | https://mathoverflow.net/users/115951 | Under ZF + DC + AD, is it known what the properties are of the Hartogs number for $\mathcal{P}(\kappa)$ for some $\kappa>\aleph_0$? | This is a bit confused, but I *think* what you're asking is:
>
> Let $S(\alpha)$ be the supremum of all the ordinals onto which $\mathcal{P}(\alpha)$ surjects. What can we say about e.g. $S(\omega\_1)$?
>
>
>
*(And your original question was whether $S(\alpha)$ always exists, which it does.)* **If** this is yo... | 6 | https://mathoverflow.net/users/8133 | 283778 | 125,485 |
https://mathoverflow.net/questions/283774 | 0 | I am looking for an English translation of Kolmogoroff, A.
*Über die analytischen Methoden in der Wahrscheinlichkeitsrechnung.* (German)
**Math. Ann.** 104 (1931), no. 1, 415–458. The title in English is "On Analytical Methods In Probability Theory". I can only find Russian and German.
| https://mathoverflow.net/users/83682 | Looking for an English translation of "On Analytical Methods In Probability Theory" by Kolmogorov | I believe that an English translation can be found in Kolmogorov's Selected Works (vol 2) pp. 62–108, 1992, Kluwer.
| 2 | https://mathoverflow.net/users/5734 | 283782 | 125,486 |
https://mathoverflow.net/questions/283770 | 4 | Let $p$ be an odd prime, and denote by $Cl\_p(H)$ the $p$-part of the ideal class group of a number field $H$. Let $\Delta:=Gal(\mathbb{Q}(\mu\_p)/\mathbb{Q})$ and $\omega : \Delta \longrightarrow \mathbb{Z}\_p^\times$ be the Teichmuller character. For a $\mathbb{Z}\_p[\Delta]$-module $C$ and an integer $i$, consider $... | https://mathoverflow.net/users/116119 | Refinement of (classical) Iwasawa main conjecture | There is an answer which follows from Kolyvagin's theory of Euler Systems, and can be found in Theorem 4.4 of
K. Rubin, *Kolyvagin's System of Gauss Sums*, in *Arithmetic Algebraic Geometry*, van der Geer, Oort, Steenbrink (eds), Birkhäuser, Progress in Mathematics **89**, 1991.
The result is (in Rubin's words) "im... | 5 | https://mathoverflow.net/users/18238 | 283784 | 125,487 |
https://mathoverflow.net/questions/283690 | 15 | Consider a continous map from $S^2$ to $C$.
Is it true that there exists 3 points equially spaced on a great circle, $x\_1,x\_2,x\_3$, such that if $w$ is the third root of unity, $f(x\_1)+wf(x\_2)+w^2f(x\_3)=0$?
More generally I'm asking this if we take nth unity roots.
Maybe I should add slight motivation: In m... | https://mathoverflow.net/users/104594 | Is this generalization of Borsuk Ulam true? Roots of unity | $\require{AMScd}$
First, suppose that $G$ acts freely on connected spaces $X$ and $Y$, and that $p\colon X\to Y$ is $G$-equivariant. We then have a diagram of fibrations
\begin{CD}
X @>>> X/G @>>> BG \\
@VpVV @VVV @VV1V \\
Y @>>> Y/G @>>> BG
\end{CD}
This gives a diagram of fundamental groups
\begin{CD}
\pi\_1(X) @... | 10 | https://mathoverflow.net/users/10366 | 283789 | 125,490 |
https://mathoverflow.net/questions/283758 | 18 | Numerical evidence suggests a conjecture that the number of
points of certain elliptic curve over $\mathbb{F}\_p$ is
either $p$ or $p+2$ for $p$ of certain form.
Let $p$ be prime of the form $p=27a^2+27a+7$ and $(a/b)$ denote
the Kronecker symbol.
For integer $k$ nonzero modulo $p$ define $E\_k / \mathbb{F}\_p : y^... | https://mathoverflow.net/users/12481 | Conjecture: The number of points modulo $p$ of certain elliptic curve is $p$ or $p+2$ for $p$ of form $p=27a^2+27a+7$ | The conjecture is true. To prove it, we can restrict to $k=1$ as explained in the comments. Let $\chi$ denote a cubic Dirichlet character modulo $p$; this exists as $p\equiv 1\pmod{3}$, and it is unique up to complex conjugation. Let $\psi$ denote the (unique) quadratic Dirichlet character modulo $p$.
The number of ... | 22 | https://mathoverflow.net/users/11919 | 283790 | 125,491 |
https://mathoverflow.net/questions/283747 | 7 | Let $BG$ denote the classifying space of a finite group $G$. For which group cohomology classes $c\in H^2(G;\mathbb{Z}/2)$ does there exist a real vector bundle $E$ over $BG$ such that $w\_2(E)=c$?
| https://mathoverflow.net/users/51107 | Stiefel-Whitney class of an orthogonal representation | Put
$$A=\{c\in H^2(BG;\mathbb{Z}/2): c = w\_2(V) \text{ for some } V\}$$
Here are some observations:
1. Let $V$ be a real representation with determinant $L$, so $w\_1(V)=w\_1(L)$. Put $W=V\oplus L\oplus L\oplus L$. We find that $\det(W)=1$ and $w\_2(W)=w\_2(V)$. It follows that $A=\{w\_2(W):\det(W)=1\}$. Moreover, ... | 8 | https://mathoverflow.net/users/10366 | 283793 | 125,492 |
https://mathoverflow.net/questions/283799 | 18 | In a recent conversation with a colleague, the following question arose:
>
> 1. What is the isomorphism class of $\mathrm{Ext}^1\_\mathbb{Z}(\mathbb{R}/\mathbb{Z},\mathbb{Z})$? That is to say, what is $\mathrm{Ext}^1\_\mathbb{Z}(\mathbb{R}/\mathbb{Z},\mathbb{Z})$ as an Abelian group?
>
>
>
I did not have any i... | https://mathoverflow.net/users/104682 | The isomorphism class of $\mathrm{Ext}^1_\mathbb{Z}(\mathbb{R}/\mathbb{Z},\mathbb{Z})$ | Writing $\mathbb{R}/\mathbb{Z} \cong \mathbb{Q}/\mathbb{Z} \oplus \bigoplus\_I \mathbb{Q}$ where $I$ indexes a Hamel basis for $\mathbb{R}$ minus one element, we have
$$\text{Ext}^1(\mathbb{R}/\mathbb{Z}, \mathbb{Z}) \cong \text{Ext}^1(\mathbb{Q}/\mathbb{Z}, \mathbb{Z}) \times \prod\_I \text{Ext}^1(\mathbb{Q}, \mathb... | 17 | https://mathoverflow.net/users/290 | 283804 | 125,496 |
https://mathoverflow.net/questions/283802 | 17 | Take a convex polyhedron $P$ in $\mathbb R^3$ and remove all the faces, i.e. leave only the edges. Call this graph $E$. Let us now try to continuously deform $E$ in $\mathbb R^3$ so that all the edges of $E$ keep their length and remain straight (like metal sticks), but allow the change of angles between the edges.
*... | https://mathoverflow.net/users/13441 | Rigidity of convex polyhedrons in $\mathbb R^3$ with faces removed | Yes, this is true.
One strategy is to use the naïve approach of counting degrees of freedom and constraints. For triangulated polyhedra one can easily show with the Euler characteristic that the expected dimension (number of variables minus number of equations) of the realization space modulo Euclidean isometries is... | 12 | https://mathoverflow.net/users/353 | 283808 | 125,498 |
https://mathoverflow.net/questions/283794 | 8 | The fundamental group of a closed orientable manifold is finitely presented, and every finitely presented group arises as the fundamental group of a closed orientable four-manifold; see [this question](https://mathoverflow.net/q/15411/21564).
Not every finitely presented group arises as the fundamental group of a cl... | https://mathoverflow.net/users/21564 | Fundamental groups of non-orientable closed four-manifolds | Let $G$ be a finitely presented group
$$ G = \langle g\_1, g\_2, \cdots, g\_n | R\_1, R\_2, \cdots, R\_m \rangle$$
One standard way to realize this as the fundamental group of a compact $4$-manifold is to construct a $5$-dimensional manifold.
1) Start with $D^5$ and attach a $1$-handle for every generator $g\_i$... | 9 | https://mathoverflow.net/users/1465 | 283825 | 125,504 |
https://mathoverflow.net/questions/283813 | 8 | Let $F$ be a number field and $\chi$ a one dimensional Artin character. That is, it is a map $\chi: Gal(\overline F/F) \to \mathbb C^\times$ and let $L(s,\chi)$ be it's L-series.
What is known about the values of $L(s,\chi)$ at negative integers? Are they always algebraic integers contained in $F(\chi)$? When are the... | https://mathoverflow.net/users/58001 | Values of Artin L-functions at negative integers | The question of *order of vanishing* is quite elementary: the L-function of a Hecke character (i.e. 1-dimensional Artin representation) over any number field has an Euler product, which is convergent and non-vanishing for $s > 1$, and it satisfies a functional equation relating $L(\chi, s)$ to $L(\bar\chi, 1-s)$; hence... | 6 | https://mathoverflow.net/users/2481 | 283833 | 125,505 |
https://mathoverflow.net/questions/283835 | 0 | Need this for probabilistic [factoring algorithm](https://mathoverflow.net/questions/283767/conjecturally-unsafe-rsa-primes-p-27a227a7).
Let $p$ be sufficiently large prime and $E$ the elliptic curve
$E /\mathbb{F}\_p: y^2=x^3+ax+b$. Let $o=\#E(\mathbb{F}\_p)$.
$\psi\_n$ denote the $n$-the division polynomial of $E$.... | https://mathoverflow.net/users/12481 | When the $o$-th division polynomial of an elliptic curve over finite vanishes only at $x$ coordinates? | If $x$ in $\mathbb{F}\_p$ is such that $\psi\_o(x)=0$ but $x^3+ax+b$ is not a square in $\mathbb{F}\_p$, let $y$ be the square root of $x^3+ax+b$ in $\mathbb{F}\_{p^2}$ and $P=(x,y)$. Then $P$ has order $d$ dividing $o$. There is also a rational point on $E$ of order $d$, so $Fr^2-1 =0$ in $E[d]$, where $Fr$ is Frobeni... | 4 | https://mathoverflow.net/users/2290 | 283840 | 125,507 |
https://mathoverflow.net/questions/283846 | 3 | Let $X=\operatorname{proj}\_nX\_n$ be a Fréchet space. What is the relation between $\operatorname{proj}\_nX\_n''$ and $X''$? This should be known but I cannot find the reference.
| https://mathoverflow.net/users/75127 | Projective description of a bidual of a Fréchet space | There is of course a canonical map $X''\to$ proj $X\_n''$ which however need not be surjective if $X$ is a *non-distinguished* Fréchet space because then the inductive limit topology on $X'=$ ind $X\_n'$ is strictly finer than the strong topology on $X'$. Examples are due to Köthe and Grothendieck and as far as I remem... | 2 | https://mathoverflow.net/users/21051 | 283863 | 125,516 |
https://mathoverflow.net/questions/283827 | 10 | Let $X$ be a smooth curve over a number field $K$ (not necessarily proper). Fix an algebraic closure $\overline{K}$ of $K$.
Let $i,i' : \overline{K}\hookrightarrow\mathbb{C}$ be two abstract embeddings (ie, as $\mathbb{Q}$-algebras). Let $X\_\mathbb{C},X\_{\mathbb{C}}'$ be the base changes of $X\_{\overline{K}}$ to $... | https://mathoverflow.net/users/15242 | Copies of topological fundamental groups inside etale fundamental groups given by different embeddings of your field into $\mathbb{C}$ | For affine hyperbolic curves, when $i$ and $i'$ agree on $K$, this happens only when $i$ and $i'$ are equal or complex conjugate of each other.
Let $f: \pi\_1^{top}(X) \to \pi\_1^{et}(X\_{\mathbb C})$ be the natural dense inclusion, and $e\_i,e\_{i'} \pi\_1^{et}(X\_{\mathbb C}) \to \pi\_1^{et}(X\_{\overline{K}})$ be... | 2 | https://mathoverflow.net/users/18060 | 283864 | 125,517 |
https://mathoverflow.net/questions/283861 | 2 | I have a question on the Total Coloring Conjecture in graph theory. This conjecture states that
$$\chi^"(G)\leq \Delta +2,$$
where $\Delta$ is the maximum degree of the graph and $\chi^"(G)$ denotes the total coloring (minimum number of colors for coloring graph such that no adjacent edges and no edge and its endpo... | https://mathoverflow.net/users/90655 | Has the Total Coloring Conjecture been proved for complete graphs? | Yes, of course.
It is known and published for decades that
>
> $n$ odd $\quad\vdash\_{n:\omega}\quad$ $\chi''(K^n) = n$,
>
>
>
and
>
> $n$ even $\quad\vdash\_{n:\omega}\quad$ $\rightarrow$ $\chi''(K^n) = n+1$,
>
>
>
and because of $\Delta(K^n)=n-1$, and $n+1 = (n-1)+2$, the conjecture is validate... | 5 | https://mathoverflow.net/users/108556 | 283867 | 125,519 |
https://mathoverflow.net/questions/283826 | 2 | Let $P$ be a finite polyhedron and $N$ be a normal subgroup of $G=\pi\_1 (P)$. It is known that there exists a covering space $(\tilde{P},p)$ so that $p\_\* \pi\_1 (\tilde{P})=N$. It follows that for the finite polyhedron $\tilde{P}$ (which is related to $P$), we have $\pi\_1 (\tilde{P})\cong N$.
My question is that... | https://mathoverflow.net/users/114580 | On the realization of a quotient group | For a finite polyhedron $P$ and finite-index normal subgroup $N$ of $G=\pi\_1P$, there is a canonical finite polyhedron $Q$ with $\pi\_1Q\cong G/N$ constructed as follows. Let $\tilde{P}\stackrel{p}{\to} P$ be the covering map corresponding to $N$, as in the question.
We now construct the polyhedron $Q$ as follows:
... | 3 | https://mathoverflow.net/users/1463 | 283869 | 125,520 |
https://mathoverflow.net/questions/283862 | 4 | Related to [this question](https://mathoverflow.net/questions/283758/conjecture-the-number-of-points-modulo-p-of-certain-elliptic-curve-is-p-or).
Let $E / \mathbb{F}\_p : y^2=x^3+2$.
Numerical evidence up to $2 \cdot 10^5$ suggests:
Conjecture: $\#E(\mathbb{F}\_p) \in \{p,p+2\}$ iff $p$ is of the form
$27a^2+27a+... | https://mathoverflow.net/users/12481 | $\#E(\mathbb{F}_p) \in \{p,p+2\}$ iff $p$ is of the form $27a^2+27a+7$ | This conjecture is also true. Let $p$ be a prime, and let me follow my proof of the [previous conjecture](https://mathoverflow.net/questions/283758/conjecture-the-number-of-points-modulo-p-of-certain-elliptic-curve-is-p-or).
If $p\not\equiv 1\pmod{3}$, then the map $x\mapsto x^3$ permutes the residues modulo $p$, hen... | 8 | https://mathoverflow.net/users/11919 | 283875 | 125,522 |
https://mathoverflow.net/questions/283873 | 6 | I was studying combinatorical group theory recently, and I came across the infinite regular rooted binary tree and its automorphism group $Aut(T^{(2)})$with the Grigorchuk subgroup.
Let me now elaborate more on a different topic, the surreal numbers. They are constructed similarly to the binary tree at first, but in... | https://mathoverflow.net/users/114143 | Automorphism of the transfinite rooted binary tree | Each connected component of the tree is either rooted at $0$ or at a number which birthday is a limit ordinal, these are exactly the vertices of degree $2$. Let $A$ be the class of all limit ordinals, with addition of $0$. For an element $\alpha \in A$ let $s\_\alpha = \{+, -\}^\alpha$ be the set of its sign expansions... | 4 | https://mathoverflow.net/users/106512 | 283876 | 125,523 |
https://mathoverflow.net/questions/283868 | 28 | First, recall that on a Riemannian manifold $(M,g)$ the Laplace-Beltrami operator $\Delta\_g:C^\infty(M)\to C^\infty(M)$ is defined as
$$
\Delta\_g=\mathrm{div}\_g\circ\mathrm{grad}\_g,
$$
where the gradient of a function $f\in C^\infty(M)$ is defined by $$\iota\_{\mathrm{grad}\_g(f)}g=df,$$
$\iota\_\bullet$ being the ... | https://mathoverflow.net/users/58125 | Why is there no symplectic version of spectral geometry? | The characteristic variety (i.e. vanishing locus of the symbol) of a symplectomorphism invariant scalar differential equation is a real projective hypersurface invariant under the group of projectivized linear symplectic transformations. This group acts transitively on the real points of projective space, so preserves ... | 20 | https://mathoverflow.net/users/13268 | 283878 | 125,525 |
https://mathoverflow.net/questions/283853 | 0 | Trying to generalize [this answered question](https://mathoverflow.net/questions/283758/conjecture-the-number-of-points-modulo-p-of-certain-elliptic-curve-is-p-or) based on limited numerical evidence.
Let $E / \mathbb{F}\_p : y^2=x^3+2$.
**Conjecture 1** Let $p=3a^2+3ab\_0+b\_0^2$ be prime and $a,b\_0$
positive in... | https://mathoverflow.net/users/12481 | Primes of the form $p=3a^2+3ab+b^2$ or $p=27a^2+27ab+7b^2$ and the number of points of $y^2=x^3+2$ modulo $p$ | These conjectures are also true, and they follow similarly as in my proofs [here](https://mathoverflow.net/questions/283758/conjecture-the-number-of-points-modulo-p-of-certain-elliptic-curve-is-p-or) and [here](https://mathoverflow.net/questions/283862/e-mathbbf-p-in-p-p2-iff-p-is-of-the-form-27a227a7).
**Proof of Co... | 7 | https://mathoverflow.net/users/11919 | 283879 | 125,526 |
https://mathoverflow.net/questions/283847 | 4 | When confronted with finding an object that is maximal with regard to some ordering relation, most of us have the reflex to use Zorn's Lemma.
I am interested in instances of proving the existence of maximal objects, where Zorn's Lemma is *explicitly of no use*. By that I mean that you can construct chains of objects ... | https://mathoverflow.net/users/8628 | Maximality without Zorn | I do not know if this is an example, but it seems related:
Michael Roddy (and others), if I remember correctly, have proven results using the fixed point property (FPP) for posets that seem to enable them to deduce the existence of maximal elements. It may be, though, that those results are themselves based on result... | 2 | https://mathoverflow.net/users/51389 | 283891 | 125,529 |
https://mathoverflow.net/questions/283871 | 5 | According to Theorem 1.7 of Mark-Tosun's [paper](https://arxiv.org/pdf/1603.07710.pdf), the Brieskorn sphere $\Sigma(2,3,6m+1)$ admits two tight contact structure $\xi\_{i}\ (i=0,1)$. They are both Stein fillable and they are contactomorphic (but not isotopic).
My question:
>
> Consider the Ozsváth-Szabó contact ... | https://mathoverflow.net/users/44651 | Ozsváth-Szabó's contact invariant on the Brieskorn sphere $\Sigma(2,3,6m+1)$ | (Note: I might have screwed up orientations, so take everything with a grain of salt.) I will write an argument for the case $m=1$, showing that the reduced contact invariant $c^{\rm red}(\xi)$ is non-zero. I think that the argument(s) can be tweaked to work for $m>1$, as well, but I won't try to write it down properly... | 3 | https://mathoverflow.net/users/13119 | 283896 | 125,532 |
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