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/359011 | 4 | I want to know whether number rings tend to the integers as the discriminant tends to infinity.
In detail, let $n$ be a natural number and let $C(n)$ be the set of all number fields $K$ of degree $n$. For $K\in C(n)$ let $K\_{\mathbb R}=K\otimes\_{\mathbb Q}\mathbb R$. The real vector space $K\_{\mathbb R}$ comes with ... | https://mathoverflow.net/users/nan | Do number rings tend to Z? | The answer is yes if and only if $n$ is prime.
If $n$ is composite, fix $1<d<n$ a divisor of $d$, $L$ a number field of degree $n$. Then there are infinitely many $K$ a degree $n/d$ extension of $L$, with discriminants tending to infinity, and they satisfy $\mathcal O\_L \subset \mathcal O\_K$, so that $B\_{R \sqrt{d... | 3 | https://mathoverflow.net/users/18060 | 359029 | 151,247 |
https://mathoverflow.net/questions/358995 | 5 | Generic polynomials, which are recalled below, play an important role in the constructive aspects of the inverse Galois problem.
**Definition.** Let $P(\mathbf{t},X)$ be a monic polynomial in $\mathbb{Q}(\textbf{t})[X]$ with $\textbf{t} = (t\_1,\dots, t\_n)$ and $X$ being indeterminates, and let $\mathbb{L}$ be the ... | https://mathoverflow.net/users/56667 | Criterion for generic polynomials | I believe there is no good way to determine in general if a polynomial $P(\mathbf{t},X)$ is generic. In fact, given a number field $K$ and a univariate polynomial $P(t,x) \in \mathbb{Q}[t,x]$, the problem of determining whether this is some $t \in \mathbb{Q}$ for which $P(t,x)$ has a root in $K$ is quite hard (and in s... | 7 | https://mathoverflow.net/users/48142 | 359032 | 151,249 |
https://mathoverflow.net/questions/359051 | 2 | Let $A\_{i, j}, B\_{i, j}, C, D \in \mathbb{Q}$, and consider the following pair of equations
$$
A\_{1, 1} x\_1 y\_1 + A\_{1, 2} x\_1 y\_2 + A\_{2, 1} x\_2 y\_1 + A\_{2, 2} x\_2 y\_2 = C
$$
$$
B\_{1, 1} x\_1 y\_1 + B\_{1, 2} x\_1 y\_2 + B\_{2, 1} x\_2 y\_1 + B\_{2, 2} x\_2 y\_2 = D.
$$
I was interested in figuring out... | https://mathoverflow.net/users/84272 | Local to global principle for a pair of bilinear equations? | In fact such system of equations *always* have a solution (at least if the coefficients are general).
Let $X$ denote the closure of your variety in $\mathbb{P}^4\_{\mathbb{Q}}$. Explicitly:
\begin{align\*}
&A\_{1, 1} x\_1 y\_1 + A\_{1, 2} x\_1 y\_2 + A\_{2, 1} x\_2 y\_1 + A\_{2, 2} x\_2 y\_2 = C z^2 \\
&B\_{1, 1} x\_... | 6 | https://mathoverflow.net/users/5101 | 359058 | 151,256 |
https://mathoverflow.net/questions/359001 | 2 | Let $X$ be a topological space and $G$ be a topological group. Let $\tilde{G}$ be the sheaf of groups defined by the sheaf of sections of the product $G$ bundle $ \pi\_1:X\times G \rightarrow X $.
**(1)** Let $T\tilde{G}\_X$ be the set of all isomorphism class of $\tilde{G}$-torsors over $X$.
**(2)** Let $H^1(X,\... | https://mathoverflow.net/users/86313 | What is the geometric description of the set of isomorphism class of $G$-torsors over a site $C$? | You should read Section 4.5 of Olsson's book *Algebraic Spaces and Stacks*.
The notion of a site is a piece of category theory with no intrinsic geometry, so it doesn't really make sense to ask for a geometric description of torsors for a general site.
However, in the concrete geometric contexts where site theory... | 3 | https://mathoverflow.net/users/56878 | 359059 | 151,257 |
https://mathoverflow.net/questions/359075 | 0 | I am looking for a differential geometric version of the proof of the Riemann--Roch theorem
for Riemann surfaces, that is, $1$-dimensional compact complex manifolds. The proofs one usually finds are given in algebraic geometric terms, and can be seen as special cases of the sheaf theoretic approach of the general Hirz... | https://mathoverflow.net/users/126606 | A differential geometric proof of the Riemann--Roch theorem for lines | The details of the heat kernel proof for Atiyah-Singer index theorem become much simpler for Riemann surfaces (using a compatible Riemannian metric): a not too difficult computation shows that the first non-trivial term of the asymptotic expansion of the heat kernel along the diagonal is given by a certain combination ... | 2 | https://mathoverflow.net/users/4572 | 359077 | 151,259 |
https://mathoverflow.net/questions/345908 | 1 | *This question is a follow-up of the [previous question](https://mathoverflow.net/questions/343118/the-maximal-order-of-an-element-in-a-coxeter-group) and especially the last comment therein.*
Let $(W,S)$ be a finite Coxeter system with reflections $T$. Let $\ell\_T$ be the reflection length. According to [1611.03442... | https://mathoverflow.net/users/66288 | The least common multiple of all degrees of a finite Coxeter group and indecomposable elements in the generalized cycle decomposition | Concerning the question:
"Is it true that the set of orders of indecomposable elements in coincides (with or without repitition) with the set of all degrees of ?"
Thomas Gobet showed that in a finite irreducible Coxeter system each quasi-Coxeter element is indecomposable (see Proposition 3.5 in <https://arxiv.org/p... | 3 | https://mathoverflow.net/users/135824 | 359079 | 151,260 |
https://mathoverflow.net/questions/359013 | 6 | I recently encountered a result about vector bundles on Abelian varieties, which I found interesting. It characterizes homogeneous (translation invariant) vector bundles on Abelian varieties. More precisely any such vector bundle is in the form of $\bigoplus\_L L\otimes U\_L $, where $U\_L$ is a unipotent vector bundle... | https://mathoverflow.net/users/127776 | Homogeneous vector bundles on Abelian varieties | I don't have access to Miyanishi's article either (at least during lockdown), but as Ulrich suggested, one can look at Mukai's paper "Duality between D(X) and $D(\hat X)$...", where he introduces the Fourier-Mukai transform. I'll denote this transform by $\mathcal{F}$. On page 159, Mukai gives a proof of the characteri... | 4 | https://mathoverflow.net/users/4144 | 359083 | 151,262 |
https://mathoverflow.net/questions/359024 | 2 | This question is a relaxed version of [this question](https://mathoverflow.net/questions/352982/is-there-a-volume-preserving-diffeomorphism-of-the-disk-with-prescribed-singular).
Let $D \subseteq \mathbb{R}^2$ be the **closed** unit disk, and let $c \ge 2$.
>
> Does there exist a diffeomorphism $f:D \to D$ with ... | https://mathoverflow.net/users/46290 | Is there a diffeomorphism of the disk with constant sum of singular values? | Take a map $f$ of the form that you propose, so that the equation reduces to the ODE
$$(\psi' + r^{-1}\psi)^2 + (\phi'\psi)^2 = c^2.$$
If we fix $c := 12/5 > 2$ and
$$\psi(r) := \frac{6}{5}r\left(1-\frac{1}{6}r^4\right),$$
the equation reduces further to
$$\phi'(r) = 2r(1-r^4/4)^{1/2}(1-r^4/6)^{-1}.$$
This has an ana... | 2 | https://mathoverflow.net/users/16659 | 359088 | 151,263 |
https://mathoverflow.net/questions/359099 | 2 | [Legendre's conjecture](https://en.wikipedia.org/wiki/Legendre%27s_conjecture), proposed by Adrien-Marie Legendre, state that there is a prime number between $n^2$ and $(n+1)^2$ for every positive integer $n$.
My conjecture: Let $n$ be a positive integer, $p\_1$, $p\_2$ and $p\_3$ be odd prime numbers, and $n\ge2p\_1... | https://mathoverflow.net/users/157345 | The equivalent proposition of Legendre's conjecture | Your conjecture for sufficiently large $n$ is implied by [Cramer's conjecture](https://en.wikipedia.org/wiki/Cram%C3%A9r%27s_conjecture). In general though, conjectures like this unless they are coming from some specific application aren't that interesting. It is very easy to make many similar conjectures.
| 7 | https://mathoverflow.net/users/127690 | 359100 | 151,266 |
https://mathoverflow.net/questions/358113 | 9 | A. Okounkov said, "symplectic resolutions are Lie algebras of the 21st century." Is there a conjecture on the classification of symplectic resolutions? Do Braverman-Finkelberg-Nakajima Coulomb branches give most known examples of symplectic singularities (and do BFN Coulomb branches have explicit descriptions)? Where c... | https://mathoverflow.net/users/12395 | Classification of symplectic resolutions | Here is an answer by Gwyn Bellamy, which he let me post here:
1) Is there a conjecture on the classification of symplectic resolutions? No, not that I am aware of. I think this is the wrong question anyway. Rather, one should first try to classify all conic symplectic singularities. There is an amazing result of Nami... | 5 | https://mathoverflow.net/users/12395 | 359102 | 151,267 |
https://mathoverflow.net/questions/359027 | -2 | **Edit: I got rid of my old definitions. Everything should be clear now**
Since no one has answered [my question](https://math.stackexchange.com/questions/3637157/how-do-prove-the-darboux-like-sum-of-a-function-defined-on-a-countable-set-with) on MSE, I’m hoping to get an answer here. I apologize if you dislike my wr... | https://mathoverflow.net/users/87856 | Prove or disprove this integral of a function, defined on a countable set with infinite limit points, converges to zero | For the concrete question you ask, the answer is yes.
Your set $A$ is in fact the set of all numbers between $0$ and $1$ with binary expansion that has no more than 3 bits equaling 1.
Let $\ell\_+ = \lceil \log\_2 n \rceil$ and $\ell\_- = \lfloor \log\_2 n \rfloor$. You can estimate
$$ n' \geq (\ell\_- - 1) + ... | 3 | https://mathoverflow.net/users/3948 | 359105 | 151,269 |
https://mathoverflow.net/questions/359057 | 10 | Matching: <https://en.wikipedia.org/wiki/Matching_(graph_theory)>
Vertex Cover: <https://en.wikipedia.org/wiki/Vertex_cover>
It is easy to see that
$$\texttt{minimum vertex cover} \leq 2 \texttt{ maximum matching}$$
I want to know that for what kind of graphs the equality is hold in the above inequality.
As a... | https://mathoverflow.net/users/153948 | What graph's minimum vertex cover equals twice the maximum matching? | **Answer.** Such a graph $G$ is a disjoint union of odd complete graphs.
Obviously such graphs satisfy the equality $$\texttt{minimum vertex cover} = 2 \texttt{ maximum matching}.\quad (\star)$$
Assume that $G=(V,E)$ satisfies $(\star)$. Denote by $k$ the size of maximal independent set in $G$, then $$k=|V|-\texttt... | 5 | https://mathoverflow.net/users/4312 | 359108 | 151,270 |
https://mathoverflow.net/questions/359082 | 1 | I am not a specialist in measure theory, so excuse me if this is simple.
Let $\mu$ be a finite measure on a set $X$ (for example, the Lebesgue measure on $[0,1]$). Integrable functions on $X$ can be defined as limits (in different senses, in particular, in some definitions as uniform limits, but this does not play a... | https://mathoverflow.net/users/18943 | Integrable functions as elements of closed absolutely convex hulls of precompact sets of indicator functions | Attempt number 2. Consider the case $f\ge 0$.
For $\alpha\in[0,1]$ let $A\_{\alpha}=\{x\in X, f(x)\ge \alpha\}$, which is measurable. For $n\in \mathbb{N}$ define $f\_n=\frac{1}{n}\sum\_{k=1}^{n}\chi\_{A\_{\frac{k}{n}}}$. It is easy to see that $f\_n\le f\le f\_n+ \frac{1}{n}$, from where convex combinations of $\chi... | 2 | https://mathoverflow.net/users/53155 | 359115 | 151,271 |
https://mathoverflow.net/questions/359107 | 3 | Define a binary $(n, M, 2e + 1)$ code to be a code $C$ having $M$ code words in $\mathbb{F}\_2^n$ whose minimum distance is $2e + 1$.
Are there any sources about using algorithms to find all given codes of certain parameters? I suspect that there are some ways to reduce the search space or some other clever techniqu... | https://mathoverflow.net/users/157352 | What is known about computing all binary error correcting codes of given parameters? | People are generally only interested in "good" codes. Which means maximum minimum distance $d$ for given $n,M$ or minimum length for given $M,d$ etc. Let $A(n,d)$ be the maximum $M$ for which an $(n,M,d)$ code exists.
Even for such codes the computational complexity of the problem is overwhelming. There are too many ... | 4 | https://mathoverflow.net/users/17773 | 359120 | 151,274 |
https://mathoverflow.net/questions/338172 | 2 | In this post I present a similar question that shows section **A19** of [1], I was inspired in it to define my sequence and question. I am asking about it since I think that the problem that arises from my question harmonizes with this.
For integers $k\geq 1$ we denote the primorial of order $k$ as $N\_k$, that is
... | https://mathoverflow.net/users/142929 | On values of $n\geq 1$ satisfying that for all primorial $N_k$ less than $n$ the difference $n-N_k$ is a prime number | Instead of looking at primorials, one can look at so-called *admissible sets*. We say that a set of integers $S = \{h\_1, \cdots, h\_k\}$ is *admissible* if for all primes $p$, $S \pmod{p}$ is not a complete set of residues. Note that for all $p > |S|$, $S \pmod{p}$ is automatically not a full set of residues.
For ex... | 3 | https://mathoverflow.net/users/10898 | 359124 | 151,275 |
https://mathoverflow.net/questions/359123 | 0 | I have a question about convergence and properties of Dirichlet series. it seems a bit interesting and different about the convergences of Dirichlet Series to me.
With $c\in [0,1]$,
$$f(n) = \pm 1,\qquad |\sum\_{n\le x} f(n)-x^c|\le 2$$
Let A(x)= $\sum\_{n\le x} f(n)$ be a Summatory Function.And F(S)=$\sum\_n f(... | https://mathoverflow.net/users/153983 | Properties of Dirichlet series | Integrating/summing by parts allows you to prove that if $\sum A(n)/n^{s+1}$ converges then $\sum f(n)/n^s$ converges, which gives you the requested convergence. In general, the region where a Dirichlet series is conditionally convergent but not absolutely convergent can be up to width 1, but no larger.
Titchmarsh's ... | 0 | https://mathoverflow.net/users/9849 | 359140 | 151,281 |
https://mathoverflow.net/questions/359074 | 3 | I am trying to understand Deligne's 'Categories Tensorielles', and therefore I need some knowledge on linear categories. Looking at Wikipedia and nLab, I found some definitions and explanations, but I could not find any references to a book that covers this topic.
Can anyone recommend a good book in which linear cate... | https://mathoverflow.net/users/157078 | Literature on linear categories | Linear categories are not an independent subject of study, they usually appear in combination with other types of categories, for example abelian and/or monoidal. For all this stuff see:
>
> P.Etingof, S.Gelaki, D.Nikshych, V.Ostrik, *Tensor Categories* (2015)
>
>
>
Note, that the concept of a linear category ... | 4 | https://mathoverflow.net/users/35349 | 359150 | 151,284 |
https://mathoverflow.net/questions/359064 | 2 | Fix $\{w\_n\}\_n$ a sequence of positive real numbers, fix positive integers $N,K$, and fix $\eta>1$. I'm looking for a sequence of integers $\{k\_n\}\_n$ optimizing the following problem:
$$
\begin{aligned}
\min \sum\_{i=1}^n \frac{w\_i}{k\_i} \\
\mbox{s.t.}\\
\sum\_{i=1}^n k\_ik & = K\\
i^{\eta} w\_i & \leq k\_{i} ... | https://mathoverflow.net/users/36886 | Maximizing the length of a sequence under constraints | Presuming $k\_i$ are constrained to be positive (which I'm assuming to be the case), this can be solved as a Mixed-Integer Second Order Cone Program (MISOCP). Specifically, the reciprocal in the objective function can be handled by use of a rotated Second Order Cone constraint for each term in the objective function.
... | 1 | https://mathoverflow.net/users/75420 | 359154 | 151,286 |
https://mathoverflow.net/questions/359157 | 2 | What is the most general version of Sato-Tate ? Like, I know when $f$ is an eigenform (lying in the space of new-forms), without CM, of weight $k$ and level $N,$ then $\frac{a(p)}{p^{(k-1)/2}}'s$ are equidistributed in $[0,1]$ with respect to a mesaure. My question is, when $f$ is arbitary modular form (without CM), or... | https://mathoverflow.net/users/100578 | A general question on Sato-Tate | No statement about modular forms is going to be the most general form of Sato-Tate, because we now know Sato-Tate for modular forms on other groups, and conjecture it in much greater generality.
For the statement you're looking for, there will also be a gap between theorem and conjecture. Anyways, the method to dedu... | 8 | https://mathoverflow.net/users/18060 | 359162 | 151,287 |
https://mathoverflow.net/questions/359160 | 5 | To compute the simplicial homotopy group of a space $X$, we find a Kan fibrant replacement $X \to Y$ and calculate for that for $Y$, which can be implemented in a computer program.
Computing homotopy groups for spheres are fundamentally hard, and I believe the problem lies in the difficulty of finding their Kan fibra... | https://mathoverflow.net/users/124549 | Kan fibrant replacement for a sphere |
>
> Computing homotopy groups for spheres are fundamentally hard, and I believe the problem lies in the difficulty of finding their Kan fibrant replacement.
>
>
>
Computing the fibrant replacement for simplicial sets is quite
easy: it is given by the [Kan fibrant replacement functor](https://ncatlab.org/nlab/sho... | 6 | https://mathoverflow.net/users/402 | 359179 | 151,291 |
https://mathoverflow.net/questions/359159 | 7 | **Question.** *Let $K$ be a field of characteristic zero (large characteristic should be fine too). Let $q,q'$ be two non-degenerate quadratic forms on $K^n$ with $n=8$. Suppose that the Lie algebras $\mathfrak{so}(q,K)$ and $\mathfrak{so}(q',K)$ are isomorphic (these are simple of type $D\_4$, 28-dimensional). Does it... | https://mathoverflow.net/users/14094 | Is a 8-dimensional quadratic form recognized by its Lie algebra, modulo equivalence and scalar multiplication? | Yes: Proposition C.3.14 in Brian Conrad's article [Reductive group schemes](http://math.stanford.edu/~conrad/papers/luminysga3.pdf) is that $SO(q)$ determines $q$ up to similarity for all $q$ of dimension $> 2$. (This was pointed out by @user74230 in a comment somewhere.)
This could be viewed as a special case of a m... | 9 | https://mathoverflow.net/users/6486 | 359203 | 151,298 |
https://mathoverflow.net/questions/359194 | 2 | The central limit theorem can be generalized to independent but not iid random variables, provided they satisfy the Lyapunov condition (which looks something like a variance bound), see <https://en.wikipedia.org/wiki/Central_limit_theorem#Lyapunov_CLT>. The Lindeberg central limit theorem provides another such conditio... | https://mathoverflow.net/users/76764 | Reference request: Donsker's theorem for non-identical, independent random variables | Such results were obtained by A. A. Borovkov and his students. See e.g.
Borovkov, A. A.
Estimates in the invariance principle. (Russian)
Dokl. Akad. Nauk SSSR 206 (1972), 1037–1039.
Borovkov, A. A.
The rate of convergence in the invariance principle. (Russian. English summary)
Teor. Verojatnost. i Primenen. 18 (19... | 1 | https://mathoverflow.net/users/36721 | 359205 | 151,299 |
https://mathoverflow.net/questions/359133 | 2 | Does anyone know if there are any standard non-asymptotic results for M-estimators? I'm looking for finite-sample guarantees. Figured maybe someone here might know.
| https://mathoverflow.net/users/62012 | Non-asymptotic results for M-estimators? | Uniform Berry--Esseen-type bounds on the rate of convergence to normality for $M$-estimators were obtained by Michel and Phanzagl -- see e.g. Refs. [14, 16, 17] in [this paper](https://projecteuclid.org/download/pdfview_1/euclid.ejs/1491897618). In the latter paper, a non-uniform Berry--Esseen-type bound for maximum li... | 2 | https://mathoverflow.net/users/36721 | 359207 | 151,300 |
https://mathoverflow.net/questions/359168 | 5 | It is quite easy to construct a dynamical system which has a physical measure with a positive Lyapunov exponent and zero entropy, just a figure $\infty$ system. By Pesin's entropy formula such a measure can not be a Sinai-Ruelle-Bowen measure. Now my question is, if there exists a system with a physical measure with po... | https://mathoverflow.net/users/23542 | Physical measures that are not SRB | Yes. The simplest construction is to let $f$ be the figure-eight system so that $\delta\_p$ is a physical non-SRB measure (where $p$ is the saddle point) and let $g$ be an Anosov diffeomorphism with SRB measure $\mu$ (a hyperbolic toral automorphism with Lebesgue measure will do the job); then consider the product syst... | 4 | https://mathoverflow.net/users/5701 | 359208 | 151,301 |
https://mathoverflow.net/questions/359204 | 9 | Let $K\subset [0,1]$ denote the usual 1/3 Cantor set. I know that $\mathbb{C}\backslash K$ has no non-constant bounded analytic function, since the singularity $K$ can be removed. However, a statement I am reading says that $\mathbb{C}\backslash K$ admits a non-constant bounded harmonic function. Why is this true? Any ... | https://mathoverflow.net/users/123187 | Complex plane minus Cantor set admits non-constant bounded harmonic function | Since the Cantor set $K$ has Hausdorff dimension $\log2/\log 3<1$, it is a removable set for bounded analytic functions, and so, as you say, there is no bounded analytic function outside of $K$. But it does not mean that there are no bounded harmonic functions outside of $K$. Actually, any point of $K$ is regular for t... | 10 | https://mathoverflow.net/users/89429 | 359210 | 151,302 |
https://mathoverflow.net/questions/359161 | 1 | I know two facts and I’ve managed to figure out how to prove one, but the other one is still a little confusing.
>
> Let $G$ be a finite solvable group and $F(G)$ is the Fitting subgroup of $G$.
>
>
> (1) $G/Z(F(G))$ is isomorphic to a subgroup of ${\rm Aut}(F(G))$;
>
>
> (2) $G/F(G)$ is isomorphic to a subgrou... | https://mathoverflow.net/users/nan | Why $G/F(G)$ is isomorphic to a subgroup of ${\rm Out}(F(G))$? | For (2), there is a general construction at work here. Given a short exact sequence of groups
$$1 \to K \to G \to Q \to 1$$
there is always a well-defined homomorphism $\varphi: Q\to \mathrm{Out}(K)$. The idea is to lift elements of $Q$ into $G$, and have them act on $K$ by conjugation. It would be instructive to w... | 5 | https://mathoverflow.net/users/95243 | 359215 | 151,304 |
https://mathoverflow.net/questions/359070 | 5 | In attempting to understand the paper "*Superrigidity, Weyl groups, and actions on the circle*" of Uri Bader, Alex Furman and Ali Shaker ([linked at Furman's page](http://homepages.math.uic.edu/~furman/preprints/supercircle061406.pdf))
I find that towards the end of the proof of Lemma 2.2 they have a situation where ... | https://mathoverflow.net/users/15482 | Seeking to understand meaning of "von Neumann spectrum" in a paper of Bader–Furman–Shaker | Let me explain what we mean by the term "von Neumann spectrum".
Before doing so, let me recall the better known *Gelfand duality*:
the functor $X\mapsto C(X)$, from
the category of compact Hausdorff topological spaces to the category of unital commutative C\*-algebras, establishes an equivalence of categories. The ... | 4 | https://mathoverflow.net/users/89334 | 359218 | 151,305 |
https://mathoverflow.net/questions/359165 | -1 | Let $ f: \mathbb{R}\longrightarrow \mathbb{R}$: compute
proximal of following mapping
$$ f(x)= \sqrt {1-x^2} $$
for $ x \geq 0 $
I know that the proximal is given by
$$ \operatorname{prox}\_{\!f} (x)= \operatorname{argmin}\_{u\in \mathbb{R}} \big\{f(u) +(1/2)\Vert u-x\Vert^2\big\}$$
| https://mathoverflow.net/users/149388 | Compute the proximal of a mapping | Too long to comment.
I assume that $x$ and $u$ are within the range $[-1,1]$ for $f$ to be well-defined.
Suppose $x=\sin(\theta)$, $0\leq \theta \leq \pi/2$. Let $u=\sin(\phi)$. In that case, the optimization problem is:
$$
\min\_{\phi}~~\cos(\phi) + \frac{1}{2}(\sin(\phi)-\sin(\theta))^2.
$$
Differentiating the c... | 0 | https://mathoverflow.net/users/155380 | 359220 | 151,306 |
https://mathoverflow.net/questions/359196 | 1 | Let $d \gg 1$. Let $G:=(U, V, E)$ be some bipartite graph such that deg$(u) \le d$ for all $u\in U$ and deg$(v) \le 3$ for all $v \in V$.
Now, is it possible to color vertices in $U$ with 3 colors such that firstly, size of each color class is roughly $|U|/3$ and secondly, at most a fraction $\beta$ of vertices in $V... | https://mathoverflow.net/users/24930 | One part of a bipartite graph has max degree 3. Partition the other part to 3 ~equal subsets s.t. just a fraction of first part see all 3 subsets | Let $G=(U,V,E)$ be a bipartite graph where $U=[n], V=\binom{[n]}{3}$, and there is an edge between $u \in U$ and $v \in V$ if and only if $u \in v$. Then $\deg(u)=\binom{n-1}{2}$ for all $u \in V$ and $\deg(v)=3$ for all $v \in V$. However, every colouring of $U$ with $3$ colours, such that each colour class has size r... | 2 | https://mathoverflow.net/users/2233 | 359230 | 151,310 |
https://mathoverflow.net/questions/359090 | 8 | I have the following question but have no idea on its proof (one direction is trivial):
>
> Let $A$ and $B$ be (bounded) positive self-adjoint operators on a complex Hilbert space $H$. Prove that
> $$\limsup\_{n \to \infty} \|A^n x\|^{1/n} \le
> \limsup\_{n \to \infty} \|B^n x\|^{1/n}$$
> holds for every $x \in H... | https://mathoverflow.net/users/42571 | A question about comparison of positive self-adjoint operators | The condition $A^n \leq B^n$ for all $n$ defines the *spectral order* on the positive part of $B(H)$, usually written $A \preceq B$. It makes the positive part of any von Neumann algebra a complete lattice. It's equivalent to saying that $P\_{[0,t]}(B) \leq P\_{[0,t]}(A)$ for all $t > 0$, where
$P\_S(A)$ is the spectra... | 5 | https://mathoverflow.net/users/23141 | 359239 | 151,315 |
https://mathoverflow.net/questions/359206 | 2 | Given a matrix $X \in \mathbb{R}^{m \times n},$ then the **spectral** norm is defined by
$$\left \| X\right\| := \max\limits\_{i \in \{1, \dots, \min\{m,n\}\} }\sigma\_i (X)$$
whereas the **nuclear** norm is defined by
$$\left \| X \right \|\_\* := \sum\limits\_{i=1}^ {\min\{m,n\}} \sigma\_i (X)$$
It is a well-... | https://mathoverflow.net/users/149742 | $\arg\max$ in the dual norm of the nuclear norm | The answer is $X^\* = uv^\*$ where $u$ and $v$ are the left and right singular vectors of $M$ associated with the largest singular value. If the largest singular value has multiplicity larger than $1$, the argsup is a convex set whose extreme points are the matrices of the form described above.
| 5 | https://mathoverflow.net/users/112954 | 359242 | 151,316 |
https://mathoverflow.net/questions/359241 | 3 | Let $S\_\omega$ denote the collection of bijections $f:\omega\to\omega$. We say that $f \in S\_\omega$ *has a fixed point* if there is $x\in \omega$ with $f(x) = x$.
It is a short exercise to show that if $f,g\in S\_\omega$ then $g\circ f$ has a fixed point if and only if $f\circ g$ has a fixed point.
Let $$E = \bi... | https://mathoverflow.net/users/8628 | Graph structure on $S_\omega$ induced by fixed points on compositions | Yes, and in addition we can choose the elements to be fixed-point-free order 2 elements in a finite symmetric group. For order 2 elements, the incidence relation means $f(x)\neq g(x)$ for every $x$.
Let $V$ be your graph (sorry $G$ sounds too group-wise to me): I view it as a graph structure on $\{1,\dots,n\}$.
St... | 4 | https://mathoverflow.net/users/14094 | 359248 | 151,318 |
https://mathoverflow.net/questions/359249 | 16 | Recall that a (unital) [Boolean ring](https://en.wikipedia.org/wiki/Boolean_ring) is a (unital) commutative ring $A$ where every element is idempotent; it follows that $A$ is of characteristic 2. There is an equivalence of categories between Boolean rings and Boolean algebras; the Boolean algebra corresponding to a Boo... | https://mathoverflow.net/users/2362 | What is a module over a Boolean ring? | **Theorem:** Given $A$ a boolean ring/boolean algebra then there is an equivalence of categories between the category of $A$-modules and the category of sheaves of $\mathbb{F}\_2$-vector spaces on Spec $A$. The equivalence sends every sheaf $\mathcal{M}$ of $\mathbb{F}\_2$-vector space to its space of section, $\Gamma(... | 19 | https://mathoverflow.net/users/22131 | 359253 | 151,320 |
https://mathoverflow.net/questions/359240 | 1 | Let $V : (-1, 1)^d \to \mathbf{R}\_+$ be a smooth function, and for $\beta > 0$, define
\begin{align}
P\_\beta ( dx ) &= \exp \left( - \beta V ( x ) \right) / z (\beta) \, dx\\
z (\beta) &= \int \exp \left( - \beta V ( y ) \right) \, dy
\end{align}
with the assumption that $V$ is sufficiently well-behaved at the bo... | https://mathoverflow.net/users/121692 | Convergence of probability measures which (asymptotically) concentrate along a submanifold | Here is an outline of the proof. Let $n:=d$. Suppose that for some real $\delta>0$ the $\delta$-neighborhood of the set $F:=\mathcal F$ can be covered by pairwise disjoint sets $U\_1,\dots,U\_k$ such for each $j=1,\dots,k$ the boundary of the set $U\_j$ is of zero Lebesgue measure and the closure $\bar U\_j$ of $U\_j$ ... | 1 | https://mathoverflow.net/users/36721 | 359255 | 151,321 |
https://mathoverflow.net/questions/359257 | 2 | Let $X$ and $Y$ be uncountable Polish spaces and let $h\colon X\to Y$ be a Borel isomorphism. Suppose that $A\subset X$ has the [Baire property](https://en.wikipedia.org/wiki/Property_of_Baire). Must $h[A]$ have the Baire property in $Y$ too?
| https://mathoverflow.net/users/157451 | Do Borel isomorphisms of Polish spaces preserve the Baire property? | No, it need not.
Take for example $X=Y=\mathbb{R}$, let $A=[0,1]$, let $B$ be some nowhere dense perfect subset of $[2,3]$, and consider a Borel isomorphism $h:\mathbb{R}\cong \mathbb{R}$ swapping $A$ and $B$ and fixing everything else.
Then for each $U\subseteq A$ the preimage $h^{-1}(U)$ is meager in $\mathbb{R}... | 2 | https://mathoverflow.net/users/8133 | 359261 | 151,323 |
https://mathoverflow.net/questions/359254 | 5 | Let $G$ be a compact simple Lie group. Choose a system of positive roots, and let $\mathrm{SU}(2) \subset G$ correspond to the highest root, and $\mathbb{Z}/2 \subset \mathrm{SU}(2)$ the centre. The centralizer of this $\mathbb{Z}/2$ inside of $G$ is a subgroup $H \subset G$ of shape $\mathrm{SU}(2) \circ K = (\mathrm{... | https://mathoverflow.net/users/78 | What is the name of the real form corresponding to the quaternionic symmetric space? | In the Crelle paper by Gross and Wallach “On quaternionic discrete series representations, and their continuations” (*J. Reine Andgew. Math.* **481** (1996) 73–123, [available here](https://eudml.org/doc/153874)), the form you mention is described in §3 and called “the quaternionic real form”.
There are other referen... | 2 | https://mathoverflow.net/users/17064 | 359263 | 151,324 |
https://mathoverflow.net/questions/358469 | 8 | *This question was [asked and bountied](https://math.stackexchange.com/q/3619818/28111) at MSE without response.*
---
Call a sentence $\varphi$ in the language of arithmetic **$Q$-like** iff $\mathbb{N}\models\varphi$ and $\{\varphi\}$ is essentially incomplete. The standard example is of course the conjunction o... | https://mathoverflow.net/users/8133 | The lattice of analogues of Robinson's $Q$ | $\mathfrak{Q}$ is the countable random distributive lattice.
Emil Jeřábek has already pointed in his comments that there are only two possibilities for $\mathfrak{Q}$. Either there are no greatest element in $\mathfrak{Q}$ and it is the countable random distributive lattice. Or there is the greatest element in $\math... | 5 | https://mathoverflow.net/users/36385 | 359268 | 151,327 |
https://mathoverflow.net/questions/359213 | 5 | [Mayer expansions and the Hamilton–Jacobi equation](https://link.springer.com/article/10.1007/BF01010398) by D. Brydges and T. Kennedy begins mentioning that many problems in statistical mechanics and QFT center on the analysis of integrals of the form:
\begin{equation}
\int d\mu\_{C}(\varphi)e^{-V(\varphi+\varphi')} \... | https://mathoverflow.net/users/150264 | Effective action, partition function and the renormalization group | These correspondences might be helpful: we have the *partition function* $Z(\beta)$, the *effective action* $W(\beta)$, the *classical action* $I\_\beta[\phi]$ for physical variables $\phi(x,\tau)$ in Euclidean time $\tau$ with period $\beta$ equal to inverse temperature. These are related by
$$ e^{-W(\beta)}=Z(\beta)=... | 4 | https://mathoverflow.net/users/11260 | 359270 | 151,328 |
https://mathoverflow.net/questions/358313 | 10 | For positive integers $n\geq 2k$, it is known that the chromatic number of the [Kneser graph](https://arxiv.org/pdf/1803.04342.pdf) $K\_{n,k}$ is $n-2k+2$. Moreover, the Schrijver graph $S\_{n,k}$ (definition in the same link), which is a subgraph of $K\_{n,k}$, also has chromatic number $n-2k+2$. The number of vertice... | https://mathoverflow.net/users/156705 | Kneser subgraph with high chromatic number | The general answer is **no**, at least when $n$ is close to $2k$.
Theorem 3 in [this paper](https://arxiv.org/abs/1201.3271) shows that a graph with odd girth $\geq 2d+1$ and chromatic number $>m$ contains more than
$$
\frac{(m+d)(m+d+1)\dots(m+2d-1)}{2^{d-1}d^d}
$$
vertices. The parameters for the Kneser graph $K\_... | 3 | https://mathoverflow.net/users/17581 | 359290 | 151,333 |
https://mathoverflow.net/questions/359308 | 3 | Let $(\mathcal{M},g)$ be a torsion free compact Riemannian manifold of dimension $n$. Hence from the metric we know there is an associated horizontal sub-bundle $H\_u F \mathcal{M}$ of the orthonormal frame bundle $F \mathcal{M}$ at a frame $u:\mathbb{R}^n\to T\_p\mathcal{M}$. Furthermore there are theorems that state ... | https://mathoverflow.net/users/68927 | Orthonormal frame bundles on a manifold | On the orthonormal frame bundle we have soldering forms $\omega\_i$ and connection forms $\omega\_{ij}$. A lift is horizontal just when $\omega\_{ij}=0$ on it. So the velocity can be described by its $\omega\_i$ components: $v\_i(t)=i\_{\tilde\gamma'(t)}\omega\_i$. The energy is $\sum\_i v\_i^2$. You don't define $w$, ... | 6 | https://mathoverflow.net/users/13268 | 359313 | 151,340 |
https://mathoverflow.net/questions/359306 | 13 | Recently, Jade Master [asked](https://twitter.com/JadeMasterMath/status/1252045020804177921) whether the tensor product of chain complexes could be viewed as a special case of Day convolution. Noting that chain complexes [may be viewed](https://math.stackexchange.com/questions/2139250/can-the-category-of-chain-complexe... | https://mathoverflow.net/users/130058 | Is the tensor product of chain complexes a Day convolution? | The answer to the question posed in the title of your post is **yes**, the tensor product of chain complexes is a Day convolution product. The important thing to note is that, to define a Day convolution monoidal structure on the $\mathcal{V}$-enriched functor category $[\mathcal{C},\mathcal{V}]$ (where $\mathcal{V}$ i... | 20 | https://mathoverflow.net/users/57405 | 359316 | 151,342 |
https://mathoverflow.net/questions/266049 | 6 | I asked this question on stackexchange (<https://math.stackexchange.com/questions/2212226/restriction-of-spin7-4-form-to-mathbbr-times-s7>) but was advised to ask again here:
I'm currently reading through Jason Lotay's paper "Associative Submanifolds of the 7-sphere" (<https://arxiv.org/pdf/1006.0361v1.pdf>) and the ... | https://mathoverflow.net/users/106816 | Restriction of "Spin(7) 4-form" to $\mathbb{R}_+\times S^7$ | One way of looking at this is that you know that $\Phi\_0$ induces the standard Euclidean metric $g\_{0}$ on $\mathbb{R}^8$ and that the restriction of this metric to $S^7$ is the round metric, i.e.
$$g\_{0}=dr \otimes dr + r^2 g\_{S^7}.$$
Moreover the volume forms of $\mathbb{R}^8$ and $S^7$ (with the induced orientat... | 1 | https://mathoverflow.net/users/86198 | 359322 | 151,346 |
https://mathoverflow.net/questions/359062 | 6 | I’m learning the Fitting subgroup these days. I’m interested in this topic and particularly in the role that it plays in the structure of groups. Many people on MSE mentioned that the Fitting subgroup/generalized Fitting subgroup controls the structure of a group. Here are some quotes.
@Stephan mentioned in the [comm... | https://mathoverflow.net/users/nan | Why do we say the Fitting subgroup/generalized Fitting subgroup control the structure of a group? | I'll try and give my point of view on the question in the title.
The key facts that answer your question are the following:
1. If $G$ is a finite solvable group, then $C\_G(F(G))=Z(F(G))$.
2. If $G$ is a finite group, then $C\_G(F^\*(G))=Z(F^\*(G))$.
The first fact is classical -- I don't know who should be credi... | 5 | https://mathoverflow.net/users/801 | 359335 | 151,352 |
https://mathoverflow.net/questions/358483 | 7 | I hope this is an appropriate question for this forum. If not, I apologize. Before stating my question (which may be found at the end of this post), I will attempt to provide sufficient context.
I am studying eigenfunctions of the Laplacian (both Dirichlet and Neumann cases) on a specified bounded domain $D\subseteq... | https://mathoverflow.net/users/126714 | Why are we interested in operators that share a basis of eigenfunctions? | I'm not sure what precisely you are looking for but let me try the guess an answer on the soft end (basic intuition stuff) of what you are asking. I'm not qualified for the hard end of this question, for this you want an expert on partial differential operators and microlocal analysis. I'm in the applied/physics corner... | 1 | https://mathoverflow.net/users/143349 | 359336 | 151,353 |
https://mathoverflow.net/questions/358673 | 6 | Does there exist a finite morphism $\pi\colon X\to \mathbb{P}\_{\mathbb{C}}^n$, that does not admit a rational section along any prime divisor $D\subset\mathbb{P}^n$ in the locus where $\pi$ is etale?
| https://mathoverflow.net/users/nan | Does a finite morphism to $\mathbb{P}^n$ necessarily split at some height one point in the etale locus? | Let $f:X\to Y$ be a finite surjective morphism of smooth (projective) varieties over complex numbers. Let $B\subset Y$ be the branch locus and let $q\in Y-B$. Let $f^{-1}(q)=\{p=p\_1,\ldots, p\_n\}$, where $n=\deg f$. Take $D\subset X$, a smooth divisor passing through $p$, but not passing through $p\_i, i>1$. Let $E=f... | 2 | https://mathoverflow.net/users/9502 | 359344 | 151,357 |
https://mathoverflow.net/questions/359262 | 3 | We have tempered distribution $K$ in $\mathbb{R}^n$ which coincides with a locally integrable function in $\mathbb{R}^n\setminus \{0\}$. We call the condition
$$\int\_{|x|>2|y|}|K(x-y)-K(x)|dx\leq B \hspace{1cm}\forall y\in \mathbb{R}^n$$
for some constant $B$, the Hörmander condition.
>
> **I want to show that if... | https://mathoverflow.net/users/127918 | Gradient condition implies Hörmander condition | I figured out proving it by using mean value theorem (thanks to @WillieWong). Observe that
\begin{align\*}
\int\_{|x|>2|y|}|K(x-y)-K(x)|dx &\leq \int\_{|x|>2|y|}|\nabla K(tx+(1-t)(x-y))||y| dx\\
&\leq \int\_{|x|>2|y|} \frac{C}{|x-(1-t)y|^{n+1}}|y|dx\\
&\leq \int\_{|x|>2|y|} \frac{C}{[|x|-(1-t)|y|]^{n+1}}|y|dx&\because ... | 2 | https://mathoverflow.net/users/127918 | 359354 | 151,361 |
https://mathoverflow.net/questions/358048 | 3 | I am a masters student and I am interested in number theory. Due to lockdown I have a lot of time and I thought of reading a research paper in Number theory which is " Many Odd zeta values are irrational by " Stephane Fischler, Johannes Sprang and Wadim Zudilin.
I have a question on page 8 just after the (3.6)
>... | https://mathoverflow.net/users/151209 | Unable to deduce an inequality in paper on odd zeta values of Fischler, Sprang and Zudilin | **1.** Let us prove the first inequality. From the definition of $c\_{k,j}$, it is clear that
$$c\_{k,j}\leq D^{3Dn} n!^{s+1-3D} (k+3n+1)^{3Dn+1}k^{-(s+1)(n+1)},$$
hence it suffices to verify that
$$(k+3n+1)^{3Dn+1}\leq 2^{3Dn}k^{3D(n+1)}.$$
As explained in the paper, $k$ is much larger than $n$, and $n$ is itself larg... | 5 | https://mathoverflow.net/users/11919 | 359357 | 151,362 |
https://mathoverflow.net/questions/359362 | 2 | I want to understand theorem 5.21 (page 224) in [this link](https://www.math.wisc.edu/~roch/mdp/roch-mdp-chap5.pdf) and here is where I don't understand: $$
\{W = t\} = \{t \text{ is the first ladder index in }R\_1, \dots, R\_t\},$$
i.e. $\{R\_t = 1, R\_1 < 1, \dots, R\_{t-1} < 1\}$.
And if anyone can offer easier wa... | https://mathoverflow.net/users/157350 | Proof of Hitting-time theorem in branching processes | In the proof of Theorem 5.21 (page 224) in the notes at your link, we find the definitions
$R\_i:=1-S\_i$, with $R\_0:=0$. By Lemma 5.17 and the three-line display on page 221 of those notes,
$$W=\tau\_0=\inf\{t\ge0\colon S\_t=0\},$$
whence
$$W=\tau\_0=\inf\{t\ge0\colon R\_t=1\}.$$
Thus,
$$W=t\iff R\_1<1,\dots,R\_{t... | 2 | https://mathoverflow.net/users/36721 | 359373 | 151,366 |
https://mathoverflow.net/questions/359297 | 0 | Given an integer $n$, and 2 real sequences $\{a\_1, \dots, a\_n\}$ and $\{b\_1, \dots, b\_n\}$, with $a\_i$, $b\_i$ > 0, for all $i$. For a fixed $m < n$ let $\{P\_1, \dots, P\_m\}$ be a partition of the set $\{1, \dots, n\}$ as in $P\_1 \cup \dots \cup P\_m$ = $\{1, \dots, n\}$, with the $P\_i$'s pairwise disjoint. I ... | https://mathoverflow.net/users/43628 | Optimal partition search | Given the monotonicity property, here is a shortest-path formulation. The nodes are $(i,k)$, where $i\in\{1,\dots,n\}$ and $k\in\{1,\dots,m\}$, plus a dummy sink node $(n+1,m+1)$. The directed arcs are from $(i,k)$ to $(j,k+1)$, where $i<j$, with the interpretation that items $i,\dots,j-1$ appear in part $P\_k$. The ar... | 1 | https://mathoverflow.net/users/141766 | 359387 | 151,371 |
https://mathoverflow.net/questions/359390 | 4 | Let $X$ be a "nice" space: metrizable, connected, locally path connected perhaps. Let $K\subset X$ be a compact set.
>
> Is there a always a compact connected $L\subset X$ such that $K\subset L$?
>
>
>
This is true if we assume local compactness: cover $K$ with a finite number of connected relatively compact o... | https://mathoverflow.net/users/53155 | Is it possible to connect every compact set? | Choose a sequence $\varepsilon\_n\to 0$ and a $\varepsilon\_n$-net $N\_n$ for each $n$.
Assume $N\_0$ is a one-point set.
For each point in $x\in N\_k$ choose a closest point in $y\in N\_{k-1}$ and connect $x$ to $y$ by a curve.
Note that we can assume that diameter of the curve is at most $\delta\_k$ for a fixed seque... | 5 | https://mathoverflow.net/users/1441 | 359395 | 151,374 |
https://mathoverflow.net/questions/359427 | 1 | I have run into the following statement in the literature (e.g. [here](https://arxiv.org/pdf/1306.4089.pdf), p.5, after Theorem 1.1): that weak convergence of positive $(1,1)$-currents on a complex manifold is equivalent to $L^1$ (I presume $L^1\_{loc}$ in the non-compact case) convergence of their $dd^c$-potentials wh... | https://mathoverflow.net/users/2234 | weak convergence of positive currents vs. $L^1$ convergence of normalized potentials | Let $X$ be a compact complex manifold and let $T\_n=\theta+dd^c \varphi\_n$ be a sequence of positive currents where $\theta$ is a fixed smooth $(1,1)$-form on $X$. Assume that $\varphi\_n$ are normalized such that $\sup\_X \varphi\_n=0$ (one could have an analogous statement choosing the normalization $\int\_X \varphi... | 2 | https://mathoverflow.net/users/5659 | 359429 | 151,380 |
https://mathoverflow.net/questions/359419 | 3 | Let $f:X \to \mathbb{R}$ be a Morse function on some compact submanifold $X \subset \mathbb{R}^n$, and assume that $p \in X$ is *not* a critical point of $f$. For some $\epsilon > 0$ let $D\_\epsilon(p)$ denote the Euclidean disk of radius $\epsilon$ around $p$. I'd like to claim that there are some small $\epsilon > 0... | https://mathoverflow.net/users/18263 | When is the Morse equivalence local? | If $p\_0$ is **not** a critical point of $f$ then the implicit function theorem states that, there exists local coordinates $(x^1,\dotsc, x^n)$, defined in an open neighborhood $U$ of $p\_0$ in $\newcommand{\bR}{\mathbb{R}}$ $\bR^n$ such that, in these coordinates we have ($m=\dim X$)
$$
x^i(p\_0)=0,\;\;\forall i,
$$
$... | 2 | https://mathoverflow.net/users/20302 | 359440 | 151,383 |
https://mathoverflow.net/questions/359414 | 1 | Suppose that $A$ is a given subset of $I=[0,1],\ $ and
$ \left\{ x\_j = \frac{j}{m} \right\}\_{j=0}^{m}\ $ is the $m$-partition of $I$, and $\nu(m)$ is the number of
$\ [x\_{i-1},x\_{i}]\ $ such that $\ [x\_{i-1},x\_{i}]\cap A \neq \emptyset\ $ for intervals $\ [x\_{i-1},x\_{i}]\ $ belonging to the $m$-partition. Also ... | https://mathoverflow.net/users/152618 | A question about dense sets | I'll construct an $A$ that satisfies the conditions of the question but is not dense in any interval. I'll use the notation in the question and abbreviate the $i$-th interval of the $m$-th partition as $J(m,i)$. There are two requirements that I need to satisfy:
(1) For every $m$, $A$ contains points from at least $\... | 2 | https://mathoverflow.net/users/6794 | 359446 | 151,386 |
https://mathoverflow.net/questions/359401 | 10 | I am in master program of mathematics, specialized in PDE and numerical analysis. Now I am trying to decide which classes to take for next semester. Of course I want to become an expert in my field, but I am also interested in Geometry. I learned some differential geometry in my bachelor class, and I want to take the n... | https://mathoverflow.net/users/151368 | Can learning Riemann surfaces be more beneficial than numerical analysis for an analyst? | [Painleve](http://www.math.harvard.edu/archive/213b_spring_05/riemann_roch_abel_th.pdf) - It came to appear that, between two truths of the real domain, the easiest path quite often passes through the complex domain.
Hadamard - It has been written that the shortest and best way between two truths of the real domain o... | 10 | https://mathoverflow.net/users/12178 | 359448 | 151,387 |
https://mathoverflow.net/questions/359437 | 6 | A (split) reductive linear algebraic group is equivalently described by combinatorial information called a root datum.
A toric variety is described by combinatorial information called a fan.
Both correspondences use the character lattice.
The reference:
<http://u.cs.biu.ac.il/~margolis/Linear%20Algebraic%20Mo... | https://mathoverflow.net/users/148524 | Relationship between fans and root data | (1) A (connected) reductive group $G$ over an algebraically closed field $k$ is described by a combinatorial object called the *based root datum* ${\rm BRD}(G)$.
(2) A *spherical homogeneous space* $Y=G/H$ is a homogeneous space on which a Borel subgroup $B$ of $G$ acts with an open Zariski-dense orbit. It is descri... | 8 | https://mathoverflow.net/users/4149 | 359453 | 151,389 |
https://mathoverflow.net/questions/359460 | 2 | I have asked [this question](https://math.stackexchange.com/questions/3644735/why-does-this-tweak-to-sers-series-expression-for-zetas-always-yield-fs) at math stack exchange, however it did not get any traction. Still curious about the answer though.
Numerical evidence suggests that:
$$\lim\_{N \to +\infty} \sum\_{... | https://mathoverflow.net/users/12489 | Does this series, related to the Hasse/Ser series for $\zeta(s)$, converge for all $s \in \mathbb{C}$? | We have to show that
$$l(s):=\sum\_{n=1}^\infty\frac1n\,S\_n(s)=s,$$
where
$$S\_n(s):=\sum\_{k=0}^n(-1)^k\binom nk\frac1{(k+1)^s} \\
=\sum\_{k=0}^n(-1)^k\binom nk\int\_0^\infty du\,u^{s-1}e^{-(k+1)u}/\Gamma(s) \\
=\int\_0^\infty du\,u^{s-1}e^{-u}\sum\_{k=0}^n(-e^{-u})^k\binom nk/\Gamma(s) \\
=\int\_0^\infty du\,u^{s-... | 6 | https://mathoverflow.net/users/36721 | 359466 | 151,392 |
https://mathoverflow.net/questions/359355 | 3 | Preparing my Linear Algebra lecture I like the determinant free approach of Axler because the proof that operators $T$ on an $n$-dimensional complex vector space have eigenvalues is so simple:
Fix any non-zero vector $x$, observe that $x,T(x),\ldots,T^n(x)$ are linearly dependent to get a non-trivial linear combinati... | https://mathoverflow.net/users/21051 | How to find eigenvalues following Axler? | Using minimality can help.
Without using the polynomial-root oracle you can find the *minimal* polynomial of $T$ by looking for the first linear dependence among $I, T, T^2, \dotsc, T^r$ (for the minimal $r$—starting with just $I,T$ and increasing $r$ until you find a linear dependence). Alternatively, find the first... | 3 | https://mathoverflow.net/users/88133 | 359467 | 151,393 |
https://mathoverflow.net/questions/359391 | 5 | Suppose $X$ and $Y$ are two infinite dimensional Banach spaces. What can we say about the set of all injective continuous linear operators between $X$ and $Y$? Is it always nonempty?
| https://mathoverflow.net/users/41137 | Injective continuous operators between Banach spaces | Of course the dimension is an obvious obstacle, but even if the space have the same cardinality of Hamel bases the answer is no. For example in the paper
**A. Avilés, P. Koszmider,**
A Banach space in which every injective operator is surjective.
*Bull. Lond. Math. Soc.* 45 (2013), no. 5, 1065–1074
the authors con... | 11 | https://mathoverflow.net/users/121665 | 359469 | 151,394 |
https://mathoverflow.net/questions/359288 | 7 | It is well known that if $X$ is a Polish space and $\mathcal{F} \subset \mathcal{M}\_+(X)$ (the set of finite positive Radon measures on $X$) is uniformly tight and bounded in mass, it is relatively compact w.r.t. to the weak topology, i.e. the coarsest topology on $\mathcal{M}\_+(X)$ w.r.t. the maps $\mu \mapsto \int\... | https://mathoverflow.net/users/142961 | Prokhorov theorem on non Polish spaces | This seems to be a pure general topology problem. Enriching the topology will necessarily destroy the Prohorov property whenever the enrichment matters. If you have two nested Hausdorff topologies and a set is relatively compact in the finer topology, then the closure is the same under both topologies and the trace top... | 2 | https://mathoverflow.net/users/35357 | 359490 | 151,398 |
https://mathoverflow.net/questions/359485 | 3 | For a Schwartz function $\psi(x)=xe^{-x^2}$ define $\varphi(x):=\psi'(x)$ and consider a family of $L^1$-dilations of $\varphi$ given by:
$$
\varphi\_t(x)=\frac{1}{t}\varphi(x/t), \qquad t>0.
$$
$\textbf{Question:}$ Is there a function $f\in L^\infty(\mathbb{R})$ such that
\begin{equation}\label{eq:1}
\liminf\_{t\rig... | https://mathoverflow.net/users/157356 | Example of a bounded function whose mean-zero mollification diverges at a point | $\newcommand{\ph}{\varphi}$
$\newcommand{\eps}{\varepsilon}$
Let $a\_k$ be a very fast-growing sequence of integers (I think $a\_k = 2^{1000k}$ should be enough). Consider the function $f$ defined as
$$f(x) = \sum\_{k = 1}^\infty \chi\_{[a\_k, 2a\_k]}.$$
I claim that there is a constant $c > 0$ such that for $\frac{1... | 2 | https://mathoverflow.net/users/104330 | 359495 | 151,400 |
https://mathoverflow.net/questions/359488 | 3 | Under Martin's Axiom (and non-CH) the Lebesgue measure is $2^\omega$-additive in the sense that unions of fewer than continuum ($2^\omega$) many null sets are measureable and null. In ZFC we may however extend the Lebesgue measure to a finitely-additive measure on the power set of $[0,1]$ and still call it atomless.
... | https://mathoverflow.net/users/157577 | Finitely additive, $\kappa$-additive atomless measures in ZFC | I claim that a non-atomic measure $\mu$ can never be $<{2^\omega}^+$-additive. Then the same applies to any finitely-additve extension.
Let $(\Omega, \frak{A}, \mu)$ be a measure space and let us assume that $\mu$ is non-atomic. It follows that there exists $A \in \frak{A}$ such that $0 < \mu(A) < \infty$. I now want... | 3 | https://mathoverflow.net/users/134910 | 359496 | 151,401 |
https://mathoverflow.net/questions/359331 | 11 | Let $f\colon X \to Y$ a surjective proper map between smooth varieties over an algebraically closed field $k$ of characteristic zero. Let $Z\subset Y$ be a closed non-reduced subscheme. Is the preimage $f^{-1}(Z)$ nonreduced?
(The situation I am interested in is the resolution of an ideal sheaf, but I do not know if ... | https://mathoverflow.net/users/48866 | Is the preimage of a nonreduced subscheme via a proper map nonreduced? | If $Z$ is not generically reduced, then its pullback is not reduced. (The argument can show that the pullback is not generically reduced, but not the way I wrote it.)
Lemma: To show that the pullback of $Z$ is not reduced, it suffices to check that there is a smooth curve $C$ mapping to $Y$ such that the pullback of ... | 3 | https://mathoverflow.net/users/18060 | 359499 | 151,402 |
https://mathoverflow.net/questions/359447 | 7 | I am looking for an estimation of the number of paths of length $n$ going from 0 to 0 on the hexagonal (or honeycomb) lattice.
I can find plenty on references on self avoiding paths, but I am looking into every paths. Is it considerably more difficult? Has anyone a reference?
| https://mathoverflow.net/users/16934 | what is the number of paths returning to 0 on the hexagonal lattice | This is answered by Ian Agol [here](https://mathoverflow.net/a/114364/81295), with the reference "[All Roads Lead to Rome-Even in the Honeycomb World](https://www.jstor.org/stable/2684723)", Brani Vidakovic, *Amer. Statist.* 48 (1994) no. 3, 234-236.
An exact formula is
$$ p(n) = \sum\_{k=0}^m \binom{2k}{k} \binom{m}... | 12 | https://mathoverflow.net/users/81295 | 359502 | 151,404 |
https://mathoverflow.net/questions/359513 | 2 | This is a follow-up question to [this one](https://mathoverflow.net/questions/359488/finitely-additive-kappa-additive-atomless-measures-in-zfc).
Is there a ZFC example of an atomless measure that is $2^\omega$-additive, meaning, fewer than continuum many null sets have measurable union that is null?
| https://mathoverflow.net/users/157577 | Atomless, c-additive measures in ZFC | The answer is negative. Suppose $(X,\mu)$ is such a measure space. By the argument in the linked question, there is a partition of $X$ into continuum-many pairwise disjoint $\mu$-null sets. This is done by building a binary tree that splits a given node into two nodes of one half the measure. Each branch corresponds to... | 1 | https://mathoverflow.net/users/11145 | 359515 | 151,408 |
https://mathoverflow.net/questions/359501 | 0 | Consider multiplication operation $$f(x\_1,\dots, x\_k)=\prod\_{i=1}^kx\_i$$ where $x\_i\in\{1,\dots, n\_i\}$ with $n\_1,\dots, n\_k\in\{1,\dots,\infty\}$.
What is the cardinality of the range?
At $k =2$ with $n\_1=n\_2$ this is the standard Erdos multiplication table problem whose estimates are in [Distinct number... | https://mathoverflow.net/users/136553 | Generalized Erdős multiplication table problem | This problem is solved for $k\leq 5$ and open for $k\geq 6$. See [Koukoulopoulos's paper](https://arxiv.org/abs/1102.3236) for more details.
| 4 | https://mathoverflow.net/users/11919 | 359516 | 151,409 |
https://mathoverflow.net/questions/23352 | 37 | A theorem (I do unfortunately not remember to whom it is due) states that there exists a finitely presented group containing a subgroup isomorphic to the additive group of rational numbers. Can somebody give an explicit construction?
| https://mathoverflow.net/users/4556 | An explicit example of a finitely presented group containing a subgroup isomorphic to $(\mathbb Q,+)$. | Francesco Matucci, James Hyde and I have just posted an [arXiv preprint](https://arxiv.org/abs/2005.02036) with a solution to this problem. We prove that $\mathbb{Q}$ embeds in the group $\overline{T}$ of piecewise-linear homeomorphisms of the real line obtained by lifting Thompson's group $T$ through the covering map ... | 41 | https://mathoverflow.net/users/6514 | 359526 | 151,411 |
https://mathoverflow.net/questions/358095 | 3 | Consider some collection of weakly dependent Gaussians $\{w\_i\}$ with a uniform bound of $r$ on the magnitude of their covariances. Are there any bounds or techniques towards:
$$E[\inf\_i|w\_i|] \le f(r)$$
for some function $f$?
Alternatively--this would be even better--are there any comparison theorems such as:
if... | https://mathoverflow.net/users/134361 | Infimum of weakly dependent Gaussian process? | I haven't come across this type of problem even though I have been very interested lately in Gaussian processes and read a few books and references. Still, I am not an expert.
I propose a bound in an overly simplified setting that is nearly sharp at its endpoints. I also started studying the possibility to make a com... | 2 | https://mathoverflow.net/users/125260 | 359532 | 151,412 |
https://mathoverflow.net/questions/359518 | 2 | Let $G$ be a reductive group scheme over some base $X$ and $P \subseteq G$ a parabolic subgroup. To a $P$-torsor $\mathscr{E}\_P$, we may associate a $G$-torsor $\mathscr{E} = G \times^P \mathscr{E}\_P$, which is $G \times \mathscr{E}\_P$ mod the relation $(gp, s) \sim (g, ps)$, with $G$ acting by $g \cdot (h, s) = (gh... | https://mathoverflow.net/users/56878 | Why is the set of parabolic reductions of a G-torsor E bijective to the set of parabolic subgroups of Aut(E)? | Thanks to Laurent Moret-Bailly for pointing out that I missed a crucial hypothesis! Now I can construct the quasi-inverse, which I'll record below in case some future person is confused by the same problem:
The hypothesis is that "$\mathscr{P} \subseteq \mathrm{Aut}(\mathscr{E})$ is an inner form of $P \subseteq G$".... | 2 | https://mathoverflow.net/users/56878 | 359534 | 151,413 |
https://mathoverflow.net/questions/359041 | 6 | In Schalg's [Classical multilinear and Harmonic analysis](https://www.cambridge.org/core/books/classical-and-multilinear-harmonic-analysis/0F3A1DE477BFF6547CB8A93F6B3588E8?__cf_chl_jschl_tk__=f8145904b8dc6b8f9f7ba2a471ef08ae91a8ccb5-1588300573-0-Af7EgtxN6uQKGgw21kmfN1GlCaZKgJFr1K_Gs5K5CjM-fkaZ0qVCEzNoFyCaNsfI7XN5SD4fRy... | https://mathoverflow.net/users/137915 | the fractional integration method of the proof of Stein-Tomas theorem? | Let's first clarify the definitions (also, there are some typos in your post, perhaps you should consider correcting them).
For $\xi\in\mathbb{R}^d$ we shall write $\xi=(\xi',\xi\_d)$ with $\xi'\in\mathbb{R}^{d-1}$.
For a tempered distribution $T$ we shall denote its distributional Fourier transform by $\widehat{T}$.... | 2 | https://mathoverflow.net/users/157356 | 359567 | 151,419 |
https://mathoverflow.net/questions/359574 | 18 | Let $k(n)$ be the $n$th connective Morava K-theory, with $k(n)\_\* = \mathbb F\_p[v]$ where $|v| = 2p^n-2$. If $X$ is a space or a spectrum (assumed bounded below), one can compute $k(n)\_\*(X)$ using either the classical Adams spectral sequence or the even more classical Atiyah-Hirzebruch spectral sequence.
Both sp... | https://mathoverflow.net/users/102519 | Are the AHSS and Adams spectral sequence the same when computing connective Morava K-theory of a space? | Tyler's comment answers my question. A bit more detail: the Postnikov tower of $k(n)$ is an Adams resolution, because the `bottom class' map $k(n) \rightarrow H\mathbb F\_p$ is onto in mod $p$ cohomology; indeed $H^\*(k(n);\mathbb F\_p) = A\_p//E(Q\_n)$.
The appendix by Greenlees and May has the details that two spe... | 12 | https://mathoverflow.net/users/102519 | 359582 | 151,422 |
https://mathoverflow.net/questions/359550 | 2 | It is well known that over a topological space $X$ (and choosing an open cover $\mathfrak{U}$) every locally constant Cech cocycle $g$ on $\mathfrak{U}$ with coefficients in a group $G$ yields a $G$-covering space $X\_g \rightarrow X$. As such, the monodromy action of this covering space gives a homomorphism from the f... | https://mathoverflow.net/users/143492 | Čech cocycles and monodromy | First, one can pull back the Čech cocycle to S^1 and work directly with S^1
instead of X.
Any two open covers have a common refinement,
so it suffices to show that the monodromy map does not change
under passing to refinements.
As already pointed out in the comments, the open cover must be cyclic: $U\_0=U\_n$.
B... | 3 | https://mathoverflow.net/users/402 | 359589 | 151,423 |
https://mathoverflow.net/questions/359302 | 4 | I have a question about some (seemingly unimportant) behavior I noticed in Collatz sequences, which I haven't been able to find a general answer to upon rough scan of the literature (please be aware that this is not my field, so there may be some ignorance here).
A length-$m$ Collatz tuple is an $m$-tuple of the form... | https://mathoverflow.net/users/129192 | A possibly easy question about latent geometry in Collatz sequences | I see the following arithmetical background of your conjecture.
Put $\Lambda=\{1/2,3\}^m$. For each $\lambda=(\lambda\_0,\dots,\lambda\_{m-1})\in\Lambda$ we define a vector $r'\_\lambda=(r\_{\lambda,0},\dots, r\_{\lambda,m-1})$ as follows. Put $I(\lambda)=\{0\le i\le m-1:\lambda\_i=3\}$. If $I(\lambda)$ is empty the... | 1 | https://mathoverflow.net/users/43954 | 359596 | 151,425 |
https://mathoverflow.net/questions/359587 | 3 | I have kind of a soft question. I've studied the basics of L. Lafforgue's proof of function field Langlands for GLn, and its use of the moduli of shtukas with two legs, and the cocharacters $[1,0,\ldots, 0]$ and $[0,0,\ldots,-1]$. I vaguely know that V. Lafforgue's approach to the general reductive group involves highe... | https://mathoverflow.net/users/120548 | Understanding moduli of shtukas of non-minuscule cocharacter | Yes, in general you need to consider all cocharacters.
$\mathrm{GL}\_n$ has the special property that the dominant coweights are all sums of minuscule cocharacters, i.e. ones of the form $[1,\cdots,1,0,\cdots,0]$ or $[0, \ldots, 0, -1,\ldots, -1]$. (For a general group, a dominant cocharacter is minuscule if its pai... | 7 | https://mathoverflow.net/users/56878 | 359603 | 151,426 |
https://mathoverflow.net/questions/359542 | 4 | Let $d$ be an integer. We denote by $m$ the Lebesgue measure on $\mathbb{R}^d$. We define $BL(\mathbb{R}^d)$ by
\begin{align\*}
BL(\mathbb{R}^d)=\{f \in L^2\_{\rm loc}(\mathbb{R}^d,m) \mid |\nabla f|\in L^2(\mathbb{R}^d,m)\},
\end{align\*}
where $L^2\_{\rm loc}(\mathbb{R}^d,m)$ denotes the space of locally square integ... | https://mathoverflow.net/users/68463 | Analyticity of the semigroup generated by a time-changed Brownian motion | I find that the answer is no. Let us work on the half line $(0,\infty)$ with Dirichlet boundary conditions at $0$; however the problems come from $\infty$. Let $L=a(x)D^2$ where $a=1/V$ is supposed to be smooth, positive and $a(0)=1$ and consider the change of variable $s=\phi(x)=\int\_0^x \frac{1}{\sqrt{a(t)}} dt$. If... | 3 | https://mathoverflow.net/users/150653 | 359618 | 151,431 |
https://mathoverflow.net/questions/348493 | 10 | Every representation of $A\_n$ of degree $n^{O(1)}$ is contained in a $O(1)$ tensor power of the defining permutation representation. Is there an analogous result for classical groups of Lie type, say $G = \text{SL}\_n(q)$ ($q$ bounded, $n$ large)? The defining action of $G$ on $F\_q^n$ gives rise to a complex (permuta... | https://mathoverflow.net/users/20598 | Representations of $\text{SL}_n(q)$ of degree $q^{O(n)}$ | A complete answer to my question is available from recent work on "character level" of Guralnick, Larsen, and Tiep: see [1] for linear and unitary groups and [2] for symplectic and orthogonal groups.
First of all, a correction to my question. There is no need to include the representation $W$, because it is isomorphi... | 5 | https://mathoverflow.net/users/20598 | 359627 | 151,433 |
https://mathoverflow.net/questions/359623 | 3 | The classical Cantor-Schroder-Bernstein Theorem says that there exists a bijective function $X\leftrightarrow Y$ if and only if there exist injective functions $X\hookrightarrow Y$ and $Y\hookrightarrow X$.
It is known that without Axiom of Choice the injectivity cannot be replaced by the surjectivity in this result.... | https://mathoverflow.net/users/61536 | Implications of the existence of a pair of surjective functions, without Axiom of Choice | No, and here is a counterexample.
Suppose that $|\Bbb R|<|[\Bbb R]^\omega|$, that is, there are more countable *subsets* of reals than reals. This is indeed possible, e.g. if all sets of Lebesgue measurable.
Since $\sf ZF$ proves there are bi-surjections (in fact, an injection from $\Bbb R$ into $[\Bbb R]^\omega$),... | 5 | https://mathoverflow.net/users/7206 | 359633 | 151,434 |
https://mathoverflow.net/questions/359604 | 2 | I was reading some papers and come up with the next definition :
>
> A function is *differentiable in the $L^p$ sense* at $x$ if there
> exists a real number $f'\_p(x)$ such that $$\bigg(\frac{1}{h}∫\_{-h}^{h}|f(x+s)−f(x)−f\_p'(x)s|^pds\bigg)^{1/p}=o(h)$$
>
>
>
And he states that many $f\_p'$ are equivalent t... | https://mathoverflow.net/users/151368 | What is the motivation of the $L^p$ differentiability? | I don't know the literature, but it maybe helpful to understand exactly what this notion of differentiability gains for us.
First, **unlike** the Sobolev notions, this does not handle functions which belong in Holder classes. For example, based on the definition the absolute value function is **not** $L^p$ different... | 8 | https://mathoverflow.net/users/3948 | 359645 | 151,436 |
https://mathoverflow.net/questions/359632 | 2 | Igor Krichever introduced an algebro-geometric construction of solutions of KP equations starting from an algebraic curve with some additional data.
George Wilson introduced the [adelic Grassmannian](https://mathoverflow.net/questions/338024/explanation-of-definition-of-george-wilsons-adelic-grassmannian), which is a... | https://mathoverflow.net/users/12395 | Can every point of Wilson's adelic Grassmannian be obtained by Krichever construction of solutions to KP equations? | Yes - the adelic Grassmannian precisely parametrizes "rational solutions of KP", which are the Krichever solutions attached to rank 1 torsion free sheaves on cuspidal genus 0 curves -- i.e. curves with $P^1$ as their bijective normalization (or subrings of the field of rational functions). Its adelic (or factorization)... | 4 | https://mathoverflow.net/users/582 | 359652 | 151,438 |
https://mathoverflow.net/questions/359660 | 3 | This is a follow-up to this question: [For what sets $X$ do there exist a pair of functions from $X$ to $X$ with the identity being the only function that commutes with both?](https://mathoverflow.net/questions/358057/for-what-sets-x-do-there-exist-a-pair-of-functions-from-x-to-x-with-the-id)
Let $X$ be a set, and $X... | https://mathoverflow.net/users/14094 | Centralizer of a single element in the monoid of self-maps of a set | Let $|X| > \mathbb{N}$ and $f \in X^X$, we construct a function $g$ that commutes with $f$ but is not a power of $f$. Let $G$ be the directed graph of $f$. Split $G$ into connected components. Call a connected component boring if it is of the form $a\_1, a\_2, \cdots$ or $\{a\_i\}\_{i \in \mathbb{Z}}$ with $f(a\_i) = a... | 7 | https://mathoverflow.net/users/156473 | 359661 | 151,441 |
https://mathoverflow.net/questions/359662 | 1 | It is known that not every locally compact Haussdorff space is normal, see for example
[here](https://mathoverflow.net/questions/53300/locally-compact-hausdorff-space-that-is-not-normal)
But it seems that the following is true, I just want to make sure I am not making any mistake:
**Lemma** Let $X$ be a locally ... | https://mathoverflow.net/users/11552 | Separating compact sets in locally compact spaces | Yes, this is known and you don't need local compactness even as long as $X$ is completely regular (locally compact spaces are completely regular).
I think that the best way to prove is to go via the Čech–Stone functor, which is available precisely for completely regular spaces. Consider $\beta X$ and note that $K, W... | 2 | https://mathoverflow.net/users/15129 | 359663 | 151,442 |
https://mathoverflow.net/questions/359651 | 6 | I've been learning about Mirzakhani's use of hyperbolic geometry to compute Weil-Petersson volumes of moduli space of curves, and the application to proving Virasoro constraints for a point. Why have these methods not been directly extended to higher dimensional target spaces? I don't know much about the structure of t... | https://mathoverflow.net/users/123002 | Mirzakhani's hyperbolic method generalized to moduli space of stable maps | Mirzakhani's computation of volumes of deformation spaces is very heavily based on the work of Greg McShane (McShane's identity). McShane's theory has been extended to other deformation spaces, see, for example
*Labourie, François; McShane, Gregory*, [**Cross ratios and identities for higher Teichmüller-Thurston the... | 3 | https://mathoverflow.net/users/11142 | 359665 | 151,444 |
https://mathoverflow.net/questions/359657 | 8 | Consider a closed, bounded and convex set $C \subset \mathbb{R}^{2}$ and denote its boundary with $\partial C$. It is very well-known that the Minkowski sum of two convex sets is convex again. What about the Minkowski sum of its boundary?
Is the Minkowski sum $\partial C + \partial C$ again a convex set and how can one... | https://mathoverflow.net/users/157696 | Is Minkowski sum of boundary convex again? | Yes, $\partial C + \partial C$ is convex since it equals $2C$. Equivalently, every point in $z \in C$ is a midpoint of two boundary points. This is obvious if $z \in \partial C$. Otherwise, let $f :S^{n-1} \to \mathbf{R}$ be the continuous function which sends $u$ to the length of the segment going from $z$ to $\partia... | 12 | https://mathoverflow.net/users/908 | 359666 | 151,445 |
https://mathoverflow.net/questions/359667 | 4 | Let $k$ be a field and $R$ be a $k$-algebra. Let $M$ and $N$ be left $R$-modules. Finally, let $\ell$ be a field extension of $k$. We thus have an $\ell$-algebra $\ell \otimes R$, and both $\ell \otimes M$ and $\ell \otimes N$ are left modules over it.
**Question**: What is the relationship between the $k$-vector spa... | https://mathoverflow.net/users/157702 | Effect of extending scalars on maps of modules | We have that $\text{Hom}\_{\ell \otimes R}(\ell \otimes M,\ell \otimes N)$ is isomorphic to $\ell \otimes \text{Hom}\_R(M,N)$ as $\ell$-vector spaces for a general (possible infinite dimensional) $k$-algebra $R$ in case $M$ has finite $k$-dimension, see for example Lemma 7.4. in the book "A first course in noncommutati... | 6 | https://mathoverflow.net/users/61949 | 359669 | 151,446 |
https://mathoverflow.net/questions/359673 | 7 | I would like to understand the conditions that support
the [Green-Tao Theorem](https://en.wikipedia.org/wiki/Green%E2%80%93Tao_theorem), which established that
the primes contain arbitrarily long arithmetic progressions.
I am wondering:
>
> ***Q***. Is it difficult to define an infinite
> set $S$ of natural numb... | https://mathoverflow.net/users/6094 | Prime-like numbers that avoid Green-Tao? | This is likely impossible.
Indeed the largest sets known to be free of arbitrarily long arithmetic progressions asymptotically satisfy $| [ A \cap [1, n] | \lesssim\_{k} n / \log^{k} n$ for all $k>1$, and it is widely believed that these examples are near maximal.
The Erdos-Turan conjecture mentioned in the commen... | 11 | https://mathoverflow.net/users/630 | 359674 | 151,447 |
https://mathoverflow.net/questions/359631 | 5 | I am trying to read Tao's *Higher order Fourier analysis* but I would be very happy to find another book on the subject. I would like to learn something about the Gowers norm and about Roth's theorem (density increment and energy increment arguments). Sorry if this question is a bit too open ended . Lecture notes, even... | https://mathoverflow.net/users/nan | Another reference for higher order Fourier analysis | You could try "Nilpotent structures in Ergodic theory" by Host and Kra, which covers this topic in greater depth.
| 2 | https://mathoverflow.net/users/157708 | 359676 | 151,448 |
https://mathoverflow.net/questions/359642 | 12 | It is well known that any smooth curve
$C\in |\mathcal{O}\_{\mathbf{P}^1\times\mathbf{P}^1}(2,2)| $ has geometric genus equal to 1, so its isomorphism class is determined by its $j$-invariant. Nevertheless we have that $\dim(\mathbf P(H^0(\mathcal{O}\_{\mathbf{P}^1\times\mathbf{P}^1}(2,2))))=8$ and $\dim(\operatorname... | https://mathoverflow.net/users/129611 | Moduli of smooth curves in $|\mathcal{O}_{\mathbb{P}^1\times\mathbb{P}^1}(2,2)| $ and their invariants | [**EDITED** to exhibit $j$ as a rational function of $J\_2,J\_3,J\_4$,
and to fix various local errors etc.]
The action of ${\rm SL\_2} \times {\rm SL\_2}$ on the $9$-dimensional space of
$(2,2)$ forms has a polynomial ring of invariants, with generators in degrees
$2,3,4$. If we write a general $(2,2)$ form $P(x\_1,... | 11 | https://mathoverflow.net/users/14830 | 359683 | 151,449 |
https://mathoverflow.net/questions/359687 | 10 | It is well known that the Abel-Jacobi map restricted to $\text{Eff}\_g(C)$ surjects onto the Jacobian $\text{Jac}(C)$, since every divisor of degree $g$ is effective.
Is there an analogous statement for Prym varieties? That is, given an unramified double cover $\widetilde C\to C$ with involution $\tau$, consider the... | https://mathoverflow.net/users/126867 | Surjectivity of the Abel-Prym map | First of all, note that your definition is not correct: when $d$ is odd, the image of your map does not land in the Prym variety -- you have to add a constant term. When this is done, the answer is yes, for the following reason. Let $X$ be the image of $\tilde{C} $ in $P:=\operatorname{Prym}(\tilde{C}/C ) $. Let me put... | 9 | https://mathoverflow.net/users/40297 | 359690 | 151,451 |
https://mathoverflow.net/questions/359693 | 3 | Given a $N×M$ random complex gaussian matrix $X$ and $N×K$ random complex gaussian matrix $Y$ I'm interested in approximating the expectation expressed as:
\begin{align}
E[trace({(aX{X^H} + I)^{ - 1}}Y{Y^H})]
\end{align}
a Is a positive given variable. I know that $XX^H$ and $YY^H$ have Wishard distribution, However. ... | https://mathoverflow.net/users/144355 | Expectation of the trace of inverse of a Gaussian random matrix | I assume the matrices $X$ and $Y$ are independent. Since the trace commutes with the expectation value, and since the expectation value of the product of independent random variables is the product of expectation values, we have
$$
F(a)=\mathbb{E}\bigl[{\rm tr}\,\bigl({(aX{X^H} + I)^{ - 1}}Y{Y^H}\bigr)\bigr]={\rm tr}\b... | 2 | https://mathoverflow.net/users/11260 | 359700 | 151,455 |
https://mathoverflow.net/questions/335912 | 3 | Informally the following theory is a kind of extension of second order arithmetic, where numbers are not limited to naturals, instead here we have formation of further numbers by setting limits on sets of numbers, so here the set of all naturals have a limit, if a set is equinumerous to a set that has a limit, then it ... | https://mathoverflow.net/users/95347 | Is ZFC interpretable in a kind of an extended form of second order arithmetic? | I think the theory presented here is equi-consistent with **ZFC** since it interpret Takeuti's system presented in his article: [Construction of the set theory from the theory of ordinal numbers.](https://projecteuclid.org/download/pdf_1/euclid.jmsj/1261415284)
All axioms 1.1 - 1.17 can be captured in the extended fo... | 1 | https://mathoverflow.net/users/95347 | 359704 | 151,456 |
https://mathoverflow.net/questions/358543 | 7 | **Edited**: Phil Tosteson suggested Thom's first isotopy lemma, but it does not seem to be in the direction that I'm trying to generalize. Let me reformulate my question again.
Let $N\subset M$ be a pair of the smooth manifolds and $\pi:M\to B$ a proper submersion to a smooth base. Assume moreover that the restrictio... | https://mathoverflow.net/users/74322 | A relative version of Ehresmann's theorem | The answer is positive. There are several proofs of Eheresmann's genuine lemma; I think that each of them can be straightforwardly generalized and gives your relative version. But you can also, alternatively, deduce the relative version from the absolute one together with a classical theorem of Cerf, as follows. After ... | 2 | https://mathoverflow.net/users/105095 | 359708 | 151,458 |
https://mathoverflow.net/questions/359684 | 23 | For a permutation $\pi=\pi\_1\pi\_2\cdots\pi\_n$ written in one-line notation, an index $i$ for which $\pi\_i > i$ is usually called an 'excedance.' To me, this seems like a mispelling of what should be 'exceedance': [many](https://www.merriam-webster.com/dictionary/exceedance) [dictionaries](https://en.wiktionary.org/... | https://mathoverflow.net/users/25028 | Why 'excedances' of permutations? | *Mea culpa.* Comtet used the term *excédence*. When writing EC1 I needed an English term for this concept. For some reason I didn't like the word exceedance. I thought it looked better without the double e, analogous to proceed and procedure. Thus I made up the word *excedance*.
| 51 | https://mathoverflow.net/users/2807 | 359724 | 151,465 |
https://mathoverflow.net/questions/359692 | 4 | I'm fairly new to thinking about homological algebra and chain complexes in their own right, *i.e* outside of isolated examples such as for constructing simplicial homology, or for computing $Ext$ groups for some Hopf algebroid.
Given an abelian category $\mathscr{A}$ with a category of chain complexes $Ch(\mathscr{... | https://mathoverflow.net/users/157519 | Model categories and chain complexes | I'll take your question as license to advertise a relatively recent paper in a slightly more specialized but concretely calculational direction: <http://nyjm.albany.edu/j/2014/20-53p.pdf>. Its title is
Six model structures for DG-modules over DGAs: Model category theory in homological action. The theme is how differen... | 6 | https://mathoverflow.net/users/14447 | 359727 | 151,467 |
https://mathoverflow.net/questions/359713 | 0 | Suppose I have some triangular array $\{X\_{n,j}\}$ of random variables, which need not be independent or identically distributed. Suppose I further know that
$$Var\left(\sum\_{j=1}^n X\_{n,j}\right)\to \sigma^2 \text{ and } \sum\_{j=1}^n \mathbb{E}|X\_{n,j}|^{2+\epsilon}\to 0,$$
for some $\epsilon>0$. Then this triang... | https://mathoverflow.net/users/154137 | Lyapunov condition for CLT for asymptotically independent sequence | The answer is no. E.g., suppose that
\begin{equation\*}
(X\_{n,j})=(X\_{n,j})\_{j=1}^n\sim(1-u^2/n)N\_n(0,I\_n/n)+(u^2/n) N\_n(0,J\_n/n),
\end{equation\*}
where $u\in(0,\infty)$, $I\_n$ is the $n\times n$ identity matrix, and $J\_n$ is the $n\times n$ matrix with all entries equal $1$, so that $N\_n(0,J\_n/n)$ is the... | 1 | https://mathoverflow.net/users/36721 | 359728 | 151,468 |
https://mathoverflow.net/questions/359643 | 11 | Suppose that $X$ is distributed uniformly in the scaled $n$-sphere $\sqrt{n} \mathbf{S}^{n-1} \subset \mathbf{R}^n$. Then apparently the distribution of $(X\_1, \dots, X\_k)$, the first $k < n$ coordinates of $X$ has density $p(x\_1, \dots, x\_k)$ with respect to Lebesgue measure in $\mathbf{R}^k$, moreover if $r^2 = x... | https://mathoverflow.net/users/121486 | Marginal density of uniform spherical distribution | $\newcommand{\R}{\mathbb{R}}
\newcommand{\x}{\mathbf{x}}
\newcommand{\X}{\mathbf{X}}$
This is to present a formalization of the answer by Carlo Beenakker, without explicit use of the delta function.
We are going to assume that $\X=(X\_1,\dots,X\_n)$ is uniformly distributed on the unit sphere $\mathbb S^{n-1}$, rath... | 5 | https://mathoverflow.net/users/36721 | 359747 | 151,475 |
https://mathoverflow.net/questions/359712 | 3 | Suppose we have a fixed function $f: \mathbb{R}^2 \rightarrow \mathbb{R}^2$ (sufficiently regular, say $C^\infty$ ). The question is: for which $f$ there exists a scalar function $g: \mathbb{R}^2 \rightarrow \mathbb{R}$ such that the product $fg = (f\_1g, f\_2g)$ is holomorphic (using the standard identificacion $\math... | https://mathoverflow.net/users/157730 | Space of holomorphic functions multiplied by smooth functions taking real values | First, if $fg$ were holomorphic and nontrivial, then $\{g = 0\}$ has to be discrete. This implies that $g$ has to be either $\geq 0$ or $\leq 0$. So we can assume WLOG $g \geq 0$.
Restricting away from its zero set, we can study the function $v = \ln g$. The Cauchy-Riemann relations becomes
$$ \begin{cases} f\_1 v... | 3 | https://mathoverflow.net/users/3948 | 359749 | 151,477 |
https://mathoverflow.net/questions/359758 | 6 | The index of a subgroup $H$ in a group $G$ is the number of distinct cosets of $H$ in $G$.
Why did someone decide to call this an 'index'? What's the rationale for this?
| https://mathoverflow.net/users/89347 | Origin of the term 'index of a subgroup' | The short answer is Cauchy, with only justification: “for short”. Burnside’s *Theory of groups of finite order* (1897) has a useful glossary stating, [p. 382](//archive.org/details/117768378/page/382) (my bold):
>
> The ratio of the order of a sub-group $H$, to the order of the group
> $G$ containing it, is called... | 13 | https://mathoverflow.net/users/19276 | 359759 | 151,480 |
https://mathoverflow.net/questions/359734 | 11 | Is there a known description of the **free category with both product and coproduct?**
That is, given a small category $C$, I want to consider a category $U C$ which has product and coproduct, a functor $C \to U C$ and such that $UC$ is universal for functor preserving both product and coproduct. The case $C = \empty... | https://mathoverflow.net/users/22131 | Free category with product and coproduct | The general problem of giving a categorical construction of the free category with finite coproducts and products (or "free sum–product category") seems to still be open, though there are several works on special cases of the problem.
Cockett–Santocanale's *[On the word problem for ΣΠ-categories, and the properties o... | 13 | https://mathoverflow.net/users/152679 | 359762 | 151,481 |
https://mathoverflow.net/questions/359761 | 1 | Suppose $h:\mathcal{X \times Y \times W}\rightarrow\mathcal{X}$, with $\mathcal{X,Y,W} \subseteq \mathbb R^d$, $d \in \mathbb N$ is Lipschitz in $x,y$, i.e.,
$$ \| h(x,y,w) - h(x',y',w) \|\_2 \le L\_h (\|x - x' \|\_2 + \| y -y'\|\_2) \quad \forall \, x,x' \in \mathcal X, \ \forall \, y,y'\in \mathcal Y. $$
We also ha... | https://mathoverflow.net/users/143945 | Lipschitz continuity of multivariable function in expected value | Assume that
$$|f(x)-f(x')|\le L\_f\|x-x'\|\_2\quad\forall \ x,x' \in \mathcal X,$$
rather than
$$|f(x)-h(x')|\le L\_f\|x-x'\|\_2\quad\forall \ x,x' \in \mathcal X.$$
Then
$$|f(h(x,y,w))-f(h(x',y,w))|\le L\_f\|h(x,y,w)-h(x',y,w)\|\_2
\le L\_fL\_h\|x-x'\|\_2,$$
whence
$$\left|\int\_{\mathcal W} p(w) f(h(x,y,w))\, dw ... | 1 | https://mathoverflow.net/users/36721 | 359765 | 151,483 |
https://mathoverflow.net/questions/283921 | 22 | There is a problem in [Richard Kenyon's list](https://gauss.math.yale.edu/%7Erwk25/openprobs/index.html) ([Wayback Machine](https://web.archive.org/web/20191209170902/https://gauss.math.yale.edu/%7Erwk25/openprobs/index.html)) which I would like to post here, because although I have thought about it from time to time, ... | https://mathoverflow.net/users/68969 | Equilaterally triangulated surfaces with prescribed boundary | We resolve Kenyon's problem in [this paper](https://arxiv.org/abs/2005.02555 "Alexey Glazyrin, Igor Pak: Domes over curves"). We discuss in Section 5 a number of further conjectures and open problems.
| 12 | https://mathoverflow.net/users/4040 | 359775 | 151,486 |
https://mathoverflow.net/questions/359715 | 2 | When I was reading a paper, I saw something like:
If $F$ and $E$ are Banach spaces with symmetric bases (precisely, they are symmetric sequence spaces), and $F$ is isomorphic to a complemented subspace of $l\_2\oplus E$, then $F=l\_2$ or $F$ is isomorphic to a complemented subspace of $E$.
The author claimed that th... | https://mathoverflow.net/users/91769 | complemented subspace of the direct sum of two Banach spaces | This is well known but not trivial. It follows primarily from Theorem 2.c.13 of [LT]. The Theorem says (applied to this situation) if every operator $T:E\to \ell\_2$ is strictly singular, then every complemented subspace of $\ell\_2\oplus E$ is of the form (up to isomorphism) $X'\oplus E'$ for some complemented subspac... | 5 | https://mathoverflow.net/users/3675 | 359781 | 151,488 |
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