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 |
|---|---|---|---|---|---|---|---|---|---|
https://mathoverflow.net/questions/348901 | 9 | I have been looking around for an answer to this question, but I have not been able to find anything. My question is:
Is it known how to tell whether two elements $b\_1, b\_2 \in B\_n$ commute?
EDIT: Is there perhaps a nicer/cleaner rule for $B\_3$ and $B\_4$?
| https://mathoverflow.net/users/150179 | When do elements in the braid group $B_n$ commute? | Krammer ("[The braid group $B\_4$ is linear](https://link.springer.com/article/10.1007%2Fs002220000088)", *Invent. Math.* **142** (2000), 451–486 ([MSN](https://mathscinet.ams.org/mathscinet-getitem?mr=1804157))) constructed
a representation $\rho: B\_n \to {\rm GL}\_N({\bf Z}[q^{\pm 1}, t^{\pm 1}])$
with $N = {n \choo... | 14 | https://mathoverflow.net/users/14830 | 349520 | 147,911 |
https://mathoverflow.net/questions/349512 | 3 | Let $M(n, k)$ be a minimal number $m$ such that there exists set $C$ ($|C|=m$) of colorings of n-element set $[n]$ with $k$ colors such that for every $k$-element subset $K$ of $[n]$ there exists coloring $c\in C$ such that $c$ assigns different colors for all elements of $K$.
* Is there any research about this objec... | https://mathoverflow.net/users/150554 | What number of colorings can guarantee that for every k-element subset there exists a coloring assigns different colors for elements from this subset? | These are known as *Perfect Hash Families*. The notation $PHF(N;n,v,k)$ denotes a set $\mathcal H$ (hash functions) with $|\mathcal H|=N$, of $v$-colorings of the base $[n]$, such that for any $X\subset [n]$ with $|X|=k$, there exists at least one hash function which assigns it different colors. Let's denote by $f(n,v,... | 4 | https://mathoverflow.net/users/2384 | 349528 | 147,917 |
https://mathoverflow.net/questions/349471 | 8 | I was thinking about de Rham cohomology in characteristic $p$, and in particular the [recent question about Poincare residues](https://mathoverflow.net/questions/349301), and I came up with the following construction.
Let $k$ be a perfect field of characteristic $p$ and let $A$ be a regular $k$-algebra. Let $\Omega^j... | https://mathoverflow.net/users/297 | A variant on characteristic $p$ de Rham cohomology | To be able to compute the iterates of the Cartier operator it is convenient to understand how $C$ interacts with the de Rham differential:
Cartier isomorphism induces an isomorphism of complexes $$(\Omega^{\*}\_A,d\_{dR})\simeq (H^{\*}(\Omega^{\bullet}\_A),\beta)$$ where $\beta$ is the Bockstein differential provided... | 6 | https://mathoverflow.net/users/39304 | 349537 | 147,922 |
https://mathoverflow.net/questions/349353 | 4 | Recently I have read two different proposals for a notion of homotopy between functors, and I am curious which contexts each best lend themselves to. The first comes from Ming-Jung Lee's 1972 paper [Homotopy for Functors](https://www.ams.org/journals/proc/1972-036-02/S0002-9939-1972-0334212-5/S0002-9939-1972-0334212-5.... | https://mathoverflow.net/users/nan | Homotopy of functors | I'm surprised this has been up for days with nobody telling Atticus the obvious. Let $I$ be the unit interval category with two objects, $0$ and $1$, and one non-identity arrow $0 \to 1$. A natural transformation $F\to G$ between functors $\mathcal C \to \mathcal D$ is the same thing as a functor $\mathcal C \times I \... | 7 | https://mathoverflow.net/users/14447 | 349538 | 147,923 |
https://mathoverflow.net/questions/349554 | 12 | Let $n$ line sets be $\mathcal{S}\_i=\{a\mathbf{h}\_i:0 \le a \le 1\}$, for $1 \le i \le n$, where $\{\mathbf{h}\_1,\cdots,\mathbf{h}\_n\}$ is a vector group of rank $r$ in the $r$-dimensional Euclidean space. Define the Minkowski sum of two sets as $\mathcal{S}\_1+\mathcal{S}\_2=\{\mathbf{s}\_1+\mathbf{s}\_2:\mathbf{s... | https://mathoverflow.net/users/149696 | The $r$-dimensional volume of the Minkowski sum of $n$ ($n\geq r$) line sets | Let $M$ be the matrix whose rows are the vectors $\boldsymbol{h\_i}$. Then the $r$-dimensional volume of $\mathcal{S}=\mathcal{S}\_1+\cdots+\mathcal{S}\_n$ is equal to the sum of the absolute values of the $r\times r$ minors of $M$. I don't know who originally showed this, but one can show that $\mathcal{S}$ can be til... | 14 | https://mathoverflow.net/users/2807 | 349558 | 147,936 |
https://mathoverflow.net/questions/265435 | 6 | The website [Formalizing 100 Theorems](http://www.cs.ru.nl/~freek/100/) by [Freek Wiedijk](http://www.cs.ru.nl/~freek/) contains a list of some theorems that were chosen at some point as good candidates for formalization (because of their complexity, their importance, etc.) This website seems to be updated very often.
... | https://mathoverflow.net/users/66044 | Are there any recent advances in formalizing the undecidability of $\mathit{CH}$? | Jesse Michael Han and Floris van Doorn recently formalized the independence of the continuum hypothesis in the [Lean theorem prover](https://leanprover.github.io). See [the Flypitch project webpage](https://flypitch.github.io/about/) for their papers and code.
| 7 | https://mathoverflow.net/users/353 | 349566 | 147,942 |
https://mathoverflow.net/questions/310323 | 2 | This was posted as a side question in [Formal definition of this ordinal?](https://mathoverflow.net/questions/310244) and was split as a separate question based upon suggestion in comments there.
Assume an ordinary ORM model (call it $C\_1$). Suppose we add an extra instruction of the form to $u:=u+\omega\_1$ (where ... | https://mathoverflow.net/users/112385 | Relation of $\omega_{\omega_1+1}^{CK}$ to some other ordinals | It seems that the relation between $p$ and $\theta$ should be fairly simple (with $p>\theta$). Because we are talking about sufficiently general programs, $p$ is just the supremum of clocking times given a single parameter $\omega\_1$.
To observe why $p>\theta$ is true, we can do the following. First set a counter v... | 0 | https://mathoverflow.net/users/112385 | 349570 | 147,943 |
https://mathoverflow.net/questions/349156 | 2 | Suppose, $\Xi$ is a collection of random variables. We call $\Xi$ $k$-independent, iff any $k$ distinct elements of $\Xi$ are mutually independent. For example, $2$-independence is pairwise independence and $|\Xi|$-independence is the full mutual independence of random variables from $\Xi$.
Let's define independence ... | https://mathoverflow.net/users/110691 | Independence depth of linearly dependent random variables | Here's a different argument.
Pick $t>0$ such that $P[|X\_i|>t]\leq 1/n$ for all $i.$ The event $|X\_i|>tn$ is a subset of the union of the events $|X\_j|>t$ for $j\neq i,$ so
$$P[|X\_i|>tn] \leq \sum\_{j\neq i} P[|X\_j|>t \text{ and }|X\_i|>tn] \leq \frac{n-1}{n}P[|X\_i|>tn].$$
Since each $X\_i$ is essentially boun... | 3 | https://mathoverflow.net/users/112284 | 349572 | 147,945 |
https://mathoverflow.net/questions/349574 | 0 | I need to create a set of numbers where any amount of them can be added together and each result will always give a unique answer, so we always know that the result was created from adding exactly these numbers and no other combination in the set can work. I'm not sure how many numbers I will need in the set at this po... | https://mathoverflow.net/users/150587 | Set of numbers with unique results from sums | The term would be "subset sum distinct sets". An initial reference to start with is ["A construction for sets of integers with distinct subset sums"](https://www.combinatorics.org/ojs/index.php/eljc/article/view/v5i1r3) by Tom Bohman.
| 0 | https://mathoverflow.net/users/47453 | 349575 | 147,947 |
https://mathoverflow.net/questions/349540 | 8 | Suppose that I have a linear system $AX=b$ with $A\in\mathbb{Z}^{n\times m}$ and $b\in\mathbb{Z}^{n}$. Assume that $AX=b$ has exactly one rational solution. Then how can I obtain this solution efficiently?
I know LU decomposition is a choice for this purpose. But it seems that the denominators may expand in the proce... | https://mathoverflow.net/users/50466 | How to obtain the rational solution of a linear system efficiently? | The following answer is only theoretical. Suppose you first reduce your matrix to [Smith normal form](https://en.wikipedia.org/wiki/Smith_normal_form) $D = S A T$, where $S$ and $T$ are integral matrices with integral inverses (the determinants of $S$ and $T$ are each either $+1$ or $-1$) and $D$ is diagonal, whose dia... | 4 | https://mathoverflow.net/users/2622 | 349579 | 147,949 |
https://mathoverflow.net/questions/349496 | 3 | Is the following true:
For any $n$ there exists $p\_0$ s.t. for any finite group $G$ of Lie type of rank $\leq n$ and characteristic $p\geq p\_0$ and any (irreducible) $\mathbb F\_p$ representation $V$ of $G$ of dimension $\leq n$ we have
$$H^1(G,V)=0$$
| https://mathoverflow.net/users/4690 | Vanishing of first co-homology with coefficients modular representations of small dimension | I think the answer to your question is yes for example with $p\_0 = n+3$.
See the following paper:
>
> "Small Representations Are Completely Reducible", Robert M. Guralnick, J. Algebra (1999).
>
>
>
Theorem A in this paper says the following:
>
> Let $k$ be a field of positive characteristic $p$. Let $G$... | 4 | https://mathoverflow.net/users/38068 | 349581 | 147,950 |
https://mathoverflow.net/questions/349057 | 19 | [This](https://en.wikipedia.org/wiki/Functional_derivative) page on Wikipedia defines the so-called functional derivative as follows: "Given a manifold $M$ representing (continuous/smooth) functions $\rho$ (with certain boundary conditions, etc.) and a functional $F: M \to \mathbb{C}$, the functional derivative of $F[\... | https://mathoverflow.net/users/150264 | Question about functional derivatives | **Premise (a long one)**: before answering your questions, I must say that, if your are searching mathematically rigorous informations, you should not rely on Wikipedia entry "[Functional derivative](https://en.wikipedia.org/wiki/Functional_derivative)" in its current status, since it is seriously flawed due to an "edi... | 20 | https://mathoverflow.net/users/113756 | 349584 | 147,951 |
https://mathoverflow.net/questions/349586 | 1 | Let $F$ be an ordered field and $f\in F[X]$ be a polynomial such that $f(x)>0$ for all $x\in K$. Is it possible that there is an extension $L\supseteq K$ of ordered fields and $y\in L$ such that $f(y)\leq 0$? My conjecture is that this is not the case but I do not find a proof for it.
| https://mathoverflow.net/users/150594 | Positivity in extensions of ordered fields | Let $K=F=\mathbb Q,f=(x^2-2)^2,L=\mathbb Q(\sqrt{2})$, so that $f(\sqrt{2})=0$.
It is more interesting to ask for $f(y)<0$. This is also possible. Let $K=F=\mathbb Q(T)$, where $T$ is an indeterminate larger than all elements of $\mathbb Q$. Let $f=(x^2-T)^2-1$ and $L=\mathbb Q(\sqrt{T})$. Then $f$ is positive on $F$... | 3 | https://mathoverflow.net/users/30186 | 349587 | 147,952 |
https://mathoverflow.net/questions/349594 | 8 | This question may be easy and indicative of my ignorance about the failure of the axiom of choice. If so, I apologize. Below assume $\mathsf{DC}$ but not $\mathsf{AC}$. Suppose we have a partial order $A = (A, \leq)$ satisfying the following:
1. $A$ is uncountable in the sense that there is no surjection of $\omega$ ... | https://mathoverflow.net/users/114946 | Aronszajn Trees when AC fails | Is it acceptable? Sure. In some sense, it is an Aronszajn tree.
The condition of being well-founded, which in the presence of $\sf DC$ is the same as saying there are no decreasing sequences, is equivalent to having a rank function. So much is true in $\sf ZF$.
So you can make sense of this tree having height $\ome... | 8 | https://mathoverflow.net/users/7206 | 349596 | 147,954 |
https://mathoverflow.net/questions/349593 | 8 | Suppose I have a braided monoidal category $\mathcal{C}$. I specifically am interested in the case where $\mathcal{C}$ is the category of finite-dimensional modules of a quantum group, say $\mathcal{U}\_q(\mathfrak{sl}\_2)$ (or a variant of it.)
The braiding $c\_{-,-}$ embeds $\mathcal{C}$ into its Drinfeld center $\... | https://mathoverflow.net/users/113402 | Drinfeld center of a braided category | No, the functor $\mathcal C \to \mathcal Z(\mathcal C)$ is not essentially surjective in general.
For example, in the case you have in mind, $\mathcal C = Rep\_q(G)$ (say $G$ a semisimple algebraic group), the Drinfeld center can be identified with the category
$HC\_q := \mathcal O^{RE}\_q(G)-mod\_{Rep\_q(G)}$
... | 9 | https://mathoverflow.net/users/7762 | 349599 | 147,955 |
https://mathoverflow.net/questions/349583 | 0 | Suppose we have a continuous-time stochastic process $s(t)$ that's mixing. What properties would a function $f$ have to have so that $f(s(t))$ would also be mixing?
I'm sure this is a well-known thing, I'm just having trouble locating a term for such functions or any relevant literature.
| https://mathoverflow.net/users/137936 | Functions that preserve the mixing of a stochastic process | $\newcommand{\R}{\mathbb R}$
$\newcommand{\P}{\mathsf P}$
$\newcommand{\F}{\mathscr F}$
$\newcommand{\S}{\mathscr S}$
$\newcommand{\T}{\mathscr T}$
$\newcommand{\Si}{\Sigma}$
Let $X:=(X\_t)\_{t\in\R}$ be a stochastic process in a state space $S$, endowed by a sigma-algebra $\S$, so that, for each $t\in\R$, $X\_t$ is a ... | 0 | https://mathoverflow.net/users/36721 | 349600 | 147,956 |
https://mathoverflow.net/questions/349580 | 1 | Suppose $A$ is some set. Let's define a pair semigroup over $A$ as $P[A] = (A\times A \cup \{0\}, \circ)$, where the $\circ$ operation is defined by the following two identities:
1. $\forall a \in P[A]$ we have $a \circ 0 = 0 \circ a = 0$
2. $\forall (a, b, c, d) \in A\times A \times A \times A$ we have $(a, b) \circ... | https://mathoverflow.net/users/110691 | What is the minimal possible size of a subset of this semigroup satisfying the following conditions? | $|S|=2n-1$ is indeed minimal.
In graph theoretic terms, you are asking for the minimal number of edges of a directed graph $G$ on $n$ vertices such that there is a directed path of length two between each pair of vertices (which vertices may also agree).
(If n>1, we may safely forget about the zero element of the sem... | 2 | https://mathoverflow.net/users/106723 | 349607 | 147,957 |
https://mathoverflow.net/questions/349606 | 2 | Hempel, in his 1987 article "Residual Finiteness for 3-Manifolds", shows that if $M$ is a compact Haken 3-manifold, then $\pi\_1(M)$ is residually finite. In the proof he starts by reducing the case to '$M$ is closed and irreducible'. Why can he so easily do that?
Thanks in advance!
| https://mathoverflow.net/users/150613 | Residual Finiteness for 3-Manifolds Hempel | These are a routine, but very valuable, pair of exercises in the theory of three-manifolds. So if you are trying to learn the material, don't read the following proof sketches until you desperately need some hints!
---
---
---
---
---
---
---
We assume that $M$ is compact and connected.
... | 6 | https://mathoverflow.net/users/1650 | 349611 | 147,958 |
https://mathoverflow.net/questions/349612 | 7 | What is the large class of operators for which one can define fractional powers? For example, we can consider an operator $A: D(A) \subset X \rightarrow X$, generator of an analytic semigroup on a Banach space $X$. Can we define the powers $(-A)^\alpha$ for $\alpha>0$ without additional assumptions? I found some refere... | https://mathoverflow.net/users/149793 | Fractional powers of an operator | This question was considered by Functional Analysis specialists around 1960 or earlier. In the case of Banach Algebras $C(X)$ (for Hausdorff compact $X$) this gets reduced often to studying the auto-homeomorphisms of $X$, and it is extra interesting when the compact space is nice. In this context, in 1961, I have redis... | 7 | https://mathoverflow.net/users/110389 | 349614 | 147,959 |
https://mathoverflow.net/questions/349489 | 0 | Given a planar polygonal linkage defined by a sequence of $n$ hinge joints $(j\_0,\,\cdots,\,j\_{n-1},j\_n = j\_0)$ with links of fixed lengths $\lbrace\|j\_{k+1}-j\_k\|=d\_k\ |\ 0\le k\lt n\rbrace$ between adjacent joints,
>
> how can the set of radii be calculated that allow for placing all joints of the linkage... | https://mathoverflow.net/users/31310 | Calculating radii allowing for circular placement of polygonal linkage's joints | This problem has an algebraic-number solution (set up a system of quadratic equalities between the squared link lengths and the squared distances between hinge points) but with unsolvable Galois groups, so there is not going to be a nice closed formula solution. See: Varfolomeev, V.V.: Galois groups of the Heron–Sabito... | 3 | https://mathoverflow.net/users/440 | 349615 | 147,960 |
https://mathoverflow.net/questions/349603 | 1 | This seems to be a trivial question, but I am genuinely confused about it.
The notion of weights as in [Deligne's Weil II](http://www.numdam.org/item/PMIHES_1980__52__137_0/) are defined in terms of eigenvalues of automorphisms that Frobenius morphisms induce on stalks. The following is a definition that is found in ... | https://mathoverflow.net/users/149738 | Computing weights of $\bar{\mathbb{Q}}_l(1)$ from the definition | First, there is no point in including $X$ in the definition. We're interested in the stalk at $x$. We can also view $x$ as a geometric point of $X\_0$. Pulling back from $X\_0$ to $X$ and then taking the stalk is the same as taking the stalk at $x$, almost by definition.
By definition, the stalk of $\mathcal G\_0$ at... | 2 | https://mathoverflow.net/users/18060 | 349616 | 147,961 |
https://mathoverflow.net/questions/349394 | 3 | **Motivation**
Hopf theorem, asserts that $C^0$-maps $f:M^n\to \mathbb{S}^n$ from an orientable, closed n-manifold into an n-sphere are classified up to homotopy by their degree $deg(f)$.
The theorem not only says that $[\mathbb{S}^n, \mathbb{S}^n] \simeq \mathbb{Z} $ but also gives us a way to compute the complexity... | https://mathoverflow.net/users/99042 | Homotopy class of maps into Stiefel manifolds | Maybe what you looking for is known under the name *generalized curvatura integra* (for the case $N> k+1$). I will formulate it not for $S^n$ but more generally for a $m$-dimensional framed manifold $M$, i.e. there is an embedding $F \colon M\to \mathbb R^{m+k}$ with trivialized normal bundle $\nu(F)\cong\varepsilon ^k... | 4 | https://mathoverflow.net/users/20999 | 349620 | 147,963 |
https://mathoverflow.net/questions/349622 | 8 | What is an example of an accessible category $\mathcal C$ which is not essentially small, such that $\mathcal C^{op}$ is finitely-accessible?
More generally, what is an example of an accessible category $\mathcal C$ (not essentially small) such that one of the following related conditions holds?
* $\mathcal C^{op}$... | https://mathoverflow.net/users/2362 | Can the dual of a finitely-accessible category be accessible? | In **Accessible Categories: The Foundations of Categorical Model Theory** by *Makkai and Paré*, there is the example of a finitely accessible self-dual category. Apparently the example is due to *Isbell*. This is the category of **sets and partial monomorphisms**. The example appears **right after Prop. 3.4.4** and rig... | 10 | https://mathoverflow.net/users/104432 | 349623 | 147,964 |
https://mathoverflow.net/questions/349628 | 1 | Can one characterize the elements $\alpha$ of $L=\overline{\mathbb F\_q(T)}\subset\Omega$, a completion of $L$ for the $\left(\frac1T\right)$-adic valuation such that for every conjugate $\beta$ of $\alpha$, there exists a continuous $\mathbb F\_q\left(\left(\frac1T\right)\right)$-isomorphism of $\Omega$ with $\sigma(\... | https://mathoverflow.net/users/33128 | Algebraic function fields with continuous automorphisms | Let $\Omega'$ be an algebraic closure of $\mathbb F\_q (( \frac{1}{T}))$. Then we can embed $\Omega'$ into $\Omega$ because every extension of $\mathbb F\_q (( \frac{1}{T}))$ is defined over $\mathbb F\_q(T)$, and $\Omega'$ is dense in $\Omega$ because it contains $L$, so therefore $\Omega$ is the completion of $\Omega... | 3 | https://mathoverflow.net/users/18060 | 349631 | 147,966 |
https://mathoverflow.net/questions/349317 | 5 | It is well-known fact that integral Dehn surgeries on $3$-sphere $S^3$ are viewed as the result on the boundary of attaching $2$-handles $B^2 \times B^2$ to the $4$-ball $B^4$.
Is there an analogue of rational surgeries relating handle attachment of rational framing? If not, which problem occurs?
| https://mathoverflow.net/users/nan | Rational surgery and attaching $2$-handles | To attach a 4-dimensional 2-handle to the 4-ball, one requires an attaching region in $S^3=\partial B^4$ and a map from the attaching region of the handle (which has a natural parametrization as $S^1\times D^2\subset \partial(D^2\times D^2)$) to the attaching region in $S^3$. The attaching region in $S^3$ is determined... | 10 | https://mathoverflow.net/users/113696 | 349635 | 147,967 |
https://mathoverflow.net/questions/349642 | 7 | I am applying for graduate school in pure mathematics and I recently got very interested in C\*-algebra.
I am definitely wrong but I get the feeling that C\*-algebras is not as popular as other areas of pure mathematics like number theory, analysis, algebraic geometry, etc. It also seems that most top ranked universi... | https://mathoverflow.net/users/150643 | Why C*-algebras is not as popular as other areas of pure mathematics? | One way to tell how active a field is is by looking at [what's appearing on the arXiv in that area](https://arxiv.org/search/?query=math.OA&searchtype=all). I think that will show you that operator algebra is a robust subject with a lot of activity.
In the comments, MaoWao points out that UC Berkeley and UCLA have ve... | 23 | https://mathoverflow.net/users/23141 | 349653 | 147,973 |
https://mathoverflow.net/questions/349329 | 10 | Background:
Given a well partial order $X$ (more commonly studied with antisymmetry dropped as well-quasi-orders, but I'm going to say well partial order to make this definition simpler, obviously the two theories are essentially the same), De Jongh and Parikh showed there's always a largest ordinal that can be reali... | https://mathoverflow.net/users/5583 | Is there a relation between type (maximum linearization) of a computable WQO and the ordinal strength of a theory needed to prove it? | Let me show that for extensions $T\supseteq\mathsf{ACA}\_0$ the usual proof-theoretic ordinal $|T|\_{WO}$ coincide with $|T|\_{WPO}$ that is the suprema of $\mathsf{o}(X)$, for recursive wpo $X$, for which $T$ proves that $X$ is a wpo. Here $|T|\_{WO}$ is the suprema of order types $\mathsf{ot}(X)$ of recursive well-or... | 5 | https://mathoverflow.net/users/36385 | 349657 | 147,976 |
https://mathoverflow.net/questions/349547 | 6 | It is a well known and lovely result that the maximum number of regions that $\mathbb R^{k}$ (with $k$ positive) can be divided into by $n$ hyperplanes is given by
$$1+n+\binom{n}{2}+\cdots+\binom{n}{k}.
$$ and occurs when they are in general position. It is clear that the minimum with distinct hyperplanes is $n+1$ (w... | https://mathoverflow.net/users/8008 | Division of space by hyper-planes | Let's denote by $S\_{k,n}$ the set of possible integers $m$, such that $\mathbb R^k$ can be divided into $m$ regions by $n$ hyperplanes. If we denote by $S^{P}\_{k,n}$ the set defined similarly but for the projective space $\mathbb {RP}^k$. We have that $S\_{k,n}=S^P\_{k,n+1}$ since every affine arrangement can be lift... | 6 | https://mathoverflow.net/users/2384 | 349669 | 147,981 |
https://mathoverflow.net/questions/349391 | 11 | Let $A$ be a set of generators of $S\_n$, or of a doubly transitive
subgroup of $S\_n$. Assume $e\in A$, $A=A^{-1}$. What is the least $k$
such that $A^k$ is doubly transitive as a set? That is, what is the least $k$ such that there is a pair $x = (i,j)$, $i,j\in \{1,\dotsc,n\}$, $i\ne j$, for which $A^k x$ is
the set ... | https://mathoverflow.net/users/398 | How many steps are required for double transitivity? | It seems that this is a lower bound of $\Omega(n^2)$.
Take an $n$ and an $a=\Theta( n) $ coprime with $n$ (with $a<n/2$). Then the permutations $\sigma=(12\dots n) $ and $\tau=(1, a+1) $ generate $S\_n$.
On the other hand, all residues modulo $n$ form a cycle where the neighbors differ by $a$. The only way to chang... | 9 | https://mathoverflow.net/users/17581 | 349678 | 147,983 |
https://mathoverflow.net/questions/349675 | 0 | I am currently thinking of function-valued random variables. In order to prove a result, I need to approximate by (function-valued) step functions. This naturally leads to the idea of chopping up the function space into finitely many small pieces.
Let $X \subset \mathbb R^d$ be compact and $V$ the set of $1$-Lipschit... | https://mathoverflow.net/users/58082 | Can I get away without using Arzela-Ascoli? | Of course you can, and this is how Arzela-Ascoli is often proved. You may fix a finite $\varepsilon/3$-net $D\subset X$ and partition $[0,1]$ onto disjoint subsets $A\_1,\ldots,A\_N$ of diameter less than $\varepsilon/3$. For any 1-Lipschitz function $f:X\rightarrow [0,1]$ we consider the function $[f]:D\rightarrow \{1... | 7 | https://mathoverflow.net/users/4312 | 349679 | 147,984 |
https://mathoverflow.net/questions/349674 | 2 | Given a real number $\alpha \in [0.5, 1.5]$, an integer number $k>1$, and a set of independent Bernoulli random variables $x\_1, \dots, x\_n$, I am interested to find a lower-bound for $F(\alpha, k)= \frac{E[\min(X, k)]}{k}$ subject to $E[X] = \alpha k$ where $X=\sum\_{i\in n} x\_i$.
My observation is that for $k\rig... | https://mathoverflow.net/users/130779 | Lower-bound for $E[\min(X, k)]$ where $X$ is sum of Bernoulli random variables with $E[X]$ being a linear function of $k$ | Let $x\_1,\dots, x\_n$ be independent Bernoulli random variables with expectations $p\_1,\dots, p\_n$ summing to $k\alpha$. Let $y\_1,\dots, y\_n$ be independent random variables with expectations $p\_1 \frac{k-1}{k},\dots, p\_n \frac{k-1}{k}$, summing to $(k-1)\alpha$. I claim that $$ \mathbb E \left[ \frac{ \min \lef... | 1 | https://mathoverflow.net/users/18060 | 349681 | 147,985 |
https://mathoverflow.net/questions/349665 | 2 | Let $x\_1,...,x\_n\in\mathbb{R}^p$ be i.i.d. random vectors with mean 0 and covariance $\Sigma\_p$. Let $S\_{n,p}=\sum\_{i=1}^nx\_ix\_i^T/n$ be the sample covariance. We consider the asymptotics of the empirical distribution of eigenvalues of $S\_{n,p}$. We let $p/n\to c\in(0,1)$ where $c$ is a constant.
It is well-... | https://mathoverflow.net/users/123075 | Marchenko-Pastur Law under general covariance structure | In the Gaussian case, you can rewrite $x\_i=R^{1/2}y\_i$ where $y\_i$ now possess iid entries.
This leads you to computing the eigenvalues of $Y^\*RY$, this is actually the problem solved by Pastur and Marchenko, see Math. USSR. Sbornik vol 1 (1967), with an explicit equation satisfied by the Stieltjes transform of $\m... | 6 | https://mathoverflow.net/users/35520 | 349685 | 147,987 |
https://mathoverflow.net/questions/349680 | 15 | I was trying to calculate $H^q(K(\mathbb{Z}, 3); \mathbb{Z})$ for some $q$ with the Serre spectral sequence associated to the fibration $K(\mathbb{Z}, 2) \to PK(\mathbb{Z}, 3) \simeq \* \to K(\mathbb{Z}, 3)$.
I obtained that:
$$
H^q(K(\mathbb{Z}, 3)) = \mathbb{Z}, 0, 0, \mathbb{Z}x, 0, 0, \mathbb{Z}\_2x^2, 0, \mathb... | https://mathoverflow.net/users/137622 | Calculation of $H^{10}(K(\mathbb{Z}, 3); \mathbb{Z})$ | I do not like naming a cohomology class $n$ because that deserves to be the name of an integer. I will use the name Hatcher does and call the generator of $H^2(K(\Bbb Z, 2); \Bbb Z)$ by the name "$a$".
The map $d\_3: E\_3^{0, 8} \to E\_3^{3,6}$ sends $d\_3(a^4) = 4a^3 x$ by the Leibniz rule. The map $d\_3: E\_3^{3,6}... | 20 | https://mathoverflow.net/users/40804 | 349686 | 147,988 |
https://mathoverflow.net/questions/349667 | 2 | Let $X\_1, \dots, X\_n$ be standard Gaussians. Let $\mathcal{S} \subseteq \{A \in 2^{\{1, \dots, n\}} : |A| = k\} $ be a family of subsets of $\{1,\dots, n\}$ with fixed size $k$. [Note that $\mathcal{S}$ may be smaller than $\{A \in 2^{\{1, \dots, n\}} : |A| = k\}$, the set of all subsets with fixed size $k$.]
Let $... | https://mathoverflow.net/users/150656 | Concentration bound on maximum subset sum of standard Gaussians | Claim: If $|\mathcal{S}| \to \infty$ as $n\to \infty$, then $Y \leq \sqrt{2k\log{|\mathcal{S}|}}$ with high probability.
Proof: Let $t = \sqrt{2k\log{|\mathcal{S}|}}$. By union bound, we have
$P(Y > t) \leq \sum\_{A \in \mathcal{S}} P(\sum\_{i \in A} X\_i > t) = |\mathcal{S}| \cdot P(N(0,k) > t)$.
Plugging in $t$... | 2 | https://mathoverflow.net/users/150656 | 349687 | 147,989 |
https://mathoverflow.net/questions/348416 | 5 | **Question**
The usual projection in $\mathbb{R}^n$ on a subspace can be defined as the point that minimizes the squared distance to the subspace. I'll call the Pythagorean theorem the easy fact that, given a point $x$, its projection $Px$ and another point $y$ in the subspace,
$$
|x-y|^2 = |x - Px|^2 + |Px-y|^2
$$
... | https://mathoverflow.net/users/947 | Pythagorean theorems for other distances | The so-called "Bregman divergences", which include both $\ell\_2$ distance and KL divergence among others, obey the Pythagorean relation. For a nice summary, see Banerjee et al, "Clustering with Bregman Divergences", *JMLR* 2005, in particular p.1741.
| 5 | https://mathoverflow.net/users/76565 | 349690 | 147,990 |
https://mathoverflow.net/questions/349684 | 3 | Let $G$ be an amenable (countable, discrete) group and let $F\_1,F\_2,...,F\_n,...$ and $G\_1,G\_2,...,G\_n,...$ be two Følner sequences. Is the product sequence (i.e. the sequence $(H\_n)$ where $H\_n$ is all elements of the form $f\_ng\_n$ for $f\in F\_n, g\in G\_n$) necessarily also a Følner sequence? If not, is thi... | https://mathoverflow.net/users/123459 | Is a product of Følner sets Følner? | **some details added January 7th**
It seems that the answer is **"no" even for virtually abelian groups**, and even
if $G\_i$ is chosen by your arch enemy. The sequence $(F\_i)\_i$ can also be chosen rather freely in the proof. This answers some subquestions of the question, but not all of them. In particular do not ... | 7 | https://mathoverflow.net/users/123634 | 349699 | 147,994 |
https://mathoverflow.net/questions/349652 | 7 | $\require{AMScd}$For a Riemannian manifold $M$, I have read authors talking about a 'Liouville measure' on the unit tangent bundle $\operatorname{T}^1(M)$ and then proceed to claim/prove that it is invariant under the geodesic flow.
See for example Mautner's $1957$ paper 'Geodesic Flows on Symmetric Riemann Spaces.'
... | https://mathoverflow.net/users/105628 | Help with definition of Liouville measure | The construction doesn’t really simplify on symmetric spaces. On $TM\cong T^\*M$ (using the metric) consider the canonical 1-form $\alpha=“\langle p,dq\rangle”$ and symplectic form $d\alpha$ and hamiltonian vector field $\xi$ of $H=\frac12\|p\|^2$: $\mathrm i\_\xi d\alpha=-dH$. Then $\alpha$ and $\xi$ restrict to a con... | 5 | https://mathoverflow.net/users/19276 | 349700 | 147,995 |
https://mathoverflow.net/questions/349682 | 10 | Suppose that $X$ is a connected $E\_{\infty}$-space, naturally $\Omega X$ is also an $E\_{\infty}$-space. Can we classify all $E\_{\infty}$-extensions of $X$ by $\Omega X$ (up to homotopy). I mean the following: we would like to classify of homotopy fiber sequences $A\rightarrow B\rightarrow C$ where $A\sim \Omega X$ a... | https://mathoverflow.net/users/136128 | Homotopy extension of $E_{\infty}$-spaces | This is probably belaboring the obvious, but just take seriously the equivalence between grouplike $E\_{\infty}$ spaces and connective spectra. See for example
[Equivalence between $E\_\infty$-spaces and connective spectra](https://mathoverflow.net/questions/90379/equivalence-between-e-infty-spaces-and-connective-sp... | 19 | https://mathoverflow.net/users/14447 | 349707 | 147,997 |
https://mathoverflow.net/questions/349589 | 16 | Let $X\_1,\cdots,X\_n$ be finite subsets of some set $Z$. Then the symmetric difference metric space:
$$d(X\_i,X\_j) = \sqrt{ |X\_i|+|X\_j|-2|X\_i\cap X\_j|}$$
can be embedded in Euclidean space. The value $|X\_i \cap X\_j| = \langle \phi(X\_i),\phi(X\_j) \rangle$ is equal to the dot product of the embeddings $\phi(X... | https://mathoverflow.net/users/nan | Are primes linearly separable? | The purely number-theoretic problem can be stated as: For every $n$, are there reals $b$ and $c\_i$ such that
$$j\text{ is prime iff }\sum\_{i=2}^n c\_i \gcd(i,j)>b\ ?$$
This is the same as the version above, after removing the $x$'s and $y$'s, shifting the indices by 1, and incorporating a factor of $y\_i$ into the $c... | 12 | https://mathoverflow.net/users/nan | 349708 | 147,998 |
https://mathoverflow.net/questions/277592 | 4 | Do the following Riemannian metrics on $GL(n,\mathbb{R})$ give us isometric structures?Do they generate the same volume forms? Is $O(n)$ a totally geodesic submanifold with respect to these metrics?
1. The metric with orthonormal frame $A\otimes A$ at each point $A\in GL(n,\mathbb{R})$
2. The metric with orthonormal ... | https://mathoverflow.net/users/36688 | Are these two structures isometric? | **Corrected answer:** Well, technically, my answers to the OP's questions have not changed, but my argument has changed because I found a silly mistake in my original argument (which I will explain below).
First, computation shows that the first metric, described by giving the orthonormal frame as $A\otimes A$, can b... | 13 | https://mathoverflow.net/users/13972 | 349710 | 147,999 |
https://mathoverflow.net/questions/349267 | 4 | Is there any known parabolic PDEs in the literature where the angle of analyticity of the associated semigroup is $<\pi/2$ ?
For example, the angle of heat semigroup in $L^2$ is exactly $=\pi/2$. I'm wondering if there is a known example where the angle is e.g., $\pi/4$ or other value $< \pi/2$.
| https://mathoverflow.net/users/146543 | Angle of analyticity of semigroup | Too long for a comment. Actually I do not think that it is written anywhere but these kind of counterexamples are usually provided by the Ornstein-Uhlenbeck semigroup, generated by $\Delta+Bx \cdot \nabla$, where $B$ is a matrix. Assuming that all eigenvalues of $B$ have negative real parts, then an invariant measure $... | 6 | https://mathoverflow.net/users/150653 | 349718 | 148,002 |
https://mathoverflow.net/questions/349722 | -4 | I would like to ask about if you know papers containing remarkable achievements that were written by contemporary physicists concerning the distribution of prime numbers (or closely related, maybe the distribution of the zeros of the Riemann zeta function) in the spirit of the modern analytic number theory.
I am aski... | https://mathoverflow.net/users/142929 | Remarkable articles about the distribution of prime numbers that were written by contemporary physicists | I would nominate Sir Michael Berry, FRS. According to Wikipedia, "He is known for the Berry phase, a phenomenon observed e.g. in quantum mechanics and optics, as well as Berry connection and curvature. He specialises in semiclassical physics (asymptotic physics, quantum chaos), applied to wave phenomena in quantum mech... | 6 | https://mathoverflow.net/users/6756 | 349725 | 148,005 |
https://mathoverflow.net/questions/348983 | 5 | Let $X$ be a topological space and $\mathrm{RO}(X)$ its complete boolean algebra of regular opens. Define well inside relation: $$U\prec V\iff\overline{U}\subseteq V.$$ Let $\mathcal C\subseteq\mathrm{RO}(X)$ be a chain such that (a) any two distinct elements of $\mathcal C$ are comparable w.r.t. $\prec$ and (b) $\math... | https://mathoverflow.net/users/22019 | A problem of non-emptiness of intersections of certain chains of regular open sets | Here is a provisional negative answer.
If $\mathcal{C}$ is a c-minimal chain then $N=\bigcap\mathcal{C}$ is closed and nowhere dense (if nonempty): if the interior is nonempty take a nonempty regular open set $O$ such that $\overline{O}\subseteq \operatorname{int}N$. Then the chain $\mathcal{C}\cup\{O\}$ is covered by ... | 5 | https://mathoverflow.net/users/5903 | 349728 | 148,007 |
https://mathoverflow.net/questions/349731 | 9 | Let $R$ be an $E\_{\infty}$-ring spectrum and $B$ be an $E\_\infty$-space. Suppose we have an $E\_\infty$-map $$ f: B \to BGL\_1(S^0)$$ such that the composite $$f\_R: B \to BGL\_1(S^0) \to BGL\_1(R) $$
is null, then a choice of null-homotopy produces Thom isomorphism which is a weak-equivalence
$$u: Mf \wedge R \sime... | https://mathoverflow.net/users/19186 | When is Thom isomorphism a ring map? | One comment is that you have to be careful, because to be an $E\_{\infty}$-map is not a property but rather additional structure.
You are exactly right in that what is needed is a null-homotopy of $B \rightarrow BGL\_{1}(R)$ as a map of $E\_{\infty}$-spaces. This is Proposition 3.16 in Omar and Toby's "[A simple uni... | 9 | https://mathoverflow.net/users/16981 | 349732 | 148,009 |
https://mathoverflow.net/questions/349705 | 1 | Let $f:R\rightarrow R$ be a concave function with a unique and finite maximum. Let $$g(x, \beta) = \beta f(\alpha \cdot x) + (1-\beta) f((1-\alpha) \cdot x), $$ where $\alpha \in [0,1/2]$ and $\beta \in [0,1/2]$. Furthermore, let $g^\*(\beta) = \max\_{x} g(x,\beta)$.
I am trying to find conditions for $f$ such that ... | https://mathoverflow.net/users/150688 | Monotonicity of maximum of convex combination of two scaled concave functions | Let $a:=\alpha\in(0,1/2)$ and $b:=\beta\in(0,1/2)$. Take any positive real $A$ and any real $B$, and let
$$f(x):=\min[x,(1+A)B-Ax]
$$
for real $x$. Then the function $f$ is concave with a unique and finite maximum (at $x=B$), and
$$g^\*(b)=g\Big(\frac B{1-a},b\Big)=
\Big(1-\frac{1-2 a}{1-a}\,b\Big) B
$$
if $a$ and $b... | 1 | https://mathoverflow.net/users/36721 | 349738 | 148,010 |
https://mathoverflow.net/questions/339943 | 5 | Currently I am working on applications of Bourgain Embedding (or similar embeddings of *finite* metric spaces to $l\_2$) to automatic feature engineering for machine learning/data science ( <http://www.orges-leka.de/automatic_feature_engineering.html>, <https://github.com/orgesleka/bourgain_embedding/blob/master/bourga... | https://mathoverflow.net/users/nan | Fast Bourgain embedding (or similar embeddings)? | There is a way to speed-up Bourgain's embedding in case if the original metric space has low "intrinsic dimension". The resulting algorithm will have a theoretical runtime $O(CN\log^2(N))$, where $C$ is a constant depending only on "intrinsic dimension". And also will show speedups over the standard Bourgain's embeddin... | 4 | https://mathoverflow.net/users/32454 | 349739 | 148,011 |
https://mathoverflow.net/questions/349742 | 3 | Let $f:\mathbb{R}^n\rightarrow\mathbb{R}$ be a non-negative, smooth, uniformly bounded function with uniformly bounded first derivative. Then $f$ defines a bounded operator on $L^2(\mathbb{R}^n)$ as well as on $H^1(\mathbb{R}^n)$, the first Sobolev space. In $L^2(\mathbb{R}^n)$ the following equality holds: for $e\_1,e... | https://mathoverflow.net/users/78729 | "Square root" of multiplication operator on Sobolev space | $\newcommand{\R}{\mathbb R}$
The square root of the multiplication operator does not exist unless the function $f$ is constant (and, obviously, the square root exists if $f$ is constant).
Indeed, suppose that there exists a function $g\_f$ such that
\begin{equation\*}
\langle fx,y\rangle\_{H^1}=\langle g\_f(x),g\_f... | 4 | https://mathoverflow.net/users/36721 | 349743 | 148,013 |
https://mathoverflow.net/questions/349715 | 4 | Let $X,Y$ be two compact metric spaces. Suppose there is a sequence of bi-Lipschitz homeomorphisms $f\_n: X\to Y$, and $c\in(0,1]$, satisfying
$$c\cdot d(x\_1,x\_2)\le d(f(x\_1),f(x\_2))\le \frac{1}{c}\cdot d(x\_1,x\_2),$$
for any $x\_1,x\_2\in X$.
Is is true that one can choose a sub-sequence $f\_{n\_i}$ that unifo... | https://mathoverflow.net/users/13441 | Choosing a convergent sub-sequence from a sequence of bi-Lipschitz homeomorphisms | Consider a subsequence $\,g\_n\,$ of $\,f\_n\,$ which is uniformly convergent. Then consider a subsequence $\,h\_n\,$ of $\,g\_n\,$ such that sequence $\,h^{-1}\_n\,$ is uniformly convergent. Then the limit function $\,h\,$ of $\,h\_n\,$ is bilipschitz with the bi-constant equal the same $\,c.$ ***Great***
>
> **... | 2 | https://mathoverflow.net/users/110389 | 349748 | 148,015 |
https://mathoverflow.net/questions/349693 | 2 | It is known that
$$
\exp\left\{k(1/1.71\cdots+o(1))\right\} < H(k) < \exp\left\{k(1/1.13862\cdots+o(1))\right\},
$$
where $H(k)$ is the $k^{th}$ [highly composite number.](https://en.wikipedia.org/wiki/Highly_composite_number)
**Question:** Does the number of divisors of the highly composite numbers $d(H(k))$ achieve... | https://mathoverflow.net/users/17773 | Divisor Function of Highly Composite Numbers | Applying Lemma 1 from Erdos [paper](https://www.renyi.hu/~p_erdos/1944-04.pdf), we are able to obtain
$$
\limsup\_{k\rightarrow\infty} \frac{\log d(n\_k)}{\frac{ \log n\_k}{\log \log n\_k}}\geq \frac16\log 2.
$$
Here's the proof. Let $N$ be a highly composite number other than 4, 36, we have the prime factorization of ... | 4 | https://mathoverflow.net/users/21090 | 349759 | 148,019 |
https://mathoverflow.net/questions/349730 | 1 | According to this link:
<https://en.wikipedia.org/wiki/Stericated_6-simplexes>
the stericated 5-simplex "scal" has 105 vertices defined as permutations of (0,0,1,1,1,1,2).
In the course of my team's research into the structure of a particular biomolecular energetic space, we have discovered that there are exactly... | https://mathoverflow.net/users/118372 | The Fano plane, stericated 6-simplex, and pentallated 6-simplex | As already was outlined in one of the answers to <https://math.stackexchange.com/questions/2070413/for-which-dimensions-does-it-exist-a-regular-n-polytope-such-that-the-distance-o/3492945?noredirect=1#3492945>, it is possible to understand the expanded simplex of any dimension as an axial stack of 3 vertex layers of th... | 1 | https://mathoverflow.net/users/118679 | 349761 | 148,020 |
https://mathoverflow.net/questions/349767 | 4 | I recently came across [this unanswered MO question](https://mathoverflow.net/questions/242324/set-of-w-continuous-operators-closed-for-the-weak-topology-or-not) an answer to which I would also be interested in. However the formulation of said question is somewhat imprecise and lacking detail in my opinion so I figured... | https://mathoverflow.net/users/116991 | Is the set of weak*-continuous operators closed in the weak*-operator topology? | The answer is "no", in general.
An easy counterexample can be found as follows: Let $X = \mathbb{F}$ and let $Y$ be a non-reflexive Banach space. Then $\mathcal{B}(Y^\*,X^\*)$ is simply the bi-dual $Y^{\*\*}$, and ${}^\*(\mathcal{B}(X,Y))$ is precisely the image $j(Y)$ of $Y$ in $Y^{\*\*}$ under the evaluation map $j... | 5 | https://mathoverflow.net/users/102946 | 349768 | 148,024 |
https://mathoverflow.net/questions/349755 | 3 | Is it true that :
$\forall f,g \in C([0,1],\mathbb R), \exists h \in C([0,1],[0,1])$ $f,g,h$ strictly increasing and $h([0,1])=[0,1]$ with $(f \circ h, g\circ h) \in C^{\infty}([0,1],\mathbb R)^2$?
| https://mathoverflow.net/users/110301 | Regularize continuous functions with bounded variation | The answer is NO:
Let $\,f:[0;1]\to[0;1]\,$ be the identity function. Let $\,g:[0;1]\to[0;1]\,$ be such that the set of $\,D\subseteq[0;1]\,$ of points $\,x\in[0;1]\,$ for which derivative of $\,g\,$ doesn't exist is dense.
Then, if $\,f\circ h\,$ is smooth then $\,h\,$ is smooth.
But then $\,g\circ h\,$ cannot b... | 5 | https://mathoverflow.net/users/110389 | 349770 | 148,025 |
https://mathoverflow.net/questions/349779 | 8 | The Wikipedia [states](https://en.wikipedia.org/wiki/Ordinal_analysis) that Gentzen proved that "in modern terms, the proof-theoretic ordinal of PA is $\varepsilon\_0$." Further down, the article defines what the "proof theoretic ordinal" of a theory means. However, I'm not sure what this means regarding PA, since PA c... | https://mathoverflow.net/users/27742 | Gentzen's result on PA | Yes, Gentzen found a single "induction instance" which is PA-unprovable: more-or-less $\varphi(\alpha)$ = "Every sentence with a proof of cut-rank $\alpha$ has a cut-free proof."
Now, this $\varphi$ is a $\Pi^0\_2$ formula. If memory serves, this is suboptimal: with some coding work this can be improved from a $\Pi^... | 10 | https://mathoverflow.net/users/8133 | 349781 | 148,026 |
https://mathoverflow.net/questions/349760 | 1 | Let $X$ be a connected topological space. Let $E$ be a $k$ dimensional sub vector bundle of the trivial vector bundle $X\times \mathbb{R}^n$. Then $E$ defines an idempotent with trace $k$ in $M\_n(C(X))$. Conversely every trace $k$ idempotent of this algebra determines a $k$ dimensional sub bundle of $n$ domensional tr... | https://mathoverflow.net/users/36688 | A kind of isomorphicity of vector bundles | The antipodal map of $S^{2}$ sends $L$ to $L^{-1}$, where $L$ is the line bundle constructed via clutching along an equatorial $S^1$ with the "identity" map $S^1\rightarrow U(1)$.
So let $\alpha$ be the automorphism of $M\_{n}(C(S^2))$ induced by the antipodal map, let $e$ be the projection corresponding to $L$, and ... | 1 | https://mathoverflow.net/users/148857 | 349790 | 148,030 |
https://mathoverflow.net/questions/349785 | 5 | I have just been told about this result, available as Exercise 2.5.10 in Ram. M. Murty's book, "Problems in Analytic Number Theory (2nd edition)". It says:
Let $\alpha>0$. Suppose $a\_n \ll n^{\alpha}$ and $A(x) \ll x ^{\delta} $ for some $\delta<1$, where $A(x) = \sum\_{n\leq x} a\_n$. Define $b\_n = \sum\_{d|n} a\_... | https://mathoverflow.net/users/480516 | On Exercise 2.5.10 in Ram. M. Murty's book, "Problems in Analytic Number Theory." | It seems there is a typo in the application of the hyperbola method. Since $b\_n=\sum\_{d\mid n}a\_d$, we have
\begin{align}
\sum\_{n\le x}b\_n &=\sum\_{n\le x}\sum\_{de=n}a\_d=\sum\_{de\le x}a\_d\\
&=\sum\_{\substack{de\le x\\ d\le y}}a\_d+\sum\_{\substack{de\le x\\ e\le x/y}}a\_d-\sum\_{de\le x\\ d\le y, e\le x/y}a\... | 4 | https://mathoverflow.net/users/112959 | 349791 | 148,031 |
https://mathoverflow.net/questions/349773 | 0 | Given $1\leq k\leq m$, $2\leq d\leq c i\ln i$ and $2\leq i\leq c'\ln(mi\ln i)$ at some $c,c'>0$ how many sequences (lower and upper bounds) are of form $$z\_1,\dots,z\_m$$ on the condition that
$$0\leq z\_1\leq\dots\leq z\_m\leq 2^d$$
$$|\{i\in\{1,\dots,m\}: z\_i\neq z\_{i+1}\}|=k$$
are there?
Is there a standard... | https://mathoverflow.net/users/136553 | Ordered $m$-tuples with fixed number of changes | Apparently, you mean $|\{i\in\{1,\dots,m-1\}: z\_i\neq z\_{i+1}\}|=k$ as $z\_{m+1}$ is undefined.
This condition implies that among $z\_1, \dots, z\_m$ there are $k+1$ district values. Since the number of subsets $\{1,\dots,m-1\}$ of size $k$ equals $\binom{m-1}{k}$, we conclude that the number of sequences of interest... | 1 | https://mathoverflow.net/users/7076 | 349796 | 148,033 |
https://mathoverflow.net/questions/349783 | 3 | Let $A$ be a noncommutative algebra over a field, say $\mathbb{C}$ or $\mathbb{R}$, and let $L$ be a bimodule over $A$. If $L$ is **invertible**, that is, if the dual right $A$-module $L^\*$ satisfies
$$
L^\* \otimes\_A L \simeq A
$$
then will $L$ necessarily be projective a left module over $A$?
| https://mathoverflow.net/users/147728 | Invertible bimodules and projectivity | Yes. If $\varphi :L^\*\otimes \_A L\rightarrow A$ is an isomorphism, there exists an element $t =\sum\limits\_{i=1}^{n} x\_i^\*\otimes x\_i$ of $L^\*\otimes \_A L$ such that $\varphi (t )=1$. For any $x\in L$, we have then $x\varphi (t)=x$, that is, $\sum\_i \langle x,x\_i^\*\rangle x\_i=x$. Now consider the homomorphi... | 9 | https://mathoverflow.net/users/40297 | 349800 | 148,034 |
https://mathoverflow.net/questions/147159 | 2 | I am looking for a proof/reference of the following simple fact, which I think it holds true.
Let $S\subset \mathbb{P}^n$ be a surface embedded by a very ample linear system. Then I know that the dual variety $S^\*$ is an irreducible hypersurface. Is it true that the generic plane section of $S^\*$ is an irreducible ... | https://mathoverflow.net/users/4096 | singularities of the dual variety of a surface | I. Shimada proved this result in [*Singularities of Dual Varieties in Characteristic 3*, Geom. Dedicata 120 (2006), 141–177](http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.74.8099&rep=rep1&type=pdf), by assuming the linear series is *"sufficiently ample"*. (**Edited:** This result seems to be classical. Here ... | 5 | https://mathoverflow.net/users/74322 | 349801 | 148,035 |
https://mathoverflow.net/questions/349776 | 7 | It is well-known that one can evaluate the sum
$$\sum\_{k =1}^{N-1} \csc^2\left(\pi \frac{k}{N} \right)=\frac{N^2-1}{3}.$$
The answer to this problem can be found here
[click here.](https://math.stackexchange.com/questions/544228/finite-sum-sum-limits-k-1m-1-frac1-sin2-frack-pim)
I am now interested in the mor... | https://mathoverflow.net/users/150549 | $\sum_{k =1, k \neq j}^{N-1} \csc^2\left(\pi \frac{k}{N} \right)\csc^2\left(\pi \frac{j-k}{N} \right)=?$ | Start from the well known formula
\begin{equation}
2^{N-1} \prod \_{k=1}^N \left[\cos (x-y)-\cos \left(x+y+\frac{2\pi k}{N}\right)\right]=\cos N(x-y)-\cos N(x+y),
\end{equation}
take logarithmic derivative
\begin{equation}
\sum \_{k=1}^N \frac{\sin(x-y)}{\cos (x-y)-\cos \left(x+y+\frac{2\pi k}{N}\right)}=\frac{N\sin N... | 6 | https://mathoverflow.net/users/82588 | 349806 | 148,038 |
https://mathoverflow.net/questions/349762 | 3 | In
>
> *Rothschild, Linda Preiss; Stein, Elias M.*, [**Hypoelliptic differential operators and nilpotent groups**](http://dx.doi.org/10.1007/BF02392419), Acta Math. 137(1976), 247-320 (1977). [ZBL0346.35030](https://zbmath.org/?q=an:0346.35030). [PDF at archive.ymsc.tsinghua.edu.cn](http://archive.ymsc.tsinghua.ed... | https://mathoverflow.net/users/145357 | Free Lie algebra and nilpotent groups in Rothschild and Stein's paper | First, the notation is a little confusing. I think you are supposed to understand that $n$ is always the number of vector fields in the fixed set $\{X\_1, \dots, X\_n\}$. But the symbol $n$ in the notation $n\_s$ is just an arbitrary letter and isn't a reference to the number of vector fields. So in your example, $n$ i... | 3 | https://mathoverflow.net/users/4832 | 349810 | 148,040 |
https://mathoverflow.net/questions/348736 | 8 | Let $A$ be a unital algebra, defined over the complex numbers. Any bimodule $M$ over $A$ must, by definition, be a left, and right, module satisfing
$$
a.(m.b) = (a.m).b, ~~~~~~~ \textrm{ for all } a,b \in A, ~ m \in M
$$
What is a "natural" or "well-motivated" example of a an object which is both a left and right modu... | https://mathoverflow.net/users/147728 | Left-right non-bimodule examples | Suppose $A$ is a non-commutative Hopf algebra. Then you can use $M=A$ with the left adjoint and right regular actions:
$$
a\cdot m = \sum\_{(a)}a\_{(1)}mS(a\_{(2)}), \ m\cdot b = mb .
$$
In particular, you can use the group algebra of a a group $G$ so that
$$
g\cdot m = gmg^{-1}, \ m\cdot h = mh, \ g,h,m\in G.
$$
| 3 | https://mathoverflow.net/users/5301 | 349814 | 148,041 |
https://mathoverflow.net/questions/349711 | 4 | Given two semisimple unital algebras $A$ and $B$, defined over $\mathbb{R}$ or $\mathbb{C}$, denote their categories of representations by $\_A\mathcal{M}$ and $\_B\mathcal{M}$ respectively. Can one describe the category of representations of $A \otimes\_{\mathbb{C}} B$ as some type of "tensor product" of the categorie... | https://mathoverflow.net/users/147728 | Category of representations of a tensor product algebra | Yes, it will be exactly Deligne's tensor product of abelian categories. See <https://ncatlab.org/nlab/show/Deligne+tensor+product+of+abelian+categories>
| 2 | https://mathoverflow.net/users/5301 | 349815 | 148,042 |
https://mathoverflow.net/questions/349819 | 2 | How can I determine all integer points of the following equation
$$y^2=x^3+10546$$
I tried [Magma](http://magma.maths.usyd.edu.au/calc/) with
```
IntegralPoints(EllipticCurve([0,10546]));
```
but got the answer that it "could not determine the Mordell-Weil group." What are my options here?
| https://mathoverflow.net/users/41145 | Integer points of one Mordell equation | This curve has rank 0 over $\mathbb{Q}$. The 2-descent fails to determine this, because the $2$-torsion subgroup of the Tate-Shafarevich group is non-trivial. Instead, one can compute the $L$-value. One can prove that $L(E,1) = 16 \Omega\_{+}$. By Kolyvagin, this implies that the rank is $0$.
Now one just needs to co... | 17 | https://mathoverflow.net/users/5015 | 349820 | 148,044 |
https://mathoverflow.net/questions/349753 | 2 |
>
> Let $E$ be a Banach space, let $T:E\to E$ have norm $1$ and let $\nu\in E^\*\setminus\{0\}$ be such that $T^\*\nu=\nu$. Under which conditions there is $e\in E$ such that $Te=e$ and $\langle e,\nu\rangle\ne 0$?
>
>
>
This is true for reflexive spaces. Indeed, WLOG $\|\nu\|=1$. Consider $D=\nu^{-1}(1)\cap \ov... | https://mathoverflow.net/users/53155 | Dual fixed point | **Part 1 of the answer.** In terms of $T$, the property you are looking for is characterized by the *mean ergodic theorem*:
**Theorem.** Let $E$ be a Banach space and let $T$ be a bounded linear operator on $E$ that is *power-bounded* in the sense that $\sup\_{n \in \mathbb{N}\_0}\|T^n\| < \infty$. Then the following... | 4 | https://mathoverflow.net/users/102946 | 349830 | 148,049 |
https://mathoverflow.net/questions/349757 | 6 | Let $n$ be a non-negative integer.
**Does there always exist a polynomial $P\_n(a,b)$ such that for all integers $a > b \geq n/2$ we have**
$$
\sum\_{k=b}^{a-1} \binom{2k+1+n}{2n+1}\binom{2a-1}{a+k} = \binom{2a-1}{a+b} P\_n(a,b)\quad ?
$$
For small values of $n$ this is easily verified using Gosper's algorithm, for exa... | https://mathoverflow.net/users/47484 | On the sum $\sum_{k=b}^{a-1} \binom{2k+1+n}{2n+1}\binom{2a-1}{a+k}$ | Actually, checking whether $p(k)$ lies in the image of $s(k) \mapsto q(k) s(k+1) - r(k)s(k)$ turns out to be not too difficult after all. We need to determine a single linear condition that spans the cokernel of the linear mapping. A convenient way to do this is by turning $q(k) s(k+1) - r(k)s(k)$ into a differential o... | 2 | https://mathoverflow.net/users/47484 | 349831 | 148,050 |
https://mathoverflow.net/questions/349812 | 2 | Let $Q(n)$ give the number of ways of writing the integer $n$ as a sum of positive integers without regard to order with the constraint that all integers in a given partition are distinct. Equation $(11)$ on [this page](http://mathworld.wolfram.com/PartitionFunctionQ.html) mentions (without proof) a recurrence relation... | https://mathoverflow.net/users/150757 | Recurrence relation for the number of partitions of an integer with distinct summands | Gauss showed that
$$ \prod\_{n\geq 1}\frac{1-q^n}{1+q^n} = 1+2\sum\_{n\geq 1}(-1)^n q^{n^2}. $$
We also have $\sum\_{n\geq 0} Q(n)q^n = (1+q)(1+q^2)\cdots$ and $\sum\_{n\geq 0} s(n)q^n = (1-q)(1-q^2)\cdots$ (Euler's pentagonal number formula). The recurrence follows from equating coefficients of $q^n$ on both sides of... | 7 | https://mathoverflow.net/users/2807 | 349835 | 148,051 |
https://mathoverflow.net/questions/349824 | 5 | An element $a$ of a Lie algebra $L$ is called a Fredholm element if the adjoint operator $\mathrm{ad}\_a:L \to L$ is a Fredholm linear map. That is: its kernel is a finite-dimensional space and its range $\mathrm{ad}\_a(L)$ is a finite-codimensional subspace.
Is there an infinite-dimensional Lie algebra with at least... | https://mathoverflow.net/users/36688 | Fredholm elements of a Lie algebra | The infinite-dimensional Lie algebra with basis $(e\_n)\_{n\in\mathbf{N}}$ and brackets $[e\_i,e\_j]=(i-j)e\_{i+j}$, over a field of characteristic zero, satisfies the required condition: every nonzero element is Fredholm.
First let me mention it's immediate that $\mathrm{ad}\_{e\_i}$ is Fredholm for every $i$. Deno... | 4 | https://mathoverflow.net/users/14094 | 349836 | 148,052 |
https://mathoverflow.net/questions/349832 | 1 | Let $[\omega]^\omega$ denote the set of infinite subsets of $\omega$.
Let $S\subseteq [\omega]^\omega$. We say that a map $c:\omega \to \{0,\ldots,n-1\}$ is a *coloring for $S$ with $n$ colors* if for all $s\in S$ the restriction $c|\_s$ of $c$ to $s$ is non-constant.
What is an example of a set $S\subseteq [\omeg... | https://mathoverflow.net/users/8628 | Subset of $[\omega]^\omega$ that can be "colored" with $3$, but not $2$ colors | Partition $\omega$ into three infinite subsets $A\_0,A\_1,A\_2$. Let $S$ consists of subsets which intersects precisely two of the $A\_i$ at infinitely many elements. It can obviously be $3$-colored. Suppose there was a $2$-coloring, with color classes $c\_0,c\_1$. Then either $c\_0$ or $c\_1$ contains infinitely many ... | 9 | https://mathoverflow.net/users/30186 | 349837 | 148,053 |
https://mathoverflow.net/questions/349799 | 2 | I have a problem with the entropy rate when two ergodic Markov processes who are independent of each other having a non-ergodic joint. More specifically let us consider two finite-state Markov processes $\mathscr{P}\_1$ and $\mathscr{P}\_2$ with transition matrices $\Pi\_1$ and $\Pi\_2$, respectively. Let us assume tha... | https://mathoverflow.net/users/75549 | Entropy rate problem of ergodic Markov process with non-ergodic joint | The easiest way to see this is by using the Shannon-McMillan-Breiman equidistribution theorem and noticing that the space of sample paths of the product chain is the product of the sample path spaces of the original chains, so that for a.e. sample path of the product chain the logarithms of the measures of the correspo... | 2 | https://mathoverflow.net/users/8588 | 349839 | 148,054 |
https://mathoverflow.net/questions/349834 | 7 | Let $A$ be a real square and invertible matrix. I would like to find
$$
s(A) = \min\_U \rho(U A),
$$
Where $U$ is orthogoal, i.e. $U U^T = I$ and $\rho(A)$ is the spectral radius, i.e. the largest eigenvalue of $A$ in absolute values.
I am interested in a numerical solution to determine $s(A)$.
| https://mathoverflow.net/users/51478 | Minimize spectral radius with orthogonal matrix | For an $n\times n$ matrix, the answer is
$$|\det A|^{\frac1n}.$$
Explanation: on the one hand, $\rho(UA)\ge|\det (UA)|^{\frac1n}=|\det A|^{\frac1n}$. On the other hand, singular value decomposition gives $A=PDQ$ where $P,Q$ are orthogonal and $D>0$ is diagonal. Then
$\rho(UA)=\rho(QUPD)$ and
$$\min\_U\rho(UA)=\min\_V\r... | 9 | https://mathoverflow.net/users/8799 | 349842 | 148,056 |
https://mathoverflow.net/questions/349671 | 7 | **Edit:** According to comment of Prof. GoodWillie we revise the question.
Put $M=GL(n,\mathbb{R})$.
We identify $M\_n(\mathbb{R})$ with $\mathbb{R}^{n^2}$:
The identification is based on the lexicographic order on the index $i,j$ in $(a\_{ij})$. For example $$ \begin{pmatrix} a\_{11}&a\_{12}\\ a\_{21}&a\_{22}\end{... | https://mathoverflow.net/users/36688 | Homotopicity of two certain sections of frame bundle of $GL(n,\mathbb{R})$ | They are homotopic when $n=2$, but not when $n>2$. Here is the argument:
Let $A=(a\_{ij})$. Then, the definitions of the two framings can be made more explicit as follows: For the first frame field, each vector field, say, $X\_{ij}$, can be thought of as an $n$-by-$n$ matrix, and the formula for the $kl$ entry of $X\... | 4 | https://mathoverflow.net/users/13972 | 349858 | 148,063 |
https://mathoverflow.net/questions/349864 | 1 | Let $A = \bigoplus\_{i \in \mathbb{Z}} A\_i$ be a strongly graded unital algebra over $\mathbb{C}$, with no zero divisors. Is it always true that
$$
m: A\_i \otimes\_{A\_0} A\_j \to A\_{i+j}
$$
is an isomorphism?
| https://mathoverflow.net/users/143172 | Strongly graded algebras with no zero divisors | Yes this is always an isomorphism of $A\_0$-bimodules.
It is a general result for strongly graded rings. It holds for an arbitrary grading group $G$ (not necessarily $\mathbb{Z}$) and does not depend on the presence of zero divisors. For a proof, see Corollary 3.1.2, p.82, from [Methods of Graded Rings](https://www.... | 4 | https://mathoverflow.net/users/85967 | 349867 | 148,069 |
https://mathoverflow.net/questions/349601 | 5 | Let $L/K$ be a cubic (or, more generally, odd-order) extension of fields of characteristic $0$. To every element $a \in L^\times$ we can associate the quadratic form
\begin{align\*}
q\_a : L &\to K \\
x &\to \operatorname{tr}\_{L/K}(ax^2)
\end{align\*}
and then take its class in the $2$-Brauer group $\operatorname{Br}(... | https://mathoverflow.net/users/105625 | Relation in Brauer group coming from trace form | $\newcommand{\tr}{\operatorname{tr}}\newcommand{\Ell}{\operatorname{Ell}}$That was quicker to solve than I expected.
Consider the trace map
$$
\tr : GW(L) \to GW(K)
$$
between the Grothendieck-Witt rings of $L$ and $K$, where if $q : V \to L$ is a quadratic form, $\tr q : V \to K$ is given by postcomposition with $\tr\... | 1 | https://mathoverflow.net/users/105625 | 349868 | 148,070 |
https://mathoverflow.net/questions/349863 | 1 | Consider a non-regular bipartite graph $G$ . We consider list edge coloring the edges of the graph by giving lists of cardinality $max(deg(v\_i),deg(v\_j))+2$ for each edge $e=v\_iv\_j$ where $deg(v\_i)$ denotes the degree of the vertex $v\_i$.
Then, is the graph $G$ properly colorable. If so, would the maximum cardi... | https://mathoverflow.net/users/100231 | A different version of list coloring | This problem can indeed be solved using online list coloring. The stronger result that you can use lists of size $\max(\deg(v\_i),\deg(v\_j))$ is proven as Theorem 3.3 in this [paper](https://www.combinatorics.org/ojs/index.php/eljc/article/view/v16i1r77/pdf) of Uwe Schauz.
| 2 | https://mathoverflow.net/users/61883 | 349870 | 148,071 |
https://mathoverflow.net/questions/349869 | 3 | Let $\lambda\_k$ and $\varphi\_k$ be the $k$-th eigenvalue and a corresponding eigenfunction of the fractional Laplacian in a *bounded* domain $\Omega \subset \mathbb{R}^N$, $N \geq 2$.
That is, $\lambda\_k$ and $\varphi\_k$ satisfy
$$
\left\{
\begin{aligned}
(-\Delta)^{\alpha/2} \varphi\_k &= \lambda\_k \varphi\_k &&... | https://mathoverflow.net/users/48967 | Courant nodal domain theorem for fractional Laplacian | As far as I can tell, although it is widely believed that the Courant–Hibert estimate holds for the fractional Laplacian, even finiteness of the number of nodal domains of the second eigenfunctions is open for a general domain. The only known result is that the "harmonic extension to the upper half-space" (in the sense... | 3 | https://mathoverflow.net/users/108637 | 349872 | 148,072 |
https://mathoverflow.net/questions/349673 | 3 | I don't know if I have misunderstood the information from the section about the heuristic related to Cramér's conjecture and the known facts about Maier's theorem, from the Wikipedia [*Cramér's conjecture*](https://en.wikipedia.org/wiki/Cram%C3%A9r%27s_conjecture#Heuristic_justification). I would like to know if there ... | https://mathoverflow.net/users/142929 | Is there a clear criterion or rule about when one can use the heuristic given by Cramér random model for prime numbers? | Terry Tao's blog has the following [post](https://terrytao.wordpress.com/2015/01/04/254a-supplement-4-probabilistic-models-and-heuristics-for-the-primes-optional/) which discusses *Probabilistic models and heuristics for the primes*. In particular, it gives examples of the use of Cramer's model, while covering much mor... | 4 | https://mathoverflow.net/users/17773 | 349881 | 148,073 |
https://mathoverflow.net/questions/349877 | 11 | Let $E$ and $F$ be two complex vector bundles over a space $X$. There's a fairly well-known binary operation called the direct sum, written $E\oplus F$, which has the property that its first Chern class is the sum of the Chern classes of the constituents: $c\_{1}(E\oplus F)=c\_{1}E+c\_{1}F$.
My question is: For $k\ge... | https://mathoverflow.net/users/148857 | A binary operation on vector bundles that adds Chern classes? | Let's work with virtual bundles. Your question is equivalent to the following:
If we fix a $k \geq 1$, does the map $BU \times BU \rightarrow K(\mathbb{Z},2k)$ representing $c\_k \otimes 1 + 1 \otimes c\_k$ factor through some map $BU \times BU \rightarrow BU$ composed with the map $BU \rightarrow K(\mathbb{Z},2k)$ r... | 6 | https://mathoverflow.net/users/134512 | 349887 | 148,075 |
https://mathoverflow.net/questions/349855 | 9 | Following the standard definitions of a algebraic space or Deligne–Mumford stack one imposed condition is that the diagonal morphism $\Delta: \mathcal{X} \to \mathcal{X} \times \mathcal{X}$
has to be representable. The latter means that if $X,Y$ are schemes, $h\_X,h\_Y$ their Yoneda representations and we have natural ... | https://mathoverflow.net/users/108274 | Representable diagonal map $\Delta: \mathcal{X} \to \mathcal{X} \times \mathcal{X}$ for DM-Stacks/algebraic spaces | The basic issue is the following
>
> **Theorem**. If the diagonal is representable (by schemes, algebraic spaces, etc.), then any morphism $S\rightarrow \mathcal{X}$ with $S$ a scheme, algebraic space, etc. is representable (by schemes, algebraic spaces, etc.).
>
>
>
For a proof, see
[MathOverflow "Diagonal... | 8 | https://mathoverflow.net/users/6348 | 349891 | 148,076 |
https://mathoverflow.net/questions/349876 | 3 | Let $X$ be a smooth projective variety. By Definition 24.4.2 in the 2003 book [Mirror Symmetry](https://www.claymath.org/library/monographs/cmim01c.pdf), $X$ is called convex if $h^1\left( \Sigma, f^\*T\_X \right) = 0$ for every genus zero stable map $f:\Sigma \to X$.
**Question:** Why is a convex variety called conv... | https://mathoverflow.net/users/146366 | Why is a convex variety called convex? | If there are no obstructions to deformation, i.e. $h^1=0$, then Kodaira's deformation theorem says that the map admits a family of deformations, forming a manifold, near that given map, with tangent space given by $H^0$. You think of genus zero curves as like lines, or line segments, and deforming a line while staying ... | 3 | https://mathoverflow.net/users/13268 | 349900 | 148,079 |
https://mathoverflow.net/questions/349899 | 0 | Let $\omega$ denote the set of non-negative integers and let $[\omega]^\omega$ be the collection of infinite subsets of $\omega$.
If $S\in [\omega]^\omega$ and $A\subseteq \omega$ we say that $A$ is *well-balanced with respect to* $S$ if $$\lim\_{n\to\infty}\frac{|A\cap S\cap \{1,\ldots,n\}|}{|S\cap\{1,\ldots,n\}|+1}... | https://mathoverflow.net/users/8628 | A balancing property of infinite subsets of $\mathbb{N}$ | By the strong law of large numbers, if $S$ is an infinite subset of $\omega$, a random subset of $\omega$ will be well-balanced with respect to $S$ with probability one.
By the countable additivity of Lebesgue measure, if $\mathfrak S$ is a countable collection of infinite subsets of $\omega$, a random subset of $\om... | 6 | https://mathoverflow.net/users/43266 | 349902 | 148,081 |
https://mathoverflow.net/questions/349880 | 12 | Let $A$ be an abelian group and let $n \geq 2$. For any connected CW complex $X$, it is standard that a fibration $f\colon E \rightarrow X$ whose fibers are homotopy equivalent to a $K(A,n)$ is fiberwise homotopy equivalent to the pull-back of the loop-space fibration over a $K(A,n+1)$ if and only if $\pi\_1(X)$ acts t... | https://mathoverflow.net/users/149707 | Classifying space for fibrations with Eilenberg-MacLane space fibers and nontrivial fundamental group actions | Mark Grant's excellent answer already resolves the question. However, let me sketch how this arises as a special case of the more general problem of classifying fibrations with a given fiber.
For any space $X$, the *homotopy automorphisms* $\operatorname{hAut}(X)$ are defined as the self-homotopy equivalences of $X$ ... | 18 | https://mathoverflow.net/users/35687 | 349904 | 148,082 |
https://mathoverflow.net/questions/349906 | 0 | **Setup**
---
Let $X\_t$ be a $d$-dimensional diffusion process solving the Ito-stochastic differential equation
$$
X\_t = x+ \int\_0^t f(X\_t,u\_t)dt + \int\_0^t \sigma dW\_t,
$$
where $x \in \mathbb{R}^d$, $u\_t$ is predictable, $f(\cdot,\cdot)$ is locally Lipschitz, $\sigma \sigma^T$ is positive-definite, and ... | https://mathoverflow.net/users/36886 | Convergence rate estimates of Monte-Carlo first-passage time estimates | Assuming $D$ is bounded, the best estimate that holds almost surely is given by the law of the iterated logarithm <https://en.m.wikipedia.org/wiki/Law_of_the_iterated_logarithm>
| 1 | https://mathoverflow.net/users/7691 | 349912 | 148,086 |
https://mathoverflow.net/questions/349664 | 3 | My question is about p-adic Hodge-Tate theory and p-adic Galois representation.
One of the important semi-linear object in p-adic Galois representation is the $\text{Breuil Module}$. There are examples of Breuil modules.
My question-
Is it easy to give or find examples of Breuil Modules?
Would it be a little bu... | https://mathoverflow.net/users/122445 | Would it be a little but good exercise to construct or find out Breuil modules? | It might be worth distinguishing here between two different but related constructions:
* *Breuil--Kisin modules*, which are finite free modules over a relatively simple base ring, namely $\mathfrak{S} = W[[u]]$ where $W$ is the Witt vectors of the residue field;
* *Breuil modules*, which are finite free modules over ... | 3 | https://mathoverflow.net/users/2481 | 349915 | 148,087 |
https://mathoverflow.net/questions/349914 | 6 | $B\_{\mathrm{cris}}\subseteq B\_{\mathrm{dR}}$ and $B\_{\mathrm{dR}}^+$ are well-known period rings in $p$-adic Hodge. I know $B\_{\mathrm{dR}}=B\_{\mathrm{dR}}^+[\frac{1}{t}]$ and $\frac{1}{t}\in B\_{\mathrm{cris}}$ where $t$ is Fontaine's $2\pi i$.
I want to ask if the natural map $B\_{\mathrm{cris}}\rightarrow \fr... | https://mathoverflow.net/users/nan | $B_{\mathrm{dR}}=B_{\mathrm{cris}}+{B_{\mathrm{dR}}^+}$? | Much more is true: the subring $B\_{\mathrm{cris}}^{\varphi = 1}$ surjects onto $B\_{\mathrm{dR}} / B\_{\mathrm{dR}}^+$, so there is an exact sequence
$$ 0 \to \mathbf{Q}\_p \to B\_{\mathrm{cris}}^{\varphi = 1} \to B\_{\mathrm{dR}} / B\_{\mathrm{dR}}^+ \to 0.$$
This is the **Bloch--Kato fundamental exact sequence** whi... | 9 | https://mathoverflow.net/users/2481 | 349916 | 148,088 |
https://mathoverflow.net/questions/349893 | 6 | I have a Borel probability measure $\pi$ on $\mathbb{R}^{n+1}$ such that $\pi\_1=\mu\_1, \ldots, \pi\_{n+1}=\mu\_{n+1}$ for some fixed Borel probability measures $\mu\_1, \ldots, \mu\_{n+1}$ (where each $\mu\_i$ is absolutely continuous with respect to Lebesgue measure). I want to construct a sequence of probability me... | https://mathoverflow.net/users/69849 | Absolutely continuous coupling of probability measures | Let me formulate and prove it in greater generality (which actually makes your question easier). Let $X$ be a metric space, and $\mu$ be a probability measure on $X\times X$ (for simplicity I consider the product of two copies of $X$ only; the general case is precisely the same). You want to obtain a sequence of measur... | 5 | https://mathoverflow.net/users/8588 | 349927 | 148,091 |
https://mathoverflow.net/questions/349818 | 6 | Let $R$ be a commutative ring with identity. Then $R$ is $\textit{catenary}$ if for each pair of prime ideal $p \subsetneq q$, all maximal chains of prime ideals $p = p\_0 \subsetneq p\_1 \subsetneq \dots \subsetneq p\_n = q$ have the same length.
In some (informal) texts the author conclude that (without further ex... | https://mathoverflow.net/users/47763 | local UFD with dimension less than or equal 3 is catenary | This result appears as Proposition II.3 in
>
> Hamet Seydi, *Anneaux henséliens et conditions de chaînes. III*, Bull. Soc. Math. France **98** (1970), 329–336. Numdam: [BSMF\_1970\_\_98\_\_329\_0](http://www.numdam.org/item/BSMF_1970__98__329_0/). DOI: [10.24033/bsmf.1706](https://doi.org/10.24033/bsmf.1706). MR: [... | 8 | https://mathoverflow.net/users/33088 | 349937 | 148,096 |
https://mathoverflow.net/questions/349933 | 4 | In [The stable moduli space of Riemannsurfaces: Mumford’s conjecture](https://annals.math.princeton.edu/wp-content/uploads/annals-v165-n3-p04.pdf), Madsen and Weiss introduce the *representing space* $|\mathcal{F}|$ of a sheaf of sets $\mathcal{F}$ on the site $\mathscr{X}$ of smooth manifolds as the geometric realizat... | https://mathoverflow.net/users/8320 | Representing spaces of $\infty$-stacks | As requested, the comment as an answer:
The case of sheaves valued in ∞-groupoids is the content of a recent preprint of (Berwick-Evans)-[Boavida de Brito]-Pavlov: [arxiv.org/pdf/1912.10544.pdf](http://arxiv.org/pdf/1912.10544.pdf).
| 6 | https://mathoverflow.net/users/6936 | 349938 | 148,097 |
https://mathoverflow.net/questions/128446 | 5 | A left-exact localization of a category is a reflective subcategory such that the reflector preserves finite limits. There are several prominent examples of such localizations, such as sheafification, and localization of module categories. Is there a general theory of such localizations?
I don't have any particular ... | https://mathoverflow.net/users/3711 | General theory of left-exact localization? | Just to tie this one up, the cited result in Borceux (Prop 5.6.1) says the following:
**Proposition:** Let $\mathcal C$ be a finitely-complete category and let $L \mathcal C$ be a reflective subcategory. Let $\mathcal W$ be the class of morphisms inverted by the reflector $r: \mathcal C \to L\mathcal C$. Then $r$ is ... | 6 | https://mathoverflow.net/users/2362 | 349946 | 148,100 |
https://mathoverflow.net/questions/349949 | 7 | In ordinary category theory it is a well-known and important fact that the $\hom$ bifunctor into $\text{Set}$ preserves limits.
I am unable to find a reference for the corresponding fact in infinity category theory nor am I able to write down a proof myself.
I am looking for a statement of the following form:
Le... | https://mathoverflow.net/users/111650 | Hom spaces in (∞, 1)-categories | This is [Cisinski](http://www.mathematik.uni-regensburg.de/cisinski/CatLR.pdf), Corollary 6.3.5. The proof is essentially to show that cocontinuous functors out of presheaf $\infty$-categories admit right adjoints given by the usual formula, so that $Hom\_C$ has a left adjoint if $C$ is cocomplete, and then to use the ... | 10 | https://mathoverflow.net/users/43000 | 349953 | 148,104 |
https://mathoverflow.net/questions/349943 | 2 | I'm a computer science student working on a research project that deals with computational study of atomic clusters. I'm using a graph based representation of the clusters using a binary connectivity matrix. My question is, are the eigenvalues invariant to a transformation in particle indices?
For example, if I have... | https://mathoverflow.net/users/150822 | Question about eigenvalues of connectivity matrices for graphs | If $P$ is a *[permutation matrix](https://en.m.wikipedia.org/wiki/Permutation_matrix)*, i.e., $P$ is binary and doubly stochastic, then $P^\top = P^{-1}$. Thus, any matrix $A$ is co-spectral to $P^\top A P$. Furthermore, if the matrix $A$ is the incidence matrix of a graph, then a straightforward exercise shows that th... | 0 | https://mathoverflow.net/users/104633 | 349956 | 148,106 |
https://mathoverflow.net/questions/349939 | 6 | Take two positive integers $m$ and $n$ and consider the rational function
$$G\_{m,n}(x,t)=\frac{d}{dx}\left(\frac1{(1-x^m)(1-tx^n)}\right)$$
and the corresponding Taylor expansion as
$$G\_{m,n}(x,t)=u\_0(t)+u\_1(t)x+u\_2(t)x^2+\cdots$$
where $u\_j(t)$ itself is expanded as a polynomial in $t$ (note: coefficients in eac... | https://mathoverflow.net/users/66131 | Limits (growth rates) of power series coefficients | Yes, the conjectured limit is true. Let $d=\gcd (m,n)$ and $m=m\_1d, n=n\_1d$. Suppose $a\_k$ denotes the number of solutions to $k=m\_1r+n\_1s$ with $r,s\geq 0$, so that
$$a\_0+a\_1x+a\_2x^2+\cdots =\frac{1}{(1-x^{m\_1})(1-x^{n\_1})}.$$
We have $\beta\_{\ell}(m,n)=Nd$ if and only if $\sum\_{i=0}^{N-1}a\_i< \ell\le \su... | 7 | https://mathoverflow.net/users/2384 | 349958 | 148,107 |
https://mathoverflow.net/questions/349382 | 7 |
>
> Let $ABCD$ be a regular tetrahedron with center $O.$ Consider two points $M,N,$ such that $\overrightarrow{NO}=-3\overrightarrow{MO}.$ Prove or disprove that
> $$NA+NB+NC+ND\geq MA+MB+MC+MD$$
>
>
>
I tried to use CS in the Euclidean space $E\_3$, but it does not help, because the minoration is too wide.
... | https://mathoverflow.net/users/150418 | For regular tetrahedron $ABCD$ with center $O$, and $\overrightarrow{NO}=-3\overrightarrow{MO}$, is $NA+NB+NC+ND\geq MA+MB+MC+MD$? | Following suggestions on Stack Exchange, we use the homothety with respect to $O$
and factor $-3$.
If $X$ is a point, then $X'$ denotes the point for which $\overrightarrow{XO}= -3 \overrightarrow{X'O}$.
So $M=N'$ and $A'$ is the midpoint of the face opposite to $A$.
One has $XY=3X'Y'$ and the desired inequality becom... | 8 | https://mathoverflow.net/users/4794 | 349974 | 148,114 |
https://mathoverflow.net/questions/349981 | 2 | We consider the equation (NLS)
\begin{eqnarray}\label{gnls}
i \epsilon\partial\_t u^{\epsilon} + \frac{\epsilon^2}{2}\Delta\_{\eta}u^{\epsilon} = \epsilon |u^{\epsilon}|^{2}u^{\epsilon}, \quad x \in \mathbb R^d
\end{eqnarray}
$d\geq 1$ The initial data is supposed to be given by a
superposition of highly oscillatory p... | https://mathoverflow.net/users/149364 | WKB expansion for NLS | **A.** Note that the time derivative $\partial u^\epsilon/\partial t$ and the spatial derivative $\partial u^\epsilon/\partial x$ are both of order $1/\epsilon$, since $u^\epsilon\propto e^{i\phi(t,x)/\epsilon}$. Since in the NLS the first-order time derivative has a prefactor $\epsilon$ and the second-order spatial de... | 4 | https://mathoverflow.net/users/11260 | 349984 | 148,119 |
https://mathoverflow.net/questions/349990 | -6 | During a business meeting, I was trying to find a continuous function $f:\mathbb{R}\to\mathbb{R}$ such that $|f^{-1}(\{y\})| = 2$ for all $y\in \mathbb{R}$, and after some experimentation I found $$f:\mathbb{R}\to\mathbb{R}, \, x\mapsto \log(x^2).$$
For which integers $n>2$ is it possible to find a continuous function ... | https://mathoverflow.net/users/8628 | Continuous function $f:\mathbb{R}\to\mathbb{R}$ with fixed size finite fibers | There is no continuous function $f:\,\mathbb{R}\to \mathbb{R}$ such that $|f^{-1}(x)|=2k$ for any $x$ (here $k$ is some fixed positive integer.)
Indeed, let $p\_1<p\_2<\ldots<p\_{2k}$ be preimages of $0$. Fix small $\delta$ so that the $4k$ intervals $(p\_i-\delta,p\_i)$ and $(p\_i,p\_i+\delta)$ are mutually disjoin... | 6 | https://mathoverflow.net/users/4312 | 349995 | 148,124 |
https://mathoverflow.net/questions/349969 | 2 | I am trying to prove the following fact.
Let $V$ be a unitary irreducible representation of $SO(n)$. How to prove that, if we reduce $V$ as unitary irreducible representation with respect to SO(n-1) then each irreducible representation of $SO(n-1)$ occurs at most once in $V$.
Kindly share your thoughts.
Thank you... | https://mathoverflow.net/users/33047 | Multiplicity of an irrep of SO(n-1) in SO(n) | The statement is true and well known. See e.g. Thms. 8.1.3 and 8.1.4 in Goodman-Wallach: Symmetry, representations and invariants, Springer GTM 255.
In fact, much more is known. Let, more generally, $H\subseteq G$ be a subgroup (everything is compact). Then it follows easily from Peter-Weyl that $G$-to-$H$ branching ... | 13 | https://mathoverflow.net/users/89948 | 350004 | 148,125 |
https://mathoverflow.net/questions/350005 | 1 | If consistent, is existence of a proper class of extendible cardinals provably equivalent to a $Σ^V\_5$ statement?
Recall that in ZFC, a cardinal $κ$ is extendible iff for every $λ>κ$ there is an elementary embedding $j$ of $V\_λ$ into some $V\_μ$ with $\mathrm{crit}(j)=κ$. (In ZF, one also requires $j(k)>λ$, but thi... | https://mathoverflow.net/users/113213 | Complexity of a proper class of extendibles | No, it is not; existence of a proper class of extendible cardinals is not provable in ZFC from any consistent $Σ^V\_5$ statement.
Assume a proper class of extendibles and let $φ$ be a true $Σ\_5$ statement (or even the conjunction of all true $Σ\_5$ statements), and let a set $S$ witness $φ$, and let $κ$ be the least... | 2 | https://mathoverflow.net/users/113213 | 350006 | 148,126 |
https://mathoverflow.net/questions/350025 | 2 | Let $A = \oplus\_{i \in \mathbb{Z}} A\_i$ be a graded algebra. We say that it is **strongly** graded if $A\_i.A\_j = A\_{i+j}$, for all $i,j \in \mathbb{Z}$. Can there be existing a graded algebra such that
$$
A\_i.A\_{-i} = A\_0, ~~ \forall i \in \mathbb{Z},
$$
but which is yet not strongly graded?
| https://mathoverflow.net/users/143172 | A weaker version of strongly graded algebras | If $A\_i \cdot A\_{-i} = A\_0$ then in particular we can write
$$
1 = \sum\_k a\_i^{(k)} \cdot a\_{-i}^{(k)},
$$
where $a\_l^{(k)} \in A\_l$. Now taking any $a\_{i+j} \in A\_{i+j}$ and multiplying it by this equality, we obtain
$$
a\_{i+j} = \sum\_k a\_i^{(k)} \cdot (a\_{-i}^{(k)} \cdot a\_{i+j}) \in A\_i \cdot A\_j.
$... | 2 | https://mathoverflow.net/users/4428 | 350027 | 148,130 |
https://mathoverflow.net/questions/349998 | 3 | Using Laurent Series of a function $f(z)$ around a point $a \in \mathbb{C}$
$$f(z) = \sum^{\infty}\_{n=-\infty} c\_n(z-a)^n \ \ \ \ (1)$$
where
$$c\_n = \frac{1} {2\pi i}\int\limits\_{\gamma}\frac {f(z)} {(z-a)^{n+1}} dz \ \ \ \ (2)$$
where $\gamma$ is a closed curve around $a$.
And choosing $\gamma$ such that $z$ ca... | https://mathoverflow.net/users/150850 | Fourier transform derivation from Laurent series | *Continuous time Fourier transform and Laurent series:*
I recall equations (1) and (2), for convenience set $a=0$, and substitute $z=e^{it/T}$. The function $g(t)=f(e^{it/T})$, with $t\in(-\pi T,\pi T)$, is periodic with period $2\pi T$, given by the Laurent series
$$g(t)=\sum\_{n=-\infty}^\infty c\_n e^{int/T},$$
wi... | 2 | https://mathoverflow.net/users/11260 | 350028 | 148,131 |
https://mathoverflow.net/questions/349897 | 6 | Let $G$ be an affine group scheme of finite type over an algebraically closed field $k$. Suppose that $V$ is a finite dimensional representation of $G$.
For every $k$-algebra $A$ we have the base change representation $V\_A = V\otimes\_kA$ of the group $G\_A = \mathrm{Spec}\, A\times\_{\mathrm{Spec}\,k}G$.
**Defin... | https://mathoverflow.net/users/147687 | Absolutely irreducible representations of affine group schemes of finite type over a field | Suppose $G$ is an affine group scheme over an algebraically closed field $k$, and $V$ is a finite dimensional representation of $G$. Let $K$ be an extension of $k$, and assume that $V\_K$ is reducible as a representation of $G\_K$. Then I claim that $V$ is reducible as a representation of $G$.
Let $r$ be an integer w... | 3 | https://mathoverflow.net/users/4790 | 350031 | 148,132 |
https://mathoverflow.net/questions/349996 | 5 | For any algebra $A$, a **character** for $A$ is a non-zero algebra map $c:A \to \mathbb{C}$. For $H$ be a Hopf algebra, a character is given by $\epsilon:H \to \mathbb{C}$ the counit of $H$. I am looking for other examples of (co-semi-simple) Hopf algebras with characters distinct of the counit.
I am really intereste... | https://mathoverflow.net/users/143172 | Characters on Hopf algebras | I think that a general example is the so-called **Larson's character**, which in a sense ties together the trace and determinant functions.
To make the long story short: Let $C$ be a cocommutative bialgebra, $V$ a vector space and $EV$, the [exterior algebra](http://mathworld.wolfram.com/ExteriorAlgebra.html). Then,... | 4 | https://mathoverflow.net/users/85967 | 350033 | 148,133 |
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.