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/323705 | 4 | This is an update to an [older question](https://mathoverflow.net/questions/323629/does-every-connected-t-2-space-x-have-a-proper-subspace-homeomorphic-to-x).
Is there a contractible $T\_2$-space $(X,\tau)$ on more than $1$ point such that no proper subspace of $X$ is homeomorphic to $X$?
| https://mathoverflow.net/users/8628 | Contractible $T_2$-space without subspace homeomorphic to it | An example of such contractible (compact metrizable) space can be constructed as follows.
Let $K$ be [the Cook continuum](http://matwbn.icm.edu.pl/ksiazki/fm/fm60/fm60123.pdf). It has the property that any continuous map $f:C\to K$ defined on a subcontinuum $C\subset K$ is either constant or the identity inclusion. T... | 2 | https://mathoverflow.net/users/61536 | 324958 | 140,007 |
https://mathoverflow.net/questions/324964 | 3 | It is well known that Kronheimer classified all hyperkähler ALE $4$-manifolds. In particular, any such manifold must be diffeomorphic to the minimal resolution of $\mathbb C^2/\Gamma$ for a finite group $\Gamma \subset SU(2)$.
Given a manifold $M$ which is diffeomorphic to the minimal resolution of $\mathbb C^2/\Gamm... | https://mathoverflow.net/users/105900 | Hyperkähler ALE $4$-manifolds | It is proved by Kronheimer ( <https://projecteuclid.org/euclid.jdg/1214443066> ) that the ALE metric is unique up to isometry if you fix the Kähler classes. (It is necessary, else you can give different volumes to the various exceptional curves and you get non-isometric metrics).
| 7 | https://mathoverflow.net/users/25309 | 324965 | 140,009 |
https://mathoverflow.net/questions/324960 | 2 | Given $y \sim N(0,\sigma^2 I)$, and $M$ that is a symmetric matrix (**not necessarily idempotent**)
* what is the distribution of ${y^T M y}$?
* is there a high probability bound on $|{y^T M y}|$?
Most bounds that I could find, such as in <http://www2.econ.iastate.edu/classes/econ500/hallam/documents/QUAD_NORM.pdf... | https://mathoverflow.net/users/136311 | Concentration inequality for quadratic form of Gaussian variables with non-idempotent matrix | By the spectral decomposition of $M$, the distribution of $y^T My$ is the same as that of $\sum\_i\sigma^2\mu\_i Z\_i^2$, where the $\mu\_i$'s are the eigenvalues of $M$ and the $Z\_i$'s are iid standard normal random variables (r.v.'s). By rescaling, without loss of generality $\sigma=1$.
In the paper by [Székely &... | 4 | https://mathoverflow.net/users/36721 | 324966 | 140,010 |
https://mathoverflow.net/questions/324913 | 1 |
>
> Let G is a linear algebraic group over algebraic closed field, B is an
> Borel subgroup of G. Does there exist g$\in$G which is only in a finite
> numbers of conjugates of B (they are also Borel subgroups) ?
>
>
>
I choose this version of condition from the book:Tauvel, Patrice, and W. T. Rupert. *Lie alge... | https://mathoverflow.net/users/40640 | Element in finite number of Borel subgroups | First of all, it's probably intended that "linear algebraic groups" are semisimple or at least reductive (and connected). For example, a solvable group might have no semisimple elements except 1 (etc.) and Venkatarama's comment won't apply. The basic object of study is a connected semisimple group, for which Steinberg'... | 2 | https://mathoverflow.net/users/4231 | 324979 | 140,013 |
https://mathoverflow.net/questions/324903 | 5 | In the first chapter of Lurie's HTT, Proposition 1.2.13.8 shows that the functors out of over (infinity)-categories reflect infinity-colimits.
For the life of me I cannot follow the proof.
Can anyone help?
| https://mathoverflow.net/users/121425 | Understanding the proof of Proposition 1.2.13.8 in Lurie's Higher Topos Theory | As a learning exercise, I will try to expand Dylan's comment into an answer. If things are still unclear, please ask in the comments. First, the $\star$ operator is defined in 1.2.8.1. The notation $K^{\triangleright}$ is defined to mean $K \star \Delta^0$ (see 1.2.8.4). The proposition you ask about features $q: T\to ... | 4 | https://mathoverflow.net/users/11540 | 324981 | 140,015 |
https://mathoverflow.net/questions/324973 | 11 | It's well known that a category $\mathcal{C}$ having (conical) limits/colimits of shape $\mathcal{D}$ is equivalent to the diagonal functor $\Delta^\mathcal{D}\_\mathcal{C}:\mathcal{C}\to\mathcal{C}^\mathcal{D}$ having a left/right adjoint respectively.
>
> Is there a similar characterization of weighted limits/co... | https://mathoverflow.net/users/92164 | Weighted (co)limits as adjunctions | Here's one way in which weighted limits are adjoints. Let $D$ be a $V$-category and $W:D\to V$ a weight, and $C$ a $V$-category. Suppose that $C$ has copowers, so for $k\in V$ and $x,y\in C$ we have $C(k\odot x,y) \cong V(k,C(x,y))$. Then there is a $V$-functor $\Delta^W:C \to C^D$ defined by $\Delta^W(c)(d) = W(d) \od... | 5 | https://mathoverflow.net/users/49 | 324986 | 140,016 |
https://mathoverflow.net/questions/284258 | 2 | Given stochastic payoff functions $X\_{1}(t) \dots X\_{K}(t)$, each having a different probability distribution on $[0,1]$, denote the expected value of $X\_i(t)$ by $\mu\_i$, and define $\mu^\* = \max\_{i \in {1\dots K}}\mu\_i$, and define the *pseudoregret* for a certain strategy $I\_t$(which associates an integer be... | https://mathoverflow.net/users/111800 | Where does the expected value in the restatement of the pseudoregret come from? | Since
$$E(X\_{I\_t}\mid I\_t)=\mu\_{I\_t},$$
by iterated expectation
$$E(X\_{I\_t})=E(E(X\_{I\_t}\mid I\_t)) = E(\mu\_{I\_t}),$$
from which we have the desired equality.
| 2 | https://mathoverflow.net/users/4600 | 324987 | 140,017 |
https://mathoverflow.net/questions/325001 | 3 | Let $G$ be a compact real Lie group. We say that $G$ has property $(\*)$ if every abstract automorphism of $G$ is continuous. A theorem of Cartan says that if $G$ has perfect Lie algebra, it has property $(\*)$. It the converse true?
I think that $G$ having property $(\*)$ is equivalent to its identity component hav... | https://mathoverflow.net/users/136781 | A converse of Cartan's automatic continuity theorem |
>
> Proposition : let $G$ be a compact Lie group (with unit component $G^\circ$) whose Lie algebra is not perfect; let $G\_\delta$ be the underlying discrete group. Then there is a semidirect decomposition $$G\_\delta=L\ltimes V,$$ where $[G^\circ,G^\circ]\subset L$, the subgroup $L$ is dense in $G$, and $V\subset Z(... | 2 | https://mathoverflow.net/users/14094 | 325008 | 140,022 |
https://mathoverflow.net/questions/325016 | 10 | Let $\Sigma$ be a two-dimensional connected smooth manifold without boundary. (Feel free to assume it is compact and orientable.)
Let $\mathcal{X}(\Sigma)$ denote the smooth vector fields on $\Sigma$ and let $\operatorname{Diff}(\Sigma)$ denote the group of diffeomorphisms of $\Sigma$. Clearly $\operatorname{Diff}(\S... | https://mathoverflow.net/users/394 | Smooth vector fields on a surface modulo diffeomorphisms | This is not finite dimensional. For example, consider non-vanishing vector fields on $T^2=\mathbb R^2/\mathbb Z^2$, transversal to vertical circles. Any such field defines a self diffeo $S^1\to S^1$ on a vertical circle, called the return map. Suppose that such a diffeo $\varphi$ has $n$ fixed points $x\_i$ (they corre... | 12 | https://mathoverflow.net/users/943 | 325017 | 140,024 |
https://mathoverflow.net/questions/324985 | 3 | Let $(a\_n)\_{n\in\mathbb N}$ and $(b\_n)\_{n\in\mathbb N}$ be sequences of $\mathbb Q\_p$ such that the function $f:z\in\mathbb Q\_p\to\sum\_{n\ge0}a\_nz^n+b\_nz^{n+1}$converges in $\{|z|\_p<1\}$. Assume that the series $\sum\_{n\ge0}a\_n+b\_n$ converges in $\mathbb Q\_p$. Can the function $f$ be continued in a larger... | https://mathoverflow.net/users/33128 | Analytic continuation of a $p$-adic function | I think the answer is "not necessarily", by the following (counter)example. First, let $b\_n=-a\_n$ for $n\ge0$. Then $\sum(a\_n+b\_n)=0$ and
$$f(z)=\sum\_{n\ge0} a\_n(1-z)z^n=\sum\_{n\ge0} (a\_n-a\_{n-1})z^n,$$
where we set $a\_{-1}=0$. Now, define
$$a\_n:=\frac{-1}{n+1}.$$
Then, $\sum a\_nz^n$ converges if and only ... | 2 | https://mathoverflow.net/users/109085 | 325020 | 140,026 |
https://mathoverflow.net/questions/325019 | 3 | I recall reading once that the sum $$\sum\_{p \,\, \small{\mbox{is a known prime}}} \frac{1}{p}$$
is less than $4$.
Does anybody here know what the **ultimate** source of this claim is?
Please, let me thank you in advance for your insightful replies...
| https://mathoverflow.net/users/1593 | Yet another question on sums of the reciprocals of the primes | It's well known that the sum of the reciprocals of the primes below $n$ tends to $\log \log n + M$, where $M$ is a small constant (the Meissel-Mertens constant). That is to say:
$$ \sum\limits\_{\small{\mbox{prime}} \, p \, < \, n} \frac{1}{p} = \log \log n + M + o(1) $$
This allows us to determine an approximate l... | 6 | https://mathoverflow.net/users/39521 | 325023 | 140,027 |
https://mathoverflow.net/questions/325026 | 3 | In Huybrechts' book *Fourier-Mukai Transforms in Algebraic Geometry* there is an exercise (Exercise 1.7 and 1.8) to prove the statement that the derived category $D^b(X)$ of a Calabi-Yau variety $X$ has no non-trivial semi-orthogonal decompositions. Yet also, it is later stated that it is known that every toric variety... | https://mathoverflow.net/users/136792 | Semi-orthogonal decompositions for Calabi-Yau varieties | For both statements properness is essential. The proof of the exercise (a theorem of Bridgeland) is based on Serre duality, and the construction of an exceptional collection (by Kawamata) is inductive with the base of induction given by the case of a weighted projective space. And in projective case no toric variety is... | 6 | https://mathoverflow.net/users/4428 | 325028 | 140,029 |
https://mathoverflow.net/questions/324917 | 1 | Consider
$$ Au:=\operatorname{div}\left(y^{\beta}\nabla u\right) \text{ for } (x,y)\in \mathbb{H} $$
and $u|\_{\mathbb{R}}(x,0)=\phi(x)$ and some $\beta\in (0,1)$. For $\phi\in L^{2}(\mathbb{R},dx)$ (in fact less is required) we have a unique solution with
$$\int\_{0}^{\infty}\|\nabla\_{(x,y)}u\|^{2}y dy<\infty$$
(see... | https://mathoverflow.net/users/99863 | Divergence form degenerate pde and Feynman Kac | This is just an extended comment, not an actual answer. In fact, I am not sure if I understand your question. Apparently what you are attempting to do is quite standard and rather general.
1. Given any Hunt process $X(t)$, any non-negative or bounded Borel function $\phi$ and any open (or even Borel) set $D$, the fun... | 3 | https://mathoverflow.net/users/108637 | 325029 | 140,030 |
https://mathoverflow.net/questions/325024 | 3 | Let $(M,g)$ be a compact Riemannian manifold with an isometric action $\rho : G \to \mathrm{Iso}(M)$ by a compact Lie group $G$. There is a natural extension of $\rho$ to $TM$ given by:
$$\psi : G \times TM \to TM$$
$$\psi(g,p,X) := (\rho(g)p,d\rho(g)X).$$
I would like to know if the set:
$$TM \supset \tilde S := \{(... | https://mathoverflow.net/users/94097 | Is this a submanifold? | Your definition implies that
$$ \tilde S = \bigcup\_{p\in M} T\_pM^{G\_p}. $$
In particular, $\pi(\tilde S) = M$, and $\tilde S$ will be a submanifold of $TM$ iff the dimension of $T\_pM^{G\_p}$ is the same for all $p\in M$.
$G$ being connected won't necessarily make this happen: Another counterexample is $S^1$ actin... | 4 | https://mathoverflow.net/users/58888 | 325038 | 140,035 |
https://mathoverflow.net/questions/325032 | 5 | Can there exist three transitive models of ZFC with the same ordinals, $M\_0,M\_1,N$, such that there are elementary embeddings $j\_i : M\_i \to N$ for $i<2$, but there is no elementary embedding from $M\_0$ to $M\_1$ or vice versa?
| https://mathoverflow.net/users/11145 | Amalgamation via elementary embeddings | Suppose there are exactly two measurable cardinals $\kappa<\lambda$. Let $i:V\to M\_0$ and $j:V\to M\_1$ be the elementary embeddings given by normal ultrafilters on $\kappa$ and $\lambda$, respectively. In $M\_0$, the two measurable cardinals are $i(\kappa)$, which lies strictly between $\kappa$ and $\lambda$, and $i(... | 7 | https://mathoverflow.net/users/6794 | 325039 | 140,036 |
https://mathoverflow.net/questions/325057 | 3 | This question extends my [earlier MO post](https://mathoverflow.net/questions/324816/oddity-of-generalized-catalan-numbers) for which I'm grateful for answers and useful comments.
The [Catalan numbers](https://en.wikipedia.org/wiki/Catalan_number) $C\_n=\frac1{n+1}\binom{2n}n$ satisfy:
$\text{$C\_{1,n}$ is odd iff $... | https://mathoverflow.net/users/66131 | "Oddity" of $q$-Catalan polynomials: Part II | For a positive integer $m$, an $m$-th primitive root $\alpha\_m$
of $-1$ [which is the root of the polynomial $1+q^m$, and when $m$ is a power of two the converse also holds: any root of $q^m+1$ is an $m$-th primitive root
of $-1$] is a root of $[k]\_q$ exactly when $2m$ divides $k$. So, $\alpha\_m$ is a root of $[N!]\... | 4 | https://mathoverflow.net/users/4312 | 325069 | 140,042 |
https://mathoverflow.net/questions/325037 | 10 | The first part of the question was [asked](https://math.stackexchange.com/questions/3138407/axiom-of-choice-and-dual-of-a-tensor-product) on Math-stackexchange.
Let $V$, and $W$ be vector spaces. By the universal property of the tensor product,
there is a canonical map from $V^\*\otimes W^\*$ into $(V\otimes W)^\*$ ... | https://mathoverflow.net/users/100552 | Axiom of choice and algebraic tensor product | I think both can be proved without choice, essentially because, in both cases, whenever you're tempted to choose a basis, you can manage with a little care to get by with a basis of a finite dimensional subspace.
For (2), if there's a linear dependence between the $v\_i\otimes w\_j$ then it involves only finitely man... | 13 | https://mathoverflow.net/users/22989 | 325070 | 140,043 |
https://mathoverflow.net/questions/325071 | 1 | Let's say $P\subset\Bbb R^d$ is some *convex* polytope with the following two properties:
1. all vertices are on a common sphere.
2. all edges are of the same length.
I suspect that such a polytope is already rigid, i.e. there is (up to scaling and rotation) only a single way to realize it geometrically. Is this tr... | https://mathoverflow.net/users/108884 | Is a polytope with vertices on a sphere and all edges of same length already rigid? | I guess you also fix the combinatorial structure. Then yes. Induct in dimension with obvious base $d\leqslant 2$. By induction proposition the facets are fixed. By Cauchy - - Alexandrov rigidity theorem the whole polytope also is fixed.
| 2 | https://mathoverflow.net/users/4312 | 325073 | 140,044 |
https://mathoverflow.net/questions/325074 | 5 |
>
> **Question:**
>
>
> Are there standard techniques available for solving the following kind of linear matrix recurrence relations:
>
>
> $$M\_1,\cdots,M\_k\ \in\ \mathbb{R}^{m\times n}$$
> $$ A\_1,\cdots,A\_k\ \in\ \mathbb{R}^{n\times n}$$
> $$M\_{i+k+1}\ =\ \sum\_{j=1}^{k}{M\_{i+j}A\_j} $$
>
>
>
I need t... | https://mathoverflow.net/users/31310 | Solving Linear Matrix Recurrences | Same method as for scalar equations works. Put matrices $M\_{n+k-1},M\_{n+k-2},...,M\_{n}$
vertically into a big matrix $X\_n$ of size $mk\times n$. Then your recurrence becomes
a one-step recurrence $X\_{n+1}=AX\_{n}$, with some $A$, whose solution is $M^n=A^nX\_0$, and this is solved by diagonalizing $A$ (or using it... | 7 | https://mathoverflow.net/users/25510 | 325078 | 140,046 |
https://mathoverflow.net/questions/325093 | 8 | How explicit are the model structures for various categories of spectra?
Naive, symmetric and orthogonal spectra are obtained via left Bousfield localization of model structures with explicit generating (acyclic) cofibrations, so we have explicit generating cofibrations for them.
I'm thinking it's too much to ask f... | https://mathoverflow.net/users/2362 | For which categories of spectra is there an explicit description of the fibrant objects via lifting properties? | You have explicit generating (acyclic) cofibrations for pretty much any model of spectra you can think of. As you point out, most begin as levelwise model structures, and so you have explicit generating (acyclic) cofibrations before left Bousfield localization, hence still have the same generating cofibrations after. Y... | 7 | https://mathoverflow.net/users/11540 | 325099 | 140,050 |
https://mathoverflow.net/questions/325046 | 11 | Let $R$ be the hyperfinite $II\_1$ factor, and let $G$ be a locally compact group.
>
> (1) Does there always exist a continuous, (faithful) outer action of $G$ on $R$?
>
> (2) If so, how does one construct one?
>
> (3) Is there any hope of classifying such actions?
>
>
>
---
(1) Here, an "action" i... | https://mathoverflow.net/users/5690 | Actions of locally compact groups on the hyperfinite $II_1$ factor | The answer to (1) (for second countable groups) is yes: every locally compact second countable group $G$ admits a countinuous, faithful outer action on the hyperfinite $II\_1$ factor. This is attributed to Blattner, and is stated explicitly in Proposition 1 of the following article:
R. J. Plymen. "Automorphic group r... | 12 | https://mathoverflow.net/users/1243 | 325100 | 140,051 |
https://mathoverflow.net/questions/325082 | 4 | Let $\kappa$ be an infinite cardinal. An ultrafilter ${\cal U}$ on $\kappa$ is said to be *uniform* if $|R|=\kappa$ for all $R\in{\cal U}$. If ${\cal U}$ is a non-principal ultrafilter on $\kappa$, denote by $b({\cal U})$ the minimal cardinality that a filter base for ${\cal U}$ can have.
If ${\cal U, V}$ are non-pri... | https://mathoverflow.net/users/8628 | Minimal cardinality of a filter base of a non-principal uniform ultrafilters | Your number $b(U)$ is usually called the "character" of the ultrafilter $U$.
In general, there may be uniform ultrafilters on the same set with different characters. For example, it is consistent with $2^{\aleph\_0}=\aleph\_2 $ that some ultrafilters have character $\aleph\_1$, others $\aleph\_2$.
Also more complic... | 8 | https://mathoverflow.net/users/14915 | 325102 | 140,052 |
https://mathoverflow.net/questions/325066 | 0 | Let $$\begin{array}{ccccccccc}
A & \rightarrow & X \\
i\downarrow & & \downarrow p \\
B & \xrightarrow{v} & S
\end{array} $$ be a commutative diagram of simplicial sets, with $p$ an inner fibration. Two solutions $f,g$ to this lifting problems are called homotopic relative to $A$ over $S$ provided that they are equiva... | https://mathoverflow.net/users/128857 | A characterization of maps that are homotopic relative to $A$ over $S$ | This fact is indeed standard. Because $\Delta^1 \times B$ is a cylinder object for $B$, the equivalent formulation (the hypothesis of the converse direction you're asking for) is just explicitly spelling out the homotopy that shows the two maps are equivalent in that fiber. A good place to learn about relative homotopy... | 0 | https://mathoverflow.net/users/11540 | 325104 | 140,053 |
https://mathoverflow.net/questions/325025 | 2 | Let $(M,g\_M)$ be a closed oriented Riemannian manifold that has a fibration structure
$$
Y \rightarrow M \overset{\pi}{\rightarrow} B
$$
where $(Y,g\_Y)$ and $(B,g\_B)$ are closed Riemannian manifolds such that $\pi$ is a Riemannian submersion.
Now we define $g\_{\epsilon}=\epsilon^{-2}\pi^\*g\_B+g\_Y$ and $\tilde g... | https://mathoverflow.net/users/105900 | Perturbation of the adiabatic limit | I think the answer is "no" in general (whenever the dimensions are such that $\eta(A\_\epsilon)\ne 0$) because you can use $\alpha\_\epsilon$ to change
the fibre-wise metric $g\_Y$ in any way you like.
So take the special case where $\dim Y\equiv 3$ mod $4$ and $\dim B\equiv 0$
mod $4$, and assume that $B$ has nonzer... | 2 | https://mathoverflow.net/users/70808 | 325107 | 140,054 |
https://mathoverflow.net/questions/325113 | 1 | I am currently trying to perform some statistical analysis on some data to see if there is any meaningful conclusion for a research project I am working on; however, I have come across a problem. There is a lot of null responses in my data set (because the person could not answer the question).
Here is the link to th... | https://mathoverflow.net/users/136843 | Performing Statistical Analysis on a Data Set With a lot of Null Responses | Here are a few links to some of the huge literature on dealing with missing data:
[Wikipedia, Missing data](https://en.wikipedia.org/wiki/Missing_data)
[Wikipedia, Expectation–maximization algorithm](https://en.wikipedia.org/wiki/Expectation%E2%80%93maximization_algorithm)
[Little--Rubin, Statistical Analysis wi... | 2 | https://mathoverflow.net/users/36721 | 325115 | 140,058 |
https://mathoverflow.net/questions/324884 | 2 | Let $(X,\tau)$ be a topological space. Denote by $Bor(X)$ the space of Borel functions $f:X\to\mathbb{R}$ where we identify two functions whenever they agree on the complement of a meager set. We endowed $Bor(X)$ with the pointwise partial order. I have read somewhere that this space is Dedekind complete. Is that true?... | https://mathoverflow.net/users/70540 | The space of Borel function modulo comeager sets is Dedekind complete | Fremlin's measure theory textbook is a good reference for these things. I am splitting things up into the Boolean algebra part and the real-valued functions part.
Complete Boolean algebras:
--------------------------
The way to show that $\mathcal{B}or(X)/\mathcal{M}(X)$ (using $\mathcal{M}(X)$ for the meagre sets)... | 3 | https://mathoverflow.net/users/61785 | 325128 | 140,064 |
https://mathoverflow.net/questions/325134 | 1 | i have a question about the following assertion:
let $n,j,u $ positive integer satisfying
$ n \geq 5,$ $ 1\leq j \leq n-1$,$ \; n+1 \leq u \leq n+j$
let $ d[n]:=\operatorname{lcm}[1,2,..,n]$ thus $u$ divide $d[n] \cdot C\_{n+j}^n$
I think I have found a proof using valuation p-adic of prime number appearing in... | https://mathoverflow.net/users/136807 | Question about arithmetic binomial coefficient | Here is a simple proof of your divisibility relation. It suffices to show that
$$\frac{\mathrm{lcm}(1,2,\dots,n+j)}{\mathrm{lcm}(1,2,\dots,n)}\quad\text{divides}\quad\binom{n+j}{n}\quad\text{for}\quad 0\leq j\leq n.$$
That is, for any prime $p$ and for $0\leq j\leq n$, we have that
$$\lfloor\log\_p(n+j)\rfloor-\lfloor\... | 1 | https://mathoverflow.net/users/11919 | 325143 | 140,073 |
https://mathoverflow.net/questions/325097 | 2 | This is related to trying to resolve the currently faulty second part of my answer to [this](https://mathoverflow.net/questions/229303/hyperimaginaries-and-continuous-logic/325027#325027) question, but is by itself a purely real analysis question.
Let $X$ be a set with a uniform structure presented by a function $f:X... | https://mathoverflow.net/users/83901 | Uniformly Converging Metrization of Uniform Structure | $f\_k$ converges uniformly for $1\leq\beta\leq 2.$
I will argue that for any $\epsilon>0$ and any path $z\_0,z\_1,\dots,z\_{k-1},z\_k,$ there is a sub-path $z\_0,z\_{i\_1},\dots,z\_{i\_{m-1}},z\_k$ such that either
$$f(z\_0,z\_{i\_1})+\dots+f(z\_{i\_{m-1}},z\_k)\leq (1+\epsilon)(f(z\_0,z\_1)+\dots+f(z\_{k-1},z\_k))\t... | 2 | https://mathoverflow.net/users/112284 | 325145 | 140,074 |
https://mathoverflow.net/questions/325158 | 3 | I am now reading the book Higher Topos Theory. In A.3.1.5, it gives the definition
of a $\mathbf{S}$-enriched model category, where $\mathbf{S}$ is a monoidal model category. But in the book model structures are introduced only on $\mathsf{Set}$-enriched categories.
>
> So, what does a model structure on a $\mathb... | https://mathoverflow.net/users/123746 | Definition A.3.1.5 of Higher Topos Theory | Your guess is correct, indeed. In general, given any monoidal category $(\mathbf V, \otimes, 1)$, and any $\mathbf V$-enriched category $\mathbf C$, one can always consider the *underlying category* $\mathbf C\_0$ as the ($\mathbf{Set}$-)category having as objects the same objects as $\mathbf C$, and as hom-sets
$$
\ma... | 5 | https://mathoverflow.net/users/119308 | 325160 | 140,080 |
https://mathoverflow.net/questions/325172 | 0 | I have an recursive sequence and want to convert it to an explicit formula.
The recursive sequence is:
$f(0) = 4$
$f(1) = 14$
$f(2) = 194$
$f(x+1) = f(x)^2 - 2$
| https://mathoverflow.net/users/136893 | Solving an recursive sequence | $$f(x+1) = f(x)^2 - 2,\;\;f(0)=4$$
$$\Rightarrow f(x)=2\cos\left(2^x\arccos 2\right)=2 \cosh \left(2^x \ln \left(\sqrt{3}+2\right)\right)$$
| 2 | https://mathoverflow.net/users/11260 | 325173 | 140,085 |
https://mathoverflow.net/questions/325153 | 3 | Let $c\in\mathbb{R}^n$ and let $X\_1,X\_2$ be two independent uniform samples on the unit cube in $\mathbb{R}^n$. Is there anything at all (in an analytic sense) we can say about the expectation $E\min\{c^T X\_1 , c^T X\_2\}$? How about if I have $k>2$ independent samples?
| https://mathoverflow.net/users/70190 | Expected minimum of a linear function on the unit cube | Let us provide an explicit (albeit complicated) expression for $EM\_k$, where
\begin{equation\*}
M\_k:=\min\_{1\le i\le k}c^TX\_i
\end{equation\*}
and $k$ is any natural number. Without loss of generality, each of the coordinates $c\_j$ of the vector $c=(c\_1,\dots,c\_n)$ is nonzero; otherwise, one can reduce the dim... | 1 | https://mathoverflow.net/users/36721 | 325175 | 140,086 |
https://mathoverflow.net/questions/325161 | 7 | If ${\cal U}$ and ${\cal V}$ are ultrafilters on non-empty sets $A$ and $B$ respectively, then the *tensor product* ${\cal U}\otimes{\cal V}$ is the following ultrafilter on $A\times B$:
$$\big\{X\subseteq A\times B: \{a\in A:\{b\in B: (a,b)\in X\}\in {\cal V}\}\in {\cal U}\big\}.$$
It is a standard exercise to verify ... | https://mathoverflow.net/users/8628 | Non-tensor-representable ultrafilters on $\omega$ | Recall that $\mathcal Z$ is a *weak $P$-point* if it is not in the closure of any countable subset of $\omega^\* \setminus \{\mathcal Z\}$. A weak $P$-point is never the tensor product of two non-principal ultrafilters.
To see this, suppose $\mathcal U$ and $\mathcal V$ are two non-principal ultrafilters on $\omega$,... | 12 | https://mathoverflow.net/users/70618 | 325179 | 140,089 |
https://mathoverflow.net/questions/325163 | 2 | Suppose we have a $C^\*$-algebra $\mathcal{U}$, Consider the $C^\*$-subalgebra generated by elements of the form $a\otimes a$, what is it isomorphic to? Is it isomorphic to $\mathcal{U}$ itself?
| https://mathoverflow.net/users/136400 | On diagonal part of tensor product of $C^*$-algebras | You get the symmetric part of the tensor product.
The map $\phi: a\otimes b \mapsto b\otimes a$ extends to an order 2 automorphism of $\mathcal{U}\otimes\mathcal{U}$. The set of fixed points for this $\mathbb{Z}/2$ action is a C\*-subalgebra $(\mathcal{U}\otimes\mathcal{U})\_s$ of $\mathcal{U}\otimes\mathcal{U}$. Eve... | 6 | https://mathoverflow.net/users/23141 | 325182 | 140,091 |
https://mathoverflow.net/questions/325186 | 28 | If $p$ is a prime then the zeta function for an algebraic curve $V$ over $\mathbb{F}\_p$ is defined to be
$$\zeta\_{V,p}(s) := \exp\left(\sum\_{m\geq 1} \frac{N\_m}{m}(p^{-s})^m\right). $$
where $N\_m$ is the number of points over $\mathbb{F}\_{p^m}$.
I was wondering what is the motivation for this definition. The su... | https://mathoverflow.net/users/103423 | Motivation for zeta function of an algebraic variety | The definition using exponential of such an ad hoc looking series is admittedly not too illuminating. You mention that the series looks vaguely logarithmic, and that's true because of denominator $m$. But then we can ask, why include $m$ in the denominator?
A "better" definition of a zeta function of a curve (more ge... | 27 | https://mathoverflow.net/users/30186 | 325191 | 140,093 |
https://mathoverflow.net/questions/325189 | 2 | Is there a bounded self-adjoint operator $H$ acting on $\ell^2(\mathbb{Z})$ such that for all sequences $u,v\in \ell^2(\mathbb{Z})$
$$ H(uv)=(Hu)v+u(Hv)$$
where uv is the pointwise product.
This is for instance not the case of $i\partial$ where $[\partial u](n)=u(n+1)-u(n-1)$
| https://mathoverflow.net/users/107004 | Existence of a bounded operator which satisfies the discrete product rule | Assuming that $uv$ means the pointwise product, the only such $H$ is the zero operator. (Self-adjointness is unnecessary.)
**Proof.** Let $n\in {\bf Z}$. Taking $u=v=e\_n$ we get $H(e\_n) = H(e\_ne\_n)= 2e\_n\cdot (He\_n)$.
Multiply both sides by $e\_n$ to get $e\_n\cdot H(e\_n) = 2e\_n\cdot H(e\_n)$ and deduce tha... | 7 | https://mathoverflow.net/users/763 | 325194 | 140,094 |
https://mathoverflow.net/questions/323532 | 1 | $$f(\omega,u):=\frac1{\omega+iu}$$
where $i$ is the imaginary unit number. We see that the integral of a Fourier transform
$$\int\_1^\infty du\int\_{-\infty}^\infty d\omega\,f(\omega,u)\,e^{-i\omega x}=2\pi i\int\_1^\infty du\,e^{-ux}=2\pi i\frac{e^{-x}}x$$
for $u>0$, $x>0$, while interchanging the order of integration... | https://mathoverflow.net/users/32660 | Interchanging Integration Order involving Fourier Transform | The subject of definite integrals for distributions was investigated in some detail by several mathematicians in the 50’s—-they used an elementary approach based on the fact that distributions are (locally) higher derivatives of continuous functions. This allows a treatment at the level of a (european) freshman calculu... | 2 | https://mathoverflow.net/users/131781 | 325196 | 140,095 |
https://mathoverflow.net/questions/325185 | 0 | The sentence *s* "In many supervised learning problems one has an output variable $y$ and a vector of input variables $x$ described via a joint probability distribution $P(x,y)$" from [wiki](https://en.wikipedia.org/wiki/Gradient_boosting)
Here implicitly declared 3 notations $\{x,y,P(x,y)\}$.
So I wonder if there ... | https://mathoverflow.net/users/136118 | Is there any math notation for `be denoted by`? | We may see $:=$ used this way.
$$
\text{Let } A\_3 := \{n \in \mathbb N\;:\; 3 | n\}
$$
That is, let the set of multples of $3$ be denoted by $A\_3$.
| 7 | https://mathoverflow.net/users/454 | 325206 | 140,100 |
https://mathoverflow.net/questions/325188 | 1 |
>
> Let $M$ be a holonomic $D\_X$-module. This means that the minimal primes in $\sqrt{\operatorname{Ann}(\operatorname{gr}M)}$ are $n=\dim X$ dimensional, for some (and any) good filtration on $M$. But what about the embedded primes?
>
>
>
I think there cannot be embedded primes, and the argument would be as fo... | https://mathoverflow.net/users/64302 | For a holonomic $D_X$-module $M$, can $\operatorname{gr}M$ have embedded primes? | Embedded primes may exist and they depend on the good filtration.
Example: Let $X=\mathbf A^1$ be the affine line. Then $D\_X=\mathbb C\langle x,\partial\rangle$. Let $M=D\_X/D\_Xx=\mathbb C[\partial]$. Choose the filtration $M\_n:=\langle 1,\partial,\ldots,\partial^n\rangle$ for $n\ge\mathbf 2$ but put $M\_1:=0$. Th... | 3 | https://mathoverflow.net/users/89948 | 325213 | 140,103 |
https://mathoverflow.net/questions/325218 | 9 | Recall that a [*filter*](https://en.wikipedia.org/wiki/Filter_(mathematics)#Filter_on_a_set) on a set $X$ is a nonempty collection $\mathcal{F}$ of subsets of $X$ such that
(i) $U\subseteq V\subseteq X$ and $U\in\mathcal{F}$ implies $V\in\mathcal{F}$, and
(ii) $U,V\in\mathcal{F}$ implies $U\cap V\in\mathcal{F}$.
... | https://mathoverflow.net/users/1384 | Reference request: filter tends to filter along map | The category that you are referring to is the category $\mathcal{F}$ in the paper [1](http://matwbn.icm.edu.pl/ksiazki/fm/fm94/fm94115.pdf).
The paper [1](http://matwbn.icm.edu.pl/ksiazki/fm/fm94/fm94115.pdf) also introduces another category $\mathcal{G}$ which is the quotient category $\mathcal{F}/\simeq$ category o... | 5 | https://mathoverflow.net/users/22277 | 325220 | 140,106 |
https://mathoverflow.net/questions/325221 | 1 | Let $\underline{c}:=\left(c\_1,\dots,c\_n\right)$ be pairwise distinct complex numbers, and let $k$ be a non-negative integer. Define the matrix $A\_{n,k}(\underline{c})$ to contain the $k$-th powers of the pairwise differences, i.e.
$$
A\_{n,k}\left(\underline{c}\right):=\left[ (c\_i-c\_j)^k \right]\_{1\leq i,j \leq n... | https://mathoverflow.net/users/23862 | Matrix of powers of pairwise differences | $$(c\_i-c\_j)^k = \sum\_{h=0}^k \binom{k}{h} (-1)^{k-h}c\_i^h c\_j^{k-h}$$
and each summand is a rank-1 matrix (since it's a function of $i$ times a function of $j$). To prove that the rank is not lower than that, consider that the vectors $\mathbf{v}\_h = (c\_i^h)\_{i=1}^n$ are independent because they form a Vander... | 6 | https://mathoverflow.net/users/1898 | 325223 | 140,108 |
https://mathoverflow.net/questions/325211 | 5 | In the paper [CLL], Carlen, Lieb, and Loss demonstrate a version of the Young inequality on the sphere $S^{N-1}$ in $\mathbb{R}^N$. For positive functions $f\_j$ on $[-1,1]$, the following bound holds:
$$
\int\_{S^{N-1}} \prod\_{j=1}^N f\_j(x\cdot e\_j) \,d\mu(x)
\leq \prod\_{j=1}^N \left(\int\_{S^{N-1}} f\_j(x\cdot e\... | https://mathoverflow.net/users/69564 | Brascamp-Lieb inequalities on the sphere | Indeed, there are extensions to spheres:
F. Barthe, D. Cordero-Erausquin, B. Maurey. *Entropy of spherical marginals and related inequalities*. J. Math. Pures Appl., 86: 89–99 (2006).
It reads as follows: If $x ∈ S^{n−1} \subset R^n$ (the standard (n − 1)-sphere), consider the projection $P\_{E\_i}(x)$, $i = 1,...,... | 4 | https://mathoverflow.net/users/48356 | 325227 | 140,109 |
https://mathoverflow.net/questions/324847 | 6 | Given a functor $F : \mathcal C \to \mathcal C$ with initial algebra $\alpha : FA \to A$, and another algebra $\xi : FX \to X$, we obtain a unique morphism $\mathsf{fold}~\xi : A \to X$ such that $\mathsf{fold}~\xi \circ \alpha = \xi \circ F(\mathsf{fold}~\xi)$.
I am looking for a sufficient condition - ideally phras... | https://mathoverflow.net/users/66017 | When is a fold monomorphic/epimorphic | Suppose $({\cal E,M})$ is a factorization system on $\cal C$ such that $\cal M$ consists of monomorphisms and $F$ preserves $\cal E$-morphisms. If $X$ is an $F$-algebra, define an *$\cal M$-subalgebra* of $X$ to be a morphism $Y\to X$ of $F$-algebras that is an $\cal M$-morphism. Then if $A$ is the initial $F$-algebra,... | 1 | https://mathoverflow.net/users/49 | 325228 | 140,110 |
https://mathoverflow.net/questions/325226 | 3 | [An essentially algebraic theoery, according to Adamek and Rosicky](https://ncatlab.org/nlab/show/essentially+algebraic+theory#definition) (second definition on nlab), consists of a many-sorted signature $\Sigma$ (consisting of function symbols on sorts $S$), a set $E$ of equations, a subset of total function symbols $... | https://mathoverflow.net/users/104294 | How to turn a limit sketch into an essentially algebraic theory? | We'll add a new sort $E$ and force it to be isomorphic (by a specified isomorphism) to the equalizer of $f$ and $g$. Then we can replace $h$ and $k$ by total functions $E \to C$, and so $p$ will have domain of definition defined by equations between total functions.
To do this we just need to add $e : \mathrm{Eq}(f, ... | 3 | https://mathoverflow.net/users/126667 | 325235 | 140,114 |
https://mathoverflow.net/questions/284033 | 10 | Let $L$ be a Lie algebra over a field of characteristic $p>0$ and $D$ a derivation of $L$. For every $x\in L$ denote by $\mathrm{ad} x$ the adjoint map $\mathrm{ad}x: L \rightarrow L, a\mapsto [x,a]$. Is the following relation true?
>
> $$D^{p-1}((\mathrm{ad} x)^{p-1}(D(x)))=(\mathrm{ad} x)^{p-1}(D^p(x))$$
>
>
>... | https://mathoverflow.net/users/17582 | An identity in Lie algebras over fields of positive characteristic | Yes, this identity holds, and both sides are equal to $-D^p(x^p)$.
We need the following facts, all of which are straightforward to verify, where $D$ is an arbitrary derivation, still working over characteristic $p>0$: $$(ad\_x)^k(y)=\sum\_{m=0}^{k}(-1)^{m+1} {k\choose m}x^m y x^{k-m}$$
$$D(x^k)=\sum\_{m=0}^{k-1}x^m ... | 5 | https://mathoverflow.net/users/128502 | 325255 | 140,120 |
https://mathoverflow.net/questions/325292 | 8 | Let the split group of type $F\_4$ act as the automorphism group of the split Albert algebra $A$. Consider the action of $F\_4\times \mathbb{G}\_m$ on $A$, given by letting $\mathbb{G}\_m$ act by scalar multiplication.
Does this action have finitely many orbits? Are the stabilizers known? I am very new to $F\_4$ and... | https://mathoverflow.net/users/136468 | Orbits of action of the split group of type $F_4$ | I work over the complexe numbers, but the story is similar over any fields (provided you choose the split octonions).
Let us identify the Albert algebra $A$ with the algebra of self-adjoint $3\*3$ matrices with octonionic coefficients.
The hyperplane given by $\mathrm{Tr}(X) = 0$ is stabilized by $F\_4 \times \mathbb{G... | 6 | https://mathoverflow.net/users/37214 | 325303 | 140,131 |
https://mathoverflow.net/questions/325297 | 8 | Let $\mathfrak{u}$ denote the ultrafilter number, which is defined to be the minimum cardinality of a subset of $\mathcal{P}(\mathbb N)$ which is a base for a nonprincipal ultrafilter on $\mathbb{N}$. Clearly $\aleph\_1\leq \frak{u}\leq 2^{\aleph\_0}$, so it is only interesting to study $\frak{u}$ under the negation of... | https://mathoverflow.net/users/123905 | A question on the ultrafilter number | The answer to your question is yes. In fact, one can force to make $\mathfrak u$ equal to any prescribed uncountable regular cardinal while making the cardinal of the continuum equal to any larger prescribed uncountable regular cardinal. This is proved in and old paper by Shelah and me:
Blass, Andreas(1-PAS); Shelah,... | 17 | https://mathoverflow.net/users/6794 | 325310 | 140,134 |
https://mathoverflow.net/questions/325287 | 7 | In *Pursuing Stacks*, Grothendieck uses the category $Cat$ of small categories to model spaces. A recurring theme is the question of whether there is a Quillen model structure supporting this homotopy theory, i.e. with weak homotopy equivalences $\mathcal W\_{min}$ -- those functors which become weak homotopy equivalen... | https://mathoverflow.net/users/2362 | Is the Thomason model structure the optimal realization of Grothendieck's vision? | The answer to question 1) is no. However, the functor
$$N Elts\_A:Psh(A)\to sSet$$
commutes with colimits and is a left Quillen equivalence whenever $A$ is a test category. We may transfer the model structure on $Psh(A)$ to $Cat$ through the left adjoint
$$c Sd N Elts\_A:Psh(A)\to Cat$$
(the proof is slight variation o... | 12 | https://mathoverflow.net/users/1017 | 325315 | 140,135 |
https://mathoverflow.net/questions/325101 | 8 | In set theory, there are several distinct notions of club sets, stationary sets, diagonal intersection, regressive function, normal filters, normal ultrafilters, etc. I am wondering if there is an abstract general theory of stationary that encapsulates all or most of these notions at the same time. Consider the followi... | https://mathoverflow.net/users/22277 | Is there an abstract theory of club sets and stationary sets? | *Stationary logic* goes very far in the abstract analysis of stationarity; [this article](https://core.ac.uk/download/pdf/81121517.pdf) by Barwise, Makkai, and Kaufmann (1978) started the subject.
See also [this MO post](https://math.stackexchange.com/questions/2371500/definition-generalized-logics-and-stationary-log... | 4 | https://mathoverflow.net/users/9269 | 325331 | 140,138 |
https://mathoverflow.net/questions/325320 | 12 | Let $F:\cal C\to D$ be an accessible functor between locally presentable categories. By Theorem 2.19 in Adamek-Rosicky *Locally presentable and accessible categories*, there exist arbitrarily large regular cardinals $\lambda$ such that $F$ preserves $\lambda$-presentable objects. It is tempting to expect that $F$ shoul... | https://mathoverflow.net/users/49 | Accessible functors not preserving lots of presentable objects | An example is given in my paper with Tibor Beke,
>
> *Abstract elementary classes and accessible categories*, Annals Pure Appl. Logic **163** (2012), 2008-2017, doi:[10.1016/j.apal.2012.06.003](https://doi.org/10.1016/j.apal.2012.06.003), arXiv:[1005.2910](https://arxiv.org/abs/1005.2910).
>
>
>
see Remark 3.2... | 13 | https://mathoverflow.net/users/73388 | 325334 | 140,141 |
https://mathoverflow.net/questions/325317 | 6 | Is there any example of a manifold with a positive isotropic curvature but it possibly obtains a negative Ricci curvature at some point and the direction? If we see the definition of the positive isotropic curvature condition, it doesn't imply the positivity of the Ricci curvature (it is true if $(M\times \mathbb{R},g+... | https://mathoverflow.net/users/136982 | Example of a manifold with positive isotropic curvature but possibly negative Ricci curvature | If you have a look into <https://arxiv.org/pdf/1711.05167.pdf> , page 4, it's written there:
"Conversely, it follows from work of Micallef and Wang [29] that
every manifold which is diffeomorphic to a connected sum of quotients of
$S^n$ and $S^{n−1} × \mathbb R$ admits a metric with positive isotropic curvature."
S... | 7 | https://mathoverflow.net/users/943 | 325339 | 140,143 |
https://mathoverflow.net/questions/325313 | 1 | Does someone know of any sort of software openly available online which can be used to compute various characteristics of Demazure modules for semisimple Lie algebras? Specifically, I'm interested in dimensions of Demazure modules for type $A$. (To clarify: I have my own implementation of the Demazure character formula... | https://mathoverflow.net/users/19864 | Openly available software to work with Demazure modules |
```
> bash-3.2$ LiE
>
> LiE version 2.2.2 created on Oct 22 2018 at 11:36:00 Authors: Arjeh M.
> Cohen, Marc van Leeuwen, Bert Lisser. Purpose: development CWI
>
>
> type '?help' for help information type '?' for a list of help entries.
> > p=Demazure(X[1,1],[1,2],A2)
> > p
1X[-1, 2] +1X[ 0, 0] +1X[ 1,-2] +1X[ 1, ... | 2 | https://mathoverflow.net/users/4794 | 325344 | 140,144 |
https://mathoverflow.net/questions/321741 | 0 | I was reading a recent theory paper in machine learning by Kenji Kawaguchi and Leslie Pack Kaelbling
<https://arxiv.org/pdf/1901.00279.pdf>
and the authors seem to suggest in section 2.2 that cross-entropy loss for classification is not twice differentiable. This seems wrong, I thought it was $C^\infty$.
What am ... | https://mathoverflow.net/users/24951 | Cross entropy loss is not twice differentiable? | Kawaguchi and Kaelbling are saying that the results by [Liang et al.](https://arxiv.org/pdf/1805.08671.pdf) are not applicable to binary classification with cross-entropy loss: in the linked article by Liang, the loss criterion $L:\mathbb{R}^2 \rightarrow \mathbb{R}$ is of the form $L(p,q) = \tilde{L}(-pq)$, where $\ti... | 0 | https://mathoverflow.net/users/137003 | 325346 | 140,145 |
https://mathoverflow.net/questions/325345 | 4 | I am looking for a locally ringed space the stalks of which are noetherian and such that the structural sheaf is not coherent over itself. Can you provide me an example of this?
Notice that one may not find such an example which is a scheme, since schemes with noetherian stalks are locally noetherian and any locally ... | https://mathoverflow.net/users/83945 | Locally ringed space with noetherian stalks and a non-coherent structural sheaf | Welcome new contributor. It is best to think about these kinds of examples on one's own. Thus, *please try this for yourself before reading the following*.
Let $(Y,Z)$ be any pair of a Noetherian scheme $Y$ and a nonempty closed subset $Z$ that is nowhere dense. For instance, let $Y$ be the following Spec of a DVR, ... | 8 | https://mathoverflow.net/users/13265 | 325351 | 140,147 |
https://mathoverflow.net/questions/325352 | 2 | Want to find $f\_n$ a sequence of continuous functions, so that for all Borel regular measure $\mu$, we have
$\int f\_n d\mu\rightarrow\int \chi\_\Delta d\mu$, as $n$ goes to infinity, where $\Delta$ is a Borel set.
| https://mathoverflow.net/users/137008 | Can the characteristic function of a Borel set be approached by a sequence of continuous function through a certain convergence in $L^\infty$? | (EDITED)
Let $\Delta$ be the rationals in $[0,1]$. If your condition is satisfied, in particular $f\_n$ converges pointwise to $\chi\_\Delta$. Let $C\_n = \{x \in [0,1]: \forall m > n, \;|f\_n(x) - f\_m(x)|\le 1/3\}$. Then $C\_n$ are closed and their union is $[0,1]$. By the Baire category theorem some $C\_n$ has nonem... | 6 | https://mathoverflow.net/users/13650 | 325355 | 140,148 |
https://mathoverflow.net/questions/325354 | 8 | Has anyone seen an inequality of this form before? It seems to be true (based on extensive testing), but I am not able to prove it.
Let $a\_1,a\_2,\ldots,a\_k$ be non-negative integers such that $\sum\_i a\_i = A$. Then, for any non-negative integer $B \le A$:
$$
\sum\_{(b\_1,\ldots,b\_k): \sum\_i b\_i = B} \prod\_i ... | https://mathoverflow.net/users/68511 | A curious inequality concerning binomial coefficients | By Cauchy–Bunyakovsky–Schwarz inequality we have
$$
\left(\sum \prod\_i \frac{\binom{a\_i}{b\_i}}{\binom{A-a\_i}{B-b\_i}}\right)\left(\sum\prod\_i \binom{a\_i}{b\_i}\binom{A-a\_i}{B-b\_i}\right)\geqslant
\left(\sum \prod\_i\binom{a\_i}{b\_i}\right)^2=\binom{A}{B}^2
.$$
Thus it suffices to prove that
$$
\sum\prod\_i... | 14 | https://mathoverflow.net/users/4312 | 325361 | 140,150 |
https://mathoverflow.net/questions/325363 | 2 | I study differential geometry and do not understand the choice of environment of $(-2,2)$ for the $\gamma$ curve in the following proof. Are there geodesics in arbitrarily small intervals? Why is this interval explicitly desired?
I would be very happy if I could get an intuitive explanation, because this proof haunts m... | https://mathoverflow.net/users/128041 | A question on the existence of geodesics on smooth manifolds | The point to these existence results is usually that at the core they boil down to questions about the existence of solutions of certain ODEs. I think a lucid explanation of this can be found in Lang's book: Fundamental of differential geometry. As the proof you cite shows we can prolong the intervall of existence by s... | 2 | https://mathoverflow.net/users/46510 | 325369 | 140,153 |
https://mathoverflow.net/questions/325242 | 8 | Let $w\in S\_n$ be a permutation. Is there a reasonable "formula" for the number of elements of the initial interval $[e,w]$ of weak (Bruhat) order from the identity to $w$?
In terms of what such a "formula" might look like: if $w$ is a Grassmannian permutation of shape $\lambda$ then we have $[e,w]\simeq[\varnothing... | https://mathoverflow.net/users/25028 | Formula for number of permutations less than a given permutation in weak order | By [Dittmer and Pak - Counting linear extensions of restricted posets](https://arxiv.org/abs/1802.06312) (Theorem 1.4), computing the size of $[e,w]$ is $\#$P-complete. Thus, a nice formula like the suggested $n \times n$ determinant filled with entries of easy-to-calculate permutation data would imply P$=$NP.
Though... | 7 | https://mathoverflow.net/users/22379 | 325372 | 140,154 |
https://mathoverflow.net/questions/325367 | 2 | If I have 2 random variables $\xi, \eta$ and $\forall n,m \ \mathbb E\xi^n\eta^m=\mathbb E\xi^n \mathbb E\eta^m$, does this imply that $\xi,\eta$ are independent? How to show it?
| https://mathoverflow.net/users/132038 | Statement about independence of random variables | The answer is no. Indeed, let $X:=\xi$ and $Y:=\eta$. Let $U$ and $V$ be any independent random variables (r.v.'s) with different distributions but with the same finite moments of all orders:
$$EU^m=EV^m=:\mu\_m$$
for all natural $m$. A standard example of the distributions of such r.v.'s $U$ and $V$ is given in [the ... | 3 | https://mathoverflow.net/users/36721 | 325373 | 140,155 |
https://mathoverflow.net/questions/325374 | 3 | Let $\Sigma^n \subseteq \mathbb{R}^{n+1}$ be a minimal complete hypersurface. Let's consider $\bar{\Sigma} := \Sigma \times \mathbb{R}^k \subseteq \mathbb{R}^{n+k +1}$. Clearly $\bar{\Sigma}$ is a minimal hypersurface as well.
Moreover it is easy to check that if $\Sigma$ is stable then $\bar{\Sigma}$ is stable as w... | https://mathoverflow.net/users/86341 | Stability of minimal hypersurface with flat directions | Yes, this is true.
I will prove it for $k=1$, and the result clearly follows by induction. If $\Sigma$ is unstable, then there is some compactly supported function $\varphi(x)$ with $\mathscr{Q}\_\Sigma(\varphi,\varphi) < -\delta <0$ (where $\mathscr{Q}$ is the second variation operator). Consider a function $\chi(y... | 3 | https://mathoverflow.net/users/1540 | 325387 | 140,159 |
https://mathoverflow.net/questions/325385 | 1 | Let $G$ be a topological group and $X$ be a topological space.
Let $\alpha$, $\beta:G\times X\to X$ be two group actions. We say that these two actions are homotopic actions if there is a continuous path of actions connecting $\alpha$ to $\beta$. This means that there is a continuous function $\Gamma:[0,1]\times G\time... | https://mathoverflow.net/users/36688 | Homotopy of group actions | It's enough to pick a contractible manifold $M$ with two non-homotopic actions. For example, let us pick $M=\mathbb{R}^2$ with two actions of $S^1$, the trivial one and the one given by rotations. These two actions are not homotopic. In fact for every action $\rho$ of $S^1$ on $\mathbb{R}^2$ we can consider the map $\m... | 6 | https://mathoverflow.net/users/43054 | 325388 | 140,160 |
https://mathoverflow.net/questions/325383 | 15 | Is there any known example of a combinatorial model category $C$ together with a set of map $S$ such that the left Bousefield localization of $C$ at $S$ does not exists ?
It is well known to exists when $C$ is left proper, and it seems that it also always exists as a left semi-model structure, but I don't known if th... | https://mathoverflow.net/users/22131 | Counter-example to the existence of left Bousfield localization of combinatorial model category | A surprisingly effective way to construct counterexamples in model category theory is to just write down all the objects and morphisms involved and try to give the resulting (finite!) diagram the structure of a model category.
Here, we know that a counterexample must fail to be left proper, so start with a diagram$\r... | 26 | https://mathoverflow.net/users/126667 | 325390 | 140,161 |
https://mathoverflow.net/questions/325397 | 15 | I'm having an hard time trying to figuring out a concrete example of an odd-dimensional closed manifold that do **not** admit any contact structure.
Can someone provide me with some examples?
| https://mathoverflow.net/users/99042 | Examples of odd-dimensional manifolds that do not admit contact structure | According to a well known result of Martinet, every compact orientable $3$-dimensional manifold has a contact structure [2], see also [1] for various proofs. On the other hand we have
>
> **Theorem.** For $n\geq 2$ there is a closed oriented connected manifold of dimension $2n+1$ without a contact structure.
>
>
... | 24 | https://mathoverflow.net/users/121665 | 325398 | 140,164 |
https://mathoverflow.net/questions/325378 | 12 | Let $p$ be a prime. Let $f \in \mathbb{F}\_p[x\_1, \ldots, x\_n]$ be a **homogenous** polynomial of degree $p$. Can $f$ have more than $(1-p^{-1}+p^{-2}) p^n$ zeroes in $\mathbb{F}\_p^n$?
Basic observations: The lower bound (for $n \geq 2$) is achieved by $\prod\_{c\in \mathbb{F}\_p} (x\_1 - c x\_2)$. If nonhomogenou... | https://mathoverflow.net/users/297 | Most points on a degree $p$ hypersurface? | As suggested by the OP, I am making my comment an answer, as it resolves his query in the affirmative.
Theorem 2.1 (Serre, 1991), cited in *[Maximum Number of Common Zeros of Homogeneous Polynomials over Finite Fields](http://arxiv.org/pdf/1705.10185.pdf)* by Peter Beelen, Mrinmoy Datta, Sudhir R. Ghorpadel, gives an... | 10 | https://mathoverflow.net/users/12218 | 325399 | 140,165 |
https://mathoverflow.net/questions/325414 | 1 | Suppose $a>1,b>0$ are real numbers. Consider the summation of the infinite series:
$$S=\frac 1 {a+b} + \frac 1 {(a+b)(a+2b)} + \cdots + \frac 1 {(a+b)\cdots(a+kb)} + \cdots$$
How can I give a tight estimation on the summation?
Apparently, one can get the upper bound:
$$S\le \sum\_{k=1} ^\infty \frac 1 {(a+b)^k}=\frac 1... | https://mathoverflow.net/users/120302 | Estimate $\frac 1 {a+b} + \frac 1 {(a+b)(a+2b)} + \cdots + \frac 1 {(a+b)\cdots(a+kb)} + \cdots$ | Maple produces a closed-form expression for the sum under consideration by
```
sum(1/product(b*j+a, j = 1 .. k), k = 0 .. infinity))/(a+b) assuming a>1,b>0;
```
$${\frac {a{{\rm e}^{{b}^{-1}}}}{a+b}{b}^{{\frac {a-b}{b}}} \left( -
\Gamma \left( {\frac {a}{b}},{b}^{-1} \right) +\Gamma \left( {\frac {a
}{b}} \right)... | 3 | https://mathoverflow.net/users/35959 | 325415 | 140,171 |
https://mathoverflow.net/questions/325407 | 2 | Let $B \subseteq C$ be an inclusion of finite dimensional (associative) algebras over a field $k$. Assume that $C$ is a free $B$-module. Let $\bigoplus\_i U\_i$ be
a decomposition of $B$ into indecomposable $B$-modules. Let $\bigoplus\_j V\_{ij}$ be a decomposition of the extension $C \otimes\_B U\_i$ into indecomposa... | https://mathoverflow.net/users/34538 | A weak Schur's lemma for non-semisimple finite dimensional algebras | No, not necessarily.
Consider the case $B=kH$, $C=kG$ of finite group algebras over a field $k$, where $H\leq G$. I'll write $\downarrow$ and $\uparrow$ for restriction and induction. This case has a couple of simplifying features. Firstly, induction is both left and right adjoint to restriction. Secondly, $kH$ is se... | 2 | https://mathoverflow.net/users/22989 | 325418 | 140,172 |
https://mathoverflow.net/questions/325405 | 4 | The Weyl law for Maass cusp forms for $SL\_2(\mathbb Z)$ was obtained by Selberg firstly and has been generalized to various caces (Duistermaat-Guillemin, Lapid-Müller and others). For example, one of the version of Weyl's law for maass wave forms for congrunce subgroup is
$$
\sum\_{t\_j\in \mathbb R\atop{|t\_j|<T}}1=\... | https://mathoverflow.net/users/45691 | The exceptional eigenvalues and Weyl's law in level aspect | You are asking about density theorems for exceptional eigenvalues. As you mention, this has been answered by Iwaniec: if $\mathcal{B}\_0(q)$ denotes an orthonormal basis of Maass forms of weight $0$, level $q$, and trivial nebentypus with spectral parameter $t\_f \in [0,\infty)$ or $it\_f \in (0,1/2)$ (the latter the e... | 3 | https://mathoverflow.net/users/3803 | 325431 | 140,174 |
https://mathoverflow.net/questions/325429 | 3 | Let $\alpha\in[0,1]$ be a fixed constant, and let
$w,x\in[0,1]^n$ be two vectors such that $\sum\_i w\_i x\_i=\alpha$.
Define $Y = \sum\_i w\_i X\_i$, where $X\_i \sim \operatorname{Bernoulli}(x\_i)$, so it holds $\mathbb{E}[Y]=\alpha\leq 1$, and assume that the $X\_i$'s are *independent* variables, .
Can we use so... | https://mathoverflow.net/users/67002 | concentration inequality for a weighted sum of independent but not identical binary variables | Without further restrictions on $w,x$, you cannot beat the Markov bound by much for $\alpha$ close to $1$ (as in your post).
Indeed, let $a:=\alpha$. Let
$x\_i=a$ for all $i=1,\dots,n$, and let $w\_1=1$ and $w\_2=\cdots=w\_n=0$. Then all the conditions in the OP on $w,x$ will hold, and
\begin{equation}
P(Y\ge1)=P... | 0 | https://mathoverflow.net/users/36721 | 325456 | 140,184 |
https://mathoverflow.net/questions/325455 | 0 | Let $\text{NPU}(\omega)$ be the set of non-principal [ultafilters][1] on $\omega$. The *Rudin-Keisler preorder* on $\text{NPU}(\omega)$ is defined by
$${\cal U} \leq\_{RK} {\cal V} :\Leftrightarrow (\exists f:\omega\to\omega)(\forall U\in{\cal U}) f^{-1}(U)\in {\cal V} .$$
It is easy to see that $\leq\_{RK}$ is refle... | https://mathoverflow.net/users/8628 | Maximal elements in the Rudin-Keisler ordering | No. First, the Rudin–Keisler preorder is directed: for any ultrafilters $\mathcal U$ and $\mathcal V$ on $\omega$, the ultrafilter
$$\mathcal U\times\mathcal V=\{X\subseteq\omega\times\omega:\{i\in\omega:\{j\in\omega:(i,j)\in X\}\in\mathcal V\}\in\mathcal U\}$$
on $\omega\times\omega$ is Rudin–Keisler above both $\math... | 7 | https://mathoverflow.net/users/12705 | 325457 | 140,185 |
https://mathoverflow.net/questions/325465 | 13 | The Lovász Local Lemma gives a probability bound in a context where there are many events that are "not quite" independent.
What other theorems exist in this genre? That is, what other theorems have a hypothesis of the form "Let events E\_1, E\_2, ... satisfy [relaxed form of independence]" and a conclusion of the fo... | https://mathoverflow.net/users/38672 | Theorems like the Lovász Local Lemma? | A large number of results for sums $W$ of possibly dependent indicators of events (that is, for sums of possibly dependent Bernoulli random variables) $X\_i$ have been obtained by the Chen--Stein method. See e.g. [Theorem 1](https://projecteuclid.org/euclid.ss/1177012015), which gives an upper bound on the total variat... | 12 | https://mathoverflow.net/users/36721 | 325467 | 140,187 |
https://mathoverflow.net/questions/325368 | 1 | I've been reading [Paolini-Stepanov arcticle](https://arxiv.org/abs/1303.5664) and in section 4, at page 6, they define a metric current from a transport:
$$T\_{\eta}(\omega)=\int\_{\Theta}[[\theta]](\omega)d\eta(\theta),$$
which satisfies
$$\mu\_{T\_\eta}\leq\int\_{\Theta}\mu\_{[[\theta]]}d\eta.$$
Where $\Theta$ is th... | https://mathoverflow.net/users/56713 | Metric 1-current decomposition | I think for any $\bar e \in \mathbb R^2$ the current $\bar e \wedge \mathcal{L}^2 \llcorner Q$ is defined in their paper via $<\bar e \wedge \mathcal{L}^2 \llcorner Q, \varphi> = \int\_Q \bar e\cdot \varphi \, dx$ for any bounded Borel $\varphi \colon Q \to \mathbb R^2$. Therefore $T\_{\eta\_1} = \bar e\_1 \wedge \math... | 2 | https://mathoverflow.net/users/44463 | 325470 | 140,188 |
https://mathoverflow.net/questions/324940 | 9 | Consider the connective $K$-theory spectrum $ku$. Let $H\mathbb{Z}$ be the Eilenberg-MacLane spectrum and $H\mathbb{F}\_p$ be the mod-$p$ Eilenberg-MacLane spectrum.
>
> Is it known what $ku^{\*}(H\mathbb{Z})$, or $ku^{\*}(H\mathbb{F}\_{p})$, is?
>
>
>
| https://mathoverflow.net/users/108963 | The connective $k$-theory cohomology of Eilenberg-MacLane spectra | Charles Rezk already answered this in the comments; I'll just expand on what he wrote. [This paper](https://faculty.math.illinois.edu/~rezk/mahowald-rezk-bc-dual-final.pdf) discusses what's now known as Mahowald-Rezk duality; this is a version of Anderson duality that takes into account "additive degeneration". I'll br... | 8 | https://mathoverflow.net/users/102390 | 325472 | 140,189 |
https://mathoverflow.net/questions/325468 | 2 | I would like to read a popular literature on the topic "Numerical methods for integro-differential equations".
Could you recommend me any articles or book with a brief overview of some methods (maybe classical one), please?
Which methods are most popular?
Thank you in advance!
| https://mathoverflow.net/users/137078 | Numerical methods for IDE | I must premise that I am not a specialist in numerical analysis, therefore I may be not right when talking about more popular methods in this field pertaining the solution of IDEs. Said that, however, I think I can be of some help.
>
> Could you recommend me any articles or book with a brief overview of some method... | 2 | https://mathoverflow.net/users/113756 | 325474 | 140,190 |
https://mathoverflow.net/questions/325409 | 3 | Let $P = N(\vec{0}, I^d)$ be a standard multivariate Gaussian distribution in $d$ dimensions. Let $Q$ be distributed the same as $P$, except that samples from $Q$ have one of their coordinates, chosen uniformly at random from $1$ to $d$, distributed as $N(\mu, \sigma^2)$ instead.
That is, $Q$ is a mixture of $d$ mult... | https://mathoverflow.net/users/57321 | Asymptotic bound on the total variation distance between a standard multivariate normal and a simple mixture | $\newcommand{\m}{\vec{\mu}}
\newcommand{\e}{\vec{e}}
\newcommand{\x}{\vec{x}}$
Let $n:=d$. Assume that $\sigma=1$ and $\mu\ne0$. Then for the densities
\begin{equation\*}
f\_Q(\x)=(2\pi)^{-n/2}\,\frac1n\,\sum\_1^n e^{-|\x-\mu\e\_i|^2/2}
\end{equation\*}
and
\begin{equation\*}
f\_P(\x)=(2\pi)^{-n/2}\,e^{-|\x|^2/2}... | 2 | https://mathoverflow.net/users/36721 | 325482 | 140,193 |
https://mathoverflow.net/questions/325309 | 8 | In Higson and Roe's Analytic K-homology, for a unital $C\*$-algebra $A$, the definitions of K-theory and K-homology have quite a similar flavor.
Roughly, the group $K\_0(A)$ is given by the Grothendieck group of homotopy classes of matrix algebra projections over $A$.
On the other hand, $K^0(A)$ is the Grothendiec... | https://mathoverflow.net/users/128876 | Comparing the definitions of $K$-theory and $K$-homology for $C^*$-algebras | KK-theory provides one way to eliminate the asymmetry in the definitions of K-theory and K-homology. A cycle in $KK(A, B)$ is a triple $(H, \rho, F)$ where $H$ is a (adjectives) Hilbert $B$-module, $\rho$ is a (adjectives) representation of $A$ on $H$, and $F$ is a (adjectives) bounded operator on $H$ such that $[F, \r... | 5 | https://mathoverflow.net/users/4362 | 325486 | 140,194 |
https://mathoverflow.net/questions/325421 | 5 | Let $\Gamma\_0(N)$ be the Hecke congruence subgroup. Let $S\_{k+1/2}(\Gamma\_0(N))$ be the space of holomorphic forms of weight $k+1/2$ on $\Gamma\_0(N)$, where $k\in\mathbb{N}$. How to prove that if $S\_{k+1/2}(\Gamma\_0(N))\neq0$ then $4\mid N$? It should be in Shimura's paper in 1973. But since I am not familiar wit... | https://mathoverflow.net/users/70741 | Why the level of a half integral weight modular form must be a multiple of 4? | The problem isn't that $S\_{k + 1/2}(\Gamma\_0(N))$ is zero if $4 \nmid N$; it's that the space is *not defined* if $4 \nmid N$.
In order to make sense of what a "half-integer weight form of level $\Gamma$" means, for some subgroup $\Gamma \subseteq SL\_2(\mathbf{Z})$, you need to have a consistent way of choosing s... | 14 | https://mathoverflow.net/users/2481 | 325490 | 140,196 |
https://mathoverflow.net/questions/325492 | 3 | Let $G$ be a random cubic graph on $n$ vertices. Let $M$ be the set of (not necessarily maximum) matchings of $G$. What is the expected size (i.e. number of edges) of an element of $M$?
In other words, what is the typical size of a matching in a typical cubic graph?
The Bethe lattice approximation gives the number ... | https://mathoverflow.net/users/125498 | Expected size of matchings in a cubic graph | This has been done before, but I can't find it. I'll outline how it can be proved, without dotting all the "$i$"s.
Consider the $d$-regular case, for $n$ vertices. The expected number of matchings with $K$ edges can be obtained by dividing two values from (for example) [Thm 1 in this paper](https://core.ac.uk/downloa... | 4 | https://mathoverflow.net/users/9025 | 325499 | 140,199 |
https://mathoverflow.net/questions/325289 | 1 | Do there exist constants $d>0$, $0<c<1$, $\delta>0$ so that for all large $n$, there exists a graph $H$ satisfying $$e\_H\ge dn^2,$$ and then no matter how we remove some edges from $H$ to get an $n$-vertex subgraph $G$, whenever $G$ has minimum degree at least $cn$, there are two disjoint vertex-subsets $A,B$ of $G$ w... | https://mathoverflow.net/users/50335 | Find large "induced" bipartite graph in a dense graph? | The answer is no (if I understood your question correctly this time).
Fix $H$ and take $G$ to be a random subgraph of $H$, where you keep each edge with probability $p$, for some constant $p>0$. For fixed sets $A$ and $B$ with at least $\tilde{\delta}n^2$ edges of $H$ between them, the number of edges of $H\setminus... | 2 | https://mathoverflow.net/users/45855 | 325507 | 140,201 |
https://mathoverflow.net/questions/325501 | 0 | Suppose random variables $X\_1$ and $X\_2$ have the same distribution under P, $Y\_1$ is an arbitrary random variable,let $Z\_1:=X\_1+Y\_1$.Can we find a r.v. $Y\_2$ which has same distribution as $Y\_1$,such that $Z\_2:=X\_2+Y\_2$ having same distribution as $Z\_1$?
I gave much reflection on the problem but still had ... | https://mathoverflow.net/users/137090 | How to find a special random variable? | Yes, if the underlying probability space is "rich enough." In particular, assume there is a uniformly distributed random variable $W$ that is independent of $X\_2.$ Let $\mu$ be the joint distribution of $X\_1$ and $Y\_1$ and let $\kappa:\mathbb{R}\times\mathcal{B} \to [0,1]$ be a regular conditional probability of $\m... | 1 | https://mathoverflow.net/users/35357 | 325508 | 140,202 |
https://mathoverflow.net/questions/325495 | 2 | I think that the following statement is true, but I do not know how to prove it.
Let $\mathfrak{g}\_1$ and $\mathfrak{g}\_2$ be two real simple Lie algebras. If $M$ is a (infinite dimensional) complex simple $\mathfrak{g}\_1\oplus\mathfrak{g}\_2$-module, then $M\cong M\_1\otimes M\_2$ where $M\_1$ and $M\_2$ are simp... | https://mathoverflow.net/users/56989 | Simple modules for direct sum of simple Lie algebras | This is true more generally for any two finite-dimensional Lie algebras over a field. Recall that a $\frak{g}$-module is the same as a module over $U (\frak{g})$, the universal enveloping algebra of $\frak{g}$, which is an *associative* algebra. Moreover, $U ({\frak{g}\_1\oplus\frak{g}\_2})\simeq U({\frak{g}\_1})\otime... | 6 | https://mathoverflow.net/users/5740 | 325510 | 140,203 |
https://mathoverflow.net/questions/325433 | 5 | I am sorry to mislead the notations: $SU(n)$ should be replaced by $PSU(n)$. I will reformulate it now.
We consider the special unitary Lie group $SU(n)$. Then its center is $\mathbb{Z}\_n$ and we define $PSU(n)=SU(n)/\mathbb{Z}\_n$. Then $\pi\_1(PSU(n))=\mathbb{Z}\_n$ and $H^3(PSU(n))=\mathbb{Z}$.
Does there exist... | https://mathoverflow.net/users/46230 | Homotopy classes of maps between special unitary Lie groups | No. Such a map would give rise to a map $PSU(4)/(\mathbb Z/2) \to SU(2)$ inducing multiplication by $d$ on $H^3$ and hence to a map $SU(4) \to SU(2)$ inducing multiplication by $2d$ on $H^3$ and such a map can not exist.
Indeed, let $\Sigma \mathbb{C}P^3 \to SU(4)$ be the axial map (see for instance I.M.James, "The t... | 6 | https://mathoverflow.net/users/33141 | 325519 | 140,207 |
https://mathoverflow.net/questions/325511 | 10 | For $k\in\mathbb{N}$, let $H\_k$ be the free Heyting algebra on $k$ variables $p\_1,\ldots,p\_k$ and $B\_k$ be the free Boolean algebra on the same $k$ variables. Thus, $B\_k$ has $2^{2^k}$ elements (corresponding to all possible truth-value tables with $k$ entries), while $H\_k$ is infinite as soon as $k\geq 1$.
Sin... | https://mathoverflow.net/users/17064 | Fibers of the morphism from the free Heyting algebra to the free Boolean algebra | $\let\eq\leftrightarrow$Notice that $\psi(A)=u$ iff $\vdash\_\mathrm{CPC}A\eq u$ iff $\vdash\_\mathrm{IPC}\neg\neg(A\eq u)$. (I will write just $\vdash$ for $\vdash\_\mathrm{IPC}$.) Thus:
* $\bot$ has a one-element fiber consisting of $\bot$: if $\vdash\neg\neg(A\eq\bot)$, then $\vdash\neg A$.
* For each $i$, $u\_i:=... | 12 | https://mathoverflow.net/users/12705 | 325521 | 140,208 |
https://mathoverflow.net/questions/325464 | 10 | Both the $\mathfrak{sl}\_2$ and $\mathfrak{sl}\_3$ quantum framed link invariants can be computed using linear skeins. The first being computed using the Kauffman bracket and the second using a similar bracket which involves trivalent graphs. The recursive rules prescribed to compute these two invariants via skeins are... | https://mathoverflow.net/users/137075 | Is the quantum $\mathfrak{sl}_3$ invariant stronger than the quantum $\mathfrak{sl}_2$ invariant? | The quantum $\mathfrak{sl}\_3$ invariant is a special case of the HOMFLY-PT polynomial, which is essentially the $\mathfrak{sl}\_N$ invariant. That polynomial has two variables $q$ and $t$. The $q$ variable corresponds to the deformation parameter, and is the same $q$ as in the $\mathfrak{sl}\_2$ and $\mathfrak{sl}\_3$... | 6 | https://mathoverflow.net/users/113402 | 325533 | 140,212 |
https://mathoverflow.net/questions/325532 | 23 | Let $A\subseteq \mathbb{R}$ be Lebesgue-measurable and let $0<\alpha<1$ be its Hausdorff dimension.
>
> For a given $0<\beta <\alpha$ can we find a subset $B\subset A$ with Hausdorff dimension $\beta$?
>
>
>
In case this is true, could you provide a reference for this statement?
**Added:** Actually I am happ... | https://mathoverflow.net/users/91098 | Existence of subset with given Hausdorff dimension | First of all, $\dim\_{H} (A) = \alpha$ iff $ H^k(A)=\infty$ for all $k<\beta$ and $H^k(A) = 0$ for all $k>\beta$. Then $H^\alpha(A) = \infty$ for all $\alpha \in (0,\beta)$.
If $A$ is *closed* then by Theorem 5.4 from [1] there is a compact $K\subset A$ such that $0<H^\alpha(K)<\infty$.
More generally, if $A$ is *... | 14 | https://mathoverflow.net/users/44463 | 325537 | 140,214 |
https://mathoverflow.net/questions/325265 | 1 | Is there a classification of all equivariant structures of the Möbius line bundle $\ell\to S^1$?.
For example the antipodal action of $\mathbb{Z}/2\mathbb{Z}$ on $S^1$ cannot be lifted to the total space $\ell$ to get an equivariant structure. But what about the general case?
In particular, let $\phi\_{\theta}$ be ... | https://mathoverflow.net/users/36688 | Classification of all equivariant structure on the Möbius line bundles | Most of my work is with actions of compact Lie groups, so forgive me if I make any errors below related to other groups.
I'm not sure how general a classification you're looking for, but I'm going to assume a framework in which I can say something: Consider a group $G$ acting on the base space $S^1$ by rotations, so ... | 4 | https://mathoverflow.net/users/58888 | 325545 | 140,217 |
https://mathoverflow.net/questions/40686 | 8 | Let P be the statement: Every ccc partial order has $\omega\_1$-precaliber; i.e., every uncountable subset $X$ of a ccc partial order $P$ has an uncountable subset $Y$ such that for every finite subset $F$ of $Y$, there is a member in $P$ below every member of $F$.
Let Q be the statement: Product of ccc partial order... | https://mathoverflow.net/users/2689 | Variants of Martin's axiom | Todorcevic and Velickovic ([Martin's axiom and partitions](http://www.numdam.org/article/CM_1987__63_3_391_0.pdf)) proved $P$ implies $\text{MA}\_{\aleph\_1}$.
| 7 | https://mathoverflow.net/users/67193 | 325557 | 140,221 |
https://mathoverflow.net/questions/325360 | 8 | In his paper "Comparing analytic assemby maps", J. Roe considers a proper and cocompact action of a countable group $\Gamma$ on a metric space $X$. He constructs the Hilbert $C^\*\_r(\Gamma)$-module $L^2\_\Gamma(X)$ as the simultaneous completion of the $\mathbb C\Gamma$-module $C\_c(X)$ under the $\mathbb C\Gamma$-val... | https://mathoverflow.net/users/17229 | Morita equivalence of the invariant uniform Roe algebra and the reduced group C*-algebra | To define the Roe algebra abstractly, we need a suitable covariant representation of the $\Gamma$-$C^\ast$-algebra $(C\_0(X), \Gamma)$ on some Hilbert space $H$.
In concrete situations, one usually takes some $L^2$-space on $X$.
However, if the action of $\Gamma$ on $X$ is not free, then we need to be more careful abou... | 4 | https://mathoverflow.net/users/66077 | 325564 | 140,223 |
https://mathoverflow.net/questions/325566 | 46 | Oftentimes open problems will have some evidence which leads to a prevailing opinion that a certain proposition, $P$, is true. However, more evidence is discovered, which might lead to a consensus that $\neg P$ is true. In both cases the evidence is not simply a "gut" feeling but is grounded in some heuristic justifica... | https://mathoverflow.net/users/8927 | Have the tides ever turned twice on any open problem? | I think that originally there was a belief (at least on the part of some mathematicians) that for an elliptic curve $E/\mathbb{Q}$, both the size of the torsion subgroup of $E(\mathbb Q)$ and its rank were bounded independently of $E$. The former is true, and a famous theorem of Mazur. But then as curves of higher and ... | 51 | https://mathoverflow.net/users/11926 | 325571 | 140,224 |
https://mathoverflow.net/questions/325543 | 9 | I know that the Cohen-Macaulay type has these two definitions:
* Let $(R,\mathfrak{m},k)$ be a Cohen-Macaulay (noetherian) local ring and $M$ a finite $R$-module of depth $t$. The number $r(M) = \dim\_k \mathrm{Ext}\_R^t(k,M)$ is called the Cohen-Macaulay type of $M$.
* Denote by $\beta\_i(M)$ the Betti numbers in a ... | https://mathoverflow.net/users/135244 | Equivalence of definitions of Cohen-Macaulay type | For the equivalence you need two more assumptions: (a) $M$ to have finite projective dimension and (b) $R$ to be Gorenstein. (a) is implicitly required in the second condition, otherwise you don't have the last nonzero Betti number. For necessity of (b), take $M$ to be $R$. Then the last betti is $1$, and the type of $... | 8 | https://mathoverflow.net/users/2083 | 325576 | 140,225 |
https://mathoverflow.net/questions/325582 | 10 | I am looking for examples of constructions for transfinite towers $(X\_{\alpha})\_{\alpha}$ generated by structures $X$ where the problem of determining whether the tower $(X\_{\alpha})\_{\alpha}$ stops growing is a non-trivial problem or the problem of determining the ordinal in which $X\_{\alpha}$ stops growing is a ... | https://mathoverflow.net/users/22277 | Examples of transfinite towers | Consider the following construction of sets of ordinals.
* $X\_0=\{0\}$,
* $X\_{\alpha+1}=$ the closure of $X\_{\alpha}$ under $\gamma\mapsto\gamma+1$ and under *countable* sums,
* $X\_\alpha=\bigcup\_{\beta<\alpha}X\_\beta$ for $\alpha$ limit.
We can consider in ZF the question of whether this tower stabilizes.
... | 11 | https://mathoverflow.net/users/30186 | 325585 | 140,227 |
https://mathoverflow.net/questions/71545 | 7 | Let $Y^3$ be a handlebody with boundary $\Sigma$. By definition, there is some associated vector $v\_{WRT}(Y^3)\in Z(\Sigma)$, the (finite dimensional) Hilbert space associated to $\Sigma$ by the Witten-Reshetikhin-Turaev TQFT. I'd like to understand what this vector is.
In short, $Z(\Sigma)$ is a space of sections o... | https://mathoverflow.net/users/35353 | What is the state in the WRT TQFT associated to a handlebody? | I can think of two cases where the Witten-Reshetikhin-Turaev vector $Z\_k(Y^3)\in Z\_k(\Sigma)$ has been connected to a Lagrangian state as you described:
-Laurent Charles and Julien Marché showed that the WRT vector of the figure eight knot complement $Z\_k(E\_K)$ is a Lagrangian state concentrating on $\mathcal{M}(... | 4 | https://mathoverflow.net/users/62201 | 325589 | 140,228 |
https://mathoverflow.net/questions/325593 | 2 | Let $X$ be a random variable s.t. for $v, b > 0$ and $C \geq 1$:
$$ P(X \geq t) \leq C\exp\left(-\frac{t^2}{2(v^2 + bt)} \right) $$
I am trying to show that $\mathbb{E}X \leq 2v(\sqrt{\pi} + \sqrt{\log C}) + 4b(1 + \log C)$.
In terms of progress I have made so far, I set $X = \int\_0^\infty P(X \geq t)dt$, and h... | https://mathoverflow.net/users/128729 | Bounding integral arising from expectation of a random variable satisfying Bernstein's inequality | For $t>0$, we have $v^2+bt\le\max(2bt,2v^2)$, whence
\begin{equation}
P(X\ge t)\le1\wedge\max[Q\_1(t),Q\_2(t)]\le[1\wedge Q\_1(t)]+[1\wedge Q\_2(t)],
\end{equation}
where $x\wedge y:=\min(x,y)$,
\begin{equation}
Q\_1(t):=C\exp\Big(-\frac{t^2}{4bt}\Big),\quad
Q\_2(t):=C\exp\Big(-\frac{t^2}{4v^2}\Big).
\end{equatio... | 4 | https://mathoverflow.net/users/36721 | 325599 | 140,230 |
https://mathoverflow.net/questions/325560 | 3 | If for any graph $H$ we define $\alpha(H)$ to be the cardinality of any maximum size indepedent set in $H$. Then under what conditions can any graph $G$ be expressed as a union of $\alpha(G)$ complete graphs?
Or equivalently if $[S]\_2$ is the set of all two element subsets of $S$, then for which class of graphs $G$ ... | https://mathoverflow.net/users/38626 | When can any graph $G$ be expressed as a union of $\alpha(G)$ complete graphs? | I don't think there is a pretty answer to this question. A graph is in this class iff it is the union of a set of cliques such that each of the cliques has a vertex not in any of the other cliques.
Given a maximum independent set $S$, you can identify said cliques as the closed neighbourhoods of the vertices in $S$. ... | 1 | https://mathoverflow.net/users/9025 | 325600 | 140,231 |
https://mathoverflow.net/questions/325581 | 6 | \*\* I simplified the question: \*\*
On bounded domains, the maximum principle implies that the solution to the heat equation is (strictly) positive, if the initial and boundary data is positive.
I would like to know whether this result is also true if applied on an unbounded domain:
Let $a>0$ and $g$ a continuo... | https://mathoverflow.net/users/nan | Properties of heat equation | The answer is no. A counterexample has the form
$$u(x,t)=\sum\_0^\infty \frac{g^{(n)}(t)}{((2n+1)!)^2}(r-R)^{2n+1},$$
where $r>R$ is the distance from the origin in $R^2$.
First one shows that this function satisfies the heat equation formally. Second,
there exists an infinitely differentiable $g\not\equiv 0$ such that... | 6 | https://mathoverflow.net/users/25510 | 325611 | 140,233 |
https://mathoverflow.net/questions/325594 | 20 | I apologize if this is off topic.
I think most of his listeners would agree with me that Gian-Carlo Rota had a wonderful style of lecture delivery. I have heard him lecture, both as an undergraduate lecturer and a technical colloquium speaker.
My efforts to find any video recordings of Rota delivering a mathematica... | https://mathoverflow.net/users/17773 | Videos of Gian-Carlo Rota Lectures | Here is an [Introduction to Geometric Probability](https://bookstore.ams.org/video-102/) by Rota --- sold by the AMS for quite a hefty sum.
This [transcript](http://www-history.mcs.st-and.ac.uk/Extras/Rota_Snapshots.html) of a 1997 lecture might convey some of the flavor without the expense. And here is [one more tra... | 10 | https://mathoverflow.net/users/11260 | 325615 | 140,235 |
https://mathoverflow.net/questions/325624 | 7 | Let $f: \mathbb{R}^m\rightarrow \mathbb{R}^{m-k}$ be a Lipschitz map.
>
> Can we get a uniform estimate on the Hausdorff dimension of the boundaries of fibres of $f$? I.e. do we have an upper bound for
> $$ \sup\_{y\in \mathbb{R}^{n-k}} \dim\_H(\partial f^{-1}(\{y\})) ?$$
>
>
>
Theorem 2.5 in [1] tells us, t... | https://mathoverflow.net/users/91098 | Hausdorff dimension of the boundary of fibres of Lipschitz maps | Unfortunately, you can always find a Lipschitz map
$$
f:\mathbb{R}^m\to\mathbb{R}^{m-k}
\quad
\text{and}
\quad
y\in\mathbb{R}^{m-k}
$$
such that $\partial f^{-1}(y)$ has positive $m$-dimensional measure so
$\dim\_H \partial f^{-1}(y)=m$.
Here is an example. Let $K\subset\mathbb{R}^m$ be a Cantor set (i.e. a set hom... | 8 | https://mathoverflow.net/users/121665 | 325627 | 140,240 |
https://mathoverflow.net/questions/270120 | 7 | In a sense, this is a follow-up to [this question](https://mathoverflow.net/questions/180723/simply-connected-4-manifolds-can-be-blown-up-and-down-to-complex-projective-plan).
By work of Freedman and Wall, it is known that if two *simply-connected* 4-manifolds $M$ and $N$ are homeomorphic, then there is $k \in \mathb... | https://mathoverflow.net/users/13767 | How can Wall's theorem be generalised to non-simply connected manifolds? | Gompf showed that the statement can indeed be generalized in [this 1984 paper](https://www.sciencedirect.com/science/article/pii/016686418490004X). He proved that any two smooth structures on a compact 4-manifold become diffeomorphic after taking the connected sum with some number of copies of $S^2\times S^2$.
| 10 | https://mathoverflow.net/users/83276 | 325639 | 140,248 |
https://mathoverflow.net/questions/325632 | 7 | This [Hausdorff dimension of the graph of an increasing function](https://mathoverflow.net/questions/304573/hausdorff-dimension-of-the-graph-of-an-increasing-function) shows that:
>
> Let $f$ be a continuous, strictly increasing function from $[0,1]$ to
> itself with $f(0)=0, f(1)=1$. Then $dim\_H \; G = 1$ where ... | https://mathoverflow.net/users/82839 | Box dimension of the graph of an increasing function | [Pietro Majer's argument that you cited](https://mathoverflow.net/questions/304573) actually shows that the upper box dimension is $1,$ and hence the lower box, the upper and lower packing, and the Hausdorff dimensions are all equal to $1.$ Also, graphs of real-valued Lipschitz functions defined on an interval (of posi... | 6 | https://mathoverflow.net/users/15780 | 325644 | 140,250 |
https://mathoverflow.net/questions/325651 | 2 | I have asked the below question [on MathSE](https://math.stackexchange.com/questions/3144414/x-rtimes-y-simeq-x-vee-x-wedge-y-for-x-a-co-h-space) (with a 200 point bounty) but have yet to receive an answer there, and so am trying here. I am happy to remove it if it is nevertheless decided that this question is not appr... | https://mathoverflow.net/users/103150 | $X \rtimes Y \simeq X \vee (X \wedge Y)$ for $X$ a co-H-Space | The proof is not hard, but more tedious than I would have thought.
There is a canonical identification
$$
X\rtimes Y = X\wedge(Y\_+)
$$
where $Y\_+$ is the effect of adding a disjoint base point to $Y$ (we have forgotten the original basepoint here).
Furthermore, $X\wedge (Y\_+)$ is a co-H space when $X$ is (smash... | 6 | https://mathoverflow.net/users/8032 | 325654 | 140,253 |
https://mathoverflow.net/questions/325570 | 5 | Let $H$ be a Hilbert space and $B(H)$ be its space of all (bounded) operators. A *nuclear functional* on $B(H)$ is a linear functional $f:B(H)\to{\mathbb C}$ that can be represented in the form
$$
f(A)=\sum\_{n=1}^\infty \lambda\_n\cdot \langle Ax\_n,y\_n\rangle,\qquad A\in B(H),
$$
where $\lambda\_n\in{\mathbb C}$, $x... | https://mathoverflow.net/users/18943 | Is a set of nuclear functionals equicontinuous in compact-open topology if it is equicontinuous on each compact set? | The answer is YES. First of all, your $N(H)$ is nothing but the predual of $B(H)$ (the ultraweakly continuous linear functionals) and the nuclear norm is nothing but the norm as linear functionals.
We assume $\sup\_{f\in F}\| f\|\le1$.
Let $P\_n \in B(H)$ be finite rank orthogonal projections
such that $P\_n \near... | 1 | https://mathoverflow.net/users/7591 | 325658 | 140,255 |
https://mathoverflow.net/questions/325597 | 8 | The dual Steenrod algebra ($p=2$) has generators $\xi\_n$ and these have conjugates that are often labeled $\zeta\_n$. I am curious about the left and right actions of the Steenrod algebra on its dual, and in particular, what the total square is. I have seen in papers that $(\xi\_n)Sq = \xi\_n + \xi\_{n-1}$ and $Sq(\xi... | https://mathoverflow.net/users/66958 | What is the total square on the dual Steenrod algebra? | I don't have a reference for you, but here is a comment on how to prove these formulas using the Kronecker pairing that you alluded to.
The Steenrod operation $Sq^m$ is dual to the element $\xi\_1^m$ in the monomial basis of the dual Steenrod algebra; the left and right actions of the Steenrod algebra on $\mathcal{A}... | 3 | https://mathoverflow.net/users/360 | 325664 | 140,257 |
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