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https://mathoverflow.net/questions/201669 | 2 | Let $V \subset H$ be Hilbert spaces with a continuous, compact and dense imbedding. Let $\{w\_j\}\_j \subset V$ be a basis of $V$ and of $H$ (so finite linear combinitions are dense) which is **not** orthogonal. Define $V\_n = \text{span}(w\_1, ..., w\_n)$.
Is it possible to find an operator $Q\_n:V \to V\_n$ satisfy... | https://mathoverflow.net/users/68727 | Existence of a projection operator onto subspace of Hilbert space | I think the answer, in general, is no. Here's a counter-example; perhaps someone else can find a simpler one. Take $V = \ell^2$, and let the inner product for $H$ restricted to $V$ correspond to the infinite tridiagonal matrix
$$
\begin{bmatrix} 2 & -1 & & \\ -1 & 2 & -1 & \\ & \ddots & \ddots & \ddots \end{bmatrix}.
$... | 2 | https://mathoverflow.net/users/70005 | 201735 | 97,305 |
https://mathoverflow.net/questions/201583 | 2 | Let $u: \mathbb{R}^2 \to \mathbb{R}.$ Suppose I have a solution to the equation
$$\Delta(u)+e^{-u} \geq 0$$ on $\mathbb{R}^2$. Let r be the radial coordinate on $\mathbb{R}^2$. Suppose that $$lim\_{r \to \infty} u \to c$$
where $c$ is some constant. Can I get some estimate on the maximum of $u$ over all of $\mathb... | https://mathoverflow.net/users/36931 | "C^0 estimate for solutions to $\Delta(u)+e^{-u} \geq 0$" | **One cannot bound $\sup u$ in terms of $c$.** To see this, set $u(x) = c + \alpha \,\varphi(\varepsilon x)$, where $\varphi\in S(\mathbb R^2;\mathbb R)$, $\varphi(0)=1$, and $\varphi\leq1$, while $\alpha>0$, $\varepsilon>0$ are parameters. Notice that
$$
\Delta u(x) + e^{-u(x)} \geq e^{-\alpha} \left( e^\alpha\alpha\... | 5 | https://mathoverflow.net/users/69194 | 201740 | 97,309 |
https://mathoverflow.net/questions/201767 | 2 | For concreteness, let us work with the language of spectra introduced in EKMM.
In Strickland's paper "Products on $MU$-modules", he proves the following. If $R$ is a q-cofibrant commutative $S$-algebra which is even, in the sense that its homotopy is concentrated in even degrees, and $x\in R\_d$ is a non-zero divisor... | https://mathoverflow.net/users/6249 | Cofiber sequence $A\vee A \to A \wedge A \to \bar{A}\wedge \bar{A}$ for a spectrum $A$ | This will not work in exactly the form that you state, as you will see if you consider the case where $A=S\vee\overline{A}$; there is an extra factor of $S$. To fix this, you should replace $A\vee A$ by the pushout of the diagram $A\xleftarrow{}S\xrightarrow{}A$, which we can call $P$. It is then true that the cofibre ... | 7 | https://mathoverflow.net/users/10366 | 201772 | 97,315 |
https://mathoverflow.net/questions/201757 | 9 | $\DeclareMathOperator\lcm{lcm}$We know that $\operatorname{lcm}(1,\dotsc,n)$ is approximately $e^n$ and we also know that $\gcd(2^a-1, 2^b-1)=2^{\gcd(a,b)}-1$.
I wonder if there exists an upper bound/lower bound/approximation for $\operatorname{lcm}(2^1-1, 2^2-1,\dotsc,2^n-1)$.
| https://mathoverflow.net/users/10286 | Do we know any bound on $\operatorname{lcm}(2^1-1, 2^2-1,\dots,2^n-1)$? | Here my guess that seems to be confirmed by numerical computation.
Let $(u\_n)\_{n \geq 0}$ be a non-degenerate Lucas sequence of the first kind with a dominant positive real root $\alpha$.
In other words, $u\_0 = 0$, $u\_1 = 1$, $u\_{n+2} = a u\_{n+1} + b u\_n$ for each $n \geq 0$ for some relatively prime integers ... | 18 | https://mathoverflow.net/users/nan | 201775 | 97,316 |
https://mathoverflow.net/questions/201782 | 1 | The following question concerns the well-known paper of Dwyer and Kan "Localization of Simplicial Categories". They define a nerve for simplicial categories (with fixed set of objects $O$), by the following construction: given $\mathcal{A}\in sO-Cat$, they set $$\mathcal{N}\mathcal{A}:=diag(k \mapsto N \mathcal{A}\_k)$... | https://mathoverflow.net/users/57280 | Homotopy invariance of Kan nerve of simplicial categories | Let $\mathcal{N}\_\*\mathcal{A}$ denote the bisimplicial set $k \mapsto N\mathcal{A}\_k$. If $\mathcal{A} \to \mathcal{B}$ is a DK-equivalence, then $\mathcal{N}\_\*\mathcal{A} \to \mathcal{N}\_\*\mathcal{B}$ is a natural weak equivalence of bisimplicial sets, except not in $k$ but rather in $l$ the "nerve simplicial d... | 4 | https://mathoverflow.net/users/12547 | 201785 | 97,320 |
https://mathoverflow.net/questions/201770 | 1 | What is known, for $N\geq3$, about the existence of **nontrivial** real-valued solutions $u=u(x)$ of the following semilinear elliptic equation:
$$
\left\{ \enspace
\begin{aligned}
&\Delta u = f(u) && \text{in ${\mathbb R}^N$,} \\
&u(x) \to 0 && \text{as $|x|\to\infty$.}
\end{aligned}
\right.
$$
Here, $f\in \mathscr... | https://mathoverflow.net/users/69194 | Nontrivial solutions of a semilinear elliptic equation | Meanwhile, I have found an answer myself. It is reasonable to assume that $u\in L^\infty(\mathbb R^N)$ and $\nabla u\in L^2(\mathbb R^N;\mathbb R^N)$. Then $F(u)\in L^1(\mathbb R^N)$ as $\operatorname{supp}F(u)$ is compact.
From Pohozaev's identity
$$
\frac{N-2}{N}\,\left\|\nabla u\right\|\_{L^2}^2 + \int\_{\mat... | 1 | https://mathoverflow.net/users/69194 | 201795 | 97,322 |
https://mathoverflow.net/questions/201709 | 2 | assuming we have two smooth function ${f\_1},{f\_2}:{R^N} \to R$,
1. under what condition, we have
${f\_1}\left( {{{\bf{x}}\_1}} \right) \ge {f\_1}\left( {{{\bf{x}}\_2}} \right) \leftrightarrow {f\_2}\left( {{{\bf{x}}\_1}} \right) \ge {f\_2}\left( {{{\bf{x}}\_2}} \right), \forall \bf{x}\_1\neq \bf{x}\_2\in{R^N}$
... | https://mathoverflow.net/users/38361 | monotonicity alike functions | Let Property A be the property described above.
Let Property B be the property that $f\_2=g \circ f\_1$ for some strictly monotonically increasing function $g:R\rightarrow R$.
Then Property A and Property B are equivalent.
To see that Property B implies Property A, suppose that $f\_1(x\_1)\geq f\_1(x\_2)$. Since ... | 2 | https://mathoverflow.net/users/8938 | 201796 | 97,323 |
https://mathoverflow.net/questions/201774 | 4 | Let $G$ be a graph with $n$ vertices and $m$ edges, and let $d=\lfloor\frac{m}{n}\rfloor$ be the rounded-down average-degree. A lemma that is attributed to Erdos says that $G$ has a non-empty induced sub-graph $H$ of minimum degree at least $d$ (in particular, the lemma is attributed to [1], but I could not find it the... | https://mathoverflow.net/users/24226 | Dense high-degree sub-graphs of dense graphs | Assuming you mean "at least a constant fraction of the edges", this is true and the proof is forced upon on us in the same way as for the original lemma. I'll change notation to have $d$ as the target minimum degree.
If we have a graph $G$ and want a subgraph of minimum degree at least $d$, then we have no choice but... | 4 | https://mathoverflow.net/users/25485 | 201797 | 97,324 |
https://mathoverflow.net/questions/201799 | 5 | Let $A, B \subseteq \mathbb{Z}$ be finite subsets of the integers. Then there exists an element in $A+B$ with a unique representation as a sum of an element in $A$ and an element in $B$, namely $\max(A+B)$ (or $\min(A+B)$).
If $A, B \subseteq \mathbb{Z}\_m$ with $|A+B| < |A|+|B|-1$ or $|A|+|B| \geq m+2$ then $r\_{A+B... | https://mathoverflow.net/users/68529 | Element with unique representation in A+B | First, a technical remark: you do not need to assume $|A|+|B|<m+2$ as this is implied by your first assumption $|A+B|\ge|A|+|B|-1$, and indeed you do not need to assume $|A+B|\ge |A|+|B|-1$ either since otherwise, as you mention, there are no elements with a unique representation in $A+B$.
To the essence of your ques... | 6 | https://mathoverflow.net/users/9924 | 201805 | 97,328 |
https://mathoverflow.net/questions/201821 | 3 | Suppose that $\mu$ and $\nu$ are sigma-finite measures on the Borel sigma-algebra over $\mathbb R$ such that $\int\_{\mathbb R}f\,d\mu=\int\_{\mathbb R}f\,d\nu$ for all nonnegative continuous functions $f$ on $\mathbb R$. Does it always follow that $\mu=\nu$? Some related information can be found at [math.stackexchange... | https://mathoverflow.net/users/36721 | Is a sigma-finite Borel measure over $\mathbb R$ determined by its values on the continuous functions? | Let $\mu$ be counting measure on the rationals and $\nu$ the translate of $\mu$ by an irrational. Then for nonnegative continuous functions $f$ on $\mathbb R$,
$$ \int f \; d\mu = \int f \; d\nu = \cases{0 & if $f \equiv 0$\cr
+\infty & otherwise\cr}$$
| 11 | https://mathoverflow.net/users/13650 | 201830 | 97,336 |
https://mathoverflow.net/questions/201825 | 4 | Consider the Shimura datum $(GSp\_{2g},\mathcal{H}\_g)$. Let $G$ be a reductive $\mathbb{Q}$-subgroup of $Sp\_{2g}$. I want to know under what condition there exists a point $x\in\mathcal{H}\_g$ such that $h\_x\colon\mathbb{S}\rightarrow GSp\_{2g,\mathbb{R}}$ factors through $Z(GSp\_{2g})G$.
There are some easy neces... | https://mathoverflow.net/users/nan | Subgroups of $Sp_{2g}$ giving rise to Shimura data | First question: No, these conditions are not sufficient. For details see: Deligne, Pierre, Variétés de Shimura: interprétation modulaire, et techniques de construction de modèles canoniques. (French) Automorphic forms, representations and L-functions (Proc. Sympos. Pure Math., Oregon State Univ., Corvallis, Ore., 1977)... | 2 | https://mathoverflow.net/users/4149 | 201835 | 97,338 |
https://mathoverflow.net/questions/201794 | 23 | While trying to carry out some technical arguments in free groups, I have encountered the following problem, to which I don't know the answer.
Let $F$ be a free group and let $g,a\_1,\ldots,a\_n \in F$. Suppose that $g$ is not equal to a product of conjugates of $a\_1,a\_2,\ldots,a\_n$, in that order. That is, there ... | https://mathoverflow.net/users/35840 | products of conjugates in free groups | The answer to your question is no; there need not be a finite quotient like that. This also answers the question of Lev Glebsky and Luis Manuel Rivera Martinez mentioned in a comment. I learned this argument from Jakub Gismatullin (who presented it in a similar form at a workshop at the Erwin-Schrödinger-Institute in A... | 27 | https://mathoverflow.net/users/8176 | 201837 | 97,339 |
https://mathoverflow.net/questions/201803 | 2 | My question will rely on a clarification of a proof, which I simply don't understand.
Let us denote by $X$ a pseudo-riemannian symmetric space and define
$$
Z\_{\mathrm{Iso}\left(X\right)}G(X) = \{\, f \in \mathrm{Iso}\!\left(X\right) \mid
gf = fg \text{ for all } g \in G(X) \,\}
$$
as the centralizer of the tra... | https://mathoverflow.net/users/70051 | Discrete subgroup of centralizer of transvections in isometries acts properly discontinuously | Question 1: This is a standard Baire category argument. Let $U$ be some small neighborhood of $c$. Since $G(X)$ acts continuously on $X$, there is a neighborhood $V$ of the identity in $G(X)$, such that $V^{-1} V c \subseteq U$. Choose a neighbourhood $V\_1$ of the identity, such that the closure $\overline{V\_1}$ is c... | 1 | https://mathoverflow.net/users/68305 | 201841 | 97,341 |
https://mathoverflow.net/questions/201804 | 3 | Is it true that when the integers of a number field are not a UFD then not every point in projective $n$-space over that field can be given by relatively prime algebraic integer coordinates?
When a point can be given by integer coordinates generating a non-principle ideal as GCD then it seems that point cannot be gi... | https://mathoverflow.net/users/38783 | Projective coordinates over a non UFD ring | The author explains what the "relatively prime" terminology means: "we fix once for all a family of ideals $\mathfrak{a}\_1,\ldots,\mathfrak{a}\_h$ representing the $h$ classes of ideals $\mathfrak{R}\_i$ in $O\_K$ and additionally require from our coordinates to satisfy $(a\_0,a\_1,a\_2) = \mathfrak{a}\_i$" for some $... | 3 | https://mathoverflow.net/users/68305 | 201844 | 97,342 |
https://mathoverflow.net/questions/201806 | 6 | Let $\Lambda$ be a lattice in $\mathbb{R}^n$. For $\bar{x} \in \mathbb{R}^n$, let
$\| \bar{x} \| = max\_{1 \leq i \leq n} \{ |x\_i| \}$, i.e. the sup norm. Let $\lambda\_1, ..., \lambda\_n$
be a successive minima of $\Lambda$ with respect to the sup norm.
I am interested in counting number of points
$N(U) = \# \{ ... | https://mathoverflow.net/users/48408 | Counting number of points in a lattice with bounded sup norm | Your expectation is correct. Let $\lambda\_{j} \leq U < \lambda\_{j+1}$. There are independent lattice vectors $v\_1,\dots,v\_j\in\Lambda$ such that $\|v\_i\|=\lambda\_i$ for $1\leq i\leq j$.
Consider the linear combinations
$ c\_1v\_1+\dots +c\_j v\_j\in\Lambda$ with integral coefficients $c\_i\in\mathbb{Z}$ satisf... | 4 | https://mathoverflow.net/users/11919 | 201846 | 97,343 |
https://mathoverflow.net/questions/201845 | 7 | Let $A \in \mathbb{R}^{m\times n}$ and $\|A\| = \sum\_{i, j} |A\_{i,j}|$.
I am looking for constants $\alpha, \beta \in \mathbb{R}$ such that
$\alpha \|A\| \leq \|A\|\_\* \leq \beta \|A\|$
The function $\|\cdot\|\_\*$ is the Schatten-1 norm, or the sum of the singular values of $A$.
| https://mathoverflow.net/users/17243 | Equivalence of entrywise 1-norm and Schatten-1 norm | The best such inequality that depends only on $m$ and $n$ is:
$$
\frac{1}{\sqrt{mn}}\|A\| \leq \|A\|\_\* \leq \|A\|
$$
The right inequality is tight when $A$ is a matrix with a $1$ in the top-left corner and zeroes elsewhere. The left inequality is tight when $A$ is the matrix all of whose entries are $1$. These exampl... | 10 | https://mathoverflow.net/users/11236 | 201851 | 97,346 |
https://mathoverflow.net/questions/201855 | 3 | Given a basis in a Banach space $X$, can one find, for every $\varepsilon>0$, an equivalent basis with basis constant at most $1+\varepsilon$?
In $L\_p[0,1]$ with $1<p<\infty$ any monotone basis is unconditional so one cannot expect better than $1+\varepsilon$ in general. What about $L\_1[0,1]$, is every basis equiva... | https://mathoverflow.net/users/70079 | Basis equivalent with a monotone basis | P.Enflo (Arkiv Mat., 11 (1973), 103-107) constructed a Banach space $X$ with a basis such that the basis constant constant of any basis in $X$ is $\ge c\_X>1$.
| 5 | https://mathoverflow.net/users/37822 | 201872 | 97,353 |
https://mathoverflow.net/questions/201873 | 2 | I hope this is not too elementary, but I have asked this question at the math.stack site, but I have obtained no answers.
Let $F(M)$ be the frame bundle of a $n$-dimensional differentiable manifold $M$, and let $H\subset TF(M)$ be a connection. A Riemannian metric on $M$ can be equivalently written as an equivariant ... | https://mathoverflow.net/users/66688 | Metric, torsion free connections on principal bundles | I don't know what you mean by 'without using at all the picture on the tangent bundle $TM$', but here is how one normally does it:
First, one shows that there are canonical $1$-forms $\omega^i$ on $F(M)$ that are semibasic for the projection $\pi:F(M)\to M$ and that satisfy the equation
$$
\pi'(v) = \omega^i(v)\,e\_i... | 5 | https://mathoverflow.net/users/13972 | 201877 | 97,354 |
https://mathoverflow.net/questions/134292 | 18 | A particle lies on the real number line at the origin. For each step taken, the particle moves from its current position a distance (and direction) chosen equi-probably from range $[-1,r]$. However, if the particle would otherwise move left of the origin, the particle is set back at the origin before the next step.
... | https://mathoverflow.net/users/35160 | Random Walk on $\mathbb{R}$ with Uniformly Distributed Steps and "Reflective" Boundary at Origin | Let $X\_1,X\_2,\dots$ be independent identically distributed random variables representing the successive jumps of the particle. Let $M\_n$ be the position of the particle at time $n\in\{0,1,\dots\}$, so that $M\_0=0$ and $M\_n=\max(0,M\_{n-1}+X\_n)$ for $n\in\{1,2,\dots\}$. Let now $F\_n(x):=\mathbb{P}(M\_n\le x)$ for... | 6 | https://mathoverflow.net/users/36721 | 201891 | 97,357 |
https://mathoverflow.net/questions/201888 | 5 | I have a question about a detail in the proof of Proposition 1.6 in "The horizontal distribution of zeros of $\zeta^\prime(s)$", K. Soundararajan, Duke J. Math. vol. 91 1998.
Throughout I will simplify by assuming the Riemann Hypothesis. (His results are more general, merely assuming no counterexamples 'nearby')
Lemm... | https://mathoverflow.net/users/6756 | The horizontal distribution of zeros of $\zeta^\prime(s)$ | The zeta function will have $\gg \log T$ zeros with ordinates in the interval $[t\_2+1,t\_2+2]$. The contribution of these zeros to the sum is of absolute value $\gg \log T$.
| 7 | https://mathoverflow.net/users/38624 | 201892 | 97,358 |
https://mathoverflow.net/questions/201893 | 1 | In the stack exchange notes found in Section 10 of [this file](http://stacks.math.columbia.edu/download/derived.pdf), it is claimed that the category $K(\mathcal{A})$ of complexes up to homotopy is a triangulated category, if $\mathcal{A}$ is additive. In the proof of Lemma 10.2 (and earlier as well), given a termwise ... | https://mathoverflow.net/users/3199 | Quotients of termwise split injections, for additive categories | The assumption that the kernel exists is built into the definition of "termwise split injection" in these notes: Definition 9.4 says
>
> A *termwise split injection* $\alpha:A^\bullet\to B^\bullet$ is a morphism of complexes such that each $A\_n\to B\_n$ is isomorphic to the inclusion of a direct summand.
>
>
>
... | 3 | https://mathoverflow.net/users/75 | 201897 | 97,361 |
https://mathoverflow.net/questions/201906 | 7 | Let $\{X\_n\}\_{n\in \mathbb{N}}$ be a direct system of compact topological spaces, meaning that we have morphisms $f\_i\colon X\_i \to X\_{i+1}$ with the necessary compatibility conditions. Is there any chance that the direct limit $X$ is compact?
Any reference is welcome
thanks
| https://mathoverflow.net/users/48866 | Direct limit of compact topological spaces | A $T\_1$ colimit $X$ of a sequence of compact spaces $X\_n$ is compact iff there is some $n$ such that the map $X\_n\to X$ is surjective. This condition is obviously sufficient; suppose that it fails. Passing to a subsequence, we may assume that for each $n$, there is a point $x\_n\in X$ that is in the image of $X\_n$ ... | 21 | https://mathoverflow.net/users/75 | 201909 | 97,368 |
https://mathoverflow.net/questions/201857 | 11 | Let $\mathbb C$ be the complex plane, $H(\mathbb C)$ the set of all entire functions, and $D(\mathbb C)$ the set
of all non-negative divisors in $\mathbb C$.
Consider the map $Z:H(\mathbb C)\to D(\mathbb C)$ which to every entire function $f\in H(\mathbb C)$ puts into correspondence its divisor of zeros.
There are n... | https://mathoverflow.net/users/25510 | Continuous Weierstrass map | The Weierstrass product has the form $W(z)=\prod E\_{N(a)}(z/a)$, where the product is over the set of the desired zeros $a$, and the integers $N(a)$ can be chosen freely; they only need to be large enough asymptotically to ensure convergence.
To avoid the problem you mentioned, we must make sure that $N(a)$ depends ... | 6 | https://mathoverflow.net/users/48839 | 201919 | 97,371 |
https://mathoverflow.net/questions/201916 | 4 | Is there a procedure for finding all matrices which commute with two given square and complex matrices?
For example, given two elements $A,B \in$ $\mathfrak{su}(4)$ is it possible to find all elements which commute with both $A$ and $B$ individually.
| https://mathoverflow.net/users/41654 | Finding commuting matrices | Generically (e.g., if one of the matrices has distinct eigenvalues, or more generally, is non-derogatory; for two commuting matrices, the conditions are even weaker), the centralizer of a single matrix just consists of the polynomials in that matrix. So if you pick a pair of commuting matrices at random, the polynomial... | 7 | https://mathoverflow.net/users/42278 | 201934 | 97,377 |
https://mathoverflow.net/questions/201926 | 14 | Let $X$ be a scheme and let $D^b\_{\text{coh}}(X)$ be the derived category of complexes of sheaves with bounded, coherent cohomologies.
We know that the category $D^b\_{\text{coh}}(X)$ has some drawbacks: when defining the derived category we forget too much information. Hence we have the concept of dg-enhancement. M... | https://mathoverflow.net/users/24965 | What is the applications of the dg-enhancements of derived categories of sheaves | It is hard to know where to begin! A general principle is that as long as you are only concerned with the derived category of a single variety, it is generally sufficient to consider it as a triangulated category; while as soon as you are interested in families of derived categories, it becomes impossible to stay in th... | 15 | https://mathoverflow.net/users/2503 | 201947 | 97,383 |
https://mathoverflow.net/questions/201918 | 3 | Let us consider a self-adjoint elliptic pseudodifferential operator $P \in OPS^2$ on a compact manifold $M$ such that $spec(P) \subset (0, \infty)$. Is the norm $(Pu, u)^{1/2}$ on $H^1(M)$ equivalent to $||u||\_{H^1(M)}$, the usual Sobolev norm? A reference would be greatly appreciated.
| https://mathoverflow.net/users/70111 | Sobolev spaces on compact manifolds | Yes. Take the functional square root $Q:=\sqrt{P}$. It is a 1st order, elliptic, selfadjoint *positive* $\psi$do on $M$ according to the results of Seeley. Then $(Pu,u)^{1/2}=\Vert Qu\Vert\_{L^2}$, $\forall u\in C^\infty (M)$. The operator $Q$ induces a continuous *bijective* linear map
$$Q:H^1(M)\to L^2(M). $$
Us... | 1 | https://mathoverflow.net/users/20302 | 201950 | 97,384 |
https://mathoverflow.net/questions/201946 | 5 | In a (necessarily non-Noetherian) integral domain $A$ of (Krull) dimension $1$, is it possible that there is an infinite collection of prime ideals $\mathfrak{p}\_i$ such that $\cap\_i \mathfrak{p}\_i \neq 0$?
Context: part of Problem 5.2.4 in Qing Liu's *Algebraic Geometry and Arithmetic Curves* asks you to show th... | https://mathoverflow.net/users/56878 | An integral domain of dimension one with a non-trivial infinite intersection of prime ideals | Let $\mathcal{O}$ be the ring of all algebraic integers in $\overline{\Bbb{Q}}$. It is a non-Noetherian integral domain of dimension $1$. For any non-zero prime $\mathfrak{p} \in \Bbb{Z}$, there are infinitely many primes $\mathfrak{p}\_i$ in $\mathcal{O}$ lying above $\mathfrak{p}$, and so $\bigcap \mathfrak{p}\_i \ne... | 12 | https://mathoverflow.net/users/21278 | 201953 | 97,386 |
https://mathoverflow.net/questions/201957 | 2 | For a (compact) Kahler manifold $M$, the **Ricci tensor** is the symmetric $2$-form
$$
r(u,v) = \text{tr}\big( w \mapsto (D\_wD\_u - D\_uD\_w - D\_{[u,w]})v\big).
$$
The **Ricci curvature** is the $2$-form
$$
r(u,v) := r(I(u),v).
$$
So the question! The first Chern class of the Levi--Civita connection for $\Omega^{... | https://mathoverflow.net/users/41562 | Ricci Curvature and the Chern Class of the Levi-Civita | Yes, this is a classic result of Chern's. If $\rho = \sum R\_{a\overline b} dz^a \wedge dz^{\overline b}$ is the Ricci form, then $\frac{i}{2\pi} \rho$ represents the first Chern Class of $M$.
Among the results of Yau's classic paper [Calabi’s conjecture and some new results in algebraic geometry](http://www.pnas.org... | 4 | https://mathoverflow.net/users/394 | 201964 | 97,390 |
https://mathoverflow.net/questions/201956 | 1 | Let $X$ be a connected projective noetherian scheme over $\mathbb{C}$, with every irreducible component of the same dimension. Let $\dim X=n \ge 2$ and $p$ be a closed point on $X$. Denote by $U$ the open subset $X \backslash p$. Let $\mathcal{F}$ be a locally free sheaf on $U$. The question is: for the open immersion ... | https://mathoverflow.net/users/54369 | Pushforward of locally free sheaves under open immersion | I think that the answer is no in general. For instance, it seems that a locally free sheaf on $U$ does not even systematically extend to a locally free sheaf on $X$.
You can find a counterexample in <http://www.mathunion.org/ICM/ICM1970.2/Main/icm1970.2.0619.0624.ocr.pdf> for $X=\mathbb{C}^3$ and $p=0$. (at the bott... | 1 | https://mathoverflow.net/users/7031 | 201965 | 97,391 |
https://mathoverflow.net/questions/201972 | 2 | This question is from the proof of Theorem 11.34 in the book: *Twenty-four Hours of Local Cohomology*.
Let $R$ and $S$ be CM local ring and $R\to S$ a local homomorphism such that $S$ is a finite generated $R$-module, $k$ and $l$ are residue fields of $R$ and $S$ respectively. Let $t = dim R - dim S$, and $\omega$ b... | https://mathoverflow.net/users/40640 | Decide two indices of Ext functor | I don't know if one can prove the isomorphisms for other $a$ and $b$; but about what we have in this proof, you can see that $t$ is an specified number, namely $\dim R - \dim S$ and the proof is about this specific number because as you can see in the book they have wrote "Thus, after shifting $t$ steps to the left, $H... | 1 | https://mathoverflow.net/users/47763 | 201986 | 97,401 |
https://mathoverflow.net/questions/201990 | 5 | I was reading [this question](https://mathoverflow.net/questions/4347/ffx-expx-1-and-other-functions-just-in-the-middle-between-linear-and-exp) on functions "in the middle" of linear and exponential growth, and it caused me to think of this function:
$$ G(x) = \left(1 + \frac{x}{f(x)}\right)^{f(x)} $$
for some non-... | https://mathoverflow.net/users/29697 | The growth rate of $\left(1+\frac{x}{f(x)}\right)^{f(x)}$ | Let's suppose $x/f(x) \to 0$. Then take logarithm. As $x \to \infty$,
$$
\log G(x) = f(x) \log\left(1+\frac{x}{f(x)}\right)
=f(x)\left(\frac{x}{f(x)}+O\left(\left(\frac{x}{f(x)}\right)^2\right)\right)
\\
= x + O\left(\frac{x^2}{f(x)}\right)
\tag{\*}$$
Now if we have the stronger $x^2/f(x) \to 0$, then
$$
G(x) = \exp\le... | 6 | https://mathoverflow.net/users/454 | 201992 | 97,403 |
https://mathoverflow.net/questions/201987 | 2 | Let $\mu$ be a Borel measure with finite variation on a locally compact abelian group $G$, let $\Gamma$ denote the dual group of $G$, and let $\hat \mu: \Gamma \to \mathbb{C}$ be the Fourier-Stieltjes transform of $\mu$.
The measure $\mu$ induces a convolution operator $T\_\mu$ on the space $L\_1(G)$ defined by $T\_\m... | https://mathoverflow.net/users/39421 | How to prove that the convolution operator associated to a discrete measure on a LCA group has natural spectrum? | Discrete measures form a unital subalgebra of the algebra of measures on $G$, isomorphic to $L^{1}(G\_d)$, where $G\_d$ is the group $G$ endowed with the discrete topology. The spectrum in a subalgebra can only be bigger, so we just need to show that the spectrum of $\mu$ considered as an element of $L^{1}(G\_d)$ is co... | 3 | https://mathoverflow.net/users/24953 | 201993 | 97,404 |
https://mathoverflow.net/questions/201979 | 6 | It is well known that every Borel set has the property of Baire. That is, for every Borel set $B$, there is an open set $U$ and a sequence of dense open sets $D\_n$ such that for every $x\in \cap\_n D\_n$, $x\in B \iff x \in U$.
Also well known: every Borel set can be represented by a Borel code, a well-founded tree ... | https://mathoverflow.net/users/6649 | Degree of unsolvability of finding a open approximation to a Borel set, given its Borel code | I think this is a well known fact.
For example, let $\alpha$ be a recursive ordinal and $B=\{g\mid g\mbox{ is a }0^{(\alpha)}\mbox{-generic real}\}$. Then $B$ is a hyperarithmetic set and so has a recursive Borel code $X$.
But no real in $B$ is recursive in $0^{(\alpha)}$ and so any open approximation must not be ... | 4 | https://mathoverflow.net/users/14340 | 201995 | 97,406 |
https://mathoverflow.net/questions/201996 | 3 | I consider non-compact domains $\Omega$ with cylindrical ends. For example, $\Omega$ has a cover $\Omega\_0 \cup (0,\infty) \times M$, where $\Omega\_0$ has finite measure and $M$ is a compact manifold.
By Lockhart-McOwen's results, we know that to study the Laplacian on such domains one needs to use weighted Sobolev... | https://mathoverflow.net/users/70138 | Laplacian on non-compact domains | There is indeed a pseudodifferential calculus that can handle this situation. It is called **SG calculus**, see e.g. [here](http://www.analysis.uni-hannover.de/~schrohe/papers/weighted_symbols.pdf), or **scattering calculus**, see e.g. [here](http://www-math.mit.edu/~rbm/iml/Chapter7.pdf). Symbol estimates (on $\mathbb... | 1 | https://mathoverflow.net/users/69194 | 202006 | 97,411 |
https://mathoverflow.net/questions/201619 | 3 |
>
> This is a crosspost of [this](https://math.stackexchange.com/questions/1199921/projective-family-of-probability-spaces) question from MSE.
>
>
>
I'm confused about the definition of a projective family of probability spaces $(S\_t,\mathscr S \_t,\mu\_t,f\_{ts})\_{s,t\in T}$. The conditions
* $f\_{tt}=1\_{S... | https://mathoverflow.net/users/53127 | Projective family of probability spaces | These condition fall out of the categorical formalism, using the category of measurable spaces, provided you view probability measures from a slightly different perspective. Think of a probability measure as an affine, weakly averaging functional which preserves limits $I^X \rightarrow I$, where $I=[0,1]$...see the pap... | 3 | https://mathoverflow.net/users/70150 | 202014 | 97,413 |
https://mathoverflow.net/questions/202013 | 3 | Let $X = \mathbb A^2$. Say we blow up the origin $(0,0)$, and then blow up the intersection of $(x=0)$ with the exceptional divisor. The resulting space is the blow-up of $\mathbb A^2$ along what subscheme?
I guess that the first blow-up is at $(x,y)$, while the second extracts the same divisor as the blow-up of $(x,... | https://mathoverflow.net/users/nan | Iterated blow-ups above a point | The question isn't quite well-defined: if we blow up $\mathbb A^2$ along $\mathfrak m = \langle x,y\rangle$ or along $\mathfrak m^{k}$ for $k>1$, we get the same space.
It's handy that the spaces you're asking about are all toric varieties. So one can compute polyhedrally. Instead of $X = \mathbb A^2$, consider the m... | 2 | https://mathoverflow.net/users/391 | 202018 | 97,414 |
https://mathoverflow.net/questions/165587 | 5 | We say that a multilinear polynomial $P(x\_1,\ldots,x\_n)$ in $n$ commuting variables over $\mathbb{R}$ has *zero trace* if
$$ \frac{d}{dt} P(t,\ldots,t) = 0. $$
Equivalently,
$$ \left(\sum\_{i=1}^n \frac{\partial}{\partial x\_i}\right) P = 0. $$
(Suggestions for a better name are welcome.)
Such functions come up, fo... | https://mathoverflow.net/users/7732 | Orthogonal basis for the multilinear polynomials with zero "trace" | The basis appears in Murali K. Srinivasan's paper Symmetric chains, [Gelfand-Tsetlin chains, and the Terwilliger algebra of the binary Hamming scheme](http://arxiv.org/abs/1001.0280), though perhaps not as explicitly. The author shows that it is the canonically defined symmetric Gelfand–Tsetlin basis (up to scaling), a... | 3 | https://mathoverflow.net/users/7732 | 202025 | 97,419 |
https://mathoverflow.net/questions/202024 | 1 | Let $A$ be a square matrix of size $n \times n$ ($n>2$) and let $B$ be $A$ if we delete the last row and column (size $(n-1) \times (n-1)$). Let $\sigma (A)$ be the least singular value of $A$ and $\sigma(B)$ the least singular value of $B$. Is it true that $\sigma(A) \leq \sigma(B)$?
I have seen the interlacing theo... | https://mathoverflow.net/users/nan | On the least singular values | No: $\begin{pmatrix} 1 & 0 & 0 \\ 0 & 0 & 1 \\0 & 1 & 0 \end{pmatrix}$
| 3 | https://mathoverflow.net/users/26935 | 202029 | 97,420 |
https://mathoverflow.net/questions/193767 | 2 | Let $P$ and $Q$ be two poset (partially ordered sets) and $\phi : P \to Q$ an order-preserving function.
I would like to know whether there is a name and perhaps a different characterizations of such functions $\phi$ that satisfy the following condition.
For each $p \in P$ the restriction $\phi\restriction\_p : p\d... | https://mathoverflow.net/users/10875 | Terminology question for maps between posets | These are just [(Grothendieck) fibrations](http://ncatlab.org/nlab/show/Grothendieck+fibration) as specialised to partially ordered sets and order preserving functions.
| 1 | https://mathoverflow.net/users/10875 | 202039 | 97,424 |
https://mathoverflow.net/questions/202021 | 5 | Let $S$ be a surface of negative Euler characteristic (for simplicity let's assume $S$ to be closed), and let $\mathcal{M}(S)$ denote the moduli space of hyperbolic surfaces homeomorphic to $S$.
Given $X \in \mathcal{M}(S)$, let $L(X)= l\_{1} \leq l\_{2} \leq l\_{3} \leq...$ denote the increasing sequence of lengths... | https://mathoverflow.net/users/37118 | Teichmuller distance between isospectral riemann surfaces | As a partial answer to question one: The diameter of $M(S)$ with respect to the Weil--Petersson metric is finite. So $D\_{WP}$ is finite.
| 3 | https://mathoverflow.net/users/1650 | 202050 | 97,428 |
https://mathoverflow.net/questions/202041 | 6 | I have been trying to determine the number of metrics of constant curvature on a surface of genus $n$, say $\Sigma$. For low values, the answer is clear, the moduli space is a point for the sphere, and is two dimensional for the torus, but the higher dimensional cases stump me, and I am unable to find the result. Any h... | https://mathoverflow.net/users/41103 | How many metrics of constant curvature exist on a Riemannian surface? | First on terminology. "Riemannian surface" is a surface already equipped with a Riemannian metric. So the question "how many metrics of constant curvature exist on a Riemannian surface" makes sense only if you state what is the relation between the metric of
constant curvature and the original metric on the Riemannian ... | 15 | https://mathoverflow.net/users/25510 | 202054 | 97,429 |
https://mathoverflow.net/questions/202061 | 0 | Let $X$ be a smooth projective variety and $A$ an ample $\mathbb{Q}$-divisor on $X$.
Is the round up $\ulcorner A\urcorner$ of $A$ ample?
I think it's true. But I do not know how to arrange an argument.
| https://mathoverflow.net/users/64057 | Is the round up of ample Q-divisor ample? | This doesn't seem right. Let $\pi : X \to \mathbb P^2$ be the blow-up of $\mathbb P^2$ at a point, and let $A = L + 2/3 E$, where $E$ is the exceptional divisor, and $L$ is the strict transform of a line through the point. Take $H$ to be the pullback of a line in $\mathbb P^2$. Then $A$ is linearly equivalent to $(H-E)... | 6 | https://mathoverflow.net/users/nan | 202064 | 97,432 |
https://mathoverflow.net/questions/202063 | 3 | Q1.
Is there a compact connected Hausdorff space (with at least two points) in which every non-empty $G\_\delta$ set has non-empty interior? (Without the requirement for connectedness, every finite $T\_1$ space is an example, and a more interesting example is the remainder of the Stone-Cech compactification of the int... | https://mathoverflow.net/users/48481 | Is there a compact connected Hausdorff space in which every non-empty $G_\delta$ set has non-empty interior? | This is a proof of Alessandro Vignati's guess: Let $G\_n$ be a decreasing sequence of nonempty open subsets of $X=\beta[0,1)\setminus[0,1)$. Choose $x\in\bigcap G\_n$ - by assumption the intersection is nonempty. Then for each $n$ choose a pair of disjoint subsets of $\beta[0,1)$, namely $U\_n\ni x$ and $V\_n\supseteq ... | 3 | https://mathoverflow.net/users/16678 | 202070 | 97,433 |
https://mathoverflow.net/questions/201914 | 4 | Is their an efficient mathematical way to estimate the distribution of minimal hamming distances for a set of random strings of length 8 over a 4-letter alphabet? E.g. given a set of 100-10,000 strings of length 8 over a 4-letter alphabet, determine for each sequence the minimal distance from the rest of the set and co... | https://mathoverflow.net/users/70110 | Estimating the distribution of minimal hamming distances within a set of strings? | There are two natural meanings for the distribution of minimum distances. One is the probability distribution for the minimum distance to one point. Another is the multiset of minimum distances to each point. They are related. This answer addresses the first sense of distribution. For a partial answer to the second, se... | 3 | https://mathoverflow.net/users/2954 | 202077 | 97,436 |
https://mathoverflow.net/questions/202005 | 5 | **TL;DR version.**
Given a commutative ring $\mathbf{k}$ and $n+m$ "generic" vectors $\mathbf{x}\_1, \mathbf{x}\_2, \ldots, \mathbf{x}\_n, \mathbf{y}\_1, \mathbf{y}\_2, \ldots, \mathbf{y}\_m$ in $\mathbf{k}^k$ (really over a polynomial ring, not over $\mathbf{k}$), how can we describe the polynomial relations that ho... | https://mathoverflow.net/users/2530 | A vector version of the Segre embedding: what is the kernel of the ring map? | This is the second fundamental theorem of classical invariant theory for $GL$ acting on vectors and covectors. This is Theorem 8.1 from Ch. 13 of the book [Lie Groups: An Approach Through Invariants and Representations](https://books.google.com/books?id=Sl8OAGYRz_AC&printsec=frontcover&source=gbs_ge_summary_r&cad=0#v=o... | 4 | https://mathoverflow.net/users/7410 | 202078 | 97,437 |
https://mathoverflow.net/questions/201981 | 2 | Let $X$ be a stratified variety, and let $i:S\hookrightarrow X$ be the inclusion of a stratum. Let $F := H^k(i^!\operatorname{IC}\_X)$. This is a local system on $S$ whose fiber at a point is isomorphic to the degree $k$ compactly supported cohomology of the stalk of $\operatorname{IC}\_X$ at that point.
Now let’s as... | https://mathoverflow.net/users/54343 | Cohomology of a local system and Deligne's weight filtration | Consider the variety with equation $y^2=tx^2$. Let $S$ be the locus with $x=0,y=0,t\neq 0$. Then clearly the sheaf at each point of $S$ is a sum of one-dimensional vector spaces from the surfaces $y=\pm \sqrt{t}x$. Monodromy swaps these two, so the action is nontrivial.
What you can get from purity in this case is th... | 1 | https://mathoverflow.net/users/18060 | 202092 | 97,446 |
https://mathoverflow.net/questions/201890 | 3 | Let $p:=[0,0,...,0,1] \in \mathbb{P}^n$ the point whose all the coordinates are zero except for the $n$-th. This defines a linear projection map $\phi:\mathbb{P}^n-p \to \mathbb{P}^{n-1}$, given by $[x\_0,...,x\_n] \mapsto [x\_0,...,x\_{n-1}]$. Let $X$ be a local complete intersection subscheme in $\mathbb{P}^{n-1}$. I... | https://mathoverflow.net/users/58203 | Linear projection from a point and local complete intersection | The ideal of $\overline{\phi^{-1}(X)}$ in an affine neighborhood of $p$ is just the graded ideal of $X$, viewed as a usual ideal. So I think for any locally complete intersection that is not a complete intersection you will be in trouble.
Let $m$ be the maximal ideal at $p$. If it is a locally complete intersection, ... | 0 | https://mathoverflow.net/users/18060 | 202100 | 97,450 |
https://mathoverflow.net/questions/202098 | 3 | from generalized inequality, we now that for $p>q$, we have $M\_p(\mathbf{x})\ge M\_q(\mathbf{x})$. now I am curious to know if we can find a constant $\alpha(p,q)$ which is only function of $p,q$ such that $M\_p(\mathbf{x})\le \alpha(p,q) M\_q(\mathbf{x})$. if these exists such constant, how can we find it preferably ... | https://mathoverflow.net/users/38361 | generalized mean inequality extension | Well, if you accept dependence on the the maximum and minimum components (which are the mean values for plus and minus infinity), Mitrinovic's "Analytic Inequalities" book has such bounds on ratios and differences of the two means, even their weighted versions. E.g.,
$$
\alpha(p,q)\leq \left(\frac{q(C^p-C^q) }{(p-q)(C^... | 3 | https://mathoverflow.net/users/17773 | 202108 | 97,453 |
https://mathoverflow.net/questions/202126 | 0 | is there a name in use for Dirichlet series without the order term, analogously to Laurent or Puiseux polynomials? Is there work known about such expressions?
$D(s) = \sum\_{0<n<N}a\_n/n^s$
The question came up when implementing Dirichlet series in a computer algebra system.
| https://mathoverflow.net/users/41285 | Dirichlet series without order term | These are called [Dirichlet polynomials](http://www.encyclopediaofmath.org/index.php/Dirichlet_polynomial).
They arise in many places in analytic number theory. For example, in approximate functional equations of $L$-functions.
| 2 | https://mathoverflow.net/users/5101 | 202130 | 97,462 |
https://mathoverflow.net/questions/201985 | 6 | I am looking for any known results about the tail sigma-algebra of a branching random walk. To be specific, let $T$ be the nodes of an infinite binary tree rooted at $r \in T$. Let $\{X\_t\})\_{t \in T}$ be i.i.d. real random variables with distribution $\mu$. For $s \in T$, let $Y\_s$ equal to the sum $\sum\_{t \in P\... | https://mathoverflow.net/users/23661 | Tail sigma-algebra of a branching random walk | Actually, your idea of looking at the generation sums can be made more explicit. Let $s\_n$ be the sum of $X\_t$ over all vertices $t$ at a level $n\ge 0$, and let $S\_n$ be the corresponding sum of $Y\_t$ (in particular, $s\_0=S\_0=X\_r$). Then
$$
S\_n = 2^n s\_0 + 2^{n-1} s\_1 + \dots + 2 s\_{n-1} + s\_n \;,
$$
or
$... | 1 | https://mathoverflow.net/users/8588 | 202137 | 97,464 |
https://mathoverflow.net/questions/201753 | 1 | The two sets are, of course, supposed infinite.
This question is related to that one
[Commutation of tensor products with inverse limits in a specific case](https://mathoverflow.net/questions/200442/commutation-of-tensor-products-with-inverse-limits-in-a-specific-case/200443#200443)
where it received a (partial) answ... | https://mathoverflow.net/users/25256 | Necessary and sufficient condition for $can : A^X\otimes_A A^Y\rightarrow A^{X\times Y}$ to be an embedding | Let $k$ be a field, $I$ and $J$ infinite sets, and $A$ the $k$-subalgebra of $$k(t)[x\_i,y\_j: i\in I,j\in J]$$
generated by
$$\{x\_i,y\_j,tx\_i,t^{-1}y\_j: i\in I, j\in J\}.$$
Then
$$(tx\_i)\_{i\in I}\otimes(t^{-1}y\_j)\_{j\in J}-(x\_i)\_{i\in I}\otimes(y\_j)\_{j\in J}$$
is a non-zero element of the kernel of th... | 6 | https://mathoverflow.net/users/22989 | 202141 | 97,467 |
https://mathoverflow.net/questions/202149 | 3 | As suggested by Joseph van Name in [Is the associated order of a minimal $T\_0$ space always total?](https://mathoverflow.net/questions/201561/is-the-associated-order-of-a-minimal-t-0-space-always-total?noredirect=1#comment501286_201561), here's a natural question on $T\_0$-spaces:
If $(X,\tau)$ is $T\_0$, is there a... | https://mathoverflow.net/users/8628 | Minimal $T_0$-spaces | Let $X$ be an uncountable set with the cofinite topology. Then I claim there is no coarser minimal $T\_0$ topology. Indeed, by Joseph Van Name's answer to the previous question, any minimal $T\_0$ topology must be the minimal topology for some total specialization order. But in any total ordering of an uncountable set,... | 5 | https://mathoverflow.net/users/75 | 202151 | 97,471 |
https://mathoverflow.net/questions/201878 | 5 | Let $(X,d,\mu)$ be a metric measure space (i.e. $(X,d)$ is a metric space and $\mu$ is a Borel measure on $X$). Let's say that $X$ is *doubling* if there exists a constant $C \geq 1$ such that $0 < \mu(B(x,2r)) \leq C\mu(B(x,r)) < \infty$ for all $x \in X$ and $r > 0$ (note the finiteness and positivity conditions whic... | https://mathoverflow.net/users/nan | Existence of doubling non-Polish metric measure spaces | Expanding comment to answer: here's an example that is perhaps somewhat trivial but does achieve what you ask.
Let $(X\_0, d, \mu\_0)$ be your favorite doubling metric measure space, and suppose $X \subset X\_0$ is a subset which is Borel but not $G\_\delta$ and has full measure. Consider $(X,d)$ as a metric space in... | 3 | https://mathoverflow.net/users/4832 | 202160 | 97,477 |
https://mathoverflow.net/questions/202134 | 2 | Consider the following situation:
* $S, T$ are standard Borel spaces (say $S = [0,1]^k$, $T = [0,1]$ if it is helpful).
* There is a a random variable $\zeta: \Omega \to S$.
* $f\_n(\zeta) \to^d \eta$, i.e. $f\_n(\zeta)$ converges in distribution to some random variable $\eta$.
Will there always exist
* A rando... | https://mathoverflow.net/users/8106 | Variant of Skorokhod's theorem | This cannot work without additional assumptions, because of the following theorem on weak convergence. Given a probability measure $P$ on $S \times T$, let $\mu = P(\cdot \times T)$ denote the first marginal. If $\mu$ is nonatomic, then there exists a sequence of measurable functions $g\_n : S \rightarrow T$ such that,... | 1 | https://mathoverflow.net/users/44169 | 202161 | 97,478 |
https://mathoverflow.net/questions/202146 | 1 | $G(k,n)$ is the complex grassmanian which is homeomorphic to the space of projections in $M\_{n}(\mathbb{C})$ with trace $k$. So we can Identify $G(k,n)$ with $$\{A\in M\_{n}(\mathbb{C})\mid A=A^{\*}=A^{2},\;\;trac(A)=k\}$$
With this matrix interpretation, we define two maps from $G(k,n)$ to $G(2k,2n)$. Our quesion i... | https://mathoverflow.net/users/36688 | Homotopy type of certain maps on complex grassmanian | Let $U$ be the matrix which sends $e\_i$ to $e\_{2i}$ and $e\_{i+n}$ to $e\_{2i+1}$ for $0\leq i<n$. Then $f(X)=Ug(X)U^{-1}$. You can join $U$ to $I$ by a path in $U(n)$, and this gives a homotopy between $f$ and $g$.
| 7 | https://mathoverflow.net/users/10366 | 202164 | 97,479 |
https://mathoverflow.net/questions/202076 | 2 | In their paper "Integral Transforms and Drinfeld Centers in Derived Algebraic Geometry" the authors show that for perfect stacks $X$ and $Y$ over $k$, and their $k$-linear $\infty$-categories of unbounded chain complexes of modules $QC(X)$ and $QC(Y)$ there are equivalences of categories $$ QC(X \times\_k Y) \simeq QC(... | https://mathoverflow.net/users/69838 | Integral transform on noncommutative spaces | Let $DGCat\_k$ denote the $\infty$-category obtained by localizing the category of dg-categories at Morita equivalences. This is presented by the Morita model structure on the category of dg-categories. Recall that the fibrant objects are the idempotent complete dg-categories, or equivalently dg-categories for which th... | 2 | https://mathoverflow.net/users/2503 | 202166 | 97,480 |
https://mathoverflow.net/questions/202122 | 1 | This question is more or less a cross post of <https://math.stackexchange.com/questions/1218660/orthogonal-functions-with-shrinking-support>.
Let $X$ be a metric space (compact, if it helps) and let $Y$ be a closed subset of $X$. Fix a Borel measure $\mu$ on $X$. My question:
>
> Is there an orthogonal set $\{u\_... | https://mathoverflow.net/users/4362 | Orthogonal functions with shrinking support | As was pointed out by ifw, we should forbid $u\_n$ from being identically $0$ (a.e.), in order to make the question interesting. Also, addressing ifw's additional observation, if we let $Y\_n$ be the closed $2^{-n}$-neighborhood of $Y$, then we need to assume $L^2(Y\_n,\mu|\_{Y\_n})$ is infinite-dimensional. (If $X$ is... | 2 | https://mathoverflow.net/users/68305 | 202169 | 97,482 |
https://mathoverflow.net/questions/202087 | 3 | I have an acoustic research problem that leads to the following integral formulation:
\begin{align}
\int\_{-\infty}^{\infty}\int\_{-\infty}^{\infty}p(\mathbf{y},\tau)\frac{\partial}{\partial y\_i}\left(n\_i \delta(f) |\nabla f|\right) g(\mathbf{x} - \mathbf{y}, t-\tau) d^3\mathbf{y} d\tau,
\end{align}
where $p$ is a ... | https://mathoverflow.net/users/70172 | acoustic dipole volume integral w/ dirac delta? | First do an integration by parts:
$$\int F(\mathbf{y})\frac{\partial}{\partial y\_i}\left(n\_i\delta(f)|\nabla f|\right)\,d^n\mathbf{y}=
-\int \delta(f)|\nabla f|n\_i\frac{\partial}{\partial y\_i}F(\mathbf{y})\,d^n\mathbf{y}.$$
Then use this [identity](https://en.wikipedia.org/wiki/Dirac_delta_function#Properties_i... | 3 | https://mathoverflow.net/users/11260 | 202186 | 97,486 |
https://mathoverflow.net/questions/202192 | 13 | The cardinality of a set is just the number of elements.
To make sense of the cardinality of a category, one has to account for the morphisms. The usual definition is the sum over the isomorphism classes of 1/#automorphisms.
>
> What's the cardinality of a higher category?
>
>
>
I.e., should I somehow take ... | https://mathoverflow.net/users/4707 | What's the cardinality of a higher category? | Note that the formula "sum 1/#automorphisms" only depends on the (maximal sub-)groupoid of the category.
So your question becomes: what's a good notion of cardinality for $n$-groupoids (a.k.a. spaces with vanishing $\pi\_k$ for all $k>n$)?
One possible answer is to take the sum over connected components of the alte... | 12 | https://mathoverflow.net/users/5690 | 202194 | 97,489 |
https://mathoverflow.net/questions/202196 | 4 | Let $X$ be a compact Hausdorff (but not necessarily metrizable) space.
Is it always true that there exists a probability Borel measure $\mu$ and an open set $U$ such that any nonempty open set $V\subset U$ has positive measure?
Which additional hypothesis would help?
| https://mathoverflow.net/users/18698 | Measure with `somewhere dense' support | No. An almost $P$-space is a topological space where every nonempty $G\_{\delta}$-set has a nonempty interior. By the answers to this [recent question](https://mathoverflow.net/q/202063/22277), there exists compact almost $P$-spaces without any isolated points. However, I claim that if $X$ is a compact $P$-space and $\... | 4 | https://mathoverflow.net/users/22277 | 202201 | 97,493 |
https://mathoverflow.net/questions/201958 | 18 | I propose to collect here open problems from the theory of continued fractions. Any types of continued fractions are welcome.
| https://mathoverflow.net/users/5712 | Open problems in continued fractions theory | Guy, Unsolved Problems In Number Theory, F21, attributes to Bohuslav Divis the conjecture that in each real quadratic field there is an irrational with all partial quotients 1 or 2; more generally, same question but with 1 and 2 replaced by any pair of distinct positive integers.
| 7 | https://mathoverflow.net/users/3684 | 202204 | 97,495 |
https://mathoverflow.net/questions/202206 | 6 | *Setup:* Let's work in the category of schemes over $\mathbb C$. Let $G$ be a finite group. Let $BG=[pt/G]$ be the classifying stack of principal $G$ bundles. This is a fiberd category over the big etale site of complex schemes. The groupoid of sections of $BG$ over $U$ is the category of principal $G$ bundles over $U$... | https://mathoverflow.net/users/40862 | What are the automorphisms of $BG$? | Yes that's correct. An element of $Aut(BG)$ gives, for each scheme $X$, a functor from the category of $G$-bundles on $X$ to itself.
First look at how the functor acts on the category of $G$-bundles on a point. This is a functor from the category with one object and automorphisms $G$ to itself. These functors are giv... | 7 | https://mathoverflow.net/users/18060 | 202209 | 97,498 |
https://mathoverflow.net/questions/202178 | 4 | I was looking for a possible reference that would answer the following question,
Let $\mathbb{Q}\_{p}$ be the $p$-adic numbers and $\mathbb{Q}\_{p}((t))$ be the field of Laurent polynomials over $\mathbb{Q}\_{p}$. Does anyone know of a reference that addresses the following question,
"Are all division algebras ove... | https://mathoverflow.net/users/70171 | Reference request for division algebras, over $\mathbb{Q}_{p}((t))$ | Here is the closest result I know. If the degree of the division algebra is a prime $q \ne p$, an affirmative answer has been given over finite extensions of $\mathbb Q\_p(t)$ by Saltman's paper [Cyclic algebras over $p$-adic curves](http://arxiv.org/abs/math/0604409).
There is also [an article of Brussels](http://l... | 1 | https://mathoverflow.net/users/6518 | 202214 | 97,500 |
https://mathoverflow.net/questions/202237 | 2 | I've seen two kinds of demonstrations of Vitali's sets being not measurable (for example, answer number 2 here: <https://math.stackexchange.com/questions/137949/the-construction-of-a-vitali-set>, I adress it specifically because proof number one I haven't seen very often).
In both cases, the fact that the Lebesgue me... | https://mathoverflow.net/users/66034 | On the mesurability of a VItali set w.r.t. a Lebesgue absolutely continuous measure | user's answer, stated another way
Say the Vitali set $V \subseteq [0,1]$ has inner measure zero, outer measure 1. We may extend Lebesgue measure to a larger $\sigma$-algebra. Let $\mathscr F$ be the Lebesgue-measurable sets in $[0,1]$, and let the extension be
$$
\mathscr G = \{ (A\cap V) \cup (B \setminus V)\;|\; A,... | 2 | https://mathoverflow.net/users/454 | 202249 | 97,510 |
https://mathoverflow.net/questions/191836 | 1 | Let $H$ be an $n\times n$ Hilbert matrix,
$$h\_{ij}=(i+j-1)^{-1}.$$
The matrix $p$-norm corresponding to the p-norm for vectors is:
$\left \| A \right \| \_p = \sup \limits \_{x \ne 0} \frac{\left \| A x\right \| \_p}{\left \| x\right \| \_p}$, $p\ge 1$.
Is there a known (or what is the) formula for $\left \| H ... | https://mathoverflow.net/users/54458 | The norm of a Finite Hilbert matrix | I think the answer is no, indeed. Even in the particular case $p=2$ the formula is not known (to my best knowledge). Concerning $\|H\|\_{2}$, is known $\|H\|\_{2}\leq\pi$. More precisely, we have the following upper bound
$$\|H\|\_{2}\leq 2w\_n\arcsin\frac{1}{w\_n}$$
where
$$w\_n=2\left[\binom{2n}{n}\right]^{-1/2n},$$
... | 2 | https://mathoverflow.net/users/56553 | 202250 | 97,511 |
https://mathoverflow.net/questions/202073 | 1 | In approximation theory, it is classical to use a result that can be considered a generalization of the Schwarz Lemma:
Let $f:[-1,1]\rightarrow\mathbb{C}$ be a function that is analytic in a domain $S$ containing all the points that are at a distance of $\leq \beta$ from $[-1,1]$, where $\beta>2$. Moreover, suppose ... | https://mathoverflow.net/users/2011 | Generalized Schwarz Lemma for near-zeros | Such estimates are standard tools in transcendental number theory. The following is a simplified version (usually one also require that derivatives are small):
Suppose that $D$ is a domain with a nice boundary $\Gamma$ (piece-wise smooth is fine). Let $f$ be a holomorphic on $D$ and continuous on $\overline{D}$. Let $a... | 5 | https://mathoverflow.net/users/1811 | 202257 | 97,514 |
https://mathoverflow.net/questions/202255 | 2 | Let $m\geq 2$ be a positive integer and consider the Kloosterman sums
$$
\mathrm{Kl}(a,b,m)=\sum\_{\substack{1\leq x\leq m\\ \gcd(x,m)=1}}\exp\left(\frac{2\pi i}{m}(ax+b\bar{x})\right),
$$
where $\bar{x}$ is the inverse of $x$ modulo $m$. The following upper bound is well-known
$$
\mathrm{Kl}(a,b,m)\leq \tau(m)\gcd(a,b... | https://mathoverflow.net/users/8419 | Estermann-Weil bound for Kloosterman sums | 1. Yes, and it follows from Estermann article.
2. Usually it is enough to estimate $\tau(m)$ as $m^\varepsilon.$
| 2 | https://mathoverflow.net/users/5712 | 202262 | 97,517 |
https://mathoverflow.net/questions/202203 | 5 | Let $K$ be an arbitrary field and $\mathfrak{g}$ a finite-dimensional Lie $K$-algebra.
Let $\mathfrak{nil}\_n\leq\mathfrak{sol}\_n\leq\mathfrak{gl}\_n$ be the Lie algebras of all ((strictly) upper-triangular) $n\!\times\!n$ matrices over $K$. By the Ado-Iwasawa theorem, $\mathfrak{g}$ admits an embedding into some $... | https://mathoverflow.net/users/11317 | Solvable Lie algebras: embedded in upper triangular matrices? | The shaded question obviously has a negative answer for an arbitrary field $K$ (e.g., if the characteristic is 0 but the field fails to be algebraically closed). Maybe it's better to rewrite the question? In any case, it's essential to start with $K$ algebraically closed. Moreover, you might as well assume $K$ has prim... | 7 | https://mathoverflow.net/users/4231 | 202264 | 97,518 |
https://mathoverflow.net/questions/202271 | 0 | Let $\Gamma$ be a smooth closed curve in the complex plane (for all practical purposes). Assume $f$ is a real-valued continuous function defined on $\Gamma$ and let $d\mu=fdm$, where $dm$ is the Lebesgue measure supported on the curve.
Is it fair to say that the [logarithmic energy](https://mathoverflow.net/question... | https://mathoverflow.net/users/48438 | Positivity of logarithmic energy of certain measures | Yes, this is true if the supports of your measures are in the unit disk. All your other conditions are redundant: you do not need the curve, if you have one, it does not have to be smooth, and the measures do not need to have densities, and do not have to be positive.
The same is true also without the restriction on ... | 2 | https://mathoverflow.net/users/25510 | 202274 | 97,521 |
https://mathoverflow.net/questions/202280 | 4 | Suppose $\langle X,\mathscr{O}\rangle$ is a topological space and let $\mathscr{O}\_x$ be the family of all open neighbourhoods of $x\in X$. Let $\mathscr{F}$ be the filter generated from $\mathscr{O}\_x$:
$$\mathscr{F}=\{Q\in2^X\mid\exists\_{V\_1,\ldots,V\_n\in\mathscr{O}\_x}\,V\_1\cap\ldots\cap V\_n\subseteq Q\} $$
... | https://mathoverflow.net/users/22019 | Convergent filters generated by (not necessarily countable) chains | Topological spaces where each point has a totally ordered local basis are known as $\textit{well-based}$ spaces. The notion of a well-based space is a generalization of the notion of a first countable space since the first countable spaces are the spaces where every point has a countable totally ordered local basis. A ... | 4 | https://mathoverflow.net/users/22277 | 202293 | 97,530 |
https://mathoverflow.net/questions/202267 | 6 | Let $G=SL\_3(\mathbb{C})$ and $X=G/B$ be the associated full flag variety. Fix a non-degenerate symmetric quadratic form $Q$ on $\mathbb{C}^3$. This gives an order $2$ automorphism $F\_Q$ of $X$, mapping a flag $(L,P)$ to $(P^{\perp},L^{\perp})$. Is there any explicit description of the quotient $X/F\_Q$ ?
| https://mathoverflow.net/users/48866 | Quotient of Flag varieties | Let's coordinatize $G/B = Proj\ \mathbb C[m\_{11}\ldots m\_{33}]/\langle trace=0, rank \leq 1\rangle$, with $F\_Q(M) = M^T$, the image of $(L,P) \mapsto L P^T$ (a column vector times row vector).
Then the $F\_Q$-invariants are generated by $n\_i := m\_{ii}, q\_{i < j} := m\_{ij}m\_{ji}, s\_{i < j} := s\_{ij}s\_{ji}$.... | 1 | https://mathoverflow.net/users/391 | 202296 | 97,531 |
https://mathoverflow.net/questions/29923 | 9 | Let $k$ be a commutative ring and $L$ a $k$-module. The tensor algebra $\otimes L$ is $\mathbb{Z}$-graded and $\mathbb{Z}\_2$-graded (an element of $L^{\otimes n}$ has degree $n$ and $\mathbb{Z}\_2$-degree $n\mod 2$); hence it is a superalgebra over $k$. This canonically induces a supercommutator $\left[\cdot,\cdot \ri... | https://mathoverflow.net/users/2530 | Strange boundary-like map on tensor algebra: what is its kernel? | My conjecture was correct. This, and more, is now proven in [*The signed random-to-top operator on tensor space (draft)*](http://www.cip.ifi.lmu.de/~grinberg/algebra/r2t.pdf) (aka arXiv preprint arXiv:1505.01201).
(Some questions do remain, such as those in §9.)
| 3 | https://mathoverflow.net/users/2530 | 202301 | 97,532 |
https://mathoverflow.net/questions/202287 | 4 | $\newcommand{\ad}{\operatorname{ad}}$As my recent post ([here](https://mathoverflow.net/questions/202163/nilpotent-operator-of-the-weyl-algebra)) did not receive any answers yet, I thought I would ask a similar question in which I'm also interested.
Let $A=$ $^{k \langle x,y\rangle }\Big/\_{(yx-xy-1)}$ be the Weyl Al... | https://mathoverflow.net/users/37392 | Locally nilpotent operators of the Weyl algebra | Dixmier gives a complete characterization of ad-locally nilpotent elements of $A=A\_1$ in his paper ["Sur les algèbres de Weyl"](http://archive.numdam.org/ARCHIVE/BSMF/BSMF_1968__96_/BSMF_1968__96__209_0/BSMF_1968__96__209_0.pdf), Bulletin de la S. M. F., tome 96 (1968), Theorem 9.1:
An element $q \in A\_1$ is ad-loca... | 4 | https://mathoverflow.net/users/3380 | 202307 | 97,534 |
https://mathoverflow.net/questions/202313 | -1 | Given $S\subseteq \mathbb{N}$, we associate to $S$ the *sum-graph* $G\_S = (\mathbb{N}, E)$ where $$E = \big\{\{m,n\}: m,n \in \mathbb{N} \text{ and } m+n\in S\big\}.$$
If $S$ is infinite, is $G\_S$ always connected?
| https://mathoverflow.net/users/8628 | Sum-graph over an infinite set | No, let $S=2\mathbb N$, the set of even numbers.
Then $2\mathbb N$ and $2\mathbb N+1$ (the set of odd nunbers) are two distinct connected components of $G\_S$. (Also, they are both complete subgraphs of $G\_S$.)
| 3 | https://mathoverflow.net/users/4600 | 202314 | 97,536 |
https://mathoverflow.net/questions/202315 | 5 | Given $S\subseteq \mathbb{N}$, we associate to $S$ the *sum-graph* $G\_S = (\mathbb{N}, E)$ where $$E = \big\{\{m,n\}: m,n \in \mathbb{N} \text{ and } m+n\in S\big\}.$$
We say that $S\subseteq \mathbb{N}$ is *meager* if $$\text{lim sup}\frac{S\cap\{1,\ldots, n\}}{n} = 0.$$
Is there a set $S\subseteq \mathbb{N}$ suc... | https://mathoverflow.net/users/8628 | Diameter of sum-graph over a non-meager set | This is now an answer for the similar construction on $\mathbb Z$ rather than $\mathbb N$. To adapt the argument to $\mathbb N$ you would want to show that the intermediate $z$'s can be chosen to be positive. This can be ensured at the first stage using the ideas of Lemma 6 of <http://arxiv.org/abs/1308.0488> (deleted ... | 3 | https://mathoverflow.net/users/25485 | 202322 | 97,538 |
https://mathoverflow.net/questions/202330 | 11 | What is the Morse number of the Poincaré homology sphere? What about the stable Morse number?
| https://mathoverflow.net/users/70301 | Morse number of the Poincaré homology sphere | It's 6: you need one critical point of index 0 and index 3, and two of index 1 and index 2.
It's at least 6, since the only 3-manifolds with lower Morse number are the 3-sphere (2) and lens spaces (4).
It's at most 6, since there it has Heegaard genus at most 2. This is true for all 3-manifolds obtained as surgery ... | 18 | https://mathoverflow.net/users/13119 | 202331 | 97,541 |
https://mathoverflow.net/questions/202332 | 2 | We say that $S\subseteq \mathbb{N}$ is *meager* if $$\text{lim sup}\frac{S\cap\{1,\ldots, n\}}{n} = 0.$$
Given $S\subseteq \mathbb{N}$, we associate to $S$ the *sum-graph* $G\_S = (\mathbb{N}, E)$ where $$E = \big\{\{m,n\}: m,n \in \mathbb{N} \text{ and } m+n\in S\big\}.$$
Is there a set $S\subseteq \mathbb{N}$ suc... | https://mathoverflow.net/users/8628 | Diameter of sum-graph over a meager set | Let $S=\{2^k-1: k \in \mathbf{N}\}=\{1\_2,(11)\_2,(111)\_2,...\}$.
Clearly $S$ is sparse.
Writing $m,n \in \mathbf{N}$ in their binary expansion, $m$ is connected to $n$ in $G\_S$ if $m>n$ and $n$ is obtained by bitwise inverting $m$. For example, $(11001)\_2$ is connected to $(00110)\_2=(110)\_2$.
This shows th... | 9 | https://mathoverflow.net/users/3380 | 202334 | 97,543 |
https://mathoverflow.net/questions/202328 | 0 | Given $S\subseteq \mathbb{N}$, we associate to $S$ the *sum-graph* $G\_S = (\mathbb{N}, E)$ where $$E = \big\{\{m,n\}: m,n \in \mathbb{N} \text{ and } m+n\in S\big\}.$$
Is there an infinite subset $S\subseteq\mathbb{N}$ such that $G\_S$ has infinitely many connected components?
| https://mathoverflow.net/users/8628 | Connected components of a sum-graph over an infinite set | Write $S=\{s\_1, s\_2, \ldots\}$ with $s\_i<s\_{i+1}$ for all $i$. Then $G\_S$ has infinitely many connected components as soon as $s\_{i+1}>2s\_i$ for all $i$. (In particular, $S=\{3^k: k\in \mathbf{N}\}$ will do.)
To see this, consider the induced subgraph $\Gamma\_{n+1}$ on the vertices $\{1,\ldots,s\_{n+1}\}$. It... | 2 | https://mathoverflow.net/users/3380 | 202342 | 97,546 |
https://mathoverflow.net/questions/202325 | 8 | In [Geometry of the iterated loop space](http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.170.2840&rep=rep1&type=pdf), Peter May proved his famous recognition theorem, which is, in a simple form, stated on page 3 as the following.
>
> There exist $\Sigma$-free operads $\mathcal{C}\_n$, $1\le n\le \infty$, s... | https://mathoverflow.net/users/32022 | Reference Request: Grouplike Algebras over the little $n$-cubes operad are $n$-fold loop spaces | May gave a proof in the case $n = \infty$ in [this followup paper](http://www.math.uchicago.edu/~may/PAPERS/13.pdf) to "Geometry of iterated loop spaces," which relies on knowing that a certain map (from a free algebra on $X$ to the free infinite loop space on $X$) is a group-completion map. On page 67 he asserted "I a... | 10 | https://mathoverflow.net/users/360 | 202345 | 97,548 |
https://mathoverflow.net/questions/202323 | 3 | Similarly to the decomposition $L\_2(\mathbb R^n) = L\_2(\mathbb{R})^{\otimes n}$ as vector spaces (and even as Hilbert spaces) , do we have $bm(\mathbb{R^n}) = bm(\mathbb{R})^{\otimes n}$
where $bm(\mathbb{R})$ refers to the space of borel measures over the reals?
The reason I was thinking of this decomposition is... | https://mathoverflow.net/users/70295 | Is the space of all borel measures on $\mathbb R^n$ isomorphic to the tensor product of spaces of borel measures on $\mathbb R$? | My answer is more a plan than a real answer. It aims to not close too early by a brutal "no-go" algebraic statement. Of course, if you ask for finite (i.e. algebraic) tensor product this is not true.
If you are thinking of completed tensor products, then, I think yes.
Of course, in order to have the *completed tens... | 1 | https://mathoverflow.net/users/25256 | 202351 | 97,551 |
https://mathoverflow.net/questions/202363 | 6 | A colleague of mine suggested the following weakening of the axiom of choice:
>
> If $\mathscr{F} := \{F\_\alpha\}$ is a *well-ordered* family of non-empty sets (i.e., there is a bijection between $\mathscr{F}$ and some von Neumann ordinal), then there is a choice function $f$ with domain $\mathscr F$ such that $f(... | https://mathoverflow.net/users/3106 | Logical strength of "choice functions exist for well-ordered families"? | This principle is known in set-theoretic literature as $\forall \kappa AC\_\kappa$, where $AC\_\kappa$ refers to the existence of choice functions for well-ordered families of size $\kappa$. It is discussed in detail in Chapter 8 of Jech's textbook "The axiom of choice". The principle $\forall\kappa AC\_\kappa$ implies... | 10 | https://mathoverflow.net/users/5984 | 202370 | 97,558 |
https://mathoverflow.net/questions/202355 | 17 | This is true for finite-dimensional spaces: the diagonal operators on a finite dimensional complex vector space form contain a dense open set and the nondiagonalizable operators have measure 0.
To be precise, let $T$ be an operator on a complex Banach space $X$ which is not finite-dimensional. For each $\lambda \in \... | https://mathoverflow.net/users/2362 | Are "most" operators on an infinite-dimensional complex Banach space "diagonalizable"? | Consider the right shift $R([x\_1, x\_2, \ldots]) = [0, x\_1, x\_2, \ldots]$ on $\ell^2$. I claim the open ball of radius $1/2$ about $R$ contains no diagonalizable operators.
Let $e\_1 = [1,0,\ldots]$.
Suppose $B$ is an operator with $\|B\| < \epsilon < 1/2$.
For any $x$ we have $\|(R + B)x \| \ge (1-\epsilon) \|x... | 21 | https://mathoverflow.net/users/13650 | 202377 | 97,561 |
https://mathoverflow.net/questions/202182 | 4 | Description: Given the following parametric quartic polynomial
$y^4 - 28 z y^3 - 14 (656 - 328 z + 83 z^2) y^2 +
4 z (-20464 + 10232 z + 3409 z^2) y +
91 (62208 - 62208 z + 41504 z^2 - 12976 z^3 + 675 z^4)$
how do I find all possible rational values of $z$ such that this polynomial is reducible.
I am performi... | https://mathoverflow.net/users/62471 | Find all possible rational values of a parametric quartic such that it is reducible | I essentially follow Igor Rivin's approach here. There are two possibilities for
how the polynomial can factor: (degree 1)$\times$(degree 3) or
(degree 2)$\times$(degree 2). Writing
$$ (y + a)(y^3 + by^2 + cy + d) = f $$ or
$$ (y^2 + ay + b)(y^2 + cy + d) = f, $$
where $f$ is your polynomial, and equating coefficents, ... | 6 | https://mathoverflow.net/users/21146 | 202378 | 97,562 |
https://mathoverflow.net/questions/202210 | 3 | I'm reading the proof of Lemma 2.11 of that book, for which Tao has an [errata](https://terrytao.wordpress.com/books/nonlinear-dispersive-equations-local-and-global-analysis) showing that the case $b=b'$ is not obvious. But I can't quite understand his explanation on how to show that case. Could anyone provide more det... | https://mathoverflow.net/users/37103 | Lemma 2.11 of Tao's Nonlinear Dispersive Equations | I have now figured out the question, so I'll record it here.
Let $L=ih(\nabla/i)$ be a constant coefficient differential operator, where $h$ is a polynomial. Recall the Bourgain norm is defined as
$$ \|u\|\_{X^{s,b}\_h} = \| \langle\xi\rangle^s\langle \tau-h(\xi) \rangle^b\hat u(\tau,\xi) \|\_{L^2}. $$
Let $\eta$... | 4 | https://mathoverflow.net/users/37103 | 202382 | 97,564 |
https://mathoverflow.net/questions/202388 | 4 | An infinity-groupoid is finitely-presented when it is equivalent to the free infinity-groupoid on a finite family of generators, possibly of different dimensions.
**Is the infinity-groupoid of a finite CW complex finitely-presented?**
It seems to me this question is relevant for homotopy type theory, in which topol... | https://mathoverflow.net/users/799 | Is the infinity-groupoid of a finite CW complex finitely-presented? | A CW structure is precisely a presentation of an $\infty$-groupoid, and so "finite CW complex" means precisely "finitely presented $\infty$-groupoid."
| 5 | https://mathoverflow.net/users/290 | 202389 | 97,566 |
https://mathoverflow.net/questions/202362 | 1 | Given an in integer $n$, and let $p$ be its smallest prime divisor (you can assume that $p$ is very large ). Let $ord\_n(i)$ denotes the order of $i$ as an element of $\Bbb Z\_n^\*$ the multiplicative group of $\Bbb Z\_n$.
**My question**:
>
> What is the best known lower bound for:
> $$\max\_{2\leq i\leq p-1}(o... | https://mathoverflow.net/users/70168 | What is the best known lower bound for: $\max_{2\leq i\leq p-1}(ord_n(i)) $? | Your quantity is at least $p-1$ since the range contains a primitive root modulo $p$ and the order modulo $n$ is at least the order modulo $p$. I don't expect you'll be able to improve this much (depending on the relative sizes of $n$ and $p$). This is obviously best possible if $n=p$ but, if there are infinitely many ... | 2 | https://mathoverflow.net/users/2290 | 202392 | 97,568 |
https://mathoverflow.net/questions/202390 | 1 | I have the following recursive formula:
\begin{align}
F(m,n) & = F(m,n - 1) + F(m - 1,n) - F(m - 1,n - 1 - m), \\
F(m,0) & = F \! \left( m,\frac{m (m + 1)}{2} \right) = 1, \\
F(m,i) & = 0 ~ \text{if} ~ i < 0 ~ \text{or} ~ i > \frac{m (m + 1)}{2}.
\end{align}
Is there a way to solve this recursive formula to obtain $ F(... | https://mathoverflow.net/users/19510 | Solving a recursive formula | These are called Mahonian numbers, according to OEIS. <https://oeis.org/A008302>
| 3 | https://mathoverflow.net/users/14915 | 202403 | 97,573 |
https://mathoverflow.net/questions/200433 | 6 | Let us consider a (not necessarily finite) Coxeter group $W$ generated by a finite set of involutions $S=\{s\_1,...,s\_n\}$ subject (as usual) to the relations $(s\_is\_j)^{m\_{i,j}}$ with $m\_{i,j}=m\_{j,i}$ and $m\_{i,j}=1$ if and only if $i=j$ (if necessary you may also assume that $m\_{i,j}<\infty$ for all $i,j$ or... | https://mathoverflow.net/users/37942 | Centralizers of reflections in special subgroups of Coxeter groups | I believe I finally found a proof. Surprisingly it does only use standard facts on Coxeter-groups (exchange condition, solving the word problem via braid-moves,...).
Let me first make the notation a bit easier:
Claim: Let $P\leq W$ be a special subgroup of $W$ generated by some subset $S'\subsetneq S$ of $S$ and $s... | 2 | https://mathoverflow.net/users/37942 | 202412 | 97,577 |
https://mathoverflow.net/questions/202411 | 5 | A group is *coherent* if its finitely generated subgroups are finitely presented. For instance, $F\_2 \times F\_2$ is a well-known example of incoherent group. My question is:
>
> Is a subgroup of $F\_2 \times F\_2$ not containing a copy of $F\_2 \times F\_2$ coherent?
>
>
>
That is to say, is containing $F\_2... | https://mathoverflow.net/users/43559 | Coherent subgroups of $F_2 \times F_2$ | This is true. I change the notation for convenience: let $F\_1,F\_2$ be two free groups (on any number of generators). Let a subgroup $H$ of $F\_1\times F\_2$ not contain any direct product of two non-abelian free groups. Then $H$ is coherent.
Proof: let $P\_i$ be the projection of $H$ on $F\_i$, and $J\_i=H\cap F\_i... | 10 | https://mathoverflow.net/users/14094 | 202413 | 97,578 |
https://mathoverflow.net/questions/202404 | 1 | Let $\Omega
\subset
\mathbb{R}^{N}$
be a smooth bounded domain , $g:\Omega\times\mathbb{R}\rightarrow\mathbb{R}$
is a Caratheodory function such that $g(x,t)=0$
for $t\leq0$
. Suppose that there exist function $a\in L^{r}$
and $d\in L^{p'}$
such that
$\left|g(x,t)\right|\leq a(x)t^{p-1}+d(x)$
with $r>N/p$
... | https://mathoverflow.net/users/70326 | Prove that $\dfrac{g(x,u_{n})}{\left\Vert u_{n}\right\Vert ^{p-1}}\rightarrow g_{0}$ weakly in $L^{\overline{p}}$ | First notice that the assumption $g(x,t)=0$ for $t\leq0$ is not essential, you can work with the estimate $|g(x,t)|\leq a(x)\,|t|^{p-1}+d(x)$ instead.
Then I take it that you have trouble with the case $1<p<N$. In this
case, you pick $(\bar p,s)$ such that $(p^\*)'<\bar{p}\leq p'$, $1\leq
s\leq p^\*$, and $\displayst... | 2 | https://mathoverflow.net/users/69194 | 202417 | 97,580 |
https://mathoverflow.net/questions/202415 | 6 | Let $X$ be a smooth projective variety over $\mathbb{C}$ satisfying $H^1(\mathcal{O}\_X)=0$. Fix $i:X \to \mathbb{P}^n$ a closed immersion and let $\mathcal{O}\_X(1)$ be the corresponding very ample line bundle. This means that the Picard group is isomorphic to the Neron-severi group of $X$. By Severi's theorem of base... | https://mathoverflow.net/users/58203 | Severi's theorem of base and Hilbert polynomial | No.
For instance, there are rational surfaces containing infinitely many $(-1)$-curves. An example in given by the blow-up $X$ of $\mathbb{P}^2$ at nine points that are the base locus of a general pencil of cubics: indeed, $\textrm{Aut}(X)$ contains a copy of $\mathbb{Z}^8$ generated by translations by differences o... | 4 | https://mathoverflow.net/users/7460 | 202420 | 97,583 |
https://mathoverflow.net/questions/202421 | 1 | Are there positive integers $\Delta, d$ such that the following statement is true?
>
>
> >
> > For every $n\in \mathbb{N}$ there is a graph $G = (V,E)$ such that $|V| = n$,
> > $\Delta(G) \leq \Delta$ (where $\Delta(G)$ is the maximum degree of $G$), and
> > $\text{diam}(G) \leq d$.
> >
> >
> >
>
>
>
| https://mathoverflow.net/users/8628 | Is the number of vertices bounded for fixed max degree and fixed diameter? | A graph of maximum degree $\leq \Delta$, and diameter $\leq d$ can have at most
$1+\Delta\sum\_{i=0}^{d-1} (\Delta-1)^i$ vertices. The graphs which attain this bound are called [Moore graphs](http://en.wikipedia.org/wiki/Moore_graph#Bounding_vertices_by_degree_and_diameter).
| 11 | https://mathoverflow.net/users/2384 | 202424 | 97,585 |
https://mathoverflow.net/questions/202302 | 2 | Given a finite field extension $L$ of $\mathbb{Q}$ of dimension $n$, there is a natural way to embed it into $\mathbb{R}^n$ such that the image of its ring of integers $\mathcal{O}\_L$ is a lattice. If $L=\mathbb{Q}[\zeta\_p]$ where $\zeta\_p$ is a primitive $p$-th root of unity, and $\sigma$ is the generator for the G... | https://mathoverflow.net/users/36987 | covering radius of a lattice from cyclotomic extension | You can use the following Magma code on the [online calculator](http://magma.maths.usyd.edu.au/calc/) :
```
for p in [3,5,7] do
G:=GramMatrix(Lattice(MaximalOrder(CyclotomicField(p))));
K:=ZeroMatrix(Integers(),p-1,p-1);
for i in [1..p-1] do
for j in [1..p-1] do
K[i,j]:=Round(G[i,j]);... | 2 | https://mathoverflow.net/users/39552 | 202425 | 97,586 |
https://mathoverflow.net/questions/202043 | 19 | A [square-free word](http://en.wikipedia.org/wiki/Square-free_word)
is a string of symbols (a "word") that avoids the pattern $XX$, where $X$ is any
consecutive sequence of symbols in the string.
For alphabets of two symbols, the longest square-free word has length $3$.
But for alphabets of three symbols, there are inf... | https://mathoverflow.net/users/6094 | Three-halves-free words (analogous to square-free) | (Not a real answer, just a conjecture)
Let $w$ be the Pansiot word on $4$ letters defined here :
[J.J. Pansiot, A propos d'une conjecture de F. Dejean sur les répétitions dans les mots, Discrete Applied Mathematics Volume 7, Issue 3, March 1984, Pages 297–311](https://www.sciencedirect.com/science/article/pii/0166... | 11 | https://mathoverflow.net/users/38477 | 202426 | 97,587 |
https://mathoverflow.net/questions/201863 | 1 | As a follow up to [this question](https://mathoverflow.net/questions/199020/volume-of-randomly-changing-sphere-follows-beta-distribution), concerning [this paper](http://dl.acm.org/citation.cfm?id=355745):
Given random variables $X\_1,\ldots,X\_N,X\_q:\Omega\rightarrow\mathbb{R}^d$, where $X\_1,\ldots,X\_N$ are indep... | https://mathoverflow.net/users/41187 | Probability Content of a random ball in R^n | As Joe Neeman pointed out in the comment on the OP:
>
> The very fact that they're aksing to approximate a function $\delta(x)$ by a constant in a small ball already assumes that the function is continuous.
>
>
>
We now approximate $\delta(x)$ on $B\_k(X\_q(\omega))$ by $\overline{p}(X\_q(\omega))=\int\_{B\_k(... | 0 | https://mathoverflow.net/users/41187 | 202429 | 97,589 |
https://mathoverflow.net/questions/202381 | 10 | It is a basic fact about the symmetric group $S\_n$ that its irreducible representations are indexed by partitions of $n$.
My question is, can the association between partitions and irreps be specified without having to explicitly construct the irreps? In other words, is there for each partition $\lambda$ some relati... | https://mathoverflow.net/users/25229 | Is there a non-explicit characterization of the Specht modules? | The characterization using Jucys-Murphy elements is this: let $z$ be a formal variable, and $X\_i$ the Jucys-Murphy elements. The product $(z-X\_1)\cdots (z-X\_n)$ is central in the group algebra of the symmetric group, so it acts on any irreducible representation by a polynomial with scalar coefficients. On the Specht... | 9 | https://mathoverflow.net/users/66 | 202436 | 97,593 |
https://mathoverflow.net/questions/202448 | 3 | Let $\Lambda$ be a lattice in $\mathbb{R}^n$. For $\bar{x} \in \mathbb{R}^n$, let
$\| \bar{x} \| = max\_{1 \leq i \leq n} \{ |x\_i| \}$, i.e. the sup norm. Let $\lambda\_1, ..., \lambda\_n$
be a successive minima of $\Lambda$ with respect to the sup norm.
Let $\lambda\_{j} \leq U < \lambda\_{j+1}$.
If we consider t... | https://mathoverflow.net/users/48408 | On successive minima and basis of a lattice | You can read about these things in Siegel-Chandrasekharan: Lectures on the geometry of numbers (Springer, 1988). In particular, Section X.5 contains an example for $n=3$ where the minimizing vectors do not form a basis of the lattice. The same section and the next one also explain that for practical purposes the succes... | 4 | https://mathoverflow.net/users/11919 | 202450 | 97,596 |
https://mathoverflow.net/questions/202414 | 2 | The Hammersley-Clifford theorem says that any positive probability distribution satisfies one of the Markov properties with respect to an undirected graph G if and only if its density can be factorized over the cliques of the graph. Any edge in a graph is a clique. Every distribution trivially satisfies the local Marko... | https://mathoverflow.net/users/70331 | Do all positive distributions on $N$ variables factor pairwise? | You're misinterpreting the Hammersley-Clifford theorem. Every edge is a clique, but not every clique is an edge. You need to factor over **all** the cliques, including the complete graph itself.
| 2 | https://mathoverflow.net/users/13650 | 202451 | 97,597 |
https://mathoverflow.net/questions/202449 | 2 | Fix an odd prime $p$. Each $f \in \mathbb{Z}/2[[x]]$ can be written as $f\_{+} + f\_{-} + f\_0$ where each exponent k of $x$ appearing in $f\_{+}$ (resp. $f\_{-}$, $f\_0$) has Legendre symbol $(k/p)$ equal to 1 (resp. -1, 0).
When $f$ is a "characteristic 2 modular form for $\Gamma\_0(p)$", the same need not hold for... | https://mathoverflow.net/users/6214 | A question about decomposing mod 2 modular forms of level p^2 | Yes. In fact, if $f$ is a characteristic zero modular form of level $p^{2}$ then $f\_{+}$, $f\_{-}$ and $f\_{0}$ are also characteristic zero modular forms of level $p^{2}$.
We have $f\_{0} = f | U(p) | V(p)$, and since $f$ has level $p^{2}$, $f | U(p)$ has level $p$ and then $f | U(p) | V(p)$ has level $p^{2}$.
Al... | 3 | https://mathoverflow.net/users/48142 | 202453 | 97,599 |
https://mathoverflow.net/questions/202466 | 2 | Let me begin by defining what a polyhedral surface is.
A **path-connected** subset $ P $ of $ \mathbb{R}^{3} $ is called a *polyhedral surface* iff it is the union of a **finite** collection $ \mathcal{C} $ of polygons (possibly non-convex) that satisfies the following three conditions:
* The intersection of any pa... | https://mathoverflow.net/users/50614 | A Jordan Separation Theorem for Polyhedral Surfaces | The magic words are: [Alexander Duality.](http://en.wikipedia.org/wiki/Alexander_duality) This was initially proved in the simplicial category, just as you state it, so all is well. Since Alexander's original (1915) paper is six pages long, you should just read it.
| 5 | https://mathoverflow.net/users/11142 | 202467 | 97,602 |
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