parent_url stringlengths 37 41 | parent_score stringlengths 1 3 | parent_body stringlengths 19 30.2k | parent_user stringlengths 32 37 | parent_title stringlengths 15 248 | body stringlengths 8 29.9k | score stringlengths 1 3 | user stringlengths 32 37 | answer_id stringlengths 2 6 | __index_level_0__ int64 1 182k |
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https://mathoverflow.net/questions/391899 | 4 | Can we prove: let $2<p<q$ be two consequtive prime numbers, then there is always a prime number in interval $(q,p+q]$ (ie. betwen $q$ and $p+q$). In other words: if $p\_{n-1}<p\_n< p\_{n+1}$ are any three consequtive primes, then $p\_{n+1}<p\_{n}+p\_{n-1}$. Bertrand postulate is a consequence of this statement, since $... | https://mathoverflow.net/users/169583 | Bertrand postulate- stronger version | By Section 4 of P. Dusart [[Math. Comp. 68(1999), 411--415](https://www.ams.org/journals/mcom/1999-68-225/S0025-5718-99-01037-6/S0025-5718-99-01037-6.pdf)], for any $x\ge3275$ there is a prime $p$ such that
$$x\le p\le x\left(1+\frac1{2\log^2x}\right)\le x\left(1+\frac1{2\log^2 3275}\right)<1.01x.$$
For any integer $... | 4 | https://mathoverflow.net/users/124654 | 392009 | 162,195 |
https://mathoverflow.net/questions/392002 | 6 | Given a polynomial $P=a\_3z^3+a\_2z^2+a\_1z+1, z >0$ with non-negative integer coefficients $a\_1, a\_2, a\_3\ne 0$, it appears if $P$ is **not** factorizable then there are **finitely** many positive integers $x, z$ such that $xz+1 \mid P(z)$, $xz+1<P(z)$. If $a\_2=a\_1=0$, the claim is true. [The Diophantine equation... | https://mathoverflow.net/users/166404 | Is $xz+1 $ a proper divisor of $a_3z^3+a_2z^2+a_1z+1$ finitely often? | The conjecture is true. That is, if the integral cubic polynomial
$$P(Z)=a\_3 Z^3+a\_2 Z^2+a\_1 Z+1$$
is irreducible in $\mathbb{Z}[Z]$ (hence also in $\mathbb{Q}[Z]$ by [Gauss's lemma](https://en.wikipedia.org/wiki/Gauss%27s_lemma_(polynomial))), then there are only finitely many positive integer solutions of the equa... | 9 | https://mathoverflow.net/users/11919 | 392018 | 162,198 |
https://mathoverflow.net/questions/392037 | 1 | Let $R$ be a Noetherian regular local ring of dimension $n$ with maximal ideal $\mathfrak{m}$. Given two systems of regular parameters $\vec{u}=\left<u\_1, \dots, u\_n\right>$ and $\vec{v}=\left<v\_1, \dots, v\_n\right>$ of $R$, does there exist an element $M$ in $\mathrm{GL}\_n(R)$ which takes $\vec{u}$ to $\vec{v}$?
... | https://mathoverflow.net/users/157738 | Systems of regular parameters of a regular local ring | Since $\vec{u},\vec{v}$ both generate $\mathfrak{m}$, there exist matrices $U,V \in \mathrm{Mat}\_{n \times n}(R)$ such that $\vec{u} = U\vec{v}$ and $\vec{v} = V\vec{u}$. Then $(UV-I\_{n})\vec{u} = \vec{0}$ so $UV-I\_{n}$ has entries contained in $\mathfrak{m}$ since $\vec{u}$ is a regular sequence; thus $UV$ is inver... | 4 | https://mathoverflow.net/users/112809 | 392040 | 162,203 |
https://mathoverflow.net/questions/392005 | 7 | Let $f: X\to Y$ be a continuous surjective map between compact metric spaces. Suppose the fibre $f^{-1}(y)$ has zero topological dimension for each $y\in Y$. Then by Hurewicz dimension lowering theorem, one has $${\rm dim}(X)\le {\rm dim}(Y)+\sup\_{y\in Y} {\rm dim}(f^{-1}(y))={\rm dim}(Y).$$ I would like whether there... | https://mathoverflow.net/users/45092 | A continuous map with zero-dimensional fibres | Yes, there is such a proof. A reference is Theorem 6.4.11 from Pears *Dimension Theory of General Spaces*, which is a great reference for dimension theory in general even though it's not as well known as other books on the topic.
The other good news is that, assuming a small lemma which might even be a definition for... | 3 | https://mathoverflow.net/users/49381 | 392042 | 162,204 |
https://mathoverflow.net/questions/392048 | 2 | Suppose we have a compact plane region $R$ (not necessarily convex or connected). I am working in a problem which involves the point $p$ in $R$ that is, in average, the closest to every other point. That is, the point in $R$ which minimizes
$$\int\_R d(p,x) dx.$$
I have been looking for references but I can't find this... | https://mathoverflow.net/users/206706 | What is the center of minimum distance of a region? | There are two closely related notions: [geometric median](https://en.wikipedia.org/wiki/Geometric_median) and [medoid](https://en.wikipedia.org/wiki/Medoid). Each minimizes the sum (or integral) of distances from the points in some set $A$. The difference is simply that medoid is additionally required to be in $A$, but... | 3 | https://mathoverflow.net/users/171662 | 392055 | 162,208 |
https://mathoverflow.net/questions/360706 | 6 | When a free action gives rise to a $G$-principal bundle
Let a (topological) group $G$ act freely on a (topological) space $X$. Assume that
$G \backslash X$ is Hausdorff. (equivalently the image of the map $G \times X \to X\times X$ is closed).
Does it mean that $X \to G \backslash X$ is a $G$-principal bundle?
... | https://mathoverflow.net/users/4690 | When a free action gives rise to a $G$-principal bundle | For general topological spaces, I found some answers in Tammo to Dieck, "Algebraic Topology" [TD], Chapter 14.
I give here a quick summary.
Define a principal G-bundle $p$ as a continuous map $p: E \to X$ and a continuous right group action $R$ of $G$ on $E$ such that:
1. $p$ is $G$-invariant, that is $p \circ R\... | 6 | https://mathoverflow.net/users/68687 | 392058 | 162,209 |
https://mathoverflow.net/questions/392035 | 3 | Let $C$ be a category equipped with a Grothendieck topology generated by a cd-structure (see <https://ncatlab.org/nlab/show/cd-structure> or Voevodsky's paper *[Homotopy theory of simplicial presheaves in completely decomposable topologies](https://arxiv.org/abs/0805.4578)* ([JPAA version](https://doi.org/10.1016/j.jpa... | https://mathoverflow.net/users/205323 | Sheaves on sites given by a (regular) cd-structure | In general sections $b \in F(B)$ and $c \in F(C)$ that agree on $F(A)$ don't induce a matching family on $\{B \to D, C \to D\}$ though. The sheaf condition for that family is that
$$
F(D) \to F(B) \times F(C) \rightrightarrows F(B \times\_D B) \times F(B \times\_D C) \times F(C \times\_D B) \times F(C \times\_D C)
$$
i... | 4 | https://mathoverflow.net/users/126667 | 392071 | 162,213 |
https://mathoverflow.net/questions/392020 | 4 | It is well known that the minimum number of monochromatic triangles in a red/blue coloring of the edges of the complete graph $K\_n$ is given by Goodman's formula
$$M(n)=\binom n3-\left\lfloor\frac n2\left\lfloor\left(\frac{n-1}2\right)^2\right\rfloor\right\rfloor;$$
see OEIS sequence [A014557](http://oeis.org/A014557)... | https://mathoverflow.net/users/43266 | The number of monochromatic triangles | Here are results, obtained via integer linear programming, for the first question for $n \le 10$ and $b$ blue edges, where $b \le \binom{n}{2}/2$:
\begin{matrix}
n\backslash b & 0 & 1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 & 9 & 10 & 11 & 12 & 13 & 14 & 15 & 16 & 17 & 18 & 19 & 20 & 21 & 22 \\
\hline
2 & 0 \\
3 & 1 & 0 \\
4 & 4 &... | 2 | https://mathoverflow.net/users/141766 | 392073 | 162,214 |
https://mathoverflow.net/questions/391640 | 5 | The metric cannot be recovered from its Hausdorff measure in general. Now, assume that $(X,d\_X)$ and $(Y, d\_Y)$ are connected compact length spaces and induce $n$-dimensional Hausdorff measures $\mathcal{H}^n\_X$ and $\mathcal{H}^n\_Y$.
Assume there exists a 1-Lipschitz map $f: (X,d\_X,\mathcal{H}^n\_X)\to (Y, d\_Y... | https://mathoverflow.net/users/90512 | Recovering the length metric from Hausdorff measure | Nan Li proved that it holds for a pair of Alexandrov spaces without boundary;
in particular, it solves the problem for Riemannian manifolds.
See [Lipschitz-Volume rigidity in Alexandrov geometry](https://arxiv.org/abs/1110.5498).
It seems that his argument can be generalized quite a bit, but one cannot expect it to w... | 1 | https://mathoverflow.net/users/1441 | 392083 | 162,218 |
https://mathoverflow.net/questions/391966 | 3 | While reading [1], I encountered with the concept "Pucci extremal operator" which is defined by:
$$M\_\Lambda^-(N):=\left(\sum\text{positive eigenvalues of }N\right)+\Lambda\left(\sum\text{negative eigenvalues of }N\right),\text{ and}$$
$$M\_\Lambda^+(N):=-M\_\Lambda^-(-N),$$
where $N\in\text{Sym}\_{n\times n}$ and $\L... | https://mathoverflow.net/users/151368 | Reference request on Pucci extremal operators | The original references are the works of Pucci [2] and [3] which, however, are written in Italian. A perhaps more accessible introduction to these kind of operators is found in the monograph [1], §2.2, pp. 14-17, by Caffarelli and Cabré: in the latter reference, particularly relevant to your questions are lemma 2.12, §... | 3 | https://mathoverflow.net/users/113756 | 392086 | 162,219 |
https://mathoverflow.net/questions/322900 | 4 | For all natural numbers $n$, let $(B\_{n},\*\_{n})$ be the algebraic structure with underlying set $\{1,\dots,n\}$ where
1. $x\*\_{n}1=x+1\mod n$,
2. $n\*\_{n}y=y$, and
3. $x\*\_{n}(y+1)=(x\*\_{n}y)\*\_{n}(x+1)$ for $x<n,y<n$.
The algebra $(B\_{n},\*\_{n})$ is called the $n$-th bad Laver table. For each $n$ and $x\... | https://mathoverflow.net/users/22277 | Why do highly composite rows on the bad Laver tables have longer periods? | This phenomenon has a simple explanation that I probably should have observed while asking this question.
Suppose that $n$ is a natural number and $x<n$ where $x$ is a highly composite number. Observe that if $a=b\mod\pi\_{n}(c)$, then $c\*\_{n}a=c\*\_{n}b$. In particular, since $x$ is a highly composite number, it i... | 1 | https://mathoverflow.net/users/22277 | 392087 | 162,220 |
https://mathoverflow.net/questions/392080 | 4 | $\DeclareMathOperator\RHom{RHom}\DeclareMathOperator\Map{Map}\DeclareMathOperator\id{id}\DeclareMathOperator\colim{colim}$Let $(\mathcal A,\mathcal M)$ be a (normalized) analytic ring defined in [Lectures on Analytic Geometry](https://www.math.uni-bonn.de/people/scholze/Analytic.pdf). We review two basic concepts there... | https://mathoverflow.net/users/176381 | Pseudocoherent analogue of compact + nuclear = dualizable? | Yes, what you write is correct, at least provided that the analytic ring $A$ has the very mild property that the internal RHom from $A[S]$ to $A$ is connective for every extremally disconnected profinite set $S$. (In other words, there should be no higher cohomology with values in $A$ on any product of two extremally d... | 6 | https://mathoverflow.net/users/3931 | 392089 | 162,222 |
https://mathoverflow.net/questions/389679 | 5 | Let $G$ be a complex reductive group acting linearly on $\mathbb{C}^n$ and let $X$ be a $G$-invariant closed subvariety of $\mathbb{C}^n$. Is $X$ the zero-set of finitely many $G$-invariant functions?
In other words, is the ideal $I \subset \mathbb{C}[x\_1, \ldots, x\_n]$ defining $X$ generated by finitely many invar... | https://mathoverflow.net/users/123207 | Invariant ideal generated by invariant elements | In general, the answer is no, as Friedrich Knop intimates in a comment. Consider the reductive group $G:=\mathbb{C}^\times = GL\_1(\mathbb{C})$ acting on $\mathbb{C}^2$ by $\alpha\cdot (x,y)\mapsto (\alpha x, \alpha y)$. Then the invariant ring $\mathbb{C}[x,y]^G$ is just $\mathbb{C}$, and the only subvarieties of $\ma... | 2 | https://mathoverflow.net/users/12419 | 392094 | 162,225 |
https://mathoverflow.net/questions/390020 | 3 | I have a holomorphic principal bundle,
$$E\xrightarrow{H} B$$ defined by an action of a contractible (non-compact) Lie group $H$ (in my case $H\cong\mathbb{C}^l$). Here E and B are complex manifolds, meaning that $H$ acts properly. We can assume that $B$ is a complete manifold.
Is there any sufficient condition to sa... | https://mathoverflow.net/users/143549 | Lifting holomorphic automorphisms along the principal bundle | An automorphism $f:B \to B$ lifts to an automorphism of $E$ if and only if the identity of $B$ lifts to an isomorphism from $E$ to $f^\*E$, i.e. if and only if $E$ and $f^\*E$ are isomorphic as principal bundles over $B$.
Defining principal $H$-bundles by transition functions, one sees that isomorphism classes of (ho... | 2 | https://mathoverflow.net/users/173096 | 392103 | 162,230 |
https://mathoverflow.net/questions/391578 | 4 | This is exercise 1.10 from Reid's Young person's guide to canonical singularites.
Let $X=\mathbb{C}^3/ \mu\_3$ where $\epsilon \in \mu\_3$ acts by
$$ (x,y,z) \to (\epsilon x, \epsilon y, \epsilon^2 z).$$
Then blowing up the origin gives us $E\_1 \cup E\_2$ where $E\_1$ is a plane and $E\_2$ is a quartic scroll.
We ... | https://mathoverflow.net/users/46923 | Resolution of 3-fold quotient singularities | The quotients $\mathbf{C}^n / G$ with $G$ finite abelian group (acting linearly) are toric varieties. I present the toric description of the resolution and the discrepancies. If one needs to, one could get differential forms and coordinates from this description. The reference [CLS] is the book "Toric Varieties" by Cox... | 3 | https://mathoverflow.net/users/111491 | 392106 | 162,231 |
https://mathoverflow.net/questions/392068 | 2 | $\DeclareMathOperator\SO{SO}$Recall that the *closed-subgroup theorem* ([Wikipedia link](https://en.wikipedia.org/wiki/Closed-subgroup_theorem)) says that a closed subgroup of a Lie group is a Lie group.
I am pretty sure that this theorem should have a "local" generalisation. I'll formulate it for $\SO(n)$.
**Gener... | https://mathoverflow.net/users/13441 | A variation of closed-subgroup theorem | See Tao's book on Hilbert's fifth problem -- <https://terrytao.wordpress.com/books/hilberts-fifth-problem-and-related-topics/> -- Theorem 3.1.7, where it is referred to as local Cartan's theorem.
| 4 | https://mathoverflow.net/users/20598 | 392107 | 162,232 |
https://mathoverflow.net/questions/392027 | 1 | I am working on a problem involving nilpotent matrices over $\mathbb{F}\_2$ and I was able to reduce it to proving that the system
\begin{equation}
\begin{cases}
A^2+ BC+ BCA+ ABC+A = I\_4 \\
AB+ABD+BCB = 0 \\
CA+DCA+CBC = 0 \\
DCB+CBD = I\_4 \\
A^3+BCA+ABC+BDC=0 \\
A^2B+BCB+ABD+BD^2=0 \\
CA^2+DCA+CBC+D^2C=0 \\... | https://mathoverflow.net/users/204792 | Proving that a system of polynomial matrix equations over $\mathbb{F_2}$ has no solution | We can solve quickly this problem using the basis Grobner theory.
Put $X=[x\_{i,j}],Y=[y\_{i,j}]$.
We consider -over a field of characteristic $2$- the algebraic system in the $128$ unknowns $x\_{i,j},y\_{i,j}$ constituted by the equations $M=X+Y,X^3=0,Y^2=Y$ -entrywise- and $x\_{i,j}^2=x\_{i,j},y\_{i,j}^2=y\_{i,j}... | 4 | https://mathoverflow.net/users/9091 | 392108 | 162,233 |
https://mathoverflow.net/questions/392105 | 1 | Let $\mathbb{Z}\_p$ be a finite field of order $p$ and $\mathbb{Z}\_p^2$ be a $2$-dimensional vector space over $\mathbb{Z}\_p$. We consider the distance $\lVert \cdot \rVert:\mathbb{Z}\_p^2\to \mathbb{Z}\_p$ defined by $\lVert {x}\rVert:=x\_1^2+x\_2^2$, where ${x}=(x\_1,x\_2)$. Suppose that $E\subset \mathbb{Z}\_p^2$ ... | https://mathoverflow.net/users/121924 | Mathematical induction and the counting function on $\mathbb{Z}_p^2$ | For $y\in E$, define
$$
\mu\_n(t\_1,\ldots,t\_n,y)=\#\{(x\_1,\ldots,x\_n)\in E^n\colon \|x\_i-x\_{i+1}\|=t\_i,
$$
where $x\_{n+1}$ is taken to be $y$.
Then your $\nu\_n(t\_1,\ldots,t\_n)$ is just $\sum\_{y\in E}\mu\_n(t\_1,\ldots,t\_n,y)$. It's easy to see that for $z\in E$,
$$
\mu\_{n+1}(t\_1,\ldots,t\_n,t\_{n+1},z)... | 1 | https://mathoverflow.net/users/11054 | 392114 | 162,235 |
https://mathoverflow.net/questions/391932 | 3 | Crossposted from
<https://math.stackexchange.com/questions/4116414/conjecture-on-bernoulli-numbers-and-binomial-coefficients>
In playing around with some formulas, I have come up with the following conjecture. I have checked it for a lot of cases, and have good reason to believe it to be true. If anyone could help,... | https://mathoverflow.net/users/204346 | Conjecture on bernoulli numbers and binomial coefficients | **UPDATE.** My earlier answer was incorrect due to miscalculation. Below I give a proof of the conjecture.
First, we notice that $k^{R(i)} = (k/2+i-1)\_i\cdot 2^i = \binom{k/2+i-1}i\cdot i!\cdot 2^i$. Similarly, $k^{D(i)} = (k/2)\_i\cdot 2^i = \binom{k/2}i\cdot i!\cdot 2^i$ and $(2i-1)!!=\frac{(2i)!}{i!2^i}=\binom{2i... | 7 | https://mathoverflow.net/users/7076 | 392119 | 162,236 |
https://mathoverflow.net/questions/392104 | 7 | $\DeclareMathOperator\sh{sh}$This question is [cross-posted from Math.SE](https://math.stackexchange.com/questions/4121521/do-strict-henselizations-satisfy-going-up) where it has gone unanswered for a week -- perhaps it is harder than I guessed. My question is this:
>
> Let $A$ be a local commutative, unital ring a... | https://mathoverflow.net/users/12419 | Does the strict henselization satisfy Going-Up? | Here is a counterexample.
Let $S$ be a smooth surface (irreducible). Let $C\_1, C\_2 \subset S$ be two disjoint smooth curves in $S$ which happen to be isomorphic. Let $X$ be the result of glueing $C\_1$ and $C\_2$ by this isomorphism. The singular locus of $X$ is a curve $C$ which is mapped onto isomorphically by $C... | 8 | https://mathoverflow.net/users/152991 | 392124 | 162,237 |
https://mathoverflow.net/questions/391477 | 3 | In recent papers
<https://arxiv.org/abs/2101.05520>
<https://arxiv.org/abs/2001.06911>
(super)integrable systems on quiver varieties for cyclic and comet-shaped quivers are constructed.
My question: are there heuristics and/or conjectures about the existence of (super)integrable systems on more general quiver v... | https://mathoverflow.net/users/198061 | (Super)integrable systems on quiver varieties | <http://arxiv.org/abs/2001.06911> referred my paper. But I meant more formal analogy between quiver varieties and Hitchin moduli spaces, such as hyper-Kaehler structure, S^1 action scaling the symplectic form, etc. I also pointed out that an analog of the Hitchin integrable system is the `affinization' morphism. It is ... | 1 | https://mathoverflow.net/users/3837 | 392129 | 162,238 |
https://mathoverflow.net/questions/392084 | 1 | There are various (equivalent?) descriptions of a universal finite-type knot invariant, e.g. <https://arxiv.org/abs/q-alg/9603010>. They take the form of formal power series valued in Feynman diagrams (certain oriented decorated trivalent graphs). Each such Feynman diagram corresponds to a configuration space integral,... | https://mathoverflow.net/users/113402 | Is there a combinatorial way to determine the coefficients of the universal finite-type invariant on a given knot? | First of all, however your universal finite type invariant $Z$ is given the question of computing the coefficient of a given diagram is somewhat ill-defined, since those diagrams are not linearly independent. However it might be that $Z$ is given as a sum over all diagrams weighted by some coefficients given by a speci... | 2 | https://mathoverflow.net/users/13552 | 392137 | 162,240 |
https://mathoverflow.net/questions/392136 | 5 | Let $\mathbb{Q}$ be the field of rational numbers, and let $\overline{\mathbb{Q}}$ be its algebraic closure. Assume $\overline{\mathbb{Z}}$ is the integral closure of $\mathbb{Z}$ in $\overline{\mathbb{Q}}$, my question is that, what can we say about the integral domain $\overline{\mathbb{Z}}$?
Explicitly:
(1) Is i... | https://mathoverflow.net/users/177957 | The integral closure $\overline{\mathbb{Z}}$ and the group $\overline{\mathbb{Z}}^{\times}$ | For (3), the answer is no, because there is no irreducible element in $\overline{\mathbb{Z}}$: for example, every element is a square.
Similarly for (1), the answer is no because there are infinite ascending chains of ideals: for example, $(x) \subset (x^{1/2}) \subset (x^{1/4}) \subset \cdots$, where $x$ is not a un... | 10 | https://mathoverflow.net/users/6506 | 392142 | 162,242 |
https://mathoverflow.net/questions/392014 | 17 | Consider the space $C\_c(\mathbb R)$ of complex-valued continuous functions of compact support. This is a vector space over $\mathbb C$, and I am not considering any topology, so the question is algebraic.
A set $A \subseteq C\_c(\mathbb R)$ is called shift-invariant if for all $f \in A$ and $t \in \mathbb R$ we also... | https://mathoverflow.net/users/11552 | Existence of translation-invariant basis on $C_c(\mathbb R)$ | No, there is no shift-invariant basis of $V=C\_c(\mathbb R).$ I'll use the formulation in [YCor's answer](https://mathoverflow.net/a/392049/164965), so we need to show that $V$ is not a free $B$-module where $B=\mathbb C[T^r:r\in \mathbb R],$ with $T^r$ acting as the translation $T\_r.$
Let $f(x)=\max(0,1-|x|).$ Supp... | 11 | https://mathoverflow.net/users/164965 | 392147 | 162,243 |
https://mathoverflow.net/questions/392143 | 7 | Let $G$ be a countable group acting minimally by homeomorphisms on a compact Hausdorff space $X$ and $\mu$ be a probability measure on $G$ whose support generates $G$ as a semigroup.
Let $\nu$ is a $\mu$-stationary measure on $X$ such that $(X,\nu)$ is a $(G,\mu)$ boundary, ie for $\mu^{\mathbb{N}}$ almost every trajec... | https://mathoverflow.net/users/182955 | Uniqueness of stationary measures for $(G,\mu)$ boundaries | A way to force a counter example is by taking $G$ to be an amenable group and choose $\mu$ such that the Furstenberg-Poisson boundary of $(G,\mu)$ is non-trivial.
Now take $X$ to be a compact model of this Furstenberg-Poisson boundary.
Then, by amenability, $X$ admits an invariant measure in addition to the $\nu$,
thus... | 3 | https://mathoverflow.net/users/89334 | 392164 | 162,250 |
https://mathoverflow.net/questions/392117 | 2 | Here I would like to prove a result that I assume is known but I am having difficulty proving. I will give the set up. This problem is really coming from functions with are 'doubly radial' or 'multi-radial'. Take $ 3 \le m \le n$ and integer and consider functions $u=u(s,t)$, which are smooth for $1<s^2+t^2<4$ and assu... | https://mathoverflow.net/users/66623 | Sobolev imbedding result; proof | This first part is an **extended comment**: I think you have your scaling wrong.
Let $u\_k = \phi(\lambda\_k(s-s\_k), \mu\_n(t-t\_k))$, where $\phi$ has compact support in the ball of radius 1.
We assume that $(s\_k, t\_k) \to (1.5,0)$ from within $\tilde{\Omega}$.
That $\mu\_k > 2/t\_k$, and $\lambda\_k > 2$, so $... | 4 | https://mathoverflow.net/users/3948 | 392171 | 162,252 |
https://mathoverflow.net/questions/392160 | 1 | If $\beta: \mathbf{U}\times \mathbf{V}\to \mathbf{W}$ is a bilinear map between real linear spaces then its derivative at a point $(u,v)$ is given by the Leibniz rule $$D\beta(u,v)(u\_0,v\_0)=\beta(u,v\_0)+\beta(u\_0,v).$$
Consequently the derivative of the multiplication map of a real linear space $\mu:\mathbb R\tim... | https://mathoverflow.net/users/69037 | The tangent map of the multiplication map of a vector bundle | Go back to the definition. Let $\gamma$ be a curve in $\mathbb{R}\times E$, we can take local coordinates on $E$ using the local trivialization, so
$$ \mathbb{R} \times E \ni (\lambda(s), v(s), b(s)) = \gamma(s) $$
The multiplication map gives
$$ \mu(\gamma(s)) = (\lambda(s) v(s), b(s)) $$
From this you find immediatel... | 3 | https://mathoverflow.net/users/3948 | 392177 | 162,253 |
https://mathoverflow.net/questions/392178 | 19 | In his [summary of his book](http://www.cambridgeblog.org/2021/04/bounded-gaps-between-primes-the-epic-breakthroughs-of-the-early-21st-century/) *Bounded gaps between primes: the epic breakthroughs of the early 21st century*, Kevin Broughan writes
>
> Which brings me to my final remark: where to next in the bounded... | https://mathoverflow.net/users/4177 | Possible contemporary improvement to bounded gaps between primes? | I think that there is indeed some possibility to lower the bound, and this is something I've looked at seriously a few times. I spent a semester (in 2019) with the Computational Number Theory Group here at BYU trying to do it, without success. Let me outline some of the difficulties we found.
**Issue 1.** The main te... | 20 | https://mathoverflow.net/users/3199 | 392183 | 162,254 |
https://mathoverflow.net/questions/392133 | 0 | This arises from an engineering problem I am working on. Let $\mathbf{c}\_i,\mathbf{a}\_i,\mathbf{b}\_i\in \mathbb{R}^{d}$ be a given set (collection) of vectors where $i\in\{1,\dots,n\}$. Define the polyhedrons (indexed over $i$)
$$
\mathcal{Q}\_i\,=\,\{\mathbf{x}\_i\in\mathbb{R}^d ~\lvert~ \mathbf{b}\_i^T\mathbf{x}\_... | https://mathoverflow.net/users/27249 | Iterations of Dantzig-Wolfe Decomposition for a Simple Linear Programming problem | You could specialize Dantzig-Wolfe by solving the subproblems (one per $i$) with an oracle other than LP. If the constraint $\mathbf{b}\_i^\top \mathbf{x}\_i \le 0$ weren't there, you could enumerate the resulting $d+1$ extreme points implied by $\mathbf{e}\_i^\top \mathbf{x}\_i \le 1$. You can also solve the subproble... | 0 | https://mathoverflow.net/users/141766 | 392188 | 162,256 |
https://mathoverflow.net/questions/392165 | 4 | Let $[\omega]^\omega$ denote the collection of infinite subsets of $\omega$. Let $$E = \big\{\{A,B\}: A,B \in [\omega]^\omega \text{ and } |A\cap B| \text{ is finite}\big\}.$$
We consider the graph $G=([\omega]^\omega,E)$. Maximal cliques in $G$, also known as [maximal almost disjoint families](https://en.wikipedia.o... | https://mathoverflow.net/users/8628 | Coloring almost-disjointness | No, $\chi(G)=\mathfrak c$, in fact $G$ contains a complete subgraph on $\mathfrak c$ vertices.
A simple way to construct one is by fixing a bijection $f\colon\Bbb Q\to\omega$ and fixing, for every $r\in\Bbb R\setminus\Bbb Q$, a sequence $(q^r\_i)\_{i\in\omega}$ of rationals numbers converging to $r$. Taking the image... | 11 | https://mathoverflow.net/users/49381 | 392193 | 162,258 |
https://mathoverflow.net/questions/392194 | 2 | The functions in question are
$$L(s)=\sum\_{k=1}^\infty \frac{\lambda(k)}{k^s}=\frac{\zeta(2s)}{\zeta(s)} \mbox{ and } L^\*(s)=\frac{1}{2}\sum\_{k=1}^\infty \frac{\lambda(k)+(-1)^{k+1}}{k^s}=\frac{L(s)+\eta(s)}{2},$$
where
* $\lambda(k) = (-1)^{\Omega(k)}$ is the [Liouville function](https://en.wikipedia.org/wiki... | https://mathoverflow.net/users/140356 | Analytic continuation and convergence of a Riemann zeta related function | Let $s\in\mathbb{C}$ be any point with $\Re s>0$. The Dirichlet series of $\eta(s)$ converges, hence $L(s)$ converges if and only if $L^\*(s)$ converges.
For $\sigma\_0>1/2$, it is also known that $L(s)$ converges in the half-plane $\Re s>\sigma\_0$ if and only if $\zeta(s)$ has no zero in that half-plane.
Combinin... | 8 | https://mathoverflow.net/users/11919 | 392202 | 162,260 |
https://mathoverflow.net/questions/392190 | 5 | $\DeclareMathOperator\SO{SO}$The following PDE defined on $\mathbb{R}^2$ $$\frac{\partial}{\partial x}\frac{\partial}{\partial y}f(x,y) = 0,$$ has solution $$f(x,y) = g(x) + h(y),$$ where $g,h : \mathbb{R} \to \mathbb{R}$ are arbitrary (nice enough) functions. I believe this to be a general solution to the equation, fo... | https://mathoverflow.net/users/171026 | General solution to an ultrahyperbolic PDE | The standard method of constructing solutions is the following:
First, observe that, if $(a,b)\in\mathbb{R}^n\times\mathbb{R}^n$ is any pair of vectors that satisfies $a\cdot b = 0$, and $h:\mathbb{R}\to\mathbb{R}$ is any smooth function, then
$$
f(x,y) = h(\,a{\cdot}x + b{\cdot}y\,)
$$
is a solution of the ultrahype... | 8 | https://mathoverflow.net/users/13972 | 392212 | 162,265 |
https://mathoverflow.net/questions/392179 | 6 | Suppose $\mu$ is a finitely additive measure on $X$ (aka “content”) with $\mu(X) < \infty$, defined on an algebra of sets $\mathcal A$. Let
$$\mu^\*(Y) = \inf \{ \mu(E) : E \in \mathcal A \wedge E \supseteq Y \}.$$
Question: Is it true that for all $Y \subseteq X$, there is an extension $\nu$ of $\mu$, where $\nu$ is a... | https://mathoverflow.net/users/11145 | Extending contents | I believe part of the question is answered in the following paper:
[Łoś, J.; Marczewski, E. Extensions of measure. Fund. Math. 36 (1949), 267–276.](https://www.impan.pl/pl/wydawnictwa/czasopisma-i-serie-wydawnicze/fundamenta-mathematicae/all/36/0/93726/extensions-of-measure)
Let $c$ be any number between (inclusive... | 6 | https://mathoverflow.net/users/209914 | 392219 | 162,267 |
https://mathoverflow.net/questions/392229 | 1 | Let $G$ be a projective plane degree $2d$ curve with equation $AC-B^2$, with $A$, $B$, $C$ of degrees $\deg A=2d-4$, $\deg B=d$, $\deg C=4$. Then, for $d>2$, a dimension count shows that $G$ is not generic.
Note that $A$ cuts an even divisor on $G$, of degree $2d(2d-4)$.
Is it true that for a smooth projective plane ... | https://mathoverflow.net/users/11100 | showing a plane curve non-generic by exhibiting an even divisor | The condition that there is a degree $2d - 4$ curve cutting out on a plane curve $X$ of degree $2d$ an even divisor is equivalent to the existence of a point
$$
\alpha \in \operatorname{Pic}^0(X)
$$
of order 2 such that the divisor class
$$
(d-2)H + \alpha \in \operatorname{Pic}^{2d(d-2)}(X)\tag{\*}
$$
is effective, wh... | 3 | https://mathoverflow.net/users/4428 | 392233 | 162,271 |
https://mathoverflow.net/questions/392239 | 8 | Let $X\subset \mathbb C^n$ be a complex hypersurface (given by $F=0$ where $F$ is a polynomial). It is known then that $X$ admits *a* Whitney stratification. This is a decomposition of $X$ into smooth submanifolds (strata) that have some adjacency properties (Whitney conditions a and b).
**Question.** Does $X$ have a... | https://mathoverflow.net/users/13441 | Is there a "minimal" Whitney stratification of a complex hypersurface? | The answer is yes for any reduced equidimensional analytic space. This is the proposition 3.2 (and remark after the proof) page 479 of [Variétés polaires II](https://webusers.imj-prg.fr/%7Ebernard.teissier/documents/VarPol2.pdf) by Bernard Teissier.
| 7 | https://mathoverflow.net/users/37214 | 392244 | 162,274 |
https://mathoverflow.net/questions/392225 | 3 | $\DeclareMathOperator\op{op}$Let $A$ be a $C^\*$-algebra. We know that $A$ admits a natural operator space structure, namely the operator space structure induced by any faithful $\*$-representation of $A$ into some $B(\mathcal H)$ for some Hilbert space $\mathcal H.$ Now consider the opposite $C^\*$ algebra $A^{\op}$ w... | https://mathoverflow.net/users/136860 | Opposite $C^*$ algebras | 1. Yes. You could consult [Pisier's Operator Space Theory book](https://www.cambridge.org/core/books/introduction-to-operator-space-theory/DE174FA28C7DBC243FBEA3911E97EA4E) sections 2.9 and 2.10.
To give details: let $H$ be a Hibert space and $\overline H$ be the complex conjugate space. This has vectors $\{ \overlin... | 5 | https://mathoverflow.net/users/406 | 392248 | 162,277 |
https://mathoverflow.net/questions/392023 | 3 | In Katz' paper Antwerp III, section 1.4 (Ka-14) one reads (we assume $n \geq 3$ integer):
''The scheme $\overline{M}\_n - M\_n$" over $\mathbb{Z}[1/n]$ is finite and étale, and over $\mathbb{Z}[1/n,\zeta\_n]$, it is a disjoint union of sections, called the cusps of $\overline{M}\_n$,''
I would be interested to see ... | https://mathoverflow.net/users/141653 | Explicit natural correspondence between cusps of $X(N)$ and isomorphism classes of level $N$ structures on Tate($q^N$) | To define this correspondence you don't want to write down either side explicitly, but rather do it abstractly. Then you can calculate what it is in explicit terms if desired. The first steps are
(1) Send each cusp in $\overline{M}\_n$ to its formal neighborhood in $\overline{M}\_n$, and then to its punctured formal ... | 1 | https://mathoverflow.net/users/18060 | 392259 | 162,281 |
https://mathoverflow.net/questions/392113 | 1 | I was asking this question at [Mathematics SE](https://math.stackexchange.com/questions/4128343/translating-limit-point-under-homeomorphism-map) but I got nothing at all. This is why I am trying this site.
We consider the topology of the extended real line. Let $h\colon [-\infty,\infty]\to\Bbb R$ and suppose $(-\inft... | https://mathoverflow.net/users/161756 | Limit points and Homeomorphism | A homeomorphism on an interval is increasing or decreasing; wlog I assume it to be increasing. There are $y\_n \to -\infty$ such that $h(y\_n)\to 0$. Write $y\_n=g(x\_n)$. Then $x\_n = g^{-1}(g(x\_n)) \to a$ since $g$ is a homeomorphism and $y\_n\to -\infty$, and $(h\circ g)(x\_n)\to 0$.
| 2 | https://mathoverflow.net/users/127871 | 392263 | 162,283 |
https://mathoverflow.net/questions/392237 | 11 | Let $(F\_k)\_{k=0}^\infty$ be the classical Fibonacci sequence, defined by the recursive formula $F\_{k+1}=F\_k+F\_{k-1}$ where $F\_0=0$ and $F\_1=1$.
For every $n\in\mathbb N$ let $\pi(n)$ be the smallest positive number such that $$\begin{cases}
F\_{\pi(n)}=F\_0 \mod n,\\
F\_{\pi(n)+1}= F\_1\mod n.
\end{cases}$$ Th... | https://mathoverflow.net/users/61536 | The Fibonacci sequence modulo $5^n$ | I claim that for even $n\in \{0,2,4,\ldots, 4\cdot 5^n-2\}$ each remainder of $F\_n$ modulo $5^n$ is realized at most twice (thus exactly twice), and the same for odd $n\in \{1,3,5,\ldots, 4\cdot 5^n-1\}$. Denoting $u=(1+\sqrt{5})/2$, $v=(1-\sqrt{5})/2$ we have Binet formula $F\_n=(u^n-v^n)/(u-v)$. If $k,m$ are even, t... | 9 | https://mathoverflow.net/users/4312 | 392265 | 162,284 |
https://mathoverflow.net/questions/392256 | 6 | I know there are some research about explicit equations for affine models in $\mathbb{A}^2$ of many modular curves over $\mathbb{Q}$, for example of $X\_i(N), X(N)$ (where $i = 0, 1, 2$) for small $N$.
These equations are very useful in order to study elliptic curves.
(Indeed there are so many papers which use them i... | https://mathoverflow.net/users/128235 | An explicit equation of the canonical morphism $X_1(N) \to X_0(N)$ | It is possible to find equations for $\pi : X\_1(N) \to X\_0(N)$ using work of Yifan Yang. In the article [Defining equations of modular curves](https://www.sciencedirect.com/science/article/pii/S0001870805001714), he explains an algorithm to obtain equations for the modular curves $X\_1(N)$ and $X\_0(N)$. Let me expla... | 6 | https://mathoverflow.net/users/6506 | 392269 | 162,286 |
https://mathoverflow.net/questions/391635 | 2 | Given a tree-graph with one of the vertices designated as the root, what is the complexity of calculating the number of vertex-pairs $\lbrace u,v \rbrace$ of which $v$ is not nearer to the root than $u$ and $u$ is not a vertex of the path from the root to $v$?
It is of course possible to determine that number in $O(n... | https://mathoverflow.net/users/31310 | Calculating number of vertex-pairs with separate common ancestor | It takes only addition to count the distinct pairs where one is the ancestor of another. Couldn’t you just subtract that from the total number of pairs?
| 2 | https://mathoverflow.net/users/8008 | 392298 | 162,296 |
https://mathoverflow.net/questions/392254 | 5 | Let $G$ be a group, and consider the action of $G$ on itself by conjugation. If we think of $G$ as a one object category, then the conjugation action can be realised as automorphisms of this category, and we may build the associated $2$ category with one object, with additional $2$ morphisms given by elements of $G$, a... | https://mathoverflow.net/users/128502 | Is there a higher categorical structure which models the (higher) conjugation actions of a group acting on itself? | Following up a bit on Fernando's comment, you look to be trying to define things a bit like the tensor square of a group (acting on itself). Look at the work by Ronnie Brown, et al, *Some computations of non-abelian tensor products of groups*, J. Algebra, 111, (1987), 177 – 202. This would give you a crossed square. Th... | 3 | https://mathoverflow.net/users/3502 | 392302 | 162,297 |
https://mathoverflow.net/questions/392308 | 7 | Let $X$ and $Y$ be *smooth* varieties over the field of complex numbers $\bf C$
(smooth integral separated schemes of finite type over $\bf C$). Let
$$f\colon X\to Y$$
be a surjective morphism such that
for any closed point $y\in Y$, the schematic fibre $f^{-1}(y)\subset X$
is isomorphic to the affine space ${\Bbb A}\_... | https://mathoverflow.net/users/4149 | Smooth morphism of smooth varieties with fibres isomorphic to an affine space | Regarding Question 1, it seems to be an open problem, known as a variant of Dolgachev–Weisfeiler Conjecture. The article [$\mathbb{A}^2$-fibrations between affine spaces are $\mathbb{A}^2$-trivial](https://arxiv.org/pdf/1704.04440.pdf) (A. Dubouloz) shows that an $\mathbb{A}^2$-fibration $f\colon X\to S$ is étale-local... | 3 | https://mathoverflow.net/users/211863 | 392322 | 162,303 |
https://mathoverflow.net/questions/392281 | 3 | Assume $x \in \mathbb{R}$. We already know that
$$\lim\_{\epsilon \to 0+} \frac{1}{x-i\epsilon} - \frac{1}{x+i\epsilon} = 2\pi i \delta\_x.$$
Here $\delta\_x$ denotes the Dirac distribution. If we consider a slightly more irregular limit
$$\lim\_{\epsilon \to 0+}\frac{\log^m(x-i\epsilon)}{x-i\epsilon} - \frac{\log^m(x+... | https://mathoverflow.net/users/114951 | What is the distribution of the following limit? | This question already has an impeccable answer but I would like to give a second one in the hope that it might be of interest to the OP. Before so doing, I will list the properties of distributions (on the line) that I require:
1. Any “reasonable” function can be regarded as a distribution (in our context, this means... | 3 | https://mathoverflow.net/users/204122 | 392324 | 162,304 |
https://mathoverflow.net/questions/392339 | 4 | I am interested in the scaling of
$$F(x\_1,x\_4)=\int\_{\mathbb R^2} e^{-\vert x\_1 -x\_2 \vert -\varepsilon \vert x\_2 -x\_3 \vert- \vert x\_3 -x\_4 \vert } \ dx\_2 dx\_3 $$
In particular, I suspect that
$$F(x\_1,x\_4) \le C \varepsilon^{-n} e^{-{\varepsilon}\vert x\_1 -x\_4\vert}$$
for some universal $C>0$ and ... | https://mathoverflow.net/users/150549 | Scaling of double convolution | $\newcommand\ep\varepsilon$$F(x\_1,x\_4)$ is $8/\ep$ times a value of the convolution of two copies of a pdf with maximum value $1/2$ and a pdf with maximum value $\ep/2$. So,
$$F(x\_1,x\_4)\le(8/\ep)\min(1/2,1/2,\ep/2)=4\min(1/\ep,1)$$
for all real $x\_1,x\_4$.
---
The straightforward integration gives
$$F(x\_1,... | 6 | https://mathoverflow.net/users/36721 | 392343 | 162,313 |
https://mathoverflow.net/questions/392284 | 20 | It is an old result that every $6$-connected graph is rigid in $\mathbb{R}^2$:
>
> Lovász, László, and Yechiam Yemini. "On generic rigidity in the plane." *SIAM Journal on Algebraic Discrete Methods* **3**, no. 1 (1982): 91-98.
> [DOI](https://doi.org/10.1137/0603009).
>
>
>
It is natural to hope that sufficie... | https://mathoverflow.net/users/6094 | Is every 1-million-connected graph rigid in 3D? | I think this is still an open problem, but recent work of [Clinch, Jackson, and Tanigawa](https://arxiv.org/abs/1911.00207) (almost) shows every $12$-connected graph is generically rigid in $\mathbb{R}^3$.
In that paper, they prove that $12$-connectivity is sufficient to force rigidity in the $C\_2^1$-cofactor matroi... | 11 | https://mathoverflow.net/users/2233 | 392348 | 162,315 |
https://mathoverflow.net/questions/391991 | 3 | Let $p$ be a prime number and $S\_p=\{(n!)^2 \bmod p, n=1,2,\dotsc,p-1\}$ the set of residues mod $p$ of squares of factorials. This set is obviously a subset of the group of quadratic residues mod p. For $p=3,5,7,13,17,23,29$ it is also a group for multiplication mod $p$, i.e. a subgroup of the group of quadratic resi... | https://mathoverflow.net/users/169583 | Quadratic character of factorials | In view of the identity $(n!)^2/((n-1)!)^2=n^2$, the set $S\_p$ generates the subgroup $Q\_p<\mathbb F\_p^\times$ of quadratic residues; thus, if $S\_p$ is a subgroup, then in fact $S\_p=Q\_p$. Clearly, a necessary and sufficient condition for this to happen is that the sequence of factorials $\{1!,\dotsc,(p-1)!\}$ hit... | 4 | https://mathoverflow.net/users/9924 | 392360 | 162,318 |
https://mathoverflow.net/questions/392359 | 2 | This is a [cross-post](https://math.stackexchange.com/questions/4131916/can-we-almost-cover-any-shape-in-the-plane-by-disjoint-disks-of-prescribed-radii).
Let $(a\_n)\_{n \in \mathbb{Z}}$ be some given, strictly increasing sequence of positive numbers, such that $\lim\_{n \to -\infty} a\_n=0,\lim\_{n \to +\infty} a\_... | https://mathoverflow.net/users/46290 | Can we almost cover any shape in the plane by disjoint/tangent disks of prescribed radii? | The answer is **yes**, in an arbitrary dimension $d$. Here is a short proof.
Suppose, contrary to the above claim, that for some Jordan measurable $\Omega$ the infimum of what is left is equal to $m > 0$:
$$ m = \inf \biggl\{\biggl|\Omega \setminus \bigcup\_{k = 1}^K B(x\_k, r\_k)\biggr| : x\_k, r\_k \text{ as in the... | 2 | https://mathoverflow.net/users/108637 | 392365 | 162,319 |
https://mathoverflow.net/questions/392340 | 3 | As in the title: given a vector field $\vec f$, are there any interesting applications (in physics, biology, or economy, or ...) of the partial differential equation
$ - \operatorname{grad} ( \operatorname{div} \vec u ) = \vec f$
with unknown vector field $\vec u$?
I am aware that the PDE is generally under-const... | https://mathoverflow.net/users/2082 | Are there applications for the PDE $ - \operatorname{grad} ( \operatorname{div} \vec u ) = \vec f$? | This is the [continuity equation](https://en.wikipedia.org/wiki/Continuity_equation) in disguise. Since $\text{curl}\,\vec{f}=0$, we can write $\vec{f}=\text{grad}\,\Phi$ for a scalar function $\Phi$, and then
$$-\text{div}\,\vec{u}=\Phi+\text{constant}.$$
If we fix the constant at zero (for example, by considering the... | 7 | https://mathoverflow.net/users/11260 | 392368 | 162,321 |
https://mathoverflow.net/questions/392286 | 3 | Consider $n$ points $\{p\_i\}\_{i=1}^n$ located inside or on a circle with radius $r$ in the plane. The question is: how to place the $n$ points so that the sum of inter-point distances,
$$J=\sum\_{i=1}^n\sum\_{j=1}^n \|p\_i-p\_j\|^a,$$
is maximized? Here, $a$ could be 1 or 2.
The intuition is that the optimal soluti... | https://mathoverflow.net/users/12630 | Maximizing the distance sum of some points inside a circle | Let 0 be the centre of your circle of radius $r$.
If $a=2$, we expand the brackets and write $\sum\_{i,j} (p\_i-p\_j)^2=2n\sum p\_i^2-2(\sum p\_i)^2\leqslant 2nr^2$, with equality if and only if all points lie on a circle and 0 is a barycentre.
If $a=1$, we again may suppose that they lie on circle (since the funct... | 2 | https://mathoverflow.net/users/4312 | 392369 | 162,322 |
https://mathoverflow.net/questions/392319 | 8 | Suppose that $I, X\_1, \ldots, X\_{d-1}$ are $n \times n$ matrices with integer entries whose $\mathbb{Z}$-span is a subalgebra of $\mathrm{Mat}\_n(\mathbb{Z})$. Suppose that, thought of as a subalgebra of $\mathrm{Mat}\_n(\mathbb{C})$, this algebra is semisimple and non-commutative. Thus, by Wedderburn's Theorem, it i... | https://mathoverflow.net/users/7709 | Is it possible for the reduction modulo $p$ of an non-commutative semisimple algebra to be commutative? | Example 5.10 of
*Towers, Matthew*, [**Endomorphism algebras of transitive permutation modules for $p$-groups.**](http://dx.doi.org/10.1007/s00013-009-3083-8), Arch. Math. 92, No. 3, 215-227 (2009)
(whose author you might know) gives a positive answer to the second question. The group action involved is of
$$G=\lang... | 4 | https://mathoverflow.net/users/22989 | 392371 | 162,324 |
https://mathoverflow.net/questions/392347 | 4 | Let $f: [0, 1] \to \mathbb R$ be a function on the unit interval. We say $y \in \mathbb R$ is a local maximum value of $f$ if $y = f(x)$ for some strict local maximum $x$ of $f$.
>
> **Question:** Does there exist a $C^1$ function $f$ on the unit interval such that the set of local maximum values of $f$ is dense in... | https://mathoverflow.net/users/173490 | $C^1$ function with a dense set of maximum values | Such a function cannot exist. Let $M \subset [0,1]$ be the set of strict local maxima of $f$, and $C \subset [0,1]$ be the critical points, that is the set of points $x \in [0,1]$ so that $f'(x) = 0$. Then $f(C)$ is a closed subset of $\mathbf{R}$. As a consequence, if $f(M) \cap (a,b)$ were dense in some interval $(a,... | 6 | https://mathoverflow.net/users/103792 | 392373 | 162,325 |
https://mathoverflow.net/questions/392375 | 2 | Suppose we have a non symmetric operad $\mathcal{O}$, a collection of sets
$\{P(n)\}\_{n\geq 0}$ and maps
$$P(n)\otimes \mathcal{O}(k\_1)\otimes\cdots \otimes \mathcal{O}(k\_n)\to P(k\_1+\cdots + k\_n)$$
satisfying some conditions on associativity when composed with the operadic composition, etc.
>
> **Question... | https://mathoverflow.net/users/161009 | Right action by an operad on a non symmetric collection | This structure is called a right module over the operad. You can have a look at the book "Modules over operads and functors" by Benoit Fresse for a thorough study of these objects.
| 6 | https://mathoverflow.net/users/10707 | 392377 | 162,327 |
https://mathoverflow.net/questions/392357 | 5 | **Notation:** For a group $G$, we write $G^n$ to denote the $n$-fold direct product of $G$ with itself.
**Problem set up:**
Consider, for a finite group $G$, and $n > 1$, the set $Q(G)\_n$ of all isomorphism classes of groups of the form $\frac{G^{n}}{H\_{n-1}}$, for any normal subgroup $H\_{n-1}$ of $G^n$ isomorph... | https://mathoverflow.net/users/173490 | Quotient groups obtained by quotienting $G^n$ by $G^{n-1}$ | I'll abbreviate $\mathbb{Z}/p^k \mathbb{Z}$ to $L\_k$. Let $G = \bigoplus\_{i=1}^M L\_i^{a\_i}$. I claim that $\bigoplus\_{i=1}^N L\_i^{b\_i}$ is in the quotient series of $G$ if and only if the Littlewood-Richardson coefficient
$$c\_{(1^a\_1),\ \ (2^{a\_2}),\ \ \dotsc,\ \ (M^{a\_M})}^{(1^{b\_1} 2^{b\_2} \cdots N^{b\_N... | 10 | https://mathoverflow.net/users/297 | 392380 | 162,329 |
https://mathoverflow.net/questions/392186 | 4 | I wonder if there is a good reference on *reaction-diffusion systems* on $\mathbb{R}^N$, that treats them as *dynamical systems*.
I have the books of *Alain Haraux – Systèmes dynamiques dissipatifs et applications*, and *Joel Smoller – Shock waves and reaction–diffusion equations*, but I look for something newer and ... | https://mathoverflow.net/users/61629 | Reaction-diffusion systems treated as dynamical systems | For your basic question (topic name) the classical reference is
Henry D. Geometric Theory of Semilinear Parabolic Equations. SpringerVerlag, 1981.
However, you will not find there general recipes for your second question and that's why. A stationary state, say $y\_{0}(x)$, can be homogeneous (given by constant func... | 5 | https://mathoverflow.net/users/85336 | 392384 | 162,332 |
https://mathoverflow.net/questions/392376 | 5 | In *Turing invariant sets and the perfect set property*, Math. Log. Quart. **66** (2020), Hamel, Horowitz and Shelah, the authors work in ZF + DC. They claim that DC can be dispensed with, asserting:
>
> if $V \models {\rm ZF}\: + $ “all Turing invariant sets have the perfect set property” and $X \in V$ is a set of... | https://mathoverflow.net/users/128500 | Dependent choices (DC) in ${\bf HOD}(\mathbb{R},X)$, where $X$ is a set of reals | Consider the "singular Solovay-style model" of John Truss from
>
> *Truss, John*, [**Models of set theory containing many perfect sets**](http://dx.doi.org/10.1016/0003-4843(74)90015-1), Ann. Math. Logic 7, 197-219 (1974). [ZBL0302.02024](https://zbmath.org/?q=an:0302.02024).
>
>
>
In that model the following ... | 5 | https://mathoverflow.net/users/7206 | 392388 | 162,333 |
https://mathoverflow.net/questions/392125 | 2 |
>
> When defining a category, do we need to have the set of composable arrows be a pullback of the domain and codomain selecting functions? What can go wrong if we use a subobject of the pullback?
>
>
>
This arose when trying to formalize pasting diagrams with a minimal amount of graph theory; I would like to ha... | https://mathoverflow.net/users/92164 | Does the set of composable arrows in a category have to be a pullback? | As mentioned in the comments, there are different possible notions of "partial category". A definition due to Freyd is called a [paracategory](https://ncatlab.org/nlab/show/paracategory); this is an "unbiased" notion, equipped with partial $n$-ary composition operations for all $n\ge 0$ (although the case $n=0$ is requ... | 2 | https://mathoverflow.net/users/49 | 392389 | 162,334 |
https://mathoverflow.net/questions/392386 | 5 | I've been done some work with scalar curvature and managed to give a simple proof for the following result:
>
> Let $M$ be a closed manifold which do not admit a metric of positive scalar curvature. Then for any Riemannian metric $g$ on $M$ it holds that
> $$\int\_M \mathrm{scal}\_g \leq 0.$$
>
>
>
In particul... | https://mathoverflow.net/users/94097 | A corollary of the non-existence of positive scalar curvature | I am suspicious of your result.
The three torus $\mathbb{T}^3$ is well-known to not admit any metric of positive scalar curvature.
Let $g\_0$ be the flat metric on $\mathbb{T}^3$. Given a positive function $u > 0$, consider the metric $g = u^{4} g\_0$. Then we have the identity
$$ - 8 \Delta\_0 u = S\_g u^5 \tag{1}... | 14 | https://mathoverflow.net/users/3948 | 392393 | 162,336 |
https://mathoverflow.net/questions/392398 | 0 | Consider the divergence form uniformly elliptic operator $\nabla \cdot a(x) \nabla$
where the coefficient $a$ are smooth and bounded and $D$ is a bounded
and smooth domain of $\mathbb R^d$
$$
\begin{cases}
\nabla \cdot a(x) \nabla f (x)=g \text{ in } D \\
f(x) = 0 \text{ in } \partial D,
\end{cases}
$$
where $g$ for ... | https://mathoverflow.net/users/52960 | Reference request: Is if possible to estimate the local behaviour of the solution of $\nabla \cdot a(x) \nabla f=g$ via constant coefficients? | As Willie Wong pointed out, there are issues with the quantity you are looking at. But you can fine-tune your question in the one dimensional case, as you cannot expect better regularity than in $1$-d case. Not trying to insult anyone's intelligence, but the solution is in that case an explicit function of $a$ and $g$.... | 1 | https://mathoverflow.net/users/40120 | 392405 | 162,340 |
https://mathoverflow.net/questions/392396 | 3 | Let $X$ be a compact Riemann surface. We fix a complex vector bundle $E$ of rank $n$ and degree $d$ (unique up to diffeomorphism). From results coming originally (I think at least) by Simpson,Donaldson and Hitchin one gets an equivalence between:
* Polystable Higgs field $(\bar{\partial\_E},\phi )$ where $\bar{\parti... | https://mathoverflow.net/users/146464 | Non Abelian Hodge theory: underlying structure holomorphic vector bundles | In general, the complex structures associated to $\nabla$ and $\nabla^h$ are different. One case where things can be made explicit is when the Higgs bundle arise from a (complex) variation of Hodge structure, c.f. Simpson, *Higgs bundles and local systems*. Then $(E,\nabla^{0,1})$ is a flat bundle associated to the loc... | 3 | https://mathoverflow.net/users/4144 | 392407 | 162,342 |
https://mathoverflow.net/questions/392123 | 2 | Let $f: \mathbb{R}^{n^2} \times \mathbb{Z}^{n} \longrightarrow \mathbb{R}$ defined by
$$
f(X,z) = \prod\_{i=1}^{n} |x\_i z|,
$$
where $x\_i$ is the $i$-th row of $X$ and $x\_iz$ is a dot product of $x\_i$ and $z$.
My question is:
Is it true that
$$
\inf\_{z \neq 0} \lim\_{X \rightarrow 0} f(e^{X+C}, z) = \lim\_{X \righ... | https://mathoverflow.net/users/136645 | Problem of analysis about matrix exponential, infimum and limit | Define $I(X)=\inf\_{z\neq0}f(e^X,z)$. Because $f$ is continuous, your question is precisely: is $I$ continuous? I claim it isn't always the case.
>
> Call $\mathcal X$ the set of matrices $X$ such that $I(X)=0$. Then
>
>
> 1. $I$ is continuous precisely on $\mathcal X$;
> 2. $\mathcal X$ is a countable intersecti... | 0 | https://mathoverflow.net/users/129074 | 392411 | 162,344 |
https://mathoverflow.net/questions/392419 | 14 | This is a reference request for the following "well-known" theorems in category theory:
1. There is an equivalence of categories between finitary monads on $\mathbf{Set}$ and finitary Lawvere theories (i.e. single-sorted finite product theories).
2. There is an equivalence of categories between monads on $\mathbf{Set... | https://mathoverflow.net/users/2841 | Reference request for Linton's theorems on equational theories | (1, 2, 3) Though Linton's *An outline of functorial semantics* does contain the essence of the results and proofs of the monad–theory correspondence (see in particular Theorems 8.1 and 9.1 – 9.3), it is true that the terminology and style make it difficult to extract the results as we would expect to see them today. I ... | 10 | https://mathoverflow.net/users/152679 | 392422 | 162,347 |
https://mathoverflow.net/questions/392431 | 35 | From time to time Mathoverflow allows soft questions because they are arguably best answered by active mathematicians and they can benefit other mathematicians/PhD students/math undergraduates. I think this is such a question.
I'm a mathematics student planning to enroll in a good math PhD program this Fall. I have a... | https://mathoverflow.net/users/213977 | Is pure mathematics useful outside of mathematics itself? | This is not really an answer to the question as asked, but I believe it's important and relevant to your problem, and too long for a comment.
I will not here express any opinion about the validity or importance of your doubts, or share any of my own beliefs about them. Instead, the point I want to make at the moment ... | 42 | https://mathoverflow.net/users/49 | 392439 | 162,355 |
https://mathoverflow.net/questions/392446 | 3 | This is a sharpening of the following problem: [$C^1$ function with a dense set of maximum values](https://mathoverflow.net/questions/392347/c1-function-with-a-dense-set-of-maximum-values).
**Problem set up:**
Let $f \colon [0, 1] \to \mathbb R$ be a function on the unit interval. We say $y \in \mathbb R$ is a loca... | https://mathoverflow.net/users/173490 | Bounded $r$- variation function with a dense set of local maximum values | If you are happy with probabilistic constructions, fractional Brownian motion $X\_t^H$ gives an answer. If it's Hurst parameter is $H \in (0, 1)$, then the paths of $X\_t^H$ have infinite $1/H$-variation with probability one, and hence they necessarily attain local maxima in a dense set of $t$, just as the usual Browni... | 3 | https://mathoverflow.net/users/108637 | 392461 | 162,363 |
https://mathoverflow.net/questions/392366 | 2 | On a smooth manifold of dimension $n$, the application value of the canonical $1$-form, the Liouville form on $T^\*(X)$, to the Hamiltonian mechanics is well known; $T^\*(X)$ is a degree $1$-Jet bundle. My question is *Do canonical forms similar to the Liouville form exist on higher degree Jet bundles?*
I ask this beca... | https://mathoverflow.net/users/167228 | Canonical forms on higher degree Jet bundles similar to the Liouville form | The $k$-jet bundle $J^k$ of $k$-jets of real valued functions on a manifold $M$ has an obvious map $J^k\to J^1$, if $k\ge 1$, smooth and diffeomorphism invariant, taking the $k$-jet of a function to its $1$-jet. The $1$-jet bundle has an obvious splitting $J^1=J^0 \oplus T^\* X$, mapping each $1$-jet to its $0$-jet and... | 1 | https://mathoverflow.net/users/13268 | 392473 | 162,366 |
https://mathoverflow.net/questions/368121 | 12 | *This question was previously [asked and bountied](https://math.stackexchange.com/questions/3769001/comparing-the-cardinalities-of-generic-mathbbrs) at MSE, unsuccessfully.*
---
I'm currently interested in the behavior of cardinalities in generic extensions of models of $\mathsf{ZF+\neg AC}$, and especially of $\... | https://mathoverflow.net/users/8133 | Comparing generic versions of $\mathbb{R}$ | The answer seems to be no. Moreover: Suppose that every set of reals has the property of Baire. Let $\mathbb{C}$ be Cohen forcing and let $P$ be any wellorderable partial order. If $(c,d)$ is generic for $\mathbb{C} \times P$, then there is no injection in $V[c,d]$ from the Cantor space of $V[c]$ to any set in $V[d]$.
... | 5 | https://mathoverflow.net/users/31807 | 392486 | 162,371 |
https://mathoverflow.net/questions/392176 | 7 | Is there a countable metric space $U$ such that any countable metric space is bi-Lipschitz equivalent to a subset of $U$? How about $c\_{00}(\mathbb{Q})$ where $\mathbb{Q}$ is the rational numbers? Thanks!
| https://mathoverflow.net/users/208887 | Does there exist a countable metric space which is Lipschitz universal for all countable metric spaces? | The affirmative answer to this problem follows from
>
> **Lemma.** For any countable dense subsets $X,Y$ in the half-line $\mathbb R\_+=[0,+\infty)$ there exists a $C^2$-smooth function $f:\mathbb R\_+\to\mathbb R\_+$ such that
>
>
> $\bullet$ $f(X)\subseteq Y\cup\{0\}$;
>
>
> $\bullet$ $f(0)=0$;
>
>
> $\bull... | 5 | https://mathoverflow.net/users/61536 | 392497 | 162,374 |
https://mathoverflow.net/questions/392464 | 10 |
>
> **Question.** Does there exist a smooth complex projective variety with infinite and perfect fundamental group?
>
>
>
A group $G$ is perfect if its Abelianisation $G/[G,\, G]$ is the trivial group.
| https://mathoverflow.net/users/99732 | Variety having infinite and perfect $\pi_1$ | The simplest example I know of such a group is given by the presentation
$$
\langle a, b, c| a^2, b^3, c^7, abc\rangle.
$$
This group $G$ is obviously perfect and (less obviously) is isomorphic to the fundamental group of a smooth complex-projective variety. The reason for the latter is that $G$ acts holomorphically, p... | 8 | https://mathoverflow.net/users/39654 | 392498 | 162,375 |
https://mathoverflow.net/questions/392490 | 5 | Suppose $\{(E\_r,d\_r)\}$ for $r>0$ forms a spectral sequence over a field $\mathbb{F}$, i.e. for any $r$, $(E\_r,d\_r)$ is a chain complex over $\mathbb{F}$ and $E\_{r+1}=H(E\_r,d\_r)$. For simplicity, suppose $E\_N=E\_{N+1}=\cdots=E\_{\infty}$ for some fixed number $N$. I have the following questions:
1. Can we con... | https://mathoverflow.net/users/169890 | Can we construct a filtered chain complex from a spectral sequence? | 1. Yes, since you're working over a field, you can decompose your spectral sequence into a big direct sum of
a) permanent cycles, and
b) for each r>1, chains supporting a d\_r, and the cycle hit by that d\_r.
It is straightforward to represent each such summand by a simple filtered chain complex, so do so, and th... | 3 | https://mathoverflow.net/users/nan | 392524 | 162,380 |
https://mathoverflow.net/questions/392523 | 2 | Let $M=O\Lambda O^\top$ be a positive semi-definite matrix, where $\Lambda\in \mathbb{R}^{p\times p}$ is a diagonal matrix with non-negative entries and $O\in \mathbb{R}^{p\times p}$ is an orthogonal matrix. Let $S\_p$ be the set of all orthogonal matrices of size $p\times p$. What is the average of $M$ over the set $S... | https://mathoverflow.net/users/99157 | Average the covariance matrix over all orthogonal matrices | I presume you want to average over the orthogonal matrices uniformly, so with the Haar measure. Then $\mathbb{E}[O\_{ik}O\_{jk}]=p^{-1}\delta\_{ij}$, hence
$$\mathbb{E}[M\_{ij}]=\mathbb{E}\left[\sum\_{k=1}^p O\_{ik}\lambda\_k O\_{jk}\right]=p^{-1}\delta\_{ij}\sum\_{k=1}^p\lambda\_{k},$$
with $\Lambda\_{ij}=\delta\_{ij}... | 5 | https://mathoverflow.net/users/11260 | 392527 | 162,381 |
https://mathoverflow.net/questions/392525 | 17 | **Problem 1.** Is it true that for every cardinal $\kappa\le\mathfrak c$ there exists a partition $(B\_\alpha)\_{\alpha\in\kappa}$ of the real line into $\kappa$ pairwise disjoint non-empty Borel subsets?
**Remark.** The answer to this problem is affirmative if $\mathfrak c\le \aleph\_2$.
If the answer to Problem 1... | https://mathoverflow.net/users/61536 | Partitions of the real line into Borel subsets | The answer to both problems is no!
If the Cohen forcing is used to add lots of reals to a countable transitive model of GCH, then in the resulting extension, any partition of the real line into Borel sets has size $\leq \aleph\_1$ or $\mathfrak{c}$. (But the continuum can be anything with uncountable cofinality.) Thi... | 14 | https://mathoverflow.net/users/70618 | 392531 | 162,384 |
https://mathoverflow.net/questions/392526 | 7 | Remember that an n-REA set is a set of the form $A\_0 \oplus A\_1 \oplus \cdots \oplus A\_n$ with $A\_n$ relatively r.e. in $A\_m, m<n$ (so $A\_0$ is r.e.) and that a degree is PA just if it computes a path through every infinite computable tree.
By Arslanov's completeness criterion no incomplete r.e. set can be of P... | https://mathoverflow.net/users/23648 | 2-REA PA degrees | Theorem 5.1 in [Recursively Enumerable Sets Modulo Iterated Jumps and Extensions of Arslanov's Completeness Criterion](https://www.jstor.org/stable/2274816?seq=1) states that Arslanov's completeness criterion holds for $n$-REA sets. The paper is from 1989 and by Jockusch, Lerman, Soare, and Solovay.
| 6 | https://mathoverflow.net/users/68448 | 392542 | 162,388 |
https://mathoverflow.net/questions/392556 | 1 | Let $A\subseteq B \subseteq B(H)$ be an inclusion of $C^\*$-algebras where $H$ is some Hilbert space. We have the following conditions:
* B is a von Neumann algebra with $A'' = B$.
* The inclusion $A \subseteq B(H)$ is non-degenerate.
* $B$ contains $\operatorname{id}\_H$, but $A$ does not.
Further, let $\omega \in... | https://mathoverflow.net/users/216007 | Show convergence of net associated to GNS-triplet associated to state on a $C^*$-algebra | We need the following fact: if a net of positive operators $(a\_{\lambda})$ is increasing and bounded then it converges in the strong operator topology. It is not difficult to check that it converges in the weak operator topology to some element $a$. Therefore $(a-a\_{\lambda})$ is a bounded net of positive operators t... | 4 | https://mathoverflow.net/users/24953 | 392557 | 162,394 |
https://mathoverflow.net/questions/392517 | 7 | Let $(M^n,g)$ be a closed smooth Riemannian $n$-manifold with positive scalar curvature (or positive Ricci curvature) and $(S^n, g\_{st})$ be the standard round $n$-sphere.
Whether there exists a non-zero degree harmonic map $f$ from $M^n$ onto $S^n$, $f:M^n\to S^n$?
| https://mathoverflow.net/users/90512 | Existence of harmonic maps onto the $n$-sphere | A simple example where the answer is 'no' is when $M=\mathbb{RP}^2$ (with, say, the standard metric of Gauss curvature $K\equiv1$, though, in dimension $2$, only the conformal structure on $M$ matters in the definition of harmonic map).
There is no non-constant harmonic map $f:\mathbb{RP}^2\to S^2$ (when $S^2$ given ... | 5 | https://mathoverflow.net/users/13972 | 392569 | 162,398 |
https://mathoverflow.net/questions/392572 | 2 | I was told that the following theorem is in Demazure, and Gabriel's book Introduction to Algebraic Geometry and Algebraic Groups, but I could not find the theorem. The theorem is that a principal $ \mathbb{G}\_{a} $-bundle over an affine variety is trivial. Does someone know a reference?
| https://mathoverflow.net/users/113893 | A principal $ \mathbb{G}_{a}$-bundle over an affine variety is trivial | If by principal $\mathbf{G}\_a$-bundle you mean a $\mathbf{G}\_a$-torsor in the fppf topology, then this can proved as follows. Since such objects are classified by elements in the fppf cohomology of your scheme with coefficients in $\mathbf{G}\_a$, the result follows from the following proposition.
**Proposition:** ... | 3 | https://mathoverflow.net/users/21278 | 392574 | 162,400 |
https://mathoverflow.net/questions/391931 | 9 | My questions may turn out to be related to Schur functors.
If $\mathfrak{g}$ is a complex semisimple Lie algebra and $\lambda$ is the highest weight of an irreducible representation $V$ of $\mathfrak{g}$, I am interested in restricting the representation to an $\mathfrak{sl}(2)$ subalgebra of $\mathfrak{g}$. For exam... | https://mathoverflow.net/users/81645 | Which representations of $\mathfrak{sl}(2)$ are homomorphic images of the tensor product of finitely many copies of $\mathbb{C}^2$? | $\DeclareMathOperator\SL{SL}\DeclareMathOperator\Sym{Sym}$**Question:** "Which representations of $\mathfrak{sl}(2)$ are homomorphic images of the tensor product of finitely many copies of $\mathbb{C}^2$?"
**Answer:** If $k$ is the field of complex numbers and if $W:=k\{e\_1,e\_2\}$ it follows that any finite dimensi... | 3 | https://mathoverflow.net/users/nan | 392578 | 162,401 |
https://mathoverflow.net/questions/392311 | 6 | I have been reading these great [notes](https://faculty.math.illinois.edu/%7Erezk/quasicats.pdf) by Charles Rezk, and one thing that has been bothering me is the join construction. To solve lifting problems in quasicategory theory we use the Leibniz construction, namely given a bifunctor $F: A \times B \to C$ that is b... | https://mathoverflow.net/users/124010 | Join as a bifunctor | All of this is explained elegantly in Appendix D.2 in Riehl & Verity's Elements of $\infty$-category theory, and actually answers all of my questions. As Alexander Campbell says, the real important concept is the join as a bifunctor of augmented simplicial sets. Taking $\pi\_0$ or the trivial augmentation is not as imp... | 1 | https://mathoverflow.net/users/124010 | 392598 | 162,409 |
https://mathoverflow.net/questions/392562 | 12 | For a finitely presented group $G$, generated by a finite set $A$, the *commutator problem* is the decision problem: given a word $w$ over the alphabet $A \cup A^{-1}$, can one decide if $w$ is a commutator, i.e. whether there exist words $x, y$ such that $w = [x, y]$ in $G$. Here $[x, y] = x^{-1}y^{-1}xy$ is the commu... | https://mathoverflow.net/users/120914 | Commutator problem vs conjugacy/word problem | Denis Osin [*Osin, Denis*, [**Small cancellations over relatively hyperbolic groups and embedding theorems**](http://dx.doi.org/10.4007/annals.2010.172.1), Ann. Math. (2) 172, No. 1, 1-39 (2010). [ZBL1203.20031](https://zbmath.org/?q=an:1203.20031).] proved that every torsion-free countable group can be embedded into a... | 14 | https://mathoverflow.net/users/7644 | 392605 | 162,413 |
https://mathoverflow.net/questions/381581 | 4 | Let $X$ be a (say, complex) toric variety acted upon by a torus $T$ and defined by a fan $\Sigma$ in the cocharacter lattice $N=\mathrm{Hom}(\mathbb{C}^\times, T)$, and let $M$ be the character lattice. For any cone $\sigma \in \Sigma$ put $M(\sigma) = \sigma^\perp \cap M$, $N(\sigma) = \mathrm{Hom}(M(\sigma), \mathbb{... | https://mathoverflow.net/users/2234 | the map on Picard groups induced by restriction to a toric subvariety | A toric Cartier divisor $D$ is given by the Cartier data $\{m\_\sigma\}\_{\sigma \in \Sigma}$ [CLS, Theorem 4.2.8] where for each affine open chart $U\_\sigma$, the toric coordinate $x^{-m\_\sigma}$ is the equation for $D \cap U\_\sigma$. One way to visualize Cartier data is to consider $\{m\_\sigma\}$ as a piece-wise ... | 2 | https://mathoverflow.net/users/111491 | 392609 | 162,415 |
https://mathoverflow.net/questions/392561 | 1 | Let $X$ be a Banach space. We say that $X$ contains $\ell\_1^n$'s uniformly iff for all $n\in\mathbb N$ there exist subspaces $X\_n\subseteq X$ with $d(X\_n,\ell\_1^n)\leq \lambda$ for some $\lambda\geq 1$. A famous theorem of Pisier's asserts that an infinite dimensional Banach space is $K$-convex iff it does not cont... | https://mathoverflow.net/users/136860 | $K$-convex Banach spaces | $Y\_n$ can be $\ell\_\infty^n$, in which case the best projection onto any $\ell\_1^k$ is of order at least $\sqrt{k}$.
| 3 | https://mathoverflow.net/users/2554 | 392611 | 162,417 |
https://mathoverflow.net/questions/392415 | 2 | Let $X: \Omega\to{\mathbb R}$ be a random variable. Is it always possible to modify it (i.e. change the value of $X$ on a subset of $\Omega$ of zero measure) so that the range of $X$ is a Borel set?
This is related to the following question: we know that $E(Y|X)$ can be written as a function of $X$, i.e. $E(Y|X)=\var... | https://mathoverflow.net/users/130379 | Modify a random variable to make its range Borel? | Assume that $(\Omega, {\mathcal F}, P)$ itself is a Lebesgue space, so it can be realized as a Polish space equipped with the completion of the Borel sets. If $X$ is a random variable, it can be changed on a set of measure zero to be a Borel measurable function. (This is an easy consequence of [1]). Thus we may assume ... | 2 | https://mathoverflow.net/users/7691 | 392627 | 162,421 |
https://mathoverflow.net/questions/392636 | 3 | Who knows the name of the following coloring of graphs, a proper vertex coloring so that for every vertex its every two neighbors receive different colors?
| https://mathoverflow.net/users/148974 | What's the name of a special vertex coloring | For a graph $G$, the *$t$-th power* of $G$ is the graph $G^t$ with the same vertex set as $G$ and where two vertices are adjacent in $G^t$ if they are connected by a path with at most $t$ edges in $G$. The *distance-$t$ chromatic number* of $G$, often denoted $\chi\_t(G)$, is the chromatic number of $G^t$. As noted by ... | 5 | https://mathoverflow.net/users/2233 | 392637 | 162,425 |
https://mathoverflow.net/questions/392633 | 4 | Let $(P,M,G)$ be a principal bundle with connection 1-form $\omega$. In all books I have seen so far, the curvature is defined by
\begin{equation}
F:=D\_{\omega}\omega \in \Omega({P,\mathfrak{g}})
\end{equation}
with
\begin{equation}
(D\_{\omega}\omega)\_p(X\_1,...,X\_k)=d \omega\_p(X\_1^H,...,X\_k^H)
\end{equation}
wh... | https://mathoverflow.net/users/209074 | Curvature of principal bundle | I'll use $\Omega \in \Omega^2(P,\mathfrak{g})$ to denote the curvature tensor of $\omega$.
One way of identifying these two expressions is through Cartan's structure equation
$$\Omega = d\omega + \frac{1}{2}\omega \wedge \omega.$$
A reference is Kobayashi-Nomitsu's book, Chapter II.5;
here we use the convention
$$d \om... | 5 | https://mathoverflow.net/users/14037 | 392646 | 162,427 |
https://mathoverflow.net/questions/392643 | 6 | Let $a$ and $b$ be two generators in a Coxeter group which do not commute. Is it possible for $ab$ to be equal to a product of generators where all instances of $b$ come before all instances of $a$?
I've tried coming up with invariants that are preserved after applying the conditions of a Coxeter group to a string of... | https://mathoverflow.net/users/147705 | Swapping non-commuting generators in Coxeter group | The answer is no. The Deletion Condition says that any expression in the generators of a Coxeter group contains a reduced expression for the same element as a subexpression. Since $a$ and $b$ don't commute, the unique reduced expression for $ab$ is $ab$, which must therefore occur as a subexpression of any longer expre... | 12 | https://mathoverflow.net/users/6771 | 392648 | 162,428 |
https://mathoverflow.net/questions/392617 | 7 | Let $\mathcal{A}$ be an injective $C^\*$-algebra irreducibly acting on a Hilbert space $\mathcal{H}$, and let $\phi$ be a completely positive idempotent from $\mathbb{B}(\mathcal{H})$ onto $\mathcal{A}$, and let $u\in\mathbb{B}(\mathcal{H})$ be a unitary.
>
> **Question:** Can we conclude that $\phi(u)\ne0$? If nec... | https://mathoverflow.net/users/25499 | Can the intersection of a unitary and an irreducibly represented injective $C^*$-algebra be $\{0\}$? | No. Take any irreducibly represented simple and injective C\*-algebra $A\subset B(H)$ and an outer automorphism $\theta$ of period $2$ (which is easy to find when $A$ is the hyperfinite II\_1 factor).
Case 1: The representation is covariant, i.e., $\exists u\in{\mathcal U}(H)$ such that $uau^\*=\theta(a)$ for $a\in A... | 6 | https://mathoverflow.net/users/7591 | 392654 | 162,430 |
https://mathoverflow.net/questions/392624 | 2 | It is well known that a separable Hilbert space $H$ decomposes as a direct integral in the presence of an abelian von Neumann algebra $\mathscr A\subseteq B(H)$.
More precisely, and quoting from Kadison & Ringrose, Theorem (14.2.1), *there is a (locally compact, complete, separable metric) measure space $(X, \mu )$ s... | https://mathoverflow.net/users/110570 | Direct integral decomposition relative to a given measure space | The answer to question 1 is "yes" and this can for instance be proven by the following concrete construction that also somehow answers question 2.
I will make the (nonrestrictive) assumption that the given representation of $C\_0(Y)$ on $H$ is nondegenerate, in the sense that $C\_0(Y) H$ is total in $H$.
Choose a c... | 2 | https://mathoverflow.net/users/159170 | 392658 | 162,431 |
https://mathoverflow.net/questions/392669 | 1 | Recall that a (unital) ring $R$ is *von Neumann regular* (VNR) if, for each $x \in R$, there exists $y \in R$ such that $x = xyx$; and *unit-regular* if such an element $y$ can be taken to be a unit.
>
> **Question.** Assume $R$ is a VNR (resp., unit-regular) ring, and let $a, b \in R$ be such that the right annihi... | https://mathoverflow.net/users/16537 | If, in a unit-regular ring, the right annihilator of $a$ equals the right annihilator of $b$, then $aR = bR$? | This seems false if $r(a)=\{x\in R\mid ax=0\}$. The ring $M\_n(F)$ of $n\times n$ matrices over a field $F$ is a counterexample for $n\geq 2$. Two matrices have the same right annihilator iff they have the same null space and $aM\_n(F)=bM\_n(F)$ if and only if they have the same image or column space. It is easy to fin... | 1 | https://mathoverflow.net/users/15934 | 392681 | 162,438 |
https://mathoverflow.net/questions/392659 | 10 | Let $\alpha\not= 0$ be such that for every $\beta<\alpha$ there is $\beta<\gamma<\alpha$, where $V\_\gamma$ is an elementary substructure of $V\_\alpha$. In other words, $V\_\alpha$ is a limit of its $V\_\beta$ elementary substructures. Then it is a simple result that $V\_\alpha$ models replacement.
My question: Let ... | https://mathoverflow.net/users/17968 | Elementary embeddings and replacement | If $\alpha$ is a limit of $2^\alpha$-supercompact cardinals,
then by the Magidor characterization of supercompactness, for each $2^\alpha$-supercompact cardinal $\kappa < \alpha$, for some $\gamma < \kappa$, there is an elementary embedding $j : V\_{\gamma}\to V\_\alpha$ with critical point arbitrarily large below $\ka... | 11 | https://mathoverflow.net/users/102684 | 392682 | 162,439 |
https://mathoverflow.net/questions/392613 | 3 | Does there exist a (sequence of) power series $\sum\_{i\geq 0} a\_{n,i} x^i$ that is $1$ at $0$ and $0$ at integers from $1$ to $n$, and such that $\sum\_{i\geq 0} \vert a\_{n,i}\vert n^i=O(n^p)$ for some real $p>0$?
| https://mathoverflow.net/users/112954 | Small power series "approximating" a Dirac | Such functions do not exist. Let us call them $f\_n$. Your conditions imply that
$f\_n(0)=1, \; f(k)=0$ for $ 1\leq k\leq n-1$ and $M(n,f\_n)\leq Cn^p$, where $M(r,f):=\max\{|f(z)|:|z|=r\}$. Now [Jensen's Formula](https://en.wikipedia.org/wiki/Jensen%27s_formula) says that
$$\sum\_{k=1}^{n-1}\log\frac{n}{k}\leq \frac{1... | 4 | https://mathoverflow.net/users/25510 | 392683 | 162,440 |
https://mathoverflow.net/questions/392678 | 7 | Whatever complex oriented multiplicative cohomology theories are, they come with two basic properties (among many others):
i) a complex oriented multiplicative cohomology theory is a contravariant functor form (nice) topological spaces to (graded) rings;
ii) there is an initial object $MU^\bullet$.
Properties i) ... | https://mathoverflow.net/users/8320 | On the comparison map $MU^\bullet(X)\otimes_{MU^\bullet(pt)}E^\bullet(pt)\to E^\bullet(X)$ for complex oriented multiplicative cohomology theories | Firstly, even in the Landweber exact case your comparison map need not be an isomorphism unless $X$ is a finite complex. It is more often the case that $E^\*(X)$ agrees with the completed tensor product $MU^\*(X)\widehat{\otimes}\_{MU^\*}E^\*$. However, I am still not sure whether that works for all $X$; exactness prop... | 6 | https://mathoverflow.net/users/10366 | 392688 | 162,442 |
https://mathoverflow.net/questions/392657 | 3 | Let $M$ be a complete, connected Riemannian manifold without boundary. Given a point $p\in M$ and a subset $K$ of $S\_p M$, the unit sphere in $T\_p M$, define the $K$-cone of directions $C(K)$ around by $C(K) := \{w \in T\_p M| \ w/|w| \in K\}$.
Given $p$ and $K$ as above, define $\text{inj}(p, K)$ to be the supremu... | https://mathoverflow.net/users/173490 | If every non null set of geodesics intersects itself in uniformly bounded finite time, is the manifold compact? | Such a manifold has bounded diameter, and is therefore compact. To see this, let $0 < T < \infty$ be the quantity you define. I claim that $\mathrm{diam} M \leq T$.
Additionally for each point $p \in M$ and unit tangent vector $v \in T\_p M$ let $\tau(v)$ be the first conjugate time along the geodesic $\gamma: t \map... | 3 | https://mathoverflow.net/users/103792 | 392691 | 162,444 |
https://mathoverflow.net/questions/366894 | 23 | Let $r\_1,r\_2,r\_3,\dotsc$ be an IID sequence of Rademacher random variables, so that $\mathbb P(r\_n=\pm1)=1/2$, and $a\_1,a\_2,\dotsc$ be a real sequence with $\sum\_na\_n^2=1$. For $S=\sum\_na\_nr\_n$, does the following inequality always hold?
$$
\mathbb P\left(\lvert S\rvert\ge1/\sqrt7\right)\ge1/2.\tag{\*}\label... | https://mathoverflow.net/users/1004 | A Rademacher ‘root 7’ anti-concentration inequality | Addressed in Theorem 1.3 in [Dvořák and Klein - Probability mass of Rademacher sums beyond one standard deviation](https://arxiv.org/abs/2104.10005) (not yet peer reviewed). It describes a computer program that verifies $\Pr[\lvert S\rvert \geq 1/\sqrt{7} - \epsilon] \geq 1/2$, with concrete $\epsilon > 0$. Giving it m... | 6 | https://mathoverflow.net/users/93621 | 392704 | 162,449 |
https://mathoverflow.net/questions/392703 | 9 | Let $K$ be a number field and $O\_K$ its ring of integers. Given any non-unit element $\alpha\in O\_K$, does there exist a unit $u\in O^\times\_K$ such that
$$
|\sigma(u \alpha)| >1 \quad \text{ for all } \sigma \in \mathrm{Hom}(K, \mathbb{C}).
$$
I do not find a counter-example; and I can not see how to use Minkowsk... | https://mathoverflow.net/users/95241 | Absolute values of non-unit algebraic integers at all infinite places | No. Let $K=\mathbb Q(\sqrt{6})$ and let $\alpha=2+\sqrt{6}\in O\_K$. All units of $O\_K$ are given by $u=\pm v^n$, where $v=5+2\sqrt{6}$ is the fundamental unit. The key here is that $\alpha$ has relatively small norm while the fundamental unit $v$ is relatively large.
Consider $u\alpha=\pm v^n\alpha$. If $n<0$, then... | 15 | https://mathoverflow.net/users/30186 | 392707 | 162,450 |
https://mathoverflow.net/questions/392534 | 8 | We know that the two functions $\{\;\cos (ax),\;2\cos (b x)\;\}$ where $\frac ab \notin \mathbb{Q}$ are independently positive (and negative) over $\frac 12$ of the domain.
Is it possible to estimate the fraction of the domain in which $\;\cos (ax)+2\cos (b x) \;$ is positive (in this case, the function is not period... | https://mathoverflow.net/users/nan | Are we able to estimate the fraction of the domain where $\cos (ax)+2\cos (b x)$ with $\frac ab \notin\mathbb{Q}$ is positive? | Since $a,b$ are incommensurable, $(ax,bx)$ is asymptotically equidistributed
in the torus $({\bf R} / 2\pi{\bf Z})^2$.
[One proof is via a continuous version of Weyl's equidistribution criterion:
for any integers $r,s$ with $(r,s) \neq (0,0)$ we have
$$
\frac1m \int\_0^m e^{i(rax+sbx)}\, dx = O\_{r,s}(1/m) \to 0
$$
as ... | 11 | https://mathoverflow.net/users/14830 | 392710 | 162,452 |
https://mathoverflow.net/questions/392645 | 6 | Given reals $A,B,X$, let $A\le\_{T/X}B$ iff $A\oplus X\le\_TB\oplus X$. For each real $X$ we can define a version of the c.e. degrees over $X$: we look at the preorder on $X$-c.e. reals given by $\le\_{T/X}$ (or, if preferred, take the corresponding partial order). Call this $\mathcal{R}\_X$; equivalently, $\mathcal{R}... | https://mathoverflow.net/users/8133 | How similar are the c.e. degrees and the CEA(Cohen) degrees? | I am reading ≅ as meaning "elementarily equivalent."
There are several questions here. I believe that some of them are accessible by known methods. The first is that if $X$ is not (low-level)arithmetically definable, then $\mathcal{R}\_X$ is not elementarily equivalent to $\mathcal{R}$. The proof should go by relativ... | 7 | https://mathoverflow.net/users/31026 | 392714 | 162,454 |
https://mathoverflow.net/questions/392674 | 3 | A Banach space $X$ has Gordon-Lewis local unconditional structure (G.L. l. u. st.) if for every finite dimensional subspace $E$ of $X$, the inclusion operator $i:E\to X$ factors through a finite dimensional space $U$ with an unconditional basis in a uniform manner, i.e., there exists a constant $C$ and operators $A:U\t... | https://mathoverflow.net/users/39421 | G.L. l. u. st. for subspaces of Banach spaces with an unconditional basis | The condition implies superreflexivity and for every $\epsilon >0$ that $Y$ has cotype $2-\epsilon$ and type $2+\epsilon$. The main open problem is whether the condition implies that $Y$ is isomorphic to a Hilbert space. This is open even if you strengthen the condition to "every subspace has an unconditional basis", w... | 4 | https://mathoverflow.net/users/2554 | 392716 | 162,455 |
https://mathoverflow.net/questions/392499 | 4 | I know that we can induce irreducible representations of $B\_n$ to $B\_{n+k}$ using the Ariki-Koike branching rule.
The irreducible representations of $B\_n \times S\_k$ are parametrised by tuples 2-partitions of $n$ and partitions of $k$.
I just haven't found an analogue to the branching rule for the these types of ... | https://mathoverflow.net/users/151546 | Inducing irreducible $B_n \times S_k$ characters to $B_{n+k}$ | Expanding on my [comment](https://mathoverflow.net/questions/392499/inducing-irreducible-b-n-times-s-k-characters-to-b-nk#comment1001600_392499), here's what the two steps look like combinatorially.
Induction from $S\_k$ to $B\_k$ sends $S^\lambda$ to $\bigoplus\_{\mu, \gamma} W(\mu,\gamma)^{\oplus c^\lambda\_{\mu,\g... | 3 | https://mathoverflow.net/users/39120 | 392720 | 162,456 |
https://mathoverflow.net/questions/392539 | 5 | My question is less concerned with the physical aspects of the nonlinear Schrödinger equation and more with the mathematical mechanics of using a Lax pair.
I am considering how to recover the defocusing nonlinear Schrödinger equation $iq\_t + q\_{xx} - 2|q|^2 q = 0$ from its Lax pair. As I understand the equation ari... | https://mathoverflow.net/users/170939 | Recovering the nonlinear Schrödinger equation from its Lax pair | Okay, here's the computation in detail. Really, this is a straightforward bracket computation, though it appears my comment earlier about a factor of 1/2 was in error (oops!) so no need to make a correction about that in your post.
The Lie bracket of $U$ with $V$ is a combination of the non-trivial Lie brackets: $[Q,... | 1 | https://mathoverflow.net/users/103158 | 392727 | 162,457 |
https://mathoverflow.net/questions/392478 | 7 | Sweedler's Hopf algebra (see [here](https://en.wikipedia.org/wiki/Sweedler%27s_Hopf_algebra)) is the lowest dimesnional ($4$-dimensional) Hopf algebra that is noncommutative and non-cocommutative. What are the next examples? Are there noncommutative, noncocommutative Hopf algebras of dimension $6,8,9,10$?
Edit: Looki... | https://mathoverflow.net/users/160055 | Low dimensional noncommutative non-cocommutative Hopf algebras | By standard results (in fin dim, over an alg closed field of zero char),
* all cocommutative HAs are group algebras (for some finite group),
* all commutative HAs are duals of group HAs (for some finite group)
(see for example: [About the classification of commutative and of cocommutative, fin. dim. Hopf algebras](... | 9 | https://mathoverflow.net/users/85967 | 392729 | 162,458 |
https://mathoverflow.net/questions/392712 | 5 | Let $p$ be a prime, let $n$ and $k$ be positive integers
and let $G$ be a group of order $p^n$.
Further, let $a\_{p^k}$ denote the number of subgroups of $G$ of index $p^k$.
If $a\_{p^k}$ is greater than 1 and not congruent to $p+1$ modulo $p^2$ --
does it follow that $p = 2$ and $G$ is either a dihedral group,
a qua... | https://mathoverflow.net/users/128342 | Number of subgroups of a $p$-group of index $p^k$ | It seems that the $p>2$ part of this was proved in
*Kulakoff, A.*, [**Über die Anzahl der eigentlichen Untergruppen und der Elemente von gegebener Ordnung in $p$-Gruppen.**](http://dx.doi.org/10.1007/BF01457969), Math. Ann. 104, 778-793 (1931). [ZBL57.0146.03](https://zbmath.org/?q=an:57.0146.03).
and the $p=2$ par... | 10 | https://mathoverflow.net/users/22989 | 392730 | 162,459 |
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