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https://mathoverflow.net/questions/445795 | 7 | These are fairly standard terms, but for the sake of completeness: An ultrafilter $\mathcal{U}$ on $\omega$ is a *p-point* if whenever $(A\_n)\_{n<\omega}$ is a partition of $\omega$ such that $A\_n \notin \mathcal{U}$ for all $n$, there is an $X \in \mathcal{U}$ such that $X \cap A\_n$ is finite for all $n$. $\mathcal... | https://mathoverflow.net/users/83901 | Can there be a p-point ultrafilter that is 'aggressively non-Ramsey'? | No, such a filter cannot exist. Suppose $\mathcal U$ is a $p$-point. For $s\in 2^{<\omega}$, let $A\_s$ consist of all $m<\omega$ such that
$$\forall n\in\mathrm{dom}(s)\ (nEm\Leftrightarrow s(n)=1).$$
Note that there is a unique $x\in 2^\omega$ so that $A\_{x\upharpoonright n}\in \mathcal U$ for all $n<\omega$. As $\m... | 3 | https://mathoverflow.net/users/125703 | 445817 | 179,654 |
https://mathoverflow.net/questions/445809 | 5 | I am studying Paley-Littlewood theorem in Harmonic analysis, and I met an exercise. I would like to construct a function $f$ as a counterexample to show that the inequality
\begin{equation}
\| f \|^2\_p \leq C \sum\_{j \in \mathbb{Z}} \| P\_j f\|^2\_p,
\end{equation}
is not true when $p<2,$ and here $C$ is a constant n... | https://mathoverflow.net/users/503783 | How to give a counterexample of this estimate related to Paley-Littlewood theorem? | This question is really about non-coincidence of different function spaces: the right hand side in your inequality is equal to the the square of the norm in (homogeneous) Besov space $\dot{B}\_{p}^{0,2}$. And by Littlewood--Paley theorem, the left-hand side is equivalent to the (square of) norm in Tribel--Lizorkin spac... | 7 | https://mathoverflow.net/users/69086 | 445818 | 179,655 |
https://mathoverflow.net/questions/445822 | 2 | Let $X\_0$ be a smooth projective variety over a finite field $\mathbb{F}\_q$. Let $X$ be the corresponding variety over the algebraic closure $\bar{\mathbb{F}}\_q$. Let $Fr\_q\colon X\to X$ be the geometric Frobenius.
**Is it true that the eigenvalues of $Fr\_q$ on $H^i(X,\mathbb{Q}\_l)$ ($l\ne char(\mathbb{F}\_q)$)... | https://mathoverflow.net/users/16183 | Eigenvalues of Frobenius in $l$-adic cohomology | This was proven by Deligne in the smooth projective case [Weil I, Thm. I.6], and later in the smooth proper case [Weil II, Cor. 3.3.9].
In general (already for smooth *quasi*-projective varieties), we don't even know whether $\dim H^i(X\_{\text{ét}},\mathbf Q\_\ell)$ is independent of $\ell$. See for instance [Katz, ... | 7 | https://mathoverflow.net/users/82179 | 445825 | 179,658 |
https://mathoverflow.net/questions/445735 | 6 | Consider the [Vandermonde's determinant](https://en.wikipedia.org/wiki/Vandermonde_matrix) computed by
$$V(x\_1,\dots,x\_m):=\det(x\_j^{i-1})\_{i,j=1}^m=\prod\_{1\leq i<j\leq m}(x\_i-x\_j).$$
The number of [plane partitions](https://en.wikipedia.org/wiki/Plane_partition) in an $n\times m\times m$ box (MacMahon) is give... | https://mathoverflow.net/users/66131 | Plane partitions as sums of determinants | I haven't worked out the details, but $V(\mathbf{J})/V(\mathbf{I})$ is
the principal specialization of a Schur function. Then $\left(
\frac{V(\mathbf{J})}{V(\mathbf{I})}\right)^2$ corresponds to a pair of
SSYT (semistandard Young tableaux), which can be merged into a plane
partition as in EC2, proof of Theorem 7.20.1. ... | 7 | https://mathoverflow.net/users/2807 | 445834 | 179,661 |
https://mathoverflow.net/questions/445841 | 2 | [This answer](https://mathoverflow.net/a/55356/148161) states that the category of finitary monads is locally presentable and monadic over the category $\mathrm{Set}^{\mathbb{N}}$. Where can I find proof of this claim?
More generally: I've looked through the literature in nlab ([finitary monad](https://ncatlab.org/nl... | https://mathoverflow.net/users/148161 | Literature about the category of finitary monads | These claims are proven more generally for the category $\mathrm{Mnd}\_f(\mathscr A)$ of finitary monads on a locally presentable category $\mathscr A$ in Lack's [On the monadicity of finitary monads](https://www.sciencedirect.com/science/article/pii/S0022404999000195). (This is the same Lack as in the linked answer.)
... | 5 | https://mathoverflow.net/users/152679 | 445844 | 179,663 |
https://mathoverflow.net/questions/445849 | 2 | **Motivation.** Today my sons played a card game, in which a fixed number $n$ of cards was lying on the table. A move consists of adding an unused card to the cards on the table, and removing a card from the table, so that after the move, there are again $n$ cards on the table. Which made me ponder the following questi... | https://mathoverflow.net/users/8628 | Inspired by a card game: finding a path through $[\mathbb{N}]^n$ | $[\mathbb{N}]^n$, with edges between $a,b\in[\mathbb{N}]^n$ if $\#(a\cap b)=n-1$, is an infinite graph in which all vertices have infinite degree. Moreover, for any two vertices $a,b$ in $[\mathbb{N}]^n$ and any finite subset $E\subseteq[\mathbb{N}]^n$ not containing $a,b$, there is a path from $a$ to $b$ which does no... | 7 | https://mathoverflow.net/users/172802 | 445852 | 179,665 |
https://mathoverflow.net/questions/445833 | 1 | In a previous post [Lift chain complex from $\mathbb{F}\_2$ to $\mathbb{Z}$](https://mathoverflow.net/questions/163346/lift-chain-complex-from-mathbbf-2-to-mathbbz) the body of the question mentions that this (lifting a chain complex from $\mathbb Z/2\mathbb Z$ to $\mathbb Z$) is always possible :
"Now, this can alwa... | https://mathoverflow.net/users/16739 | How to lift a chain complex from $\mathbb{Z}/2\mathbb{Z}$ to $\mathbb{Z}$ | I presume “constructive” means a computational algorithm is desired.
Assuming the chain complex $F$ over $\def\Z{{\bf Z}}\Z/2$ is bounded from below, we are going to construct by induction on $n$ a basis $E\_n=A\_n⊔B\_n⊔C\_n$ of $F\_n$ with the following properties:
* $A\_n$ is a basis of exact elements in $F\_n$;
... | 4 | https://mathoverflow.net/users/402 | 445854 | 179,666 |
https://mathoverflow.net/questions/445861 | 2 | When I look at the count of distinct least prime factors for a range of consecutive integers, I am seeing the same minimum number appear again and again. I am wondering if this number represents the true minimum.
Consider a range of consecutive integers defined by $R(x+1,x+c) = x+1, x+2, x+3, \dots, x+c$ with $C(x+1,... | https://mathoverflow.net/users/15915 | Estimating the minimum number of distinct least prime factors found in range of $c$ consecutive integers | I think that the statement "I am finding that for any $x$, any $c$, the mimimum $C(x+1,x+c)$" should be reformulated, clarifying that if we set the value of $x$, then $c$ is free to run on its domain and vice versa, by specifying also which is the aforementioned domain of the pair $(x,c)$ (since the closed interval of ... | 1 | https://mathoverflow.net/users/481829 | 445873 | 179,673 |
https://mathoverflow.net/questions/445874 | 0 | Let $n$ be a positive integer, and $r:=\frac{p}{q}<1$ where $\mathrm{gcd}(p,q)=1$.
I am interested in the product $n\cdot r$
Whenever $n$ is a multiple of $q$, a property of rational numbers is that
$$
\{n\cdot r\}=0
$$
where $\{.\}$ represents the fractional part of a number.
I am interested in studying the loca... | https://mathoverflow.net/users/393675 | On the existence of locals that are global in $np\;\mathrm{mod}\;q$ | Local minima which are not global minima will exist for all rational numbers $r=\frac{p}{q}$ as long as $p\neq 1,q-1$. Indeed, by [Bézout's identity](https://en.wikipedia.org/wiki/B%C3%A9zout%27s_identity), there is some $n$ such that $\{nr\}=\frac{1}{q}$, and except in those two cases above, both $(n-1)r$ and $(n+1)r$... | 2 | https://mathoverflow.net/users/30186 | 445876 | 179,674 |
https://mathoverflow.net/questions/445789 | 2 | Let $f:X\rightarrow Y$ be a morphism of smooth projective varieties, and $NE(X),NE(Y)$ the Mori cones of curves of $X$ and $Y$. Assume that $NE(X)$ is finitely generated. Then is $NE(Y)$ finitely generated as well?
If $f$ is birational I think the answer is positive. Let $C\subset Y$ be an irreducible curve and $\Gam... | https://mathoverflow.net/users/14514 | Mori cones and projective morphisms | It suffices to assume that $f$ is surjective (equivalently, dominant). Then for $C \subset Y$ any irreducible curve there exists an irreducible curve $D \subset X$ such that $f(D) = C$. (A schemy proof: let $D$ be the closure in $X$ of any closed point in $f^{-1}(c)$, where $c$ is the generic point of $C$.) It follows ... | 3 | https://mathoverflow.net/users/519 | 445878 | 179,675 |
https://mathoverflow.net/questions/445891 | 1 | Let $\mathcal H(n)$ be the set of $n\times n$ Hermitian matrices, and $\mathcal S(n) \subset \mathcal H(n)$ be the subset of density matrices, i.e., $A \in \mathcal S(n)$ iff $A$ is Hermitian, positive semidefinite and of trace one. Does there exist $c>0$ such that the following inequality
$$\big| \operatorname{tr} (... | https://mathoverflow.net/users/493556 | On a matrix trace inequality | The answer is **false**.
Counterexample: $L$ is the 2x2 all-$1$ matrix, $\epsilon ↘ 0$,
$B=\left[\begin{array}{cc}
1-\epsilon & \sqrt{\epsilon (1-\epsilon)} \\ \sqrt{\epsilon (1-\epsilon)} & \epsilon
\end{array}\right]$.
Then the LHS converges to $1$ and $1-B\_{11}$ converges to $0$, so there can't be such $c$.
... | 1 | https://mathoverflow.net/users/125498 | 445901 | 179,682 |
https://mathoverflow.net/questions/423178 | 12 | Consider the following two player game based on the Euclidean algorithm: Positions are given by $(a,b)$ in $\mathbb N^2\setminus\{(0,0)\}$ (where $\mathbb N=\{0,1,2,\ldots\}$) defining a greatest common divisor in $\mathbb N$. Moves are given by
$(a,b)\longrightarrow (\min(a,b),\max(a,b)-k\min(a,b))$ for
$k$ in $\{1,\l... | https://mathoverflow.net/users/4556 | Euclid's algorithm as a combinatorial game | Following up on Peter Taylor's answer, here's a fairly complete annotated bibliography.
What you describe as the variant came first:
A. J. Cole and A. J. T. Davie, A game based on the Euclidean algorithm and a winning strategy for it, Math. Gaz. 53 (1969) 354-357. <https://doi.org/10.2307/3612461>
The analysis o... | 8 | https://mathoverflow.net/users/14807 | 445905 | 179,683 |
https://mathoverflow.net/questions/445900 | 4 | Recall that a Boolean algebra is a complemented distributive lattice. The set of subspaces of a vector space comes very close to being a boolean algebra. It satisfies all the required properties, except being distributive. Even then it is a [modular lattice](https://ncatlab.org/nlab/show/modular+lattice), which is almo... | https://mathoverflow.net/users/54507 | Boolean algebra of the lattice of subspaces of a vector space? | I'm assuming "bounded-lattice homomorphism" means what I would call "0,1-homomorphism", i.e., it takes top to top and bottom to bottom? Then there is no adjoint, left or right.
Let $L$ be the lattice of subspaces of a two-dimensional vector space $V$ (over any field). I claim that there is no 0,1-homomorphism from $L... | 6 | https://mathoverflow.net/users/23141 | 445914 | 179,688 |
https://mathoverflow.net/questions/445906 | 2 | Suppose $X$ and $Y$ are two non-negative, independent random variables such that $X \succcurlyeq\_{st} Y$. That is, $X$ first-order stochastically dominates $Y$. Suppose that $X$ and $Y$ have smooth CDFs that admit a density.
Let $a, b > 0$ be two constants such that $a > b$. Is the following true?
$$a X + b Y \suc... | https://mathoverflow.net/users/78761 | Weighted sum of two random variables ranked by first order stochastic dominance | $\newcommand\de\delta$Indeed, a counterexample is as follows: $a=2$, $b=1$,
$$X\sim\frac12(\de\_0+\de\_2),\quad Y\sim\frac12(\de\_0+\de\_1),$$
where $\de\_x$ is the Dirac measure supported on the singleton set $\{x\}$.
Then $X\succcurlyeq\_{st}Y$, but
$$a X + b Y \not\succcurlyeq\_{st} a Y + b X,$$
because $P(aX+bY\g... | 4 | https://mathoverflow.net/users/36721 | 445916 | 179,689 |
https://mathoverflow.net/questions/445884 | 3 | Let $M = (\mathbb{Z}\_p)^2$ be a Galois representation, with Galois action given by $\rho: G\longrightarrow SL\_2(\mathbb{Z}\_p)$. I am trying to understand how sensitive the Galois cohomology group $H^1(G, M)$ (or more interesting to me, $H^1\_f(G, M)$) is to perturbations in $\rho$.
For example, say I replace $\rho... | https://mathoverflow.net/users/174655 | Deformations of Galois cohomology | The answer to your example question is "No".
Let $G$ be isomorphic to $\widehat{\mathbf{Z}}$, and $\phi$ the topological generator of $G$ (corresponding to $1 \in \widehat{\mathbf{Z}}$). Then one computes $H^0(G, M) = M^{\phi = 1}$ and $H^1(G, M) = M / (\phi - 1)M$, where $\phi$ is the generator of $G$.
Now $M \con... | 4 | https://mathoverflow.net/users/2481 | 445918 | 179,690 |
https://mathoverflow.net/questions/445932 | 2 | Let $ G $ be a quasisimple finite group. Let $ d\_{min} $ be the minimum dimension of a nontrivial irrep of $ G $. Must it be the case that the image of all (nontrivial) dimension $ d\_{min} $ irreps of $ G $ are conjugate in $ SU(d\_{min}) $?
For example all $ SL(2,5) $ subgroups of $ SU(2) $ are conjugate. This is ... | https://mathoverflow.net/users/387190 | Image of minimal degree representation of quasisimple group unique up to conjugacy | Looking at the Atlas, I'd say that the smallest counterexample is probably the Mathieu group $M\_{11}$. There are three $10$-dimensional irreducible characters, not conjugate in $SU(10)$.
Edit: Actually, $L\_2(8)$ is smaller, with four 7 dimensional irreducibles. Three of them are conjugate in $SU(7)$ but the fourth ... | 4 | https://mathoverflow.net/users/460592 | 445933 | 179,695 |
https://mathoverflow.net/questions/445843 | 2 | I am looking for a proof (or references) for the following result:
>
> If $P\in \mathbb{Z}[x]$ then there exists a nonzero polynomial $Q\in \mathbb{Z}[x]$
> such that
> $$H(PQ)\leq M(P)$$
> where
> $H(R)=\max\{|a\_0|,|a\_1|,\dots,|a\_n|\}$ and $
> M(R)=|a\_n|\prod\_{k=1}^n\max\{1,|\alpha\_k|\}$ are respectively the... | https://mathoverflow.net/users/94262 | Given $P\in \mathbb{Z}[x]$ there is a nonzero $Q\in \mathbb{Z}[x]$ such that $H(PQ)\leq M(P)$ | In Chapter 3 of Jonas Jankauskas's dissertation thesis, entitled ["Heights of Polynomials"](https://epublications.vu.lt/object/elaba:1917005/1917005.pdf), we learn that the inequality
$$\min\_{Q \in \mathbb{Z}[x] \setminus \{0\}} H(PQ) \le \lfloor M(P)\rfloor$$
where $P \in \mathbb{Z}[x]$ and $\lfloor r \rfloor$ de... | 7 | https://mathoverflow.net/users/84349 | 445935 | 179,696 |
https://mathoverflow.net/questions/445944 | 6 | Let $f\in C^{\infty}([-1,0])$ be real-valued and suppose that
$$ \left| \int\_{-1}^{0} f(t)\,e^{\lambda t} \,{\rm d} t \right| \leq e^{-\sqrt{\lambda}},$$
for all $\lambda > 0$. Does it follow that $f(t)$ is zero on $-\epsilon \leq t\leq 0$, for some $\epsilon>0$?
My hunch is that this is actually not true but fa... | https://mathoverflow.net/users/50438 | On an asymptotic integral decay | I'll change $[-1,0]$ to $[0,1]$ (so $t\to -t$), which seems easier on the brain.
$f(t)=e^{-1/t}$ is a counterexample: For $0<t<1/\sqrt{\lambda}$, we have $f\le e^{-\sqrt{\lambda}}$, so this part of the integral satisfies the desired bound, and if $t>1/\sqrt{\lambda}$, then $e^{-\lambda t}\le e^{-\sqrt{\lambda}}$.
| 11 | https://mathoverflow.net/users/48839 | 445948 | 179,700 |
https://mathoverflow.net/questions/445949 | 2 | In Cox, Little and Schenck's book **Toric Varieties** they show that a toric variety $ X\_{\Sigma} $ over a field of characteristic zero is complete if and only if the support is all of $ N\_{\mathbb{R}} $. This proof was very specific to varieties over fields of characteristic zero. When I looked at Oda and Fulton's b... | https://mathoverflow.net/users/470753 | Is a toric variety over a field of positive characteristic complete if and only if the support is all of $ N_{\mathbb{R}} $? | The answer is yes. Fulton's proof is in fact valid positive characteristic as well. In the proof of "full support $\Rightarrow$ completeness" he uses the valuative criterion which, as you noticed, is true for all characteristic. For the opposite direction you only need to replace the zero characteristic language from F... | 4 | https://mathoverflow.net/users/1508 | 445956 | 179,703 |
https://mathoverflow.net/questions/445816 | 3 | Let $G$ be a second countable profinite group, $g\in G$ and $g^G:=\{hgh^{-1}~|~h\in G\}$ the conjugacy class of $g$ in $G$. Theorem 3.2 in Wesolek's [Conjugacy class conditions in locally compact second countable groups](https://www.ams.org/journals/proc/2016-144-01/S0002-9939-2015-12645-7/) gives a characterization of... | https://mathoverflow.net/users/492970 | Open conjugacy classes in a second countable profinite group | Yes, there exist profinite groups $G$ with a conjugacy class of empty interior and consisting of elements of finite order, generating $G$ as an abstract group.
Let $H$ be a nonabelian finite simple group, and $S$ a nontrivial conjugacy class in $H$. Consider $G=H^\mathbf{N}$ and $T=S^\mathbf{N}$. Then $T$ is a single... | 2 | https://mathoverflow.net/users/14094 | 445975 | 179,711 |
https://mathoverflow.net/questions/445951 | 3 | It seems well-known that any smooth plane quartic can be written as the vanishing of $Q\_0Q\_2 -Q\_1^2$. Is there a good way to work out these quadratic factors $Q\_0,Q\_1,Q\_2$? For example, given the Klein quartic $X^3Y + Y^3Z + Z^3X = 0$ what would these quadratics be?
If not, is there at least some computational ... | https://mathoverflow.net/users/160814 | Writing a smooth plane quartic as the vanishing of $Q_0Q_2 - Q_1^2$ for quadratic $Q_0,Q_1,Q_2$ | The condition on your coefficients is an underdetermined system of quadratic equations. So specializing some of them (for instance setting them $0$), you can try to get a system with a finite number of (complex) solutions, and the chances are good that they are actually rational. Such systems can be handled with SageMa... | 7 | https://mathoverflow.net/users/18739 | 445982 | 179,715 |
https://mathoverflow.net/questions/445927 | 1 | Given two von-Neumann algebra factors $\mathcal M,\mathcal N$, is $\mathcal M\cap\mathcal N$ a factor?
And how about the intersection of infinitely many factors?
**Notes:**
* I know that the intersection is a von-Neumann algebra. (This is immediate from the definition of a von-Neumann algebra as a SOT-closed alge... | https://mathoverflow.net/users/101775 | Intersection of von-Neumann algebra factors | The answer is no. There are subfactors $N\subset M$ with finite Jones index $[M:N]$ with $N^{\prime}\cap M=\mathbb{C}\oplus \mathbb{C}$. For example, consider a type $II\_1$ factor $P$ and let $\alpha$ be an outer automorphism of $P$. Put $M= P\otimes M\_2(C)$ and $N$ be the algebra of all diagonal matrices $(x,\alpha(... | 2 | https://mathoverflow.net/users/164194 | 445985 | 179,716 |
https://mathoverflow.net/questions/445643 | 22 | Ever since Simpson's paper [Sim], it was observed that many different cohomology theories arise in the following way: we begin with our space $X$, we associate to it a stack $X\_\text{stk}$ (which depends on the chosen cohomology theory), and then the cohomology of $X$ coincides with the quasi-coherent cohomology of $X... | https://mathoverflow.net/users/131975 | Is there a ring stacky approach to $\ell$-adic or rigid cohomology? | This is an interesting question.
First, I think the [PS] reference does not give the "correct" Betti stack. In my notes on 6 functors, I define a different stack $X\_B$ such that $D\_{\mathrm{qc}}(X\_B)$ is equivalent to (the left-completed version of) $D(X,\mathbb Z)$, for any locally compact Hausdorff space $X$. It... | 13 | https://mathoverflow.net/users/6074 | 445987 | 179,717 |
https://mathoverflow.net/questions/445947 | 7 | I am interested in studying fluid dynamics and am searching for a good introductory textbook. I know just the very basics of fluids on the physics side. For mathematical prerequisites, I have completed a course on integration theory, and have a basic understanding of functional analysis and PDE, though by no means am I... | https://mathoverflow.net/users/498931 | Textbook suggestions for rigorous fluid dynamics | A few possibilities:
* JC Robinson, JL Rodrigo, & W Sadowski (2016) Classical theory of the three-dimensional Navier-Stokes equations.
* OA Ladyzhenskaya (1963) The mathematical theory of viscous incompressible flow.
| 4 | https://mathoverflow.net/users/119114 | 445988 | 179,718 |
https://mathoverflow.net/questions/445973 | 1 | This question is a related question see this post [Vague convergence VS Laplace transform convergence](https://mathoverflow.net/questions/445665/vague-convergence-vs-laplace-transform-convergence). But now we assume that
\begin{equation}
\int\_0^\infty e^{-sx}\mu\_n(dx)\to \int\_0^\infty e^{-sx}\mu(dx),
\end{equation}... | https://mathoverflow.net/users/147009 | Which kind of convergence can we get from Laplace transform convergence? | $\newcommand{\thh}{\theta}$In general, you cannot get the vague convergence here. E.g., suppose that $\mu=0$ and $\mu\_n(dx)=x^2\,1(n<x<n+1)\,dx$. Then for each real $s>0$
\begin{equation\*}
0\le\int\_0^\infty e^{-sx}\mu\_n(dx)=\int\_n^{n+1} e^{-sx}x^2\,dx
\le e^{-sn}(n+1)^2\to0
\end{equation\*}
(as $n\to\infty$), so... | 1 | https://mathoverflow.net/users/36721 | 445994 | 179,719 |
https://mathoverflow.net/questions/445934 | 6 | Let $\mathcal A$ be an additive 1-category, equipped with some class of weak equivalences $\mathcal W$. Let $\mathcal A[\mathcal W^{-1}]$ be the localization of $\mathcal A$ at $\mathcal W$ (so $\mathcal A[\mathcal W^{-1}]$ is an $\infty$-category). Let's assume that $\mathcal W$ is stable under finite direct sums, so ... | https://mathoverflow.net/users/2362 | When is an $\infty$-categorical localization of an additive 1-category enriched in topological abelian groups? | The answer is indeed always.
The fact that $\mathcal A$ is an additive 1-category makes it canonically a module over $Proj\_\mathbb Z$, the 1-category of finitely generated projective $\mathbb Z$-modules.
Your assumption on $W$ makes this action compatible with $W$, and because localization is a product-preserving ... | 3 | https://mathoverflow.net/users/102343 | 446007 | 179,722 |
https://mathoverflow.net/questions/446014 | 6 | Is there a known example of a set $S$ of Diophantine equations such that
1. $S$ is computable;
2. it is a theorem that every equation in $S$ has (at most) finitely many solutions;
3. the function that maps an element of $S$ to its set of solutions is uncomputable?
There are some famous finiteness theorems in number... | https://mathoverflow.net/users/3106 | Hilbert's tenth problem for equations with finitely many solutions | Yes (assuming that you're representing finite sets in an appropriately canonical manner). The proof of the MRDP theorem gives a stronger result: there is a computable function $f$ such that, for every $e$, the c.e. set $W\_e$ is equal to the $f(e)$th Diophantine set $D\_{f(e)}$ (in some fixed standard enumerations of e... | 8 | https://mathoverflow.net/users/8133 | 446016 | 179,725 |
https://mathoverflow.net/questions/446011 | 3 | Consider an $\left(\infty, 1\right)$-category $\mathcal{C}$ with a terminal object $1$. (I'm particularly interested in the case where $\mathcal{C}$ is a topos.) It is known that the forgetful functor $U$ from the under-category ${1}/{\mathcal{C}}$ to $\mathcal{C}$ creates colimits of diagrams with weakly contractible ... | https://mathoverflow.net/users/503911 | Does the forgetful functor from a pointed $\left(\infty, 1\right)$-category only create weakly contractible colimits? | A short answer is that if $C$ is the terminal category, then $1/C \to C$ is an equivalence, and hence creates all colimits.
Less flippantly, we can take $C$ to be the ordinary topos of sets. Then the forgetful functor creates $I$-shaped colimits if $I$ is connected.
If $C$ is cocomplete, then the colimit of $F$ in ... | 3 | https://mathoverflow.net/users/360 | 446018 | 179,726 |
https://mathoverflow.net/questions/446009 | 16 | Reading about ultraproducts in model theory and in Banach spaces leads to two distinct definitions. E.g., for an ultrapower given by an ultrafilter $\mu$ on $\mathbb{N}$, both notions of ultrapower are quotients of the space of sequences $\sigma: \mathbb{N} \to \mathbb{R}$.
**In model theory.** Two sequences $\sigma,... | https://mathoverflow.net/users/153883 | Ultraproducts of Banach spaces versus model theoretic ultraproduct | The ultraproduct of Banach spaces is the ultraproduct in the sense of metric structures in continuous logic. For a nice survey on this topic, see [Model theory for metric structures](https://faculty.math.illinois.edu/%7Ehenson/cfo/mtfms.pdf) by Ben Yaacov, Berenstein, Henson, and Usvyatsov. The ultraproduct of metric s... | 18 | https://mathoverflow.net/users/2126 | 446021 | 179,728 |
https://mathoverflow.net/questions/446025 | 13 | I know not all groups can be realized as the automorphism group of a group. For example, it is well-known that no group can have $\mathbb Z/n\mathbb Z$, with $n > 1$ odd, as automorphism group. Now I'm wondering the same question about automorphism groups of rings, is there any result about this?
Since the inverse Ga... | https://mathoverflow.net/users/133679 | Is every group the automorphism group of a ring? | **EDIT:** Apologies for the delay. I was too tired yesterday for anything more than a few comments. So let me turn this into a proper answer for everyone's sake, with a few more details added, and recap the discussion from yesterday.
**1. Case: $G$ finite**
It follows from Artin's theorem that every finite group $G... | 11 | https://mathoverflow.net/users/1849 | 446028 | 179,731 |
https://mathoverflow.net/questions/418993 | 3 | **THE QUESTION**
>
>
> >
> > Let $(X,\mathcal{X})$ be a standard Borel space, $T \colon X \to X$ a measurable map, and $\mu$ a $T$-mixing probability measure.
> >
> >
> > Is it necessarily the case that for all $A \in \mathcal{X}$ and $\varepsilon>0$, there exists $N\_{A,\varepsilon} \in \mathbb{N}$ such that f... | https://mathoverflow.net/users/15570 | Does mixing automatically imply this seemingly stronger "uniform modulo re-ordering" version of mixing? | The answer to my question is **yes!!** [The assumption that $(X,\mathcal{X})$ is a standard Borel space is not needed.]
A beautiful proof of a modified version of the statement has been provided by user65023, who says that it is "inspired by results in Blum-Hanson (1960): [On the mean ergodic theorem for subsequences... | 0 | https://mathoverflow.net/users/15570 | 446034 | 179,732 |
https://mathoverflow.net/questions/446033 | 3 | Consider the matrix $$A:=\left(
\begin{array}{cccc}
0 & a & 0 & 0 \\
f & 0 & b & 0 \\
0 & e & 0 & c \\
0 & 0 & d & 0 \\
\end{array}
\right)$$
I noticed that if I square this matrix then the eigenvalues of $A^2$ are two-fold degenerate. Does anyone see how this follows? I don't want an explicit computation but rat... | https://mathoverflow.net/users/496243 | Eigenvalues two-fold degenerate | For the eigenvalues of the matrix powers the following identity holds:
>
> If $A$ is a square ($d \times d$) matrix with associated eigenvalues $\lambda\_1,\dots,\lambda\_d$, then the eigenvalues of $A^n$ are
> $$\lambda\_1^n,\dots,\lambda\_d^n$$.
>
>
>
This can be shown by considering the eigenvalue/eigenvect... | 1 | https://mathoverflow.net/users/483817 | 446035 | 179,733 |
https://mathoverflow.net/questions/427090 | 2 | Let $(X, T, \mathcal F, \mu)$ be a nonatomic standard probability space equipped with a measure preserving transformation $T$. We say $T$ is *uniformly weak mixing* if for every $\varepsilon > 0$, there exists some $N > 0$ such that for all measurable sets $A, B \in \mathcal F$
$$| \frac{1}{n} \sum\_{k=1}^n \mu(T^{-k... | https://mathoverflow.net/users/173490 | Uniformly weak mixing transformations | @John Griesmer answered this question: "I don't think there is a system that satisfies your definition of uniformly weak mixing. Such a system must be ergodic, and therefore admit Rohlin towers. Given $>0$ and a Rohlin tower $\{,,…,^{−1}\}$ with $=()$, you can set $==\cup \cup \ldots \cup ^{/2}$ and find that $\frac{1}... | 1 | https://mathoverflow.net/users/503770 | 446036 | 179,734 |
https://mathoverflow.net/questions/445810 | 0 | Let $\mathcal{T}$ be a triangulated category that is generated by one object, say $A$ in the sense that the smallest triangulated subcategory containing $A$ and closed under coproducts and isomorphisms is $\mathcal{T}$ itself. Denote by $K\_0(\mathcal{T})$ the Grothendieck group of $\mathcal{T}$. Is $K\_0(\mathcal{T})$... | https://mathoverflow.net/users/45397 | Generators of triangulated category and Grothendieck groups | Fernando Muro has already answered this in the comments, but perhaps a reference would help. This is all spelled out in Neeman's book on triangulated categories. In particular, see Definition 4.5.8 and Proposition 4.5.11, and their proofs.
| 3 | https://mathoverflow.net/users/11540 | 446039 | 179,736 |
https://mathoverflow.net/questions/446040 | 6 | I have the following recurrence relation and boundary condition?
$$
f(n,m) = \frac{\alpha n}{n+m} f(n-1,m) + \frac{\beta m}{n+m} f(n,m-1) + 1
$$
$$
f(n,0) = \frac{1-\alpha^{n+1}}{1-\alpha}, f(0,m) = \frac{1-\beta^{m+1}}{1-\beta}
$$
Is it possible to get a exact solution for this recurrence relation?
| https://mathoverflow.net/users/503932 | How to solve recurrence relation with 2 variables？ | Let
$$g(n,m)=\sum\limits\_{i=0}^{n}\sum\limits\_{j=0}^{m}\binom{i+j}{j}\binom{n-i+m-j}{m-j}\alpha^i\beta^j$$
I conjecture that
$$g(n,m)=\binom{n+m}{m}f(n,m)$$
Here is the PARI prog to verify this conjecture:
```
f(n, m) = if(n==0 || m==0, (m==0)*(1 - a^(n+1))/(1-a)+(n==0)*(1 - b^(m+1))/(1-b)-(n==0 && m==0), a*n/(n+m... | 3 | https://mathoverflow.net/users/231922 | 446045 | 179,738 |
https://mathoverflow.net/questions/446048 | 5 | This question is a more precise version of [this question.](https://mathoverflow.net/questions/446033/eigenvalues-two-fold-degenerate)
Let's assume we have the matrix
$$\left(
\begin{array}{ccccc}
0 & a & 0 & 0 & 0 \\
f & 0 & b & 0 & 0 \\
0 & e & 0 & c & 0 \\
0 & 0 & d & 0 & r \\
0 & 0 & 0 & g & 0 \\
\end{arra... | https://mathoverflow.net/users/496243 | Matrices with same eigenvalues | To answer say a previous question, call such tridiagonal matrix $T$; the matrices $T$ and $-T$ (of dimension $n$) are unitarily congruent (they have the same spectra), that is there is an 'alternating' $\pm 1$ diagonal matrix $U$ such that $U^\*TU=-T$.
As you noticed when you square $T$ and rearrange it by a permuta... | 6 | https://mathoverflow.net/users/121643 | 446055 | 179,740 |
https://mathoverflow.net/questions/445895 | 2 | **I. First Set**
Before going to Elkies' nonic, we start with something a bit simpler. There is a list of j-function formulas in this [MSE post](https://math.stackexchange.com/a/4580319/4781). For example, for prime levels $p = 5,7,13,$ we have,
$$j=\frac{(x^2+10x+5)^3}x$$
$$j=\frac{(x^2+5x+1)^3(x^2+13x+49)}x$$
$$j... | https://mathoverflow.net/users/12905 | On Elkies' $\text{9T32}$ nonic and a shared property with j-function formulas | This has relatively little to do with the j-invariant itself. If you take any rational function $f(x)\in k(x)$ and let $G:=Gal(f(x) - t/k(t))$ (also referred to as the monodromy group of $f$), then by elementary Galois theory, $Gal(f(x)-f(y) / k(y))$ is a point stabilizer in $G$ (in the usual action on the roots). In y... | 1 | https://mathoverflow.net/users/127660 | 446059 | 179,742 |
https://mathoverflow.net/questions/446053 | 8 | Let $\Omega$ be an open subset of $\mathbb R^n$ for $n \geq 2$, and $p \in \Omega$.
Let $k$ be a positive integer. Suppose that $f: \Omega \setminus \{p\} \to \mathbb R$ is in $C^k$, and $\lim\_{x \to p} D^k f (x)$ exists.
**Question:** Is it true that $f$ admits an extension to $\Omega$ that is in $C^k (\Omega)$?
... | https://mathoverflow.net/users/173490 | An extension problem | $\newcommand\R{\mathbb R}\newcommand{\Om}{\Omega}$This is to detail the [comment by Fedor Petrov](https://mathoverflow.net/questions/446053/an-extension-problem#comment1152071_446053).
Without loss of generality, $p=0$ and $\Om$ is an open ball in $\R^n$ centered at $p=0$. Assume, slightly more generally, that $f\col... | 4 | https://mathoverflow.net/users/36721 | 446086 | 179,747 |
https://mathoverflow.net/questions/221232 | 9 | Let $d$ be a fundamental discriminant and let $\chi$ be the associated primitive real character of modulus $\vert d \vert$. Assuming GRH, Littlewood proved that as $\vert d \vert$ grows large,
$$L(1, \chi) \leq (2 + o(1)) e^\gamma \log \log(\vert d \vert).$$
Granville and Soundararajan provide a treasure-trove of inf... | https://mathoverflow.net/users/3545 | Effective bound of $L(1,\chi)$ | Explicit upper and lower bounds for $L(1,\chi)$, conditional on the generalised Riemann hypothesis, are given in Theorem 1.5 of the paper ["Conditional bounds for the least quadratic non-residue and related problems"](https://doi.org/10.1090/S0025-5718-2015-02925-1) by Youness Lamzouri, Xiannan Li and Kannan Soundarara... | 4 | https://mathoverflow.net/users/3803 | 446087 | 179,748 |
https://mathoverflow.net/questions/446052 | 3 | There are two versions of Cauchy identity for Schur functions, namely
$$
\sum\_{\lambda}s\_\lambda(\underline x)s\_\lambda(\underline y) = \prod\_{i,j=1}^n\frac 1{1-x\_iy\_j}\ ,\qquad {\rm (1)}
$$
and
$$
\sum\_{\lambda}s\_\lambda(\underline x)s\_{\lambda'}(\underline y) = \prod\_{i,j=1}^n (1+x\_iy\_j)\ .\qquad {\rm (2)... | https://mathoverflow.net/users/178792 | Cauchy identity for Jack functions | The dual Cauchy identity for Jack polynomials also exists, and is better expressed in terms of the $P$-normalized Jack polynomials:
$$
\sum\_{\lambda} P^{(a)}\_ \lambda(\underline x)P^{(1/a)} \_ {\lambda'}(\underline y) = \prod\_{i,j=1}^n\bigl(1+x\_iy\_j\bigr)\ .
$$
(Cf., e.g., formula (2.6) in [these notes by I.G. M... | 5 | https://mathoverflow.net/users/178792 | 446099 | 179,750 |
https://mathoverflow.net/questions/446043 | 1 | [*Bring's curve*](https://en.wikipedia.org/wiki/Bring%27s_curve) or *Bring's surface* with genus 4 and $5!=120$ automorphisms can be given by the homogeneous equations,
$$x\_1+x\_2+x\_3+x\_4+x\_5 = x\_1^2+x\_2^2+x\_3^2+x\_4^2+x\_5^2 = \\x\_1^3+x\_2^3+x\_3^3+x\_4^3+x\_5^3 = 0$$
This is also a property of the Bring q... | https://mathoverflow.net/users/12905 | Bring's curve $\sum_{i=1}^5 x_i^k = 0$ for $k = 1,2,3$ and an analogue $\sum_{i=1}^6 y_i^k = 0$ for $k = 1,2,4,7$ | The equations $\sum\_{i=1}^6 y\_i^k = 0$ for $k=1,2,4,7$ do not cut out a curve because the $k=1,2,4$ equations imply the one for $k=7$. (The 7th power sum is a polynomial in the power sums of degree 1 through 6, and there is no way to get 7 as a sum of numbers in $\{3,5,6\}$.) So you have some surface with 720 automor... | 1 | https://mathoverflow.net/users/14830 | 446104 | 179,751 |
https://mathoverflow.net/questions/445931 | 6 | $\DeclareMathOperator\SL{SL}\DeclareMathOperator\SO{SO}\DeclareMathOperator\O{O}$I’m studying the group $\O(5,5,\mathbb{Z})$, the indefinite orthogonal matrices with integer entries. In particular, I want to know about its homology/cohomology. It doesn’t seem like much is known about this, so I’m starting with the firs... | https://mathoverflow.net/users/503849 | Computing a Commutator Subgroup | With the latest update, indicating that the quadratic form in question is, up to reordering, $\sum x\_i x\_{-i}$ (here we number the basis elements as $e\_1,\ldots,e\_5,e\_{-5},\ldots,e\_{-1}$), the answer is affirmative.
This can be seen as follows:
1. Note that the group in question is the group of $\mathbb{Z}$-p... | 8 | https://mathoverflow.net/users/5018 | 446109 | 179,753 |
https://mathoverflow.net/questions/446107 | 5 | How many digits of $\sqrt{2}$ are known to date, in base 10 and in base 2? I am trying to produce the largest sequence known to date, and would like to sense if I can do it either alone or with hiring someone. I should have no problems producing 1 trillion, but almost sure I can't produce 1 quadrillion in any reasonabl... | https://mathoverflow.net/users/140356 | How many digits of $\sqrt{2}$ are known to date? | In one sense, *all* of the base-2 digits (or, I guess, "bits") of $\sqrt{2}$ are known because we have a closed-form formula, according to the [OEIS](https://oeis.org/A004539): $$\begin{align} a(n) &= \frac{1}{2} - \frac{2\arctan(\cot(2^{-\frac{3}{2}+n}\pi))}{\pi} + \frac{\arctan(\cot(2^{-\frac{1}{2}+n}\cdot\pi))}{\pi}... | 17 | https://mathoverflow.net/users/9840 | 446118 | 179,755 |
https://mathoverflow.net/questions/446114 | 10 | I'm specifically assuming that we have replacement instead of collection; collection breaks things (because then there is a set that contains a map from $n$ to $C$ for every $n\in\mathbb N$, and you can look within that set to get an injection from an infinite subset of $C$ to some infinite set, so $\aleph(C)$ couldn't... | https://mathoverflow.net/users/152182 | Is it consistent with ZFC(A) for the Hartogs number of a proper class to be $\aleph_0$? | Yes. If you start with infinitely many urelements and then take the sets whose kernel is finite (the kernel of a set is the set of urelements in its transitive closure), the resultant inner model will satisfy Replacement. A more generalized argument is included in my dissertation (<https://arxiv.org/abs/2303.14274>, Th... | 10 | https://mathoverflow.net/users/504023 | 446122 | 179,759 |
https://mathoverflow.net/questions/446126 | 7 | Given two finite presentations of torsion-free groups, is there an algorithm to determine whether the given groups are isomorphic or not?
I have found results for narrower classes (for example, they are briefly reviewed in [The isomorphism problem for finitely generated fully residually free groups](https://www.scien... | https://mathoverflow.net/users/148161 | Is the isomorphism problem solvable for torsion-free groups? | Novikov's centrally-symmetric group $\mathfrak{A}\_P$ is a torsion-free group with undecidable word problem, constructed in [1]. Novikov did not prove it is torsion-free but, as Adian points out in [Adi1957b] (p. 76 of [my translation](https://arxiv.org/abs/2208.08560), and referenced as in there; he calls the group $F... | 9 | https://mathoverflow.net/users/120914 | 446131 | 179,761 |
https://mathoverflow.net/questions/445953 | 4 | Say that a function $f \in \omega^\omega$ witnesses that an $\omega$-REA set $A = \bigoplus\_{i \in \omega} A^{[i]}$ is non-arithmetic if $A^{[\leq f(n)]} \not\leq\_T 0^n$. Say that $A$ is effectively non-arithmetic if there is such an $f \leq\_a A$ (i.e. arithmetic in $A$)?
Is every non-arithmetic $\omega$-REA set e... | https://mathoverflow.net/users/23648 | Effectively non-arithmetic $\omega$-REA degrees | Unfortunately, I believe I can show there are non-arithmetic $\omega$-REA degrees that aren't effectively $\omega$-REA.
To this end we need to build $A$ to be $\omega$-REA with $A >\_a 0$ to satisfy
$$R\_{n,i}: \phi\_i(A^{(n)})\downarrow \implies (\exists m,k)\left(\phi\_i(A^{(n)};m) =k \implies A^{[\leq k]} \leq\_... | 1 | https://mathoverflow.net/users/23648 | 446137 | 179,764 |
https://mathoverflow.net/questions/445642 | -4 | Newtons law for gravity states that:
$$F\_{12} = \frac{G m\_1 m\_2} {|x\_1-x\_2|^2}$$
The function :
$$k(x,y):=\exp(-| x-y|^2)$$
is known to be a positive definite function, called [the RBF-kernel](https://en.wikipedia.org/wiki/Radial_basis_function_kernel).
It follows that
$$k(x\_1,x\_2) := \exp( -|x\_1-x\... | https://mathoverflow.net/users/165920 | Inverse square-law as a positive definite kernel? | Please look pages 141-147 of the book:
S. Saitoh and Y. Sawano, Theory of Reproducing Kernels and Applications,
Developments in Mathematics 44, Springer (2016).
2023.5.4.20:36
| 2 | https://mathoverflow.net/users/504063 | 446143 | 179,766 |
https://mathoverflow.net/questions/446147 | 3 | Is the following statement true?
For all $n$ large enough, there exists an $M\_n \in [-1,1]^{n \times n}$ such that the smallest singular value of $M\_n$, $\sigma\_n(M\_n) \gtrsim \sqrt{n}$.
If $n$ is such that there exists a Hadamard matrix of order $n$, we can take $M\_n$ to be that.
On Terry Tao's [blog](https... | https://mathoverflow.net/users/316923 | Existence of a matrix with bounded entries and large smallest singular value | The OP wrote $[-1, 1]$ but refers to Hadamard matrices, whose entries are more strongly in $\{ -1, 1 \}$. Assuming that the former is meant, we can do this with the [sine transform matrix](https://en.wikipedia.org/wiki/Discrete_sine_transform). Take
$$M\_{ab} = \sin \frac{\pi a b}{n+1} \ \text{for} 1 \leq a,b \leq n.$$... | 7 | https://mathoverflow.net/users/297 | 446159 | 179,767 |
https://mathoverflow.net/questions/445980 | 1 | I am interested in a concrete case of a representation in de Finetti's Theorem for expressing the distribution of exchangeable binary sequence in terms of product measures. Namely, let
$$X\_1,\ldots,X\_n$$
be a random $0-1$ string with exactly $k$ $1$'s. How could one obtain a concrete respresentation of the measure in... | https://mathoverflow.net/users/24494 | Concrete representation in de Finetti's Theorem | The distribution over $(X\_1, \ldots, X\_n)$ you are considering is extremally exchangeable, i.e. it is a vertex of the simplex of $n$-exchangeable distributions. It can not be extended to an $(n+1)$-exchangeable distribution $(X\_1, \ldots, X\_{n+1})$ of which it is the marginal. One consequence of De Finetti's theore... | 2 | https://mathoverflow.net/users/4080 | 446162 | 179,769 |
https://mathoverflow.net/questions/446139 | 8 | In his famous list of [Problems in Low-Dimensional Topology](https://math.berkeley.edu/%7Ekirby/problems.ps.gz), Kirby states the following as Problem 3.14 (B), which is attributed to Thurston:
>
> Conjecture: Suppose $G$ (an arbitrary group I suppose) acts properly
> and discontinuously on a contractible 3-manifol... | https://mathoverflow.net/users/69681 | Problem 3.14 from Kirby's list | This problem is answered in the literature, with a caveat.
As indicated in the comments, it follows from the orbifold theorem + geometrization conjecture (to handle the case of orbifolds without fixed points).
The caveat is that the action needs to be assumed smooth (or PL). Otherwise, there exists wild involutions s... | 11 | https://mathoverflow.net/users/1345 | 446165 | 179,771 |
https://mathoverflow.net/questions/446169 | 2 | Given (possibly non-Abelian) groups $H,G$ with $H \subseteq G$ and $f \in G$, I write $\langle H, f \rangle$ for the subgroup of $G$ generated by $H \cup \{f\}$.
Write $T(H)$ for the free product of $H$ with a multiplicative copy $y^{\mathbb{Z}}$ of the group of integers. We can represent a generic element $t(y)$ of ... | https://mathoverflow.net/users/45005 | Elements in group extensions which cancel unary terms in the language of groups | No. Let $G = \langle x, y \rangle$ be a $2$-generated non-free group with $x$ of infinite order such that $F = \langle x, g\rangle \cong F\_2$ for some $g \in G$. Then we get a counterexample by taking $H = \langle x \rangle$ and $f = y$. Indeed, since $G$ is not free, there is some nontrivial word $w \in F\_2$ such th... | 6 | https://mathoverflow.net/users/20598 | 446173 | 179,774 |
https://mathoverflow.net/questions/446172 | 3 | I am trying to understand the asymptotics of Haar Moments on general compact Lie groups (in particular, subgroups of $\mathrm{SU}(n)$). I have learned that closed form formulae for these moments are only available for some classical groups, namely the orthogonal, unitary and symplectic groups. However, my understanding... | https://mathoverflow.net/users/57449 | Asymptotics of Haar moments on general Lie groups | The generalization of Weingarten calculus to compact Lie groups is studied in [Expectation values of polynomials and moments on general compact Lie groups](https://arxiv.org/abs/2203.11607), see section 4.
| 2 | https://mathoverflow.net/users/11260 | 446174 | 179,775 |
https://mathoverflow.net/questions/446142 | 2 | Let $E,E'$ be a pair of elliptic curves defined over $\mathbb{Z}$. Let $T\_p[E], T\_p[E']$ be their associated ($p$-adic) Tate modules. These are Galois representations for the absolute Galois group of $\mathbb{Q}$, which we denote by $G$, denote them by $\rho,\rho'$, respectively.
Let $\overline{\rho},\overline{\rho... | https://mathoverflow.net/users/174655 | Galois cohomology of Tate modules | Let $S$ be a finite set containing all places where an elliptic curve $E$ has bad reduction as well as $p$ and $\infty$. Write $T$ for $T\_pE$ and $G\_S$ for the Galois group of the maximal extension of $\mathbb{Q}$ unramified outside $S$. Then, for any $k>0$ the exact sequence $0\to T \to^{[p^k]} T \to E[p^k]\to 0$ is... | 5 | https://mathoverflow.net/users/5015 | 446180 | 179,779 |
https://mathoverflow.net/questions/446182 | 17 | Any finitely presented simple group has solvable word problem, and hence recursive Dehn function. I'm curious though how wild these recursive functions could possibly be.
One concrete question is whether there are even any examples known of a finitely presented simple group whose Dehn function is exponential. The onl... | https://mathoverflow.net/users/164670 | Dehn functions of finitely presented simple groups | To answer the vaguer question: I think there is no known bound on the Dehn functions of finitely presented simple groups. Recall:
**Boone–Higman Embedding Theorem.**
A finitely presented group has solvable word problem if and only if it can be embedded in a recursively presented simple group.
In their paper, they a... | 13 | https://mathoverflow.net/users/24447 | 446193 | 179,780 |
https://mathoverflow.net/questions/316474 | 31 | It is well-known that any positive rational number can be written as the sum of finitely many distinct unit fractions. This is easy since
$$\frac1n=\frac1{n+1}+\frac1{n(n+1)}\quad\text{for all}\ n=1,2,3,\ldots.$$
In 2015 I thought that this easy fact should have a further refinement which is somewhat sophisticated. Not... | https://mathoverflow.net/users/124654 | Can we write each positive rational number as $\frac1{p_1-1}+\ldots+\frac1{p_k-1}$ with $p_1,\ldots,p_k$ distinct primes? | This conjecture is true (as is the version for $p-h$ for any $h\neq 0$).
The proof is too long to reproduce here, but the preprint is at <https://arxiv.org/abs/2305.02689>
EDIT:
Quick summary as requested: My previous paper (<https://arxiv.org/abs/2112.03726>) essentially shows that for any set $A$ which is large... | 16 | https://mathoverflow.net/users/385 | 446198 | 179,781 |
https://mathoverflow.net/questions/8247 | 122 | I'm collecting advanced exercises in geometry. Ideally, each exercise should be solved by one trick and this trick should be useful elsewhere (say it gives an essential idea in some theory).
If you have a problem like this please post it here.
**Remarks:**
* I have been collecting such problems for many years. Th... | https://mathoverflow.net/users/1441 | One-step problems in geometry | Let $A$ be a set of intially labelled points in $\mathbb{R}^d$. We may take any line containing at least $k$ labelled points and label any point on this line. For which minimal size $|A|$ (as a function of $d, k$) it may occur that we can label (by performing finitely many such operations) every point of $\mathbb{R}^d$... | 1 | https://mathoverflow.net/users/4312 | 446218 | 179,788 |
https://mathoverflow.net/questions/446157 | 21 | The canonical inclusion $\beta\mathbb{Q}\setminus \mathbb{Q} \hookrightarrow \beta\mathbb{Q}$ is not the Stone-Čech compactification of $\beta\mathbb{Q}\setminus \mathbb{Q}$. Even so, this doesn't necessarily mean that $\beta(\beta\mathbb{Q}\setminus\mathbb{Q})$ and $\beta\mathbb{Q}$ are not homeomorphic, just that thi... | https://mathoverflow.net/users/150060 | Are $\beta \mathbb{Q}$ and $\beta(\beta\mathbb{Q}\setminus\mathbb{Q})$ homeomorphic? | Let me summarize the discussion in the comments as an answer. Let $\chi(x, Y)$ be the character of $x$ in $Y$ i.e. the least cardinality of a local basis of the point $x$ in space $Y$.
**Proposition 1.** If $p\in \beta X\setminus X$ then $\chi(p, \beta X)$ is uncountable.
*Proof:* If it werre $\chi(p, \beta X) = \a... | 7 | https://mathoverflow.net/users/150060 | 446233 | 179,794 |
https://mathoverflow.net/questions/446015 | 11 | Apologies for a naive question (especially for Iwasawa theorists): it is well-known
and trivial to prove that the usual (elementary) construction of $p$-adic L functions
attached to odd Dirichlet characters leads to a function which is identically zero
(experts even have a highbrow explanation for this) for a "silly re... | https://mathoverflow.net/users/81776 | $p$-adic L function of an odd Dirichlet character | **Theorem**. Let $p > 2$ be prime, and let $\chi$ be a Dirichlet character (non-trivial, and of prime-to-$p$ conductor, for
simplicity). Choose $t \in \mathbb{Z} / (p-1)\mathbb{Z}$ such that
$(-1)^t = \chi(-1)$. Then there exists a continuous function $L\_{p,
t}(\chi, -) : \mathbb{Z}\_p \to \mathbb{C}\_p$ such that for... | 8 | https://mathoverflow.net/users/2481 | 446243 | 179,797 |
https://mathoverflow.net/questions/446222 | 1 | Is there a commutative subalgebra $A\subset B(H)$ containing the 1 dimensional scalars with the following property:
>
> The algebra $A$ has trivial intersection with the set of commutator elements $xy-yx$
>
>
>
| https://mathoverflow.net/users/36688 | A subalgebra of $B(H)$ which does not contain a commutator element | From the results of
*Brown, Arlen; Pearcy, Carl*, [**Structure of commutators of operators**](https://doi.org/10.2307/1970564), Ann. Math. (2) 82, 112-127 (1965). [ZBL0131.12302](https://zbmath.org/?q=an:0131.12302).
an element of $B(H)$ is a commutator $xy-yx$ if and only if it is not of the form $\lambda I + C$ f... | 10 | https://mathoverflow.net/users/766 | 446252 | 179,800 |
https://mathoverflow.net/questions/446257 | 7 | Let $W(\alpha)$ denote the set of all (countable) ordinals *writable* by [Ordinal Turing Machines](https://arxiv.org/abs/math/0502264) with a single (countable) parameter $\alpha$, i.e. each computation starts with a single ($\alpha$-th) cell on the input tape marked with a non-zero symbol (all other cells are marked w... | https://mathoverflow.net/users/122796 | Gaps in the ordinals writable by Ordinal Turing Machines with a single countable parameter | The answer is yes. For example, take $\beta=\omega\_1$, the first uncountable ordinal. Since there are only countably many programs, there can be only countably many writable ordinals relative to $\beta$ as input. But there are uncountably many ordinals $\gamma$ below $\beta$, and so most of them are not writable from ... | 8 | https://mathoverflow.net/users/1946 | 446274 | 179,804 |
https://mathoverflow.net/questions/445815 | 4 | I'm reading *"BGG category $\mathcal{O}$"* by Humphreys.
In section 3.12 we look into the projective modules over $\mathfrak{sl}(2,\mathbb{C})$. If $\lambda\in\mathbb{Z}$ is a weight and $\mu=-\lambda-2$ is the other weight linked to $\lambda$ then from calculation and the BGG reciprocity we get that the projective c... | https://mathoverflow.net/users/496537 | Structure of projective indecomposable modules for $\mathfrak{sl}_2$ | Denoting the highest weight vectors of
$M(\lambda)$ and $M(\mu=-\lambda-2),\, \lambda\in\mathbb{Z}\_{>0}$ as $v$ and $v\_\mu$, we choose a preimage $w\in P(\mu)$ of $v\_\mu$ with $h\cdot w =\mu w$. Then for nontrivial extension the vector $x\cdot w$ does not vanish, hence $x\cdot w=A\_1 y^{\lambda}\cdot v$.
The actio... | 0 | https://mathoverflow.net/users/102775 | 446277 | 179,805 |
https://mathoverflow.net/questions/446246 | 9 | The Division Paradox is the fact that there are models of ${\sf ZF \neg C}$ in which a set can be partitioned into a set that is bigger than it — equivalently, in which there are sets $X$ and $Y$ such that $|X| < |Y|$ yet there is a surjection from $X$ onto $Y$. For example, there are models of ${\sf ZF \neg C}$ in whi... | https://mathoverflow.net/users/504195 | Class-theoretic division paradox | The answer is yes.
First, let us observe the following lemma. Let us work in GBc, that is, Gödel-Bernays set theory with the axiom of choice, but only choice for sets, and not global choice.
**Lemma.** If global choice fails, then there is a proper class $A$ with no class injection $F:\newcommand\Ord{\text{Ord}}\Or... | 11 | https://mathoverflow.net/users/1946 | 446286 | 179,807 |
https://mathoverflow.net/questions/446263 | 4 | **I. Degree 8**
Assume the $j\_i$ to be free parameters. The following octics in $x$ belong to $8T43,$ have group $\text{PGL}(2,7)$, and order $2\times168 = 336.$
\begin{align}
{j\_1}\; &=\frac{(x^2 + 5x + 1)^3(x^2 + 13x + 49)}x\\
{j\_2}^2 &=\frac{j\_2\,(-7x^4 - 196x^3 - 1666x^2 - 3860x + 49)+(x^2 + 14 x + 21)^4}x\... | https://mathoverflow.net/users/12905 | Finding a sextic analogue to the solvable octic $\frac{(x + 1)^6(x^2 + x + 7)}x = -k^3$ where $e^{(\pi/3)\sqrt{d}}\approx k^3+41.999999999999\dots$ | Didn't think about eta quotients, but if the suggested analogies of ramification types are anything to go by, then
$$j\_2 = \frac{(x+1)^4(x^2+6x+25)}{x}$$
(which matches the prescribed discriminant exactly), and
$$j\_3^2 = \frac{2(x+1)^3(x^3+32x^2+231x+400) j\_3 + 5^4(x+1)^6}{64x}$$
(or possibly some reparametrization ... | 4 | https://mathoverflow.net/users/127660 | 446287 | 179,808 |
https://mathoverflow.net/questions/445495 | 4 | Let $S\_1,S\_2,S\_3$ be three simple closed curves on the 2-sphere $\mathbb{S}^2$. (With no smoothness or rectifiability assumption)
For each $i$, let $M\_i$ denote the minimal surface (i.e. disc) bounded by $S\_i$, as provided by Douglas. Note that $M\_i$ is contained in the interior of $\mathbb{S}^2$ in $\mathbb{R}... | https://mathoverflow.net/users/69681 | Is the intersection of such a triple of minimal surfaces in the 3-ball a single point? | ~~This isn't an answer, but is too long for a comment.~~
**Your question has a negative answer in general -- see below.**
One initial comment: There is no reason in general for there to be a unique Douglas-Rado disk (i.e. an area minimizer in the class of disks) so you can't really speak of "the the minimal surface... | 2 | https://mathoverflow.net/users/127803 | 446290 | 179,809 |
https://mathoverflow.net/questions/446299 | 2 | Suppose $R$ is a discrete valuation ring with fraction field $K$. Let $X\subset \mathbf{P}^n\_{C\_K}$ be a closed subscheme, flat over $C\_K$, a smooth projective curve over $K$.
Let $C\_R$ be a flat regular proper model for $C$ over $R$ and take $\mathcal{X}$ the scheme-theoretic closure of $X$ in $\mathbf{P}^n\_{C\... | https://mathoverflow.net/users/501361 | Flat scheme-theoretic closure | Explicitly, lets let $R = \mathbb{C}[x]\_{(x)}$ so $K = \mathbb{C}(x)$. You can then let $C\_K = {\bf P}^1\_K$ and $C\_R = {\bf P}^1\_R$. $C\_R$ has a chart that looks like $\mathbb{C}[x]\_(x)[y]$. This is just a localization of $\mathbb{C}[x,y]$. Blowup $(x,y)$ in the latter, and localize (equivalently, blowup $(x,y)$... | 4 | https://mathoverflow.net/users/3521 | 446301 | 179,811 |
https://mathoverflow.net/questions/446297 | 1 | Let $\Omega\subseteq\mathbb{R}^n$ open and convex. It is elementary that if $u\in C^2(\Omega)$ then
$$u \text{ is convex}\iff D^2u\geq0 \ \text{ in } \Omega$$
I was looking for a similar characterization for $u\in C(\Omega)$. If I remember correctly it can be formalized using distribution theory, but I was wondering if... | https://mathoverflow.net/users/490711 | Viscosity characterization of convex functions | You can always consider the second (distribution) derivative of a continuous function and require that it is non-negative, which means that for all $T\in \mathbb R^n$ and all $\phi\in \mathscr D(\Omega;\mathbb R\_+)$
\begin{multline}
\sum\_{1\le j,k\le n}\langle \frac{\partial^2u}{\partial x\_j\partial x\_k}(x)T\_k T\_... | 1 | https://mathoverflow.net/users/21907 | 446304 | 179,813 |
https://mathoverflow.net/questions/446307 | 10 | $$C\_{n} = \sum\_{i=1}^n (-1)^{i-1} \binom{n-i+1}{i} C\_{n-i}$$
Are there any good combinatorial proofs or algebraic proofs of this?
| https://mathoverflow.net/users/504256 | Proving an identity about Catalan numbers | $C\_n$ is the number of *Catalan sequences* $(x\_1,\ldots,x\_{2n})$ of $\pm 1$ with zero sum and non-negative prefix sums $x\_1+\ldots+x\_k$, for $k=1,\ldots,2n$. Note that any such sequence contains an index $j\in \{1,2,\ldots,n\}$ for which $x\_j=1$, $x\_{j+1}=-1$. Call $(j,j+1)$ a special pair. Then ${n-i+1\choose i... | 18 | https://mathoverflow.net/users/4312 | 446311 | 179,815 |
https://mathoverflow.net/questions/446298 | 2 | Let $p$ be a prime and ${\mathbb F}\_p$ the finite field with $p$ elements. There is a canonical ring map ${\mathbb Z} \to {\mathbb F}\_p \cong {\mathbb Z}/ p {\mathbb Z}$. Denote the image of $n$ by $[n]\_p$.
Now consider the set of algebraic integers $\overline {\mathbb Z} \subset {\mathbb C}$, which is the set of ... | https://mathoverflow.net/users/46433 | A ring map from algebraic integers to algebraic closure of $\mathbb F_p$ | This is basic ramification theory that you can find in any textbook on algebraic number theory; for instance [Neukirch]. As in my comments, I will show slightly more than nonemptiness of $\operatorname{Hom}(\overline{\mathbf Z},\overline{\mathbf F}\_p)$, namely determine this set via Galois theory.
Fix an algebraic c... | 6 | https://mathoverflow.net/users/82179 | 446312 | 179,816 |
https://mathoverflow.net/questions/439348 | 1 | This question is a copy of one I asked in the [Math StackExchange forum](https://math.stackexchange.com/questions/4623074/what-lattices-beyond-the-laminated-lattices-particularly-in-%E2%89%A4-24d-belong-to-a) a few days ago. I don't know if it qualifies as a research-level question, but it may be something beyond most ... | https://mathoverflow.net/users/155650 | What lattices beyond the laminated lattices (particularly in $\le 24D$) belong to a slightly expanded category that includes "descendants" of Λ13_mid? | I finally found the answer to this question in a 1993 paper by W. Plesken and M. Pohst, *Constructing integral lattices with prescribed minimum. II* ( <https://www.ams.org/journals/mcom/1993-60-202/S0025-5718-1993-1176715-1/S0025-5718-1993-1176715-1.pdf> ). They are the lattices in Figure 1 of that article that are to ... | 0 | https://mathoverflow.net/users/155650 | 446318 | 179,818 |
https://mathoverflow.net/questions/446281 | 2 | I have to prove this:
Let $\alpha\in(0,1)$ and $f\in L^q(a,b)$, $1\leq q<\frac 1\alpha$, and $\mathcal{I}\_{a+}^\alpha f=0$. Then $f(x)=0$ for almost all $x\in (a,b)$.
Where $(\mathcal{I}\_{a+}^\alpha f)(x):=\frac{1}{\Gamma(\alpha)} \int\_{a}^{x} f(y)(x-y)^{\alpha-1}$.
Can Somebody help me?
Thanks
| https://mathoverflow.net/users/501039 | If the Riemann-Liouville fractional integral of $f$ is zero then $f=0$ a.e | $\newcommand{\al}{\alpha}\newcommand{\Ga}{\Gamma}$Let $g:=\Ga(\al)\mathcal{I}\_{a+}^\al f$, so that $g=0$ on $(a,b)$. Then for any $z\in(a,b)$
\begin{equation}
\begin{aligned}
0&=\int\_a^z dx\,(z-x)^{-\al}g(x) \\
&=\int\_a^z dx\,(z-x)^{-\al}\int\_a^x dy\,f(y)(x-y)^{\al-1} \\
&=\int\_a^z dy\,f(y)\int\_y^z dx\,(z-x)... | 2 | https://mathoverflow.net/users/36721 | 446320 | 179,819 |
https://mathoverflow.net/questions/446276 | 4 | Let $U\_1\subset \mathbb R^3$ be a simply connected bounded open set with a smooth boundary and let $U\_2$ be a neighborhood of $U\_1$. Does there exist a real-analytic diffeomorphism $\psi: U\_2 \to W\_2$ for some $W\_2\subset \mathbb R^3$ such that
$$ \psi(U\_1) \subset B \subset \psi(U\_2)$$
where $B$ is the unit ba... | https://mathoverflow.net/users/50438 | Finding a real-analytic diffeomorphism | The answer is positive. First, the fact that $U\_1$ is simply connected implies that every connected component $S\_i$ ($1\le i\le n$) of its smooth boundary $\partial U\_1$ is diffeomorphic to the $2$-sphere (not completely obvious, but true). Second, after the Schönfliess theorem in dimension $3$, each of these $2$-sp... | 3 | https://mathoverflow.net/users/105095 | 446322 | 179,821 |
https://mathoverflow.net/questions/446334 | 4 | If $A$ is an abelian variety over a finite field $\mathbf{F}\_q$, then $A(\mathbf{F}\_q)$ (resp. $A(\overline{\mathbf{F}}\_q)$) is a finite (resp. infinite torsion) group, but $A(\mathbf{F}\_q(t))$ is a finitely generated and possibly infinite abelian group.
This question concerns a similar situation for étale group ... | https://mathoverflow.net/users/501361 | Étale group schemes and specialization | If $k \to \ell$ is any regular field extension (i.e. $k$ is algebraically closed in $\ell$), then $X(k) \to X(\ell)$ is a bijection when $X \to \operatorname{Spec} k$ is étale. Indeed, it suffices to show this when $X = \operatorname{Spec} K$ for some finite separable field extension $k \to K$, and then the statement i... | 8 | https://mathoverflow.net/users/82179 | 446346 | 179,829 |
https://mathoverflow.net/questions/446341 | 10 | Define $P(x)$ to be positive if $P(x)>0$ for $x>0$.
I can prove that a quadratic positive polynomial is the ratio of 2 polynomials with non negative coefficients, for example $\displaystyle x^2-x+1/3=\frac{x^6+1/27}{x^4+x^3+2/3 x^2+1/3 x+1/9}$, and similarly for every $x^2-x+c$ where $c>1/4$. The full proof is not ha... | https://mathoverflow.net/users/2480 | Is every positive polynomial the ratio of 2 positive coefficient polynomials? | It is well-known. It is even known that you may take $R=(1+x)^m$ for large enough $m$. See, for example, [John Scholes's solution to Problem 11 of the 38th IMO 1997 shortlist](https://prase.cz/kalva/short/soln/sh9711.html). Note that by the real fundamental theorem of algebra, the general case reduces to the case $\deg... | 17 | https://mathoverflow.net/users/4312 | 446347 | 179,830 |
https://mathoverflow.net/questions/445656 | 2 | Let $(a\_k)\_{k \geq 1}$ be random variables taking values on a finite subset $B$. Assume that
$$
(1) \quad \Pr\Big (\lim\_{n\rightarrow +\infty}d(\frac{1}{n}\sum\_{k=1}^n 1\_{[a\_k = b]}, [v\_\ell(b,\theta\_0),v\_u(b,\theta\_0)])= 0\Big)=1 \quad \forall b\in B
$$
where $d$ stays for distance (i.e., absolute value diff... | https://mathoverflow.net/users/42412 | Bound the probability that a point belongs to a set | As I wrote before in a comment, the only upper bound on $\Pr(\theta\_0\in\Theta\_n)$ under these very general conditions is the trivial bound $1$.
Indeed, suppose that for some $b\_\*\in B$
\begin{equation}
\Pr(a\_1=a\_2=\cdots=b\_\*)=1
\end{equation}
and for all $b\in B$
\begin{equation}
v\_\ell(b,\theta\_0)=1(b=b... | 2 | https://mathoverflow.net/users/36721 | 446349 | 179,831 |
https://mathoverflow.net/questions/446324 | 1 | Let $I(Y;X)$ denote the mutual information between $Y$ and $X$. If we have $I(Y;X\_{i}) < B$ for all $i \quad (1 \leq i \leq N)$, could we also get the upper-bound of $I(Y; X\_{1}, X\_{2}, ..., X\_{N})$?
| https://mathoverflow.net/users/478341 | Upper bound of $I(Y; X_{1}, ..., X_{N})$ when we have $I(Y;X_{i}) < B$ for all $i$ $(1 \leq i \leq N)$ | Fix any integer $N\ge2$. Let $X:=(X\_1,\dots,X\_N)$. There is no upper bound $U\_N(B)$ on $I(Y;X)$ such that $U\_N(B)\to0$ whenever $\max\_{i=1}^N I(Y;X\_i)< B\downarrow0$. That is, each of the $I(Y;X\_i)$'s may be arbitrarily small while $I(Y;X)$ is not small. Informally, each $X\_i$ may contain little or no informati... | 3 | https://mathoverflow.net/users/36721 | 446352 | 179,832 |
https://mathoverflow.net/questions/446344 | 4 | Recently I learned from the Stacks project that for every abelian category ${\mathcal A}$, there is a natural isomorphism $K\_0({\mathcal A})\cong K\_0(D^{b}(\mathcal A))$.
When we set $\mathcal A$ to be the abelian category $M(X)$ of coherent sheaves on a Noetherian scheme $X$,we get that $G\_0(X)\cong K\_0(D^{b}(M(X)... | https://mathoverflow.net/users/477848 | Can higher G-theory of Noetherian schemes be computed by derived categories? | Yeah this is true, as Neeman explains this follows from his more general theorem of the heart for the bounded derived category with the standard t-structure.
You can see a modern treatment ( with a shorter proof in the language of $\infty$-categories ) in Barwick's paper
*Barwick, Clark*, [**On exact (\infty)-categ... | 7 | https://mathoverflow.net/users/44499 | 446353 | 179,833 |
https://mathoverflow.net/questions/446338 | 2 | Given a surface $X:f(x,y,z)=0\subset \mathbb{A}^{3}\_{\mathbb{C}}$ with only ADE singularities, how does one determine the correct singularity type of $X$ by computing the normal forms?
Does a similar procedure also exist in positive characteristic?
| https://mathoverflow.net/users/211978 | Normal forms of ADE singularities | This may not be exactly what you are asking about, but what follows has some references. I'm guessing you are already aware of:
* V. I. Arnolʹd, *Critical points of smooth functions, and their normal forms*. Uspehi Mat. Nauk 30 (1975), no. 5(185), 3–65.
There are a couple different ways. I think you are indicating ... | 5 | https://mathoverflow.net/users/3521 | 446354 | 179,834 |
https://mathoverflow.net/questions/445853 | 3 | **One sided Shift**
Let be $M$ separable metric space. Consider $X=M^{\mathbb{N}}$ the sequence space equipped with the product metric $d(x,y)=\sum\_{i=1}^\infty |x\_i-y\_i|/2^i$ . Define the shift map $\sigma:X\to X$ by putting in each sequence $x=(x\_i)$, $\sigma(x)\_i=x\_{i+1}$. Then we have that the shift has the... | https://mathoverflow.net/users/98969 | Is the weighted shift strong frequently hypercyclic? | The answer to the question is no. If the sequence $\alpha = (\alpha\_{n})\_{\geqslant 1}$ satisfies the condition $c<\alpha\_{n}<c^\prime$ there is no other condition on a sequence $\alpha$ for the answer to be affirmative. Let's assume that the answer to the question is affirmative. That is, there exists $\alpha = (\a... | 2 | https://mathoverflow.net/users/36917 | 446356 | 179,835 |
https://mathoverflow.net/questions/446357 | 3 | When I was reading an article by CHUN-GANG JI (A SIMPLE PROOF OF A CURIOUS CONGRUENCE BY ZHAO), he mentioned in the acknowledgement
the following identity
$$\sum\_{i+j+k=p,\text{ } i,j,k\gt 0}{p\choose i}{p\choose j}{p\choose k}={3p\choose p}-3{2p\choose p}+3$$
Can anybody give me a lead to a combinatorial or algeb... | https://mathoverflow.net/users/481754 | A combinatorial identity involving binomial coefficients | Without the inequality on the $i, j, k$, the sum on the left would be $\binom{3p}{p}$. This is an immediate consequence of the Chu-Vandermonde convolution identity
$$\sum\_{i+j = k} \binom{x}{i}\binom{y}{j} = \binom{x+y}{k}$$
which is treated in many places, for example in Concrete Mathematics by Graham, Knuth, and... | 6 | https://mathoverflow.net/users/2926 | 446359 | 179,836 |
https://mathoverflow.net/questions/446215 | 7 | The question, which came up in a conversation with my advisor Ola Kwiatkowska, is pretty much in the title:
Let $Z\subseteq[0,1]^3$ be zero-dimensional. Is it possible for $[0,1]^3\setminus Z$ to be embeddable in $\Bbb R^2$?
| https://mathoverflow.net/users/49381 | Can you remove a zero dimensional subspace from a cube and obtain a planar space? | I think the answer is negative in any dimension $n\geq 2$.
**Theorem:**
$\mathbb R^n$ can not be covered by a zero-dimensional set and a set homeomorphic to a subspace of $\mathbb R^{n-1}$.
**Proof:**
Suppose for contradiction that $\mathbb R^n=A\cup B$ where A is zero-dimensional and $i:B\to\mathbb R^{n-1}$ is an ... | 6 | https://mathoverflow.net/users/128723 | 446364 | 179,839 |
https://mathoverflow.net/questions/443751 | 5 | Let $V$ be a Banach space and $B(V)$ be the bounded operators on $V$. Now, the space $B(V)$ has a norm, the strong topology (which is the topology of pointwise convergence) and it has a stronger topology called ultrastrong or $\sigma$-strong topology. It is known that these two topologies coincide on norm-bounded sets.... | https://mathoverflow.net/users/501466 | Is the sigma-strong topology generated by bounded sets? | The finest topology that coincides with $\tau\*$ ($\sigma$-strong in this case) on $\tau$-bounded (norm-bounded in this case) subsets is the mixed topology $\gamma(\tau,\tau^\*)$, introduced by [A. Wiweger, Linear spaces with mixed topology. Studia Mathematica 20 (1961), 47--68](https://eudml.org/doc/217004); see 2.2.2... | 6 | https://mathoverflow.net/users/7591 | 446375 | 179,843 |
https://mathoverflow.net/questions/446374 | 8 | The broad theme that underlies this question is: to what extent can the study of finite groups be reduced to the study of $p$-groups?
I imagine that it is possible for a pair of nonisomorphic finite groups $G$ and $H$ to have isomorphic Sylow subgroups. That is to say, for every prime $p$, $G$ and $H$ have isomorphic... | https://mathoverflow.net/users/502468 | Nonisomorphic finite groups with isomorphic Sylow subgroups | To answer your question 2, there are very few pairs of finite simple groups with the same order. There's $PSL\_3(4)$ and $PSL(4,2)$, and there's $P\Omega\_{2n+1}(q)$ and $PSp\_{2n}(q)$ for $q$ odd. None of these are Sylow isomorphic in your sense.
As a contribution to your question 1, finite groups can be not only Sy... | 11 | https://mathoverflow.net/users/460592 | 446381 | 179,846 |
https://mathoverflow.net/questions/446116 | 8 | $\DeclareMathOperator\SO{SO}$I asked this initially in math stack exchange, but thought to ask it here since it is more advanced and related to my research topic. I study optimization on Lie groups from a Riemannian geometry viewpoint. Naturally, one of the Lie groups I use the most is $\SO(3) \subset \mathbb{R}^{3 \ti... | https://mathoverflow.net/users/141449 | Is it true $\left\|\log(RS)\right\|≤\left\|\log(R)+\log(S)\right\|$ for all $R,S \in \mathrm{SO}(3)$, where $\|\cdot\|$ is the Frobenius norm? | Now that I have written out the completely elementary proof above for the quaternions and for $\mathrm{SO}(3)$, I feel that I should point out that the statement $|\log(ab)|\le |\log(a) + \log(b)|$, suitably interpreted, is true for any compact connected Lie group $G$ endowed with a biïnvariant Riemannian metric. The p... | 7 | https://mathoverflow.net/users/13972 | 446385 | 179,848 |
https://mathoverflow.net/questions/446384 | 6 | This might seems like a bit of philosophical question and so maybe if I keep reading a bit more, I might get my answer. But, I ask nonetheless.
In my attempt to study Shimura varieties, I came across the definition of a locally symmetric space associated to connected reductive group $G/\mathbb{Q}$ as follows:
>
>... | https://mathoverflow.net/users/157428 | Definition of locally symmetric space of reductive groups | There is a very natural, intrinsic definition of a "symmetric space", as a manifold (Riemannian or Hermitian) with an extra symmetry of a certain prescribed type. It is then a *theorem*, not a definition, that all such objects have the form $G(\mathbb{R}) / A\_\infty^\circ K\_\infty^\circ$ for a reductive group $G$. Yo... | 12 | https://mathoverflow.net/users/2481 | 446402 | 179,852 |
https://mathoverflow.net/questions/446407 | 0 | I'm working on a problem that involves vectors and scalar values, and I'm looking for a closed-form solution. I hope someone can help me with this or provide insights into how to approach it. Here's the problem:
Given vectors $\mathbf{W}$, $\mathbf{S}$, and $\mathbf{T}$, and a scalars $q$.
The goal is to find a sca... | https://mathoverflow.net/users/113891 | Seeking closed-form solution for vector equation | The sum of the entrywise product of two vectors is just their dot product. So this question is asking how to find $x$ so that
$$
(q+x)(\mathbf{W} \cdot \mathbf{T}) = q\mathbf{W} \cdot\mathbf{S}.
$$
This is just a scalar equation; rearranging and solving gives
$$
x = \frac{q\mathbf{W} \cdot(\mathbf{S} - \mathbf{T})}{\ma... | 2 | https://mathoverflow.net/users/11236 | 446410 | 179,854 |
https://mathoverflow.net/questions/446235 | 7 | It seems that Professor Lennart Carleson gave a series of Lectures at UCLA in 1985. For example, one could find several mentions about these lectures in the book by Garnett & Marshal (see for example, notes at the end of chapters II and III). I would like to know whether is aware of theses lectures; preferably availabl... | https://mathoverflow.net/users/62739 | Carleson's lectures at UCLA | I have contacted the mathematics department at UCLA, and then received the following information from Professor Garnett (emphasis mine):
>
> For nine or ten winter quarters beginning 1985 Lennart Carleson gave a lecture course at UCLA. The topics included Potential Theory, Harmonic Measure, Complex Dynamics, Iterat... | 8 | https://mathoverflow.net/users/36721 | 446411 | 179,855 |
https://mathoverflow.net/questions/446387 | 7 | In the article "Automorphism groups of simple algebras and group algebras" (1978), Janusz conjectures the following:
>
> The group algebra $\mathbb{Q} G$ for a non-trivial finite group has an outer automorphism (=non-inner automorphism)?
>
>
>
>
> Question: Is this conjecture solved ?
>
>
>
| https://mathoverflow.net/users/61949 | Do rational group algebras have an outer automorphism? | This conjecture was proved in Feit, Walter; Seitz, Gary M.
On finite rational groups and related topics.
Illinois J. Math. 33 (1989), no. 1, 103–131. The proof relies on the classification since this easily reduces to the case of simple groups.
| 10 | https://mathoverflow.net/users/15934 | 446413 | 179,856 |
https://mathoverflow.net/questions/446383 | 8 | Let $\Omega$ be a bounded open set in $\mathbb{R}^n$ with $C^\infty$ boundary, $n\ge 3$. Define
$$V(z)=\int\_\Omega \frac{1}{|z-y|^{n-2}}dy$$
Is it true that $V(z) \in C^{\infty}(\partial \Omega)$?
---
Motivation: Classical Holder estimate says that if $f \in C\_c^\alpha(\mathbb{R}^n)$, then the Newtonian potenti... | https://mathoverflow.net/users/51546 | Regularity of Newtonian potential along smooth boundary | Sure. By a smooth dyadic decomposition it suffices to show that convolutions of the form
$$ \varepsilon^{-n} \int\_\Omega \varphi\left(\frac{y-z}{\varepsilon}\right)\ dy$$
for $0 < \varepsilon \lesssim 1$ and $\varphi$ a fixed bump function are smooth on $\partial \Omega$ uniformly in $\varepsilon$ (multiply by $\varep... | 12 | https://mathoverflow.net/users/766 | 446421 | 179,859 |
https://mathoverflow.net/questions/446420 | 1 | Diagonally dominant matrices are required in many linear algebra algorithms such as the Gauss-Seidel algorithm.
Some matrices can be made diagonally dominant by permuting its rows and others cannot.
Given an $n$ by $n$ integer matrix $A$, what is the complexity of deciding the existence of rows permutation that res... | https://mathoverflow.net/users/8784 | Diagonally dominant matrix via rows permutation | Create a bipartite graph $G$ with $2n$ vertices, say $u\_1,\ldots,u\_n, v\_1,\ldots,v\_n$, where the edge set of $G$ represents matrix entries that could potentially appear on the diagonal, if the rows of $A$ were permuted to make it diagonally dominant. In other words, the bipartite graph has an edge from $u\_i$ to $v... | 2 | https://mathoverflow.net/users/8049 | 446422 | 179,860 |
https://mathoverflow.net/questions/446433 | -2 | Let $X$ be a finite CW-complex of $n$.
1. For $i\geq 2$, $\pi\_i (X)$ is a $\mathbb{Z}\pi\_1 (X)$-module.
2. for $i\geq 2$, $H\_i (\tilde{X})$ is a finitely generated $\mathbb{Z}\pi\_1 (X)$-module, where $\tilde{X}$ is the universal covering of $X$.
I know that the connection between homotopy groups and homology gr... | https://mathoverflow.net/users/114476 | Relation between $\mathbb{Z}\pi_1 (X)$-module $\pi_n (X)$ and $\mathbb{Z}$-module $H_n (X)$ or $\mathbb{Z}\pi_1 (X)$-module $H_n (\tilde{X})$ | Not sure if you're already aware of an exact sequence which implies Hurewicz's theorem, but if $X$ is a space with $\pi\_i(X)=0$ for $1<i<n$ then we have
$$H\_{n+1}(X)\to H\_{n+1}(\pi)\to \pi\_n(X)\_{\pi}\to H\_n(X)\stackrel{\psi}{\to} H\_n(\pi)\to0$$ where $\pi=\pi\_1(X)$ is the fundamental group and $H\_\*(\pi)$ is d... | 4 | https://mathoverflow.net/users/12310 | 446435 | 179,865 |
https://mathoverflow.net/questions/446442 | 11 | Let $B$ be some compact, path connected $n$-manifold without boundary such that its cobordism class is trivial, so that there exists some other $n+1$ manifold $M$ with $\partial M= B$. While there is not a unique choice for $M$ (for example $\mathbb{S}^1$ can be seen as the boundary of the connected sum of $n\geq 0$ to... | https://mathoverflow.net/users/504402 | Lower bounds for Betti numbers of a manifold given its boundary? | Let me assume that both $M$ and $B$ are orientable.
From the long exact sequence of the pair $(M,B)$, Poincaré–Lefschetz duality, and the universal coefficient theorem, for every $k$ we get an exact sequence (I'll use rational or real coefficients throughout):
$$
H\_{n-k}(M)^{\vee} \cong H^{n-k}(M)\cong H\_{k+1}(M,B) \... | 9 | https://mathoverflow.net/users/13119 | 446446 | 179,868 |
https://mathoverflow.net/questions/446428 | 4 | The fundamental theorem of surfaces states that if symmetric matrices $g\_{ij}$, $l\_{ij}\colon U\subset R^2\to R$, where $U$ is open and $g\_{ij}$ is positive definite satisfy the Gauss and Codazzi equations, then there exists a surface $X\colon U\to R^3$ with $g\_{ij}$, $l\_{ij}$ as the first and second fundamental f... | https://mathoverflow.net/users/68969 | Approximate isometric embeddings of surfaces | I think that the answer is 'yes' if $U$ is simply-connected, because there is a way to construct a candidate 'approximate surface' from 'approximate solutions' of Gauss and Codazzi, but a more useful answer would be one that gave you estimates of how close the metric and second fundamental form of the approximate surfa... | 5 | https://mathoverflow.net/users/13972 | 446447 | 179,869 |
https://mathoverflow.net/questions/446444 | 6 | I am interested to use the mapping degree for simplicial maps between (oriented) abstract simplicial complexes. What I mean by "elementary": My preference would be to use a definition of mapping degree that is *not* based on homology and *not* based on the geometric realization of the complexes.
As I am new to this a... | https://mathoverflow.net/users/156936 | Abstract simplicial complexes - Reference for an elementary definition of mapping degree for simplicial maps? | I think the right generality to restrict to is the following:
Let $n$ be a positive integer, and let $X$ and $Y$ be $n$-dimensional oriented simplicial complexes, with the following properties:
1. Every $n-1$-face is contained in exactly two $n$-faces (with opposite orientations relative to the $n-1$-face, i.e. if ... | 9 | https://mathoverflow.net/users/39747 | 446450 | 179,871 |
https://mathoverflow.net/questions/446443 | 0 | I am struggling to model my problem correctly since multiple days.
Maybe someone can give me a hint.
I have two levels, both with a fixed number of slots (i.e. 200 each).
The items I want to put on the slots are ordered and earlier items have to be positioned on earlier slots.
Imagine a fixed queue of vehicles driv... | https://mathoverflow.net/users/477317 | Precedence constraints in assignment problem | Here are linear constraints for the bottom:
$$\sum\_{p\in P^{bot}}px\_{pv} \le \sum\_{p\in P^{bot}}(p-1)x\_{pu} +|P^{bot}|\sum\_{p\in P^{top}}x\_{pu}\quad \forall (v, u)\in A$$
Equivalently,
$$\sum\_{p\in P^{bot}}px\_{pv} \le \sum\_{p\in P^{bot}}(p-1-|P^{bot}|)x\_{pu}+|P^{bot}|\quad \forall (v, u)\in A$$
The constraint... | 0 | https://mathoverflow.net/users/141766 | 446456 | 179,872 |
https://mathoverflow.net/questions/446461 | 2 | For a prime $p$, let $\varphi\_p\colon\mathbb Z\to\mathbb Z/p\mathbb Z$ denote the canonical homomorphism from the integers onto the group of order $p$.
>
> Given an integer $n\ge 3$, what is the smallest $\varepsilon=\varepsilon(n)>0$ such that for any subset $A\subset\mathbb Z$ with $|A|=n$, there exists a prime ... | https://mathoverflow.net/users/9924 | Prime divisors of $\prod(a_i-a_j)$ | Maybe I misunderstand the question, but doesn't the set $A=\{i\cdot n!\,|\,1\le i\le n\}$ have $\lvert\varphi\_p(A)\rvert=1$ for all $p$ for which $\varphi\_p$ is not injective on $A$?
| 4 | https://mathoverflow.net/users/18739 | 446470 | 179,876 |
https://mathoverflow.net/questions/446445 | 7 | Let $\Omega \subset \mathbb{R}^n$ be a compact domain and for given functions $g: \partial \Omega \times [0,T] \to \mathbb{R}$ and $h: \Omega \to \mathbb{R}$ consider the heat equation
$$
\begin{cases}
\frac{\partial u}{\partial t} - \Delta u &= 0 \qquad \text{on } \Omega \times (0,T]\\
u &= g \qquad \text{on } \part... | https://mathoverflow.net/users/295304 | Existence of solutions to the heat equation on nonsmooth domains | $\newcommand\R{\mathbb R}\newcommand{\Om}{\Omega}$It is convenient to reverse the time by the substitution $t\leftrightarrow T-t$ and also rescale it by a factor of $2$. So, the boundary value problem becomes the following:
\begin{align}
\frac{\partial u}{\partial t} +\frac12 \Delta u &= 0 \quad \text{on } D, \tag{1}\... | 7 | https://mathoverflow.net/users/36721 | 446471 | 179,877 |
https://mathoverflow.net/questions/446482 | 6 | For a positive integer $N$, we define $$\Gamma(N)=\big \{ \begin{bmatrix} a & b \newline c & d\end{bmatrix}\in \operatorname{SL}\_2(\mathbb{Z}): \begin{bmatrix} a & b \newline c & d\end{bmatrix}\equiv \begin{bmatrix} 1 & 0 \newline 0 & 1\end{bmatrix}(\operatorname{mod}N)\big\}$$
Then, [Diamond, Shurman] defines the s... | https://mathoverflow.net/users/157428 | Definition of modular curve associated to $\Gamma(N)$ | This is a subtle issue (which has come up before on this site several times, see e.g. [is the modular curve X(N) defined over Q?](https://mathoverflow.net/questions/192156/is-the-modular-curve-xn-defined-over-q?rq=1) for a related question).
Your $S(N)$ is naturally a scheme over $\mathbb{Z}[1/N, \zeta\_N]$. Your $X\... | 10 | https://mathoverflow.net/users/2481 | 446485 | 179,882 |
https://mathoverflow.net/questions/446462 | 4 | Let $k$ be a perfect field. I am looking for a $k$-algebra $R$ with the following properties.
* $R$ is of finite type over $k$ and is a domain;
* for all ${\mathfrak p}\in{\rm Spec}(R)$, the local ring $R\_{\mathfrak p}$ is Cohen-Macaulay and ${\rm emb.\, dim.}(R\_{\mathfrak p})-{\rm dim}(R\_{\mathfrak p})\leq1$;
* f... | https://mathoverflow.net/users/17308 | Example of a certain type of Cohen-Macaulay ring | There is no such example. A Cohen-Macaulay local ring with $\operatorname{embdim}(R)-\dim(R) \le 1$ is a hypersurface, which is in particular a complete intersection. Indeed, we may pass to the completion to suppose $R$ is complete. Then by Cohen's structure theorem, $R \cong S/I$ where $(S,\mathfrak{n})$ is a regular ... | 6 | https://mathoverflow.net/users/155965 | 446488 | 179,884 |
https://mathoverflow.net/questions/446453 | 6 | If a real analytic variety $V$ in $\mathbb{R}^n$ is both bounded and contractible, is it true that $V$ must be a single point?
Here a real analytic variety is the set of zeros of a real analytic function $f: \mathbb{R}^n \rightarrow \mathbb{R}$.
This is certainly true if $V$ is a compact real analytic manifold (wit... | https://mathoverflow.net/users/54756 | Contractible real analytic varieties | It is a consequence of Sullivan's work
*Sullivan, D.*, Combinatorial invariants of analytic spaces, Proc. Liverpool Singularities-Sympos. I, Dept. Pure Math. Univ. Liverpool 1969-1970, Lect. Notes Math. 192, 165-168 (1971). [ZBL0227.32005](https://zbmath.org/?q=an:0227.32005).
that every compact $k$-dimensional rea... | 4 | https://mathoverflow.net/users/39654 | 446497 | 179,888 |
https://mathoverflow.net/questions/446493 | 0 | Let $c>0$ and let $Y$ be the space of all distributions of compact support in $(-1,1)$ with singular support at $\{0\}$. Let $X$ be **subspace** of $Y$ such that for any $\phi \in X$ there holds:
$$ \int\_{t^2>r^2} \phi'(t)^2\,dt \leq c\,\int\_{t^2>r^2-r^4} \phi(t)^2\,dt \quad \forall\, r\in (0,1).$$
Is it true that $X... | https://mathoverflow.net/users/50438 | Finite dimensionality of a subspace | $\newcommand\de\delta$The answer is no. E.g., let $X$ be the linear span of the set $\{\de,\de',\de'',\dots\}$, where $\de$ is the Dirac delta distribution supported on $\{0\}$. Then $X$ satisfies your condition but is infinite dimensional.
| 1 | https://mathoverflow.net/users/36721 | 446502 | 179,890 |
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