parent_url stringlengths 37 41 | parent_score stringlengths 1 3 | parent_body stringlengths 19 30.2k | parent_user stringlengths 32 37 | parent_title stringlengths 15 248 | body stringlengths 8 29.9k | score stringlengths 1 3 | user stringlengths 32 37 | answer_id stringlengths 2 6 | __index_level_0__ int64 1 182k |
|---|---|---|---|---|---|---|---|---|---|
https://mathoverflow.net/questions/303900 | 26 | Many zeta functions and L-functions which are generalizations of the Riemann zeta function play very important roles in modern mathematics (Kummer criterion, class number formula, Weil conjecture, BSD conjecture, Langlands program, Riemann hypothesis,...).
Euler was perhaps the first person to consider the zeta funct... | https://mathoverflow.net/users/123463 | Why did Euler consider the zeta function? | This history is described in [Euler and the Zeta Function](https://www.jstor.org/stable/2319041?seq=1#page_scan_tab_contents) by Raymond Ayoub (1974). In his early twenties, around 1730, Euler considered the celebrated problem to calculate the sum $$\zeta(2)=\sum\_{n=1}^\infty \frac{1}{n^2}.$$ This problem goes back to... | 43 | https://mathoverflow.net/users/11260 | 303903 | 132,996 |
https://mathoverflow.net/questions/303937 | 1 | How do you prove $\lim\_{k\to\infty,k\in\mathbb{N}}{n\choose k}2^{1-{k\choose2}}=1$, where $n=\max\{n\in\mathbb{N}:{n\choose k}2^{1-{k\choose2}}<1\}$? The expression ${n\choose k}2^{1-{k\choose2}}$ comes up in a lower bound for the Ramsey number $R(k,k)$, and I want to fill in the steps in the derivation of the asympto... | https://mathoverflow.net/users/110883 | $\lim_{k\to\infty}{n\choose k}2^{1-{k\choose2}}$, where $n=\max\{n\in\mathbb{N}:{n\choose k}2^{1-{k\choose2}}<1\}$ | Let $N(k) = \max \{n \in \mathbb N: \; {n \choose k} 2^{1-{k\choose 2}} < 1 \}$.
Note that $$\frac{{n+1 \choose k}}{{n \choose k}} = \frac{n+1}{n+1-k}$$
so it suffices to prove that $N(k)/k \to \infty$ as $k \to \infty$.
But for any finite $c > 1$,
$$ \log {c k \choose k} \sim (c \log c - (c-1) \log (c-1)) k = o(k^2)$... | 4 | https://mathoverflow.net/users/13650 | 303938 | 133,005 |
https://mathoverflow.net/questions/303935 | 4 | It is known that: $ ``\forall x \exists H\_x"$, where $H\_x$ is defined as the set of all sets hereditarily subnumerous to $x$, is a theorem of $\text{ZF}$. The formal definition of $H\_x$ is:
$$H\_x=\{y| \forall z \in TC(\{y\}) (\exists f (f:z \to x \wedge f \text{ is injective}))\}$$, where $TC(k)$ stands for the "... | https://mathoverflow.net/users/95347 | Does a hereditary size set exist for every set in ZF-Regularity+Aczel's anti-foundation axiom? | The answer is yes.
The reason is that in AFA, every set is determined by the isomorphism type of the $\in$ relation on its transitive closure. So if a set $y$ hereditarily injects into another set $x$, then the transitive closure of $y$ is determined by a certain graph relation on the nodes coming from the graph of ... | 8 | https://mathoverflow.net/users/1946 | 303939 | 133,006 |
https://mathoverflow.net/questions/303915 | 2 | Is there a **complete** Riemannian metric on the cylinder such that the metric is **not flat** but the cylinder is foliated by closed geodesics with the same length?
A possibility non complete metric with the above properties is introduced in the "Remark" of the following answer:
[A curvature description for center... | https://mathoverflow.net/users/36688 | A foliation of the cylinder by closed geodesics of the same length when the metric is complete but non flat | The standard non-flat metric on a torus $T^2$ is foliated by closed geodesics. It can be unwrapped to give an example of what you want.
Specifically, let $\gamma\_1$ and $\gamma\_2$ be the standard generators for $\pi\_1(T^2)$, where there is a foliation of $T^2$ by closed geodesics freely homotopic to $\gamma\_1$ (b... | 3 | https://mathoverflow.net/users/126126 | 303945 | 133,007 |
https://mathoverflow.net/questions/303920 | -1 | I would know if it is possible to create a natural parallel transport on a Riemannian manifold $M$ with the following procedure.
Define a minimum length curve (m.l.c.) $\gamma$ from $P$ to $Q$ requiring to minimize the functional of length $L(\gamma)$ and show that for every two points $P$ and $Q$ in $M$ exists such ... | https://mathoverflow.net/users/103977 | Parallel transport with minimum length curve | Your assumption that minimal geodesics exist and are unique, in the complete case, is equivalent to $M$ being simply connected and having no conjugate points. This is already quite a restrictive class of manifolds.
In your above comment, you suggest that, in the case where multiple minimal geodesics between points ex... | 3 | https://mathoverflow.net/users/126126 | 303951 | 133,009 |
https://mathoverflow.net/questions/303944 | 4 | Let $f\colon X\to Y$ be a finite morphism of quasiprojective varieties (I’m most interested in the case that $f$ is normalization and the varieties are over $\Bbb C$). Is the following statement true (at least locally on $Y$)?
>
> There are embeddings $i$ and $j$ of $X$ and $Y$, resp., into smooth quasiprojective v... | https://mathoverflow.net/users/36720 | Embedding a finite morphism into a finite morphism of smooth varieties | This is always possible. I will first explain the local construction, and then sketch a proof of how to globalise.
**Local problem.** Let $A \to B$ be a finite map of finite type $k$-algebras. Does there exist a finite map $A' \to B'$ of smooth $k$-algebras admitting a surjection to $A \to B$?
*Construction.* Suppo... | 7 | https://mathoverflow.net/users/82179 | 303961 | 133,013 |
https://mathoverflow.net/questions/303959 | 2 |
>
> Could You give a poof, comment or reference for the inequality as follows:
>
>
> $$\sum\_{k=1}^n(-1)^k\frac{\sin kx}{k^{\alpha}} < 0$$
> for all $n=1,2,3,\ldots$ and $0<x<\pi$ and $\alpha \ge 1$
>
>
>
* See also:
[$\sum\_{k=1}^n\frac{\sin kx}{k^\alpha} >0\quad\text{for all}\ n=1,2,3,\ldots\ \text{and}\ 0... | https://mathoverflow.net/users/122662 | $\sum_{k=1}^n(-1)^k\frac{\sin kx}{k^{\alpha}} < 0 ?$ with $\alpha \ge 1$ and $n=1, 2,\cdots$ | Plug in $\pi-x$ in the already proven inequality, i.e. the one without the $(-1)^k$ one use that
$$
sin(k(\pi-x))=sin(k\pi-kx)=-(-1)^ksin(kx)
$$
| 1 | https://mathoverflow.net/users/100908 | 303965 | 133,014 |
https://mathoverflow.net/questions/303967 | 1 |
>
> Suppose we want to assign $n$ items to $m$ customers ($n \geq m$). Each assignment of an item $i$ to a customer $j$ has an associated cost $c(i,j)$. Find an assignment that maximizes the total cost. Here, we must assign every item; therefore, a customer might receive more than one item. Moreover, we require that ... | https://mathoverflow.net/users/75053 | An variation of an assignment problem in combinatorics: assign items to customers | That is a classical assignment problem, that can also be solved with the Kuhn-Munkres algorithm, for which generalizations to [rectangular assignment matrices](https://dl.acm.org/citation.cfm?id=362945) exist.
| 3 | https://mathoverflow.net/users/31310 | 303974 | 133,015 |
https://mathoverflow.net/questions/300415 | 4 | Assume that there exists at least one NP-intermediate decision problem (which, by Ladner's theorem, is equivalent to P being distinct from NP).
Do there exist two NP-intermediate decision problems, $A$ and $B$, such that neither one is polynomial-time reducible to the other?
| https://mathoverflow.net/users/39521 | Incomparable NPI decision problems | [Balcazar and Diaz](https://www.sciencedirect.com/science/article/pii/0020019082901132) proved that if $P \ne NP$ then there exists an infinite number of non-comparable languages in $NP$.
| 4 | https://mathoverflow.net/users/8784 | 303979 | 133,016 |
https://mathoverflow.net/questions/303925 | 10 | The category of fibrations in a combinatorial model category is accessible, accessibly embedded in the arrow category. How about the cofibrations?
More generally, let $C$ be a locally presentable category and let $(L,R)$ be a weak factorization system on $C$. If $(L,R)$ is cofibrantly-generated (i.e. there is a set $... | https://mathoverflow.net/users/2362 | Are cofibrations accessible? | A cofibrantly generated $(L,R)$ does not need to have $L$ accessible, see Example 3.5
in my paper "On combinatorial model categories."
Also, $L$ accessible does not imply that $(L,R)$ is cofibrantly generated, even accessible.
Take regular monos in Boolean algebras. This $L$ is accessible but $(L,R)$ cannot be accessib... | 9 | https://mathoverflow.net/users/73388 | 303986 | 133,023 |
https://mathoverflow.net/questions/303530 | 8 | Let $f:\mathbb{R} \rightarrow [0,\infty)$ be a continuous probability density function on $\mathbb{R}$ such that
\begin{equation}
\int\_{\mathbb{R}} |x| f(x)\, dx < \infty,
\end{equation}
and assume that $f$ has a strict global maximum $x\_0$, that is $f(x) < f(x\_0)$ for all $x \neq x\_0$.
For any fixed $q \in (0,1]$... | https://mathoverflow.net/users/99197 | q-Means and the mode of a distribution | Finally, I discovered that this problem was deeply investigated by the great french mathematician Maurice Fréchet, who introduced in his remarkable essay [Les éléments aléatoires de nature quelconque dans un espace distancié](http://www.numdam.org/article/AIHP_1948__10_4_215_0.pdf) the general notion of q-mean, where $... | 1 | https://mathoverflow.net/users/99197 | 303987 | 133,024 |
https://mathoverflow.net/questions/303972 | 2 | Let $X$ be a smooth projective complex variety, and $D=\cup\_{j=1}^m D\_j$ a simple normal crossings divisor on $X$. Then we have an exact sequence
$$0\to \Omega\_X^1\to \Omega\_X^1(\log D)\to \oplus\_{j=1}^m i^\*\_j\mathcal{O}\_{D\_j}\to 0,$$
where $i\_j:D\_j\to X$ is the inclusion. Note that $H^0(i\_j^\*\mathcal{O}\_... | https://mathoverflow.net/users/64302 | Admissible global residues on smooth variety with normal crossings divisor | Jason has given an essentially complete answer, which I'm just repeating it here, so that the question can be considered answered. (I think the sequence is OK, however. E.g. for $D=\{xy=0\}$ locally, $Res$ sends $fdx/x+gdy/y\to (f(0,y), g(x,0))$, and this clearly surjects.) The image of $H^0(\Omega\_X^1(\log D)\to \mat... | 5 | https://mathoverflow.net/users/4144 | 303990 | 133,026 |
https://mathoverflow.net/questions/304005 | 1 | Suppose that we have tree spaces $A, B, C$ (let say CW-complexes). Are given two pairs of maps:
$$f\_{0},g\_{0}: A\rightarrow B $$
$$f\_{1}, g\_{1}: A\rightarrow C $$
such that $f\_{0},g\_{0}$ are homotopic and $f\_{1},g\_{1}$ are also homotopic.
Is it true that $colim [B\leftarrow^{f\_{0}} A\rightarrow^{f\_{1}} C... | https://mathoverflow.net/users/103287 | pushout and homotopy | No. Take $A=[0,1]$, $B=S^1$, $C=\text{point}$. There is only one possible choice for $f\_1$ and $g\_1$. There are many possible choices for $f\_0$ and $g\_0$, but they are all homotopic. Choose $f\_0$ to be constant, and choose $g\_0$ to be surjective. Then the pushout for $f$ is $S^1$, and the pushout for $g$ is a poi... | 6 | https://mathoverflow.net/users/10366 | 304006 | 133,031 |
https://mathoverflow.net/questions/303940 | 10 | Let $V$ be an $n$-dimensional Euclidean vector space with inner product $\langle\cdot,\cdot\rangle$ and $\Phi$ an irreducible crystallographic root system in $(V,\langle\cdot,\cdot\rangle)$.
**Question 1**: Is there a root-theoretic formula for the number of facets of $\mathrm{ConvHull}(\Phi)$? Surely this must be kn... | https://mathoverflow.net/users/25028 | How many facets does the convex hull of all the roots of a root system have? | These are studied under the name **root polytopes** by various authors in particular in the context of abelian ideal in root posets. See for example Cellini [Triangulations of root polytopes](http://arxiv.org/abs/1612.06143) and Cellini-Marietti [Root polytopes and Borel subalgebras](https://arxiv.org/abs/1203.0756) wh... | 4 | https://mathoverflow.net/users/21291 | 304016 | 133,034 |
https://mathoverflow.net/questions/303970 | 4 | Or more generally, are L-functions injective in some strips $a<\Re(s)<b$, where $0\leq a<b \leq 1$?
| https://mathoverflow.net/users/41499 | Is Riemann zeta function injective in some strips $a<\Re(s)<b$, where $0\leq a<b \leq 1$? | The Riemann zeta function function is not injective in any strip $a<\Re(s)<b$ with $\frac{1}{2}<a<b$.
For $b\leq 1$ this follows, for example, from Theorem 11.10 in Titchmarsh: The theory of the Riemann zeta function (see also the remarks after the theorem in the book).
The theorem (which is probably due to Bohr) i... | 5 | https://mathoverflow.net/users/11919 | 304018 | 133,035 |
https://mathoverflow.net/questions/303806 | 10 | I am looking for a reference (better a book) that contain integral Bochner formulas for domains with boundary (I need it for 1-forms and functions only).
For example I will need the following formula:
$$\int\limits\_\Omega |\Delta f|^2
-|\mathrm{Hess}f|^2
+\langle\mathrm{Ric}(\nabla f),\nabla f\rangle
=\int\limits\_{... | https://mathoverflow.net/users/1441 | Bochner formula in different forms | Here's a reference to a paper; Theorem 3 gives a general formula for differential forms with no constraints on the boundary values. I'm not sure if this shows up in a book yet.
**Raulot, S.; Savo, A**. A Reilly formula and eigenvalue estimates for differential forms. *J. Geom. Anal*. **21** (2011), no. 3, 620--640. [... | 4 | https://mathoverflow.net/users/121820 | 304028 | 133,038 |
https://mathoverflow.net/questions/304022 | 6 | Originally [asked](https://math.stackexchange.com/questions/2831043/is-restriction-a-closed-map) on MSE.
Let $X$ be a normal (or even metrizable) topological space and let $Y$ be a closed subset of $X$. Let $C(X)$ be the linear space of all continuous scalar functions on $X$ endowed with the compact-open topology. Co... | https://mathoverflow.net/users/53155 | Is restriction a closed map? | The answer to the main question is negative:
Consider the compact subset $X=[0,1]\cup \{2\}$ of the real line and let $Y=\{2\}$ be a singleton in $X$. In the function space $C(X)$ consider the closed convex balanced subset
$$B:=\{f\in C(X):\sup\_{x\in [0,1]}|f(x)|\le 1,\;f(0)=0,\;f(2)=\int\_0^1 f(t)dt\}.$$ It is easy... | 7 | https://mathoverflow.net/users/61536 | 304029 | 133,039 |
https://mathoverflow.net/questions/304044 | 0 | Motivation: I am working on a research problem and have been stuck for a while. I hope someone can help, as it requires only linear algebra. :)
Let $H$ be a real, invertible and positive semi-definite matrix, in the sense that its symmetric part $S$ is positive semi-definite. Consider the matrix
$$ G = (I+\alpha H\_d... | https://mathoverflow.net/users/108472 | Stability of a matrix product | It's true for all $\alpha \ge 0$. $G$ has the same eigenvalues as $(1+\alpha H\_d)^{1/2} H (1+\alpha H\_d)^{1/2}$, which is positive semidefinite.
| 3 | https://mathoverflow.net/users/13650 | 304049 | 133,046 |
https://mathoverflow.net/questions/303600 | 17 | $\newcommand{\End}{\operatorname{End}}$
$\newcommand{\GL}{\operatorname{GL}}$
$\newcommand{\Cof}{\operatorname{cof}}$
Let $V$ be a $d$-dimensional **real** vector space. ($d \ge 4$). Fix an **odd integer** $2 \le k \le d-2$. Define
$H\_{>k}=\{ A \in \End(V) \mid \operatorname{rank}(A) > k
\}$. Consider the map
$$
\p... | https://mathoverflow.net/users/46290 | An explicit reconstruction of a matrix from its minors | Assume for simplicity that $kl=d+1$.
Then I claim that $V \otimes \det V$ appears as a summand of $\left( \bigwedge^k V\right)^{\otimes l}$ with multiplicity $l-1$.
The maps $\left( \bigwedge^k V\right)^{\otimes l} \to V \otimes \det V$ are easier to write down - the $i$'th maps sends $$ (v\_{1,1} \wedge \dots \wed... | 7 | https://mathoverflow.net/users/18060 | 304051 | 133,047 |
https://mathoverflow.net/questions/304039 | 3 | See [here](https://mathoverflow.net/questions/303581/is-the-category-2-vect-monoidal-closed) for the notation.
I'm trying to do this by myself but everytime I approach the problem I end up buried under computations.
>
> What is a pair of adjoint 1-cells in the 2-category of vector spaces, where
>
>
> * 0-cells ... | https://mathoverflow.net/users/7952 | Adjoints in the 2-category of 2-vector spaces | As in the previous question, I continue to suggest that everything is much cleaner if you think in terms of bimodules. So we'll ask the more general question: in the 2-category $\text{Bim}(k)$ of $k$-algebras, $k$-bimodules, and bimodule homomorphisms, what is an adjoint pair?
The answer is worked out in [this blog ... | 9 | https://mathoverflow.net/users/290 | 304058 | 133,051 |
https://mathoverflow.net/questions/287879 | 15 | There is a known proof of the identity $$\sum\_{k=0}^\infty \frac1{(2k+1)^2}=\frac{\pi^2}8\,\,\,(\*)$$ (equivalent to $\sum \frac1{n^2}=\frac{\pi^2}6$) by expanding $\arcsin x$ as a power series $$\arcsin x=\sum\_{k=0}^\infty \frac{(2k-1)!!}{(2k)!!(2k+1)}x^{2k+1},$$
substituting $x=\sin t$ and integrating for $t$ from ... | https://mathoverflow.net/users/4312 | Higher arcsin wanted | In $[1]$ Borwein and Chamberland show that for $|x|\le 2$
\begin{align\*}
\arcsin^{2N+1}(x/2) &= (2N+1)!\sum\_{k=0}^\infty \frac{G\_N(k){2k\choose k}}{2(2k+1)4^{2k}} x^{2k+1},
& N = 0,1,\ldots \tag{1} \\
\arcsin^{2N}(x/2) &= (2N)!\sum\_{k=1}^\infty \frac{H\_N(k)}{{2k\choose k}k^2} x^{2k}
& N = 1,2,\ldots \tag{2}
\end... | 11 | https://mathoverflow.net/users/22085 | 304061 | 133,052 |
https://mathoverflow.net/questions/304059 | 10 | Suppose that the smooth manifold $ M $ has the n-sphere for its universal cover (in the topological sense). Does there exist a Riemannian metric on $ M $ (not necessarily compatible with the covering map) for which all sectional curvatures of $ M $ are constant, equal to 1?
| https://mathoverflow.net/users/48208 | Must a manifold covered by $ S^n $ admit a metric of constant positive sectional curvature? | Hitchin showed ([here](https://www.sciencedirect.com/science/article/pii/0001870874900218), I think) that there is an exotic sphere which admits no metric of positive scalar curvature. This manifold certainly has the sphere as its topological universal cover, but its sectional curvatures can't all be positive let alone... | 14 | https://mathoverflow.net/users/4362 | 304062 | 133,053 |
https://mathoverflow.net/questions/304026 | 1 | Is a von Neumann algebra always separable in the $\sigma$-weak topology? If not, give a counterexample. Under what conditions will it be separable?
| https://mathoverflow.net/users/125816 | About separability of von Neumann algebras | You can find, in any decent textbook on von Neumann algebras, a proof that if $A \subseteq B(\mathcal{H})$ is a von Neumann algebra, and the Hilbert space $\mathcal{H}$ is separable, then the unit ball of the von Neumann algebra is separably metrizable in the $\sigma$-weak topology. This implies that $A$ is $\sigma$-we... | 4 | https://mathoverflow.net/users/61785 | 304068 | 133,056 |
https://mathoverflow.net/questions/304023 | 7 | The question is a special case of a [previous question](https://mathoverflow.net/questions/303964/is-c-inftym-a-projective-frechet-c-inftyn-module-for-a-smooth-map).
Let $M$ be a compact smooth manifold, then it is clear that $C^{\infty}(M)$ is a Frechet algebra with pointwise multiplication and a collection of semi-... | https://mathoverflow.net/users/24965 | Is $C^{\infty}(E)$ a projective Frechet $C^{\infty}(M)$-module for a $C^{\infty}$-fiber bundle $E\to M$ with compact fiber? | The answer is Yes. Basically, one can realize $C^\infty(E)$ as a direct summand in a larger projective $C^\infty(M)$-module. Perhaps this really is trivial to experts, but to me the argument occurred only after staring sufficiently long at related arguments in this article, which came up in your other question:
>
>... | 9 | https://mathoverflow.net/users/2622 | 304075 | 133,060 |
https://mathoverflow.net/questions/304093 | 0 | Is there a non-semistable bundle of rank $3$ and degree $1$ which is an extension of a stable bundle of rank $2$ degree $1$ by a stable bundle of rank $1$ degree $0$?
| https://mathoverflow.net/users/nan | semistability of an extension of bundles | On an elliptic curve $C$, take $$V=E\_2(p) \oplus \mathcal{O}\_C,$$ where $p \in C$ is a point and $E\_2(p)$ is the unique non-split extension $$0 \to \mathcal{O}\_C \to E\_2(p) \to \mathcal{O}\_C(p) \to 0. $$
The rank $2$ subbundle $W=E\_2(p)$ strictly destabilizes $V$, because $$\mu(W)=\frac{1}{2} > \frac{1}{3} = \... | 3 | https://mathoverflow.net/users/7460 | 304094 | 133,066 |
https://mathoverflow.net/questions/304021 | 2 | For any set $X$ set $[X]^2 = \{\{x,y\}: x,y\in X, x\neq y\}$. If $n\geq 2$ is an integer, we endow $\mathbb{Z}$ with a graph structure in the following way. If $x,y\in \mathbb{Z}^n$ we say $x,y$ are *direct neighbors* if there is $k\in\{1,\ldots,n\}$ such that $|x\_k-y\_k|= 1$, and $x\_i = y\_i$ for all $i\in \{1,\ldot... | https://mathoverflow.net/users/8628 | $\omega$-Hamilton paths in $\mathbb{Z}^n$ | In "*E. Vazonyi*, Über Gitterpunkte des mehrdimensionalen Raumes, Acta Litt. Sci. Szeged 9, 163-173 (1939).", it is shown that there is such a path for every $n$ (as well as a Hamilton double ray, i.e. a Hamilton path that is infinite on both sides).
Since the publication is in German, I include a short proof sketch... | 4 | https://mathoverflow.net/users/97426 | 304099 | 133,068 |
https://mathoverflow.net/questions/304101 | 4 | Let $P,Q$ be two distributions on a finite set $X$. Consider the following metric\*
$$ d(P,Q) = \frac12\max\_{\emptyset\neq A\subseteq X} \|P(\cdot\mid A)-Q(\cdot\mid A)\|\_1. $$
Obviously, the total variation metric $\frac12\|P-Q\|\_1$ is majorized by $d(P,Q)$.
Question: has anyone encountered $d(P,Q)$ in the lite... | https://mathoverflow.net/users/12518 | A metric stronger than total variation | I think it is just called (scaled) **supremum $L\_1$ norm** and it is mostly studied in Bayesian nonparametric estimation literature, especially posterior consistency. The following paper investigate conditional density yet it is obvious that corresponding probability measure is exactly what you wrote down.
>
> De ... | 1 | https://mathoverflow.net/users/25437 | 304107 | 133,071 |
https://mathoverflow.net/questions/304104 | 7 | I've been trying to learn some of the basic language of infinity-category theory (in the sense of Lurie), and in particular, to understand which basic statements in (1-)categories have analogues in the infinity-categorical setting. Specifically, I'm trying to understand how equivalences behave in infinity categories, m... | https://mathoverflow.net/users/126183 | Uniqueness of quasi-inverses in infinity categories | A possibly simpler way of proving what you are after is using *marked simplicial set*.
Recall that marked simplicial sets are pairs $(X,S)$ where $X$ is a simplicial set and $S\subseteq X\_1$ is a set of 1-simplices of $X$ containing all degenerate 1-simplices. If $X$ is a simplicial set we will denote the minimal an... | 4 | https://mathoverflow.net/users/43054 | 304108 | 133,072 |
https://mathoverflow.net/questions/304084 | 3 | I'm reading through [this paper by Pouchin](https://arxiv.org/pdf/0801.4290.pdf), and in it he makes a clam that some function space is isomorphic to the Hecke algebra. I'm trying to understand this and could really use some help.
Let $G = GL(n,\mathbb{F}\_q)$ and let $B$ be the subgroup of upper triangular matrices.... | https://mathoverflow.net/users/126175 | Understanding the Hecke Algebra via Different Constructions | $G$-invariant functions $ X \times X$ to $\mathbb C$ are the same thing as functions $G \times G$ to $\mathbb C$ invariant under the right action of $B \times B$ and the left, diagonal action of $G$.
But functions invariant under the left diagonal action of $G$ have the form $f(g, h) = f' ( g^{-1} h)$ for a unique f... | 2 | https://mathoverflow.net/users/18060 | 304109 | 133,073 |
https://mathoverflow.net/questions/303222 | 7 | Let $G$ be a complex reductive algebraic group and $X$ be a smooth compact complex curve. It's easy to see that the space of vacua in B-twisted $N=4$ SUSY Yang--Mills theory is $\mathfrak{h}^\*[2]/W$ (where $\mathfrak{h}$ is the Cartan subalgebra of $\mathfrak{g}$ and $W$ is the Weyl group).
There is a theorem due t... | https://mathoverflow.net/users/nan | Implications of gauge symmetry breaking on the spectral side of geometric Langlands? | We discussed some conjectural implications in Section 4.2 of the paper. I wouldn't say that the category of sheaves with nilpotent singular support was necessarily the "right" category to consider from a gauge theoretic point of view. Instead I'd say that if one considers the equivalence for the whole category of coher... | 6 | https://mathoverflow.net/users/126190 | 304111 | 133,075 |
https://mathoverflow.net/questions/303902 | 6 | In Joel David Hamkins's "Well-founded Boolean Ultrapowers as Large Cardinal Embeddings", it is mentioned that if $U \in \mathbf{V}$ is an ultrafilter of a complete Boolean algebra $\mathbb{B}$ and $U$ is non-generic over $\mathbf{V}$, then either $U$ misses a countable maximal antichain of $\mathbb{B}$ in $\mathbf{V}$,... | https://mathoverflow.net/users/29231 | Boolean ultrapower of V[G] by G | I share your view that this is a subtle point. To illustrate it, my co-author Dan Seabold and I had pointed to the case of adding a Cohen subset to $\omega\_1$ (see example 44 in [Boolean ultrapowers](http://jdh.hamkins.org/boolean-ultrapowers/) paper, the paper you mention). If you use the natural tree order $2^{<\ome... | 11 | https://mathoverflow.net/users/1946 | 304115 | 133,077 |
https://mathoverflow.net/questions/304113 | 2 | A colleague in spatial statistics was looking at a map with about 600 regions. For the application she's considering, the induced adjacency matrix had some undesirable properties (where two regions are neighbors if they share a border). Instead, she calculated the row-standardized adjacency matrix and somewhat surprisi... | https://mathoverflow.net/users/125275 | When does a row standardized adjacency matrix have a real spectrum? | If the adjacency matrix is $A,$ the "row-standardized" matrix is $DA$, where $D$ is a diagonal matrix all of whose diagonal entries are positive, so has a positive diagonal square root $D^{1/2}$. Now,
$$DA = D^{1/2} D^{1/2} A D^{1/2} D^{-1/2},$$ so your matrix is similar to
$$D^{1/2} A D^{1/2},$$ which is a symme... | 10 | https://mathoverflow.net/users/11142 | 304116 | 133,078 |
https://mathoverflow.net/questions/304129 | 0 | I wonder if there is any literature on the following problem
$$\begin{array}{ll} \underset{X \in \mathbb R^{m\times n}}{\text{minimize}} & \displaystyle\sum\_{i,j} C\_{i,j} X\_{i,j}\\ \text{subject to} & \displaystyle\sum\_{i} X\_{i,j} = \displaystyle\sum\_{j} X\_{i,j} = 1\\ & X\_{i,j} \geq 0\end{array}$$
The close... | https://mathoverflow.net/users/34972 | Relax a rectangular linear assignment problem | Your constraints are incorrect if $m \ne n$. Presuming $m < n$, then the constraints would be
$ \sum\_{i}X\_{i,j} \le 1, \sum\_{j}X\_{i,j}=1, X\_{i,j}\geq 0$, with the obvious modification for $m > n$.
This can be converted into a Square Assignment Problem by adding dummy "i" indices with zero costs (or dummy "j" in... | 0 | https://mathoverflow.net/users/75420 | 304133 | 133,083 |
https://mathoverflow.net/questions/303973 | 9 | Motivated by [this question](https://mathoverflow.net/questions/302468/local-ring-all-whose-non-maximal-ideals-are-finitely-generated), I would like to ask:
>
> If all non-prime ideals in a ring are finitely generated, then is the ring Noetherian? Can we at least say anything in the local case?
>
>
>
Note tha... | https://mathoverflow.net/users/nan | Rings with all non-prime ideals finitely generated | No.
If $\mathbb Z\_{p^{\infty}}$ is a Prufer group for prime $p$, then its endomorphism ring is isomorphic to the ring $\mathbb Z\_p$ of $p$-adic integers. Hence $\mathbb Z\_{p^{\infty}}$ is a $\mathbb Z\_p$-module, and we can form the idealization $R:=\mathbb Z\_p\oplus \mathbb Z\_{p^{\infty}}$. The ideals of this r... | 8 | https://mathoverflow.net/users/75735 | 304149 | 133,091 |
https://mathoverflow.net/questions/304148 | 0 | I apologise if this is trivial or well known to be impossible:
Can one find a finite set of integers $2\leq a\_1<a\_2<\ldots<a\_m<\infty$
such that for the function defined as
$$
f\_{a\_1,\ldots,a\_m}(x)=\max\left\{\left|\frac{\sin (a\_1 x)}{\sin x}\right| ,\left|\frac{\sin (a\_2 x)}{\sin x}\right| , \cdots,\left|\fr... | https://mathoverflow.net/users/17773 | The minimum of the maximum of a sequence of sinc functions | It looks like this is impossible. Correct me if I am wrong.
For example, $f\_{a\_1,\cdots,a\_m}(\pi/2)\leq 1$, so the minimum cannot exceed 1.
| 1 | https://mathoverflow.net/users/104791 | 304150 | 133,092 |
https://mathoverflow.net/questions/304122 | 3 | Suppose that we have a map between two pushout diagrams of topological spaces
$$ [A\_{1}\leftarrow A\_{0}\rightarrow A\_{2} ] \rightarrow [B\_{1}\leftarrow B\_{0}\rightarrow B\_{2} ]$$
such that for any $i\in\{0,1,2\}$, $A\_{i}\rightarrow B\_{i}$ is a trivial cofibration of topological spaces. And suppose that all spa... | https://mathoverflow.net/users/103287 | Pushout of spaces | Take $B\_0$ to be an annulus. Let $B\_1$ be the space obtained by collapsing the left half of the inner circle to a point, and let $B\_2$ be obtained by collapsing the right half of the inner circle to a point. Let $A\_0$, $A\_1$ and $A\_2$ all be the outer circle of $B\_0$. Then the $A$-pushout is a circle whereas the... | 6 | https://mathoverflow.net/users/10366 | 304157 | 133,094 |
https://mathoverflow.net/questions/304152 | 4 | We say that a topological space $(X,\tau)$ is *flexible*, if for every closed discrete subset $D\subseteq X$ and every map $f: D\to X$ there is a continous map $f^X:X\to X$ such that $f^X|\_D = f$.
$\mathbb{R}$ with the Euclidean topology is flexible.
Is every [homogeneous](https://en.wikipedia.org/wiki/Homogeneous... | https://mathoverflow.net/users/8628 | Are homogeneous $T_2$-spaces flexible? | I like this question a lot.
The answer is no. Let $X$ consist of two disjoint copies of the real line. This is $T\_2$ and homogeneous, in the sense that for any two points, there is a homeomorphism taking the first to the second — all points look alike.
But it is not flexible, since if $D$ consists of two points $... | 6 | https://mathoverflow.net/users/1946 | 304158 | 133,095 |
https://mathoverflow.net/questions/304156 | 4 | **Definition.** $h(n\_1,n\_2)$ is the least number $m$ such that, if the edges of $K\_m$ are colored with two colors, $1$ and $2,$ then for some color $i\in\{1,2\}$ there is a set $W\subseteq V(K\_m)$ such that $|W|=n\_i$ and every triangle in $W$ has an odd number of edges of color $i;$ **in other words,** for some $i... | https://mathoverflow.net/users/43266 | Another funny kind of Ramsey number | The statement $h(4,5)=8$ is true. It can be checked by enumerating all 8-vertex graphs (a total of 12346) and check them one by one.
For those who want to double-check, there are 48 graphs in which there is exactly one instance of $W$, and 43 in which there are two.
| 4 | https://mathoverflow.net/users/125498 | 304168 | 133,097 |
https://mathoverflow.net/questions/303839 | 17 |
>
> I am looking for a comment, reference, remark, or proof of three conjectures as follows:
>
>
> **Conjecture 1:** *Let $x$ be an odd positive integer. Then there exist two integers $n, m \ge 2$ so that $$x=P\_{n+m}-P\_n-P\_m,$$ where $P\_n$ is the $n^{th}$ prime.*
>
>
>
My calculations the conjecture 1 true... | https://mathoverflow.net/users/122662 | Is every odd positive integer of the form $P_{n+m}-P_n-P_m$? | The **conjecture 3** is mentioned [here](https://mathoverflow.net/questions/111196/name-of-a-conjecture-on-difference-of-prime-numbers) and [here](https://arxiv.org/pdf/1206.0149.pdf), in which it is called the Maillet conjecture.
The **conjecture 2** is true.
It can be seen by rewriting $n=P\_c−P\_a−P\_b$ as $P\_... | 16 | https://mathoverflow.net/users/125498 | 304173 | 133,098 |
https://mathoverflow.net/questions/303704 | 4 | I want to prove that function
\begin{equation}
f(x)=\frac{1}{x+1} \int\limits\_0^x \log \left(1+\frac{1}{x+1+t} \right)~dt
\end{equation}
is quasi-concave. One approach is to obtain the closed form of the integral (provided below) and then prove that the result is quasi-concave. I tried this but it seems to be diffi... | https://mathoverflow.net/users/123067 | Quasi-concavity of $f(x)=\frac{1}{x+1} \int_0^x \log \left(1+\frac{1}{x+1+t} \right)~dt$ | For $x>-1/2$, we have
$$
f(x)=\ln 4-\frac{x+2}{x+1}\, \ln \frac{x+2}{x+1}
-\frac{2 x+1}{x+1} \ln\frac{2 x+1}{x+1}
$$
and hence
$$
f'(x)=\frac1{(x+1)^2}\ln \left(1+\frac{1-x}{2 x+1}\right),
$$
which is $>0$ for $x\in(-1/2,1)$ and $<0$ for $x>1$. So, $f$ increases on $(-1/2,1]$ and decreases on $[1,\infty)$. So, $f$ is... | 1 | https://mathoverflow.net/users/36721 | 304175 | 133,099 |
https://mathoverflow.net/questions/304180 | 2 | I'm searching for a good reference that prove the descending chain propriety for compact Lie groups (i.e. every sequence $K\_1\supset K\_2\supset...$ of closed subgroups $K\_i$ of $G$ is eventually constant).
Thank you!
| https://mathoverflow.net/users/123935 | Descending chain property for compact Lie goups | Am I missing something, or does it follow from the fact that a closed subgroup is either a union of connected components or of lower dimension, and by compactness the number of connected components is finite, while by Lieness, the dimension is finite?
| 3 | https://mathoverflow.net/users/11142 | 304181 | 133,100 |
https://mathoverflow.net/questions/304186 | 5 | It is well-known that an essential closed curve on a hyperbolic surface (possibly with boundary) is homotopic to a unique closed geodesic. Moreover, if the curve under consideration is simple, then so is the geodesic homotopic to it. A reference is "A primer on mapping class groups" of Farb and Margalit (propositions 1... | https://mathoverflow.net/users/nan | How many simple closed geodesics in a given primitive homology class? | The thrice punctured sphere has no simple closed geodesics. The four-times punctured sphere has a unique simple geodesic in each homology class. In general, it is a result of I. Rivin that the number of simple closed geodesics of length bounded above by $L$ grows like $L^{6g - 6 + 2 c},$ where $c$ is the number of punc... | 6 | https://mathoverflow.net/users/11142 | 304190 | 133,104 |
https://mathoverflow.net/questions/304064 | 4 | I have already asked that Question on Cross Validated:
[Link](https://stats.stackexchange.com/questions/353956/how-to-find-the-optimal-convergence-rate?)
Suppose there is some data $X\_{1},X\_{2},\ldots,X\_{n}$. We further suppose that there is some parameter $\theta$, for which we want to do statistical inference.
A... | https://mathoverflow.net/users/110069 | How to find the optimal convergence rate? | $\newcommand{\al}{\alpha}
\newcommand{\be}{\beta}
\newcommand{\de}{\delta}
\newcommand{\De}{\Delta}
\newcommand{\ep}{\varepsilon}
\newcommand{\ga}{\gamma}
\newcommand{\Ga}{\Gamma}
\newcommand{\la}{\lambda}
\newcommand{\si}{\sigma}
\newcommand{\Si}{\Sigma}
\newcommand{\thh}{\theta}
\newcommand{\om}{\omega}
\newcommand{\... | 3 | https://mathoverflow.net/users/36721 | 304193 | 133,107 |
https://mathoverflow.net/questions/304178 | 5 | For a fixed positive integer $n$, the Diophantine equation
$$x^2 + y^2 + z^2 = n$$
was studied by Gauss in Disquisitiones Arithmeticae. As is known, this equation is intimately connected to the quaternion algebra $B\_{-1,-1}$, characterized by the relations
$$
i^2 = -1, j^2 = -1, k^2 = (ij)^2 = -1.
$$
More prec... | https://mathoverflow.net/users/22733 | Solutions to the Diophantine equation $x^2+3y^2+3z^2=n$ | One of the beautiful (and sometimes flummoxing) aspects of the theory of ternary quadratic forms is that you can find answers and questions coming from many different points of view! Using modular forms will give you an answer along the lines that Henri Cohen suggests. I'd like to put in a pitch for a quaternionic appr... | 7 | https://mathoverflow.net/users/4433 | 304196 | 133,109 |
https://mathoverflow.net/questions/304184 | 1 | For the vector space $M\_{n,n}(\mathbb{C})$ of $n\times n$ matrices we know that the subset
$$M\_{2r}:= \{A\in M\_{n,n}(\mathbb{C}) \mid \mbox{rank} (A) = 2r \}$$
is a manifold of dimension $2n(2r)-(2r)^2$ (see [this question](https://math.stackexchange.com/questions/518202/what-is-the-codimension-of-matrices-of-r... | https://mathoverflow.net/users/126227 | Dimension (manifold) of matrices with exact $r$ positive and $r$ negative eigenvalues | A Hermitian matrix of rank $r$ can be represented uniquely as $U D U^\ast,$ where $U$ is an $n\times r$ matrix with orthogonal rows of unit length, and $D$ is an $r\times r$ matrix (this is the singular value decomposition, if you want to think of it that way). The (complex) dimension of the space of $U$ is $n-1 + n-2 ... | 5 | https://mathoverflow.net/users/11142 | 304197 | 133,110 |
https://mathoverflow.net/questions/303781 | 1 | I am reading [Orbifolds as stacks?](https://arxiv.org/abs/0806.4160)
Given Lie groupoids $\mathcal{G}$ and $\mathcal{H}$ there is a notion of what is called a bibundle from $\mathcal{G}$ to $\mathcal{H}$ which is supposed to be a ageneralized notion of a morphism of Lie groupoids.
>
> (rough) Definition : A bibun... | https://mathoverflow.net/users/118688 | Composition of bibundles | I will give an example as to why composition of bibundles cannot simply be done as pullback, as well as the relation to perhaps more familiar geometric constructions.
Firstly, note that if $M$ is a manifold and $\mathbf{B}G$ is a one-object Lie groupoid (where the automorphisms of the one object are the Lie group $G$... | 1 | https://mathoverflow.net/users/4177 | 304203 | 133,113 |
https://mathoverflow.net/questions/304195 | 5 | Let $E = C\_c^0(\mathbb{R}^n;\mathbb{R}^m)$ be the space of compactly supported continuous functions on $\mathbb{R}^n$ with values on $\mathbb{R}^m$. There is a natural norm on this space: given $\varphi \in E$, we put
$$ \Vert \varphi \Vert = \sup\_{x \in \mathbb{R}^m} \Vert \varphi(x) \Vert.$$
For each compact set... | https://mathoverflow.net/users/85934 | Continuity and sequential continuity of a linear functional | The space $E:=C\_c^0(\mathbb{R^n})$ is the inductive limit in the TVS category of the spaces $(E\_K,\|\cdot\|\_{K,\infty})$. As a consequence, a sequence converges on $E$ if and only if it is included in a space $E\_K$ and converges there; a linear form on $E$ is continuous if and only if it is continuous on every $E\_... | 1 | https://mathoverflow.net/users/6101 | 304222 | 133,120 |
https://mathoverflow.net/questions/304182 | 3 | Let $F/\mathbb Q$ be a finite normal extension of the rational numbers. Let $V$ be an $F$-vector space of countably infinite dimension, and set $L=GL\_F(V)$. Put moreover $L^\*$ be the set of all mappings fixing elementwise at least one subspace of finite codimension in $V$.
1) Is $U=L' L^\*/L^\*$ simple modulo its c... | https://mathoverflow.net/users/66046 | On the general linear group of a vector space of infinite dimension | Everything can be answered relying on the paper *Rosenberg, Alex
The structure of the infinite general linear group.
Ann. of Math. (2) 68 1958 278-294* ([MR link](https://mathscinet.ams.org/mathscinet-getitem?mr=102528), [JSTOR](https://www.jstor.org/stable/1970248); sorry these are under paywalls.)
Namely, $L$ is p... | 5 | https://mathoverflow.net/users/14094 | 304226 | 133,122 |
https://mathoverflow.net/questions/304223 | 4 | Let $G$ be a group generated by $a,b$ (for the sake of simplicity). Consider the element
$$S=a+b+a^{-1}+b^{-1}\in{\mathbb C}[G],$$
which may also be interpreted as an operator in $l^2(G)$ (by left regular representation). Obviously $\|S\|\le 4$, and by the old result of Kesten
$\|S\|=4$ iff $G$ is amenable.
Now, sup... | https://mathoverflow.net/users/9833 | A question about spectral properties of a non-amenable group | Linnell proved in his paper "Zero divisors and $\ell^2(G)$ C. R. Acad. Sci. Paris Sér. I Math. 315 (1992), no. 1, 49-53." that all elements of the group ring of a right orderable groups, containing Abelian and free groups, are nonzero divisors, i.e. if $0\neq\alpha\in\mathbb C[G]$, then for all $0\neq\beta\in\ell^2(G)$... | 5 | https://mathoverflow.net/users/84700 | 304227 | 133,123 |
https://mathoverflow.net/questions/304234 | 14 | The Latex command `\ldots` is often used to denote "and so forth". For instance,
$$
\pi \approx 3.1415\ldots
$$
When a sentence ends with an `\ldots`, one is faced with the conundrum of how to properly terminate it. The "cleanest" way would of course be to add a terminating "period", thus signaling the reader that ... | https://mathoverflow.net/users/103722 | In a publication, should an `\ldots` always be followed by a period? | I believe papers in English should be written in full and correct English sentences, with any mathematical formula being part of a sentence. Therefore, I believe that the correct form in this case would be
>
> we have $\pi=3.1415\ldots\;$. However,
>
>
>
-- not
>
> we have $\pi=3.1415\ldots$ However,
>
... | 18 | https://mathoverflow.net/users/36721 | 304235 | 133,124 |
https://mathoverflow.net/questions/304245 | 3 | For any set $X$ set $[X]^2 = \{\{x,y\}: x,y\in X, x\neq y\}$. If $n\geq 2$ is an integer, we endow $\mathbb{Z}$ with a graph structure in the following way. If $x,y\in \mathbb{Z}^n$ we say $x,y$ are *direct neighbors* if there is $k\in\{1,\ldots,n\}$ such that $|x\_k-y\_k|= 1$, and $x\_i = y\_i$ for all $i\in \{1,\ldot... | https://mathoverflow.net/users/8628 | Orientability of $\mathbb{Z}^n$ | Color the vertices of $\mathbb{Z}^n$ by black and white according to their parity. For all but one "direction" (by direction I mean the direction of $x-y$), orient the edges from white to black. Orient the edges of the remaining direction from black to white. Then, for any $x∈\mathbb{Z}^n$ with neighbour $y$, there is ... | 9 | https://mathoverflow.net/users/125498 | 304249 | 133,128 |
https://mathoverflow.net/questions/304240 | 4 | Let $[a,b]\subset[-1,1]$ and $f\in C^{\infty}([a,b],\mathbb{R})$ a function which satisfies for all $n\in\mathbb{N}$ that $\sup\_{x\in[a,b]}|f^{(n)}(x)|\leq n!$.
Does there exist a function $g\in C^{\infty}([-1,1],\mathbb{R})$ which satisfies $g\big|\_{[a,b]}=f$ and for all $n\in\mathbb{N}$ that $\sup\_{x\in[-1,1]}|g... | https://mathoverflow.net/users/124237 | Extension of a certain type of (very) smooth functions to a larger interval | No, this won't work in general. These assumptions make $f,g$ real analytic (by Taylor's theorem), so the only $g$ that could possibly work would be the holomorphic continuation
$$
g(x) = \sum\_{n=0} a\_n x^n, \quad a\_n = \frac{f^{(n)}(0)}{n!} .
$$
However, this need not satisfy your bounds on all of $[-1,1]$ if all we... | 3 | https://mathoverflow.net/users/48839 | 304252 | 133,130 |
https://mathoverflow.net/questions/304239 | 0 | Let $(G, <)$ be a totally ordered group, and let $<$ be left-invariant. Let $G$ act (freely?) on a partially ordered set $(S, <)$, such that this group action preserves the ordering:
$$ s\_1 < s\_2 \Rightarrow gs\_1 < gs\_2;$$
$$ g\_1 < g\_2 \Rightarrow g\_1s < g\_2s$$
>
> Can the partial ordering $(S, <)$ be ext... | https://mathoverflow.net/users/119181 | Ordered group acting freely on partially ordered set | Yes, such a partial order can be extended to a total order: we can amend the proof of the Szpilrajn extension theorem which essentially establishes the result in the case $G=1$.
Starting with the given partial order $\leq\_S$ on $S$, consider partial orders $\leq$ on $S$ satisfying
>
> (1) $s\leq\_S t$ implies $s... | 4 | https://mathoverflow.net/users/22599 | 304256 | 133,132 |
https://mathoverflow.net/questions/304242 | 1 | Let
* $(M,d)$ be a separable metric space
* $E$ be a $\mathbb R$-Banach space
* $\alpha\in(0,1]$
Moreover, let $$\left\|f\right\|\_{C^{0+\alpha}(K,\:E)}:=\sup\_{x\in K}\left\|f(x)\right\|\_E+\sup\_{\substack{x,\:y\:\in\:K\\x\:\ne\:y}}\frac{\left\|f(x)-f(y)\right\|\_E}{{d(x,y)}^\alpha}\;\;\;\text{for }f:M\to E$$ for... | https://mathoverflow.net/users/91890 | Borel $\sigma$-algebra on the space of Hölder continuous functions | It's not true.
For brevity, let me take $M=[0,1]$ (or $(0,1)$ if you prefer), $E = \mathbb{R}$, $X = C^{0+\alpha}(M,E)$, $\mathcal{B}$ its $\sigma$-algebra, and $\mathcal{F}$ the $\sigma$-algebra appearing on the right side of your desired equation. I claim that $|\mathcal{F}| = \mathfrak{c}$ while $|\mathcal{B}| \ge... | 1 | https://mathoverflow.net/users/4832 | 304260 | 133,134 |
https://mathoverflow.net/questions/304262 | 1 | Let $M$ be a positive integer, and let $X\_1, X\_2, \dots$ be i.i.d. r.v.'s having uniform distribution in the set $\{1, \dots, M\}$. For given $N$, let $Z\_N$ denote the number of *distinct* elements of the set $\{X\_1, \dots, X\_N\}$.
Question: What is the distribution of $Z\_N$?
(In the end, I want to estimate ... | https://mathoverflow.net/users/46852 | What is this distribution? | [Birthday problem:](http://www.randomservices.org/random/urn/Birthday.html)
The probability density function of the number of distinct values $Z$ for population size $M$ and sample size $N$ is given by
$$P(Z = j) = \binom{M}{j} \sum\_{k=0}^j (-1)^k \binom{j}{k} \left(\frac{j - k}{M}\right)^N, \quad j \in \{1, 2, \ldo... | 3 | https://mathoverflow.net/users/11260 | 304264 | 133,136 |
https://mathoverflow.net/questions/304236 | 11 | **Question 1.** Suppose that $0^{\#}$ exists. Are there set generic filters $g,h$ over $L$ ($g,h \in V$) such that for any $f$ ($\in V$) which is generic over $L$ we have that $L[g] \cup L[h] \not \subseteq L[f]$?
In other words: Does $0^{\#}$ imply the failure of the upward directedness of the set generic universe o... | https://mathoverflow.net/users/57114 | Does $0^{\#}$ imply the failure of upward directedness in the set generic universe over $L$? | Note that we can use $g,h$ to code any real of $V$ in a recursive fashion (recursive in the Cohen reals added by $g$ and $h$). In particular we can use them to code $0^{\#}$ -- a real that cannot be added to $L$ via set forcing. This shows that $0^{\#}$ does indeed imply the failure of upward directedness of the set ge... | 6 | https://mathoverflow.net/users/57114 | 304265 | 133,137 |
https://mathoverflow.net/questions/304165 | 2 | I know that a necessary and sufficient condition for the positivity of a quartic polynomial of many variables is in general difficult. I have a somewhat special case, maybe here more can be said. Let $x\in {\mathbb R}^{n \times m}$ be a real $n\times m$ matrix, $n<m$, and $i=1,\ldots,n$ and $a=1,\ldots,m$. I'm interest... | https://mathoverflow.net/users/41312 | Non-negativity condition for special quartic | I think $W:=W\_{abcd}$ is not well-defined. Indeed, in the case $n=1$ if you want to talk about $W$ as a quadratic form on $m\times m$ matrices, you will have to write $Q(x)=Tr(YY^\top W)=Y^\top WY$, where $Y$ is the vector of all quadratic monomials in $x\_{1a}$, $1\leq a\leq m$.
But then in general $W$ is not uniqu... | 1 | https://mathoverflow.net/users/11100 | 304272 | 133,140 |
https://mathoverflow.net/questions/304274 | 5 | For a connected, finite graph $G$, let $\lambda\_1, \ldots, \lambda\_n$ denote the nonzero eigenvalues of the graph Laplacian. We define $\zeta\_G = \Sigma\_{i = 1}^n \lambda\_i^s$.
Then Kirkoffs Matrix-Tree theorem can be reformulated as saying that $e^{ - \zeta\_G'(0)} / |G| = \tau(G)$, where $\tau$ is the number o... | https://mathoverflow.net/users/41873 | What is $e^{- \zeta_{\Delta} '(0)}$ for a $\Delta$ the Laplacian of a manifold? | If I am not mistaken, this cannot be true in general. For example, one may scale the metric for $\mathbb{S}^{1}$ proportionally and the determinant of the Laplacian remains the same independent of the length of the circle. For dimension 2, the determinant of the Laplacian obtains its maximum when $M$ has [constant sect... | 3 | https://mathoverflow.net/users/18850 | 304286 | 133,146 |
https://mathoverflow.net/questions/304285 | 4 | We consider a simplex $A\_0A\_1...A\_n$ in $n$-dimensional Euclidean $\Bbb E^n.$
Denote by
$Vol(A\_0A\_2A\_3...A\_n)=V\_1,$
$Vol(A\_0A\_1A\_3...A\_n)=V\_2,$
$...$
$Vol(A\_0A\_2A\_3...A\_{n-1})=V\_n$
and
$Vol(A\_1A\_2A\_3...A\_n)=V.$
Assume that
$$V\_1^2+V\_2^2+...+V\_n^2=V^2.$$
>
> **My questio... | https://mathoverflow.net/users/126268 | Converse of Pythagorean theorem in n-dimensional Euclidean space | This is false even in three dimensions as can be shown by a simple computation. The philosophical reason is that you are assuming one condition (vanishing of a scalar function on the space of tetrahedra) and hoping to deduce several. There are several converses of pythagoras in three space but they reqire bundling your... | 8 | https://mathoverflow.net/users/125971 | 304289 | 133,147 |
https://mathoverflow.net/questions/303950 | 2 | I hope this question belongs here. The situation in this question is quite particular and specific.
I am trying to weak some of theory to measure the degree of some function on the Jacobian of a genus $2$ curve which is a rational function when restricted to certain curves inside the Jacobian.
I am currently using... | https://mathoverflow.net/users/91023 | Some curves on the Jacobian of a genus $2$ curve and their image under certain maps (char $p$) | Here I just check that when $\Theta\_n$ is a curve in $J$, $\kappa\_4$ cannot be a constant different from $0$ or $\infty$ when restricted to $\Theta\_n$.
Let $H$ be a hyperelliptic curve of genus $2$ with a rational point $\infty$ defined over $\mathbb{F}\_q$ with $(x,y)$ its generic point and consider its Jacobian ... | 0 | https://mathoverflow.net/users/91023 | 304297 | 133,151 |
https://mathoverflow.net/questions/304299 | 10 | Let $\mathsf C\_n$ denotes the statement:
*for any family $\mathcal F$ of $n$-element sets there exists a choice function (i.e., a function $f:\mathcal F\to\bigcup\mathcal F$ such that $f(F)\in F$ for all $F\in\mathcal F$).*
It is known that $\mathsf C\_2\Rightarrow \mathsf C\_4$ in ZF.
This fact suggests intro... | https://mathoverflow.net/users/61536 | The partial preorder on $\mathbb N$ generated by the finite axioms of choice | In Jech's "The Axiom of Choice", at the end of Chapter 7, he formulates the following condition on two natural numbers $n>m$:
>
> (S) There is *no* decomposition of $n$ into $p\_1+\ldots+p\_s=n$ such that $p\_i>m$ is a prime number for all $i$.
>
>
>
And he goes on to prove that if $\mathsf{C}\_k$ holds for al... | 12 | https://mathoverflow.net/users/7206 | 304305 | 133,152 |
https://mathoverflow.net/questions/304302 | 1 | An anti- affine group $G$ is defined to be an algebraic group with no global sections. Examples include abelian varieties and non trivial extensions of abelian varieties by torus (in characteristic $\neq$ 0 these are the only ones). What are the examples of such groups in characteristic 0, is there an 'obvious way' of ... | https://mathoverflow.net/users/117654 | A non trivial example of an anti -affine algebraic group | Brion's paper [Anti-affine algebraic groups](https://arxiv.org/abs/0710.5211) gives the complete classification over fields whose closure is separable. The examples are not very easy to describe.
| 2 | https://mathoverflow.net/users/13268 | 304306 | 133,153 |
https://mathoverflow.net/questions/304304 | 11 | Let $S$ be a scheme, and $SH(S)$ the stable motivic category over $S$. Which objects of $SH(S)$ are dualizable with respect to the smash product?
All I can find on this question is an old abstract of Röndigs stating that for a $k$ a perfect field, every smooth projective scheme over $k$ is dualizable in $SH(Spec(k))$... | https://mathoverflow.net/users/2362 | Which motivic spectra are dualizable? | If $X$ is noetherian of dimension $>0$, there is always a compact object of $SH(X)$ which is not dualizable: e.g. $j\_\sharp$ of the sphere spectrum where $j:U\to X$ is any dense open immersion with non-empty complement (in each connected component). However, smooth and proper $X$-schemes always provide a source of dua... | 15 | https://mathoverflow.net/users/1017 | 304313 | 133,155 |
https://mathoverflow.net/questions/304336 | 5 | Compactness and sequential compactness are not equivalent in a general topological space.
For the weak topology on a normed space, they are:
**Theorem (Eberlein-Smulian)** Let $X$ be a normed space and let $A$ be a subset of $X$. Then $A$ is weakly compact if and only $A$ is weakly sequentially compact.
I want ... | https://mathoverflow.net/users/126292 | Inequivalence of Compactness and Sequential Compactness for the Weak* Topology? | Let $X=C[0,\omega\_1]$. Consider $A = \{\delta\_\alpha\colon \alpha < \omega\_1\}$ in $X^\*$. This is a homeomorphic copy of $\omega\_1$ which is a sequentially compact but not compact space. (Here $\delta\_x$ stands for the evaluation functional at $x$.)
As for your second question, take $X = C[0,\omega\_1]\oplus \e... | 4 | https://mathoverflow.net/users/15129 | 304337 | 133,161 |
https://mathoverflow.net/questions/304332 | 1 | We assume that all the lines in the plane are each colored with one of two colors: red or blue. Given angle $\alpha.$
>
> **My question 1.** Is there possible to get two lines with the same color and angle between them is angle $\alpha?$
>
>
> **My question 2.** Is there possible to get a regular $n$-polygon with... | https://mathoverflow.net/users/126268 | Coloring lines in plane | (1)
We have problems only when parallel lines are colored in the same color.
(If in at least one direction we have two colors, then make a choice of a line making the angle $\alpha$ with it...)
So we assume this "bad case" already happens.
We separate the cases $\alpha \not\in \Bbb Q\cdot \pi$ and
$\alpha\in \Bbb Q\... | 1 | https://mathoverflow.net/users/122945 | 304343 | 133,162 |
https://mathoverflow.net/questions/304054 | 3 | I am trying to prove the following $\mathrm{Conn}^{\mathrm{reg}}(X) = \mathrm{Conn}(X) \cap \mathrm{Mod}\_{rh}(\mathcal{D}\_X)$.
Here an integrable connection on a smooth algebraic variety $X$ is a $\mathcal{D}\_X$ module which is locally free over $\mathcal{O}\_X$ of finite rank. Regular integrable connections and reg... | https://mathoverflow.net/users/91572 | Regular integrable connections and regular holonomic modules | This is actually just Theorem 6.1.6 (Curve Testing Criterion) combined with Defintion 5.3.2 (the definition of regular integrable connection).
| 1 | https://mathoverflow.net/users/36720 | 304346 | 133,165 |
https://mathoverflow.net/questions/275210 | 2 | **َContact manifold**
>
> A $(2n + 1)$-dimensional manifold $M$ is said to be a contact manifold if it
> admits a global 1-form $\eta$ such that $\eta\wedge (d\eta)^n\neq 0$.
>
>
>
There is an equivalent definition by J. W. Gray:
>
> A manifold admit an almost contact structure if the structural group
> ... | https://mathoverflow.net/users/90655 | An equivalent definition for contact pair manifolds | As Jarek said, the two definitions that you give are definitely *NOT* equivalent: the definition by Gray is about *almost* contact structures which is a much weaker notion than a proper contact structure!
Maybe you would be interested in an *almost* contact pair like $TM$ splits into $\mathbb{R}\oplus E\_1\oplus \mat... | 2 | https://mathoverflow.net/users/67031 | 304362 | 133,177 |
https://mathoverflow.net/questions/304367 | 4 | We assume that all the circles in the plane are each colored with one of two colors: red or blue.
>
> **My question 1.** Does there always exist an equilateral triangle such that its circumcircle and its incircle have the same color?
>
>
> **My question 2.** Does there always exist an equilateral triangle such th... | https://mathoverflow.net/users/126268 | Coloring circles in plane | For Question 1, notice first that you can color the set of positive real numbers with two colors so that $x$ and $2x$ always have different colors: Color $x$ according to the parity of $\lfloor\log\_2x\rfloor$. Now assign to each circle the color that you just gave its radius. Since the circumcircle of an equilateral t... | 5 | https://mathoverflow.net/users/6794 | 304369 | 133,181 |
https://mathoverflow.net/questions/304381 | 1 | For any simple, undirected graph $G$, let $L(G)$ denote its [line graph](https://en.wikipedia.org/wiki/Line_graph).
$G=(\mathbb{Z}, E)$ with $E = \{\{k, k+1\}:k\in \mathbb{Z}\}$ has the property that $G\cong L(G)$.
Is there a connected infinite graph $G$ such that every vertex has more than $2$ neighbors, and $G\c... | https://mathoverflow.net/users/8628 | Infinite connected graphs isomorphic to their line graph | There is an embedding of the complete graph on countably many vertices $K\_{\aleph\_0}$ into its line graph $LK\_{\aleph\_0}$ as the clique associated to one of the vertices. This induces embeddings $L^n K\_{\aleph\_0} \to L^{n+1}K\_{\aleph\_0}$ for all $n$. The iterated union of $L^n K\_{\aleph\_0}$ under all these em... | 2 | https://mathoverflow.net/users/18060 | 304386 | 133,186 |
https://mathoverflow.net/questions/304366 | 5 | We consider a stochastic process $\left(X\_{t}\right)\_{t\geq 0}$, defined as an integral process, s.t. $$X\_{t}=\int\_{0}^{t}u\_{s}\,dB\_{s}^{H}.$$
With a fractional Brownian motion $B^H\_{t}$.
If $H\neq\frac{1}{2}$, the stochastic integral can not be defined in the classical Itô sense, due to Bichteler-Dellacherie th... | https://mathoverflow.net/users/110069 | fractional Brownian Motion driven stochastic integrals | Yes, RPT allows you to define a notion of stochastic integral against fBm for a class of integrands that is larger than what Young's theory allows. Assuming that you're really interested in solving SDEs, so that $u$ locally looks again like $B^H$, you can go down to $H > {1\over 4}$. Below that things break down, and t... | 6 | https://mathoverflow.net/users/38566 | 304393 | 133,188 |
https://mathoverflow.net/questions/304355 | 13 | The well known [Dirichlet's unit theorem](https://en.wikipedia.org/wiki/Dirichlet%27s_unit_theorem) states that the unit group of a maximal order in a quadratic number field is *finitely generated* of rank blah blah blah. I think it's pretty naive to expect a most radical generalization to hold, namely:
>
> **Gener... | https://mathoverflow.net/users/81055 | Example of a ring with non-finitely generated unit group? | Let $A$ be a ring that is finitely generated as an abelian group. Let $I$ be the subgroup of torsion elements. By finite generation, $I$ is finite. On the other hand, $I$ is a two-sided ideal of $A$, and $A/I$ is a ring that is torsion-free as an abelian group. By Borel and Harish-Chandra, $(A/I)^\times$ is finitely-ge... | 14 | https://mathoverflow.net/users/40821 | 304395 | 133,189 |
https://mathoverflow.net/questions/304389 | 5 | Consider a metric space $(\Bbb M,d).$
If $X,Y,Z\in \Bbb M.$ We define cosin of angle by
$$\cos(\angle YXZ)=\frac{d(X,Y)^2+d(X,Z)^2-d(Y,Z)^2}{2d(X,Y)\cdot d(X,Z)}.$$
If we have four points $A,$ $B,$ $C$ and $D$ in $\Bbb M$ satify the indentity
$$d(A,B)\cdot d(C,D)+d(A,D)\cdot d(B,C)=d(A,C)\cdot d(B,D).$$
Then ... | https://mathoverflow.net/users/126268 | Cyclic quadrilateral in metric space | For a counterexample to 2, we can take the points $A,B,C,D$ at $0,30,60,90$ degrees on the unit circle, and then add a bump to perturb the $x$-axis. In more detail:
Consider the surface $z=f(x,y)^2$, where
$$f(x,y)=\max\left(0,\ 0.1 - \left(x-0.5\right)^2 - y^2 \right),$$
and distances from shortest paths on the surf... | 4 | https://mathoverflow.net/users/nan | 304401 | 133,192 |
https://mathoverflow.net/questions/304391 | 2 | Let $G$ be a torsion free group with identity $e$. For a subset $X$ of $G$, denote by $X^\#$ the set $X\setminus\{e\}$. Let $A$ be a finite subset of $G$ containing $e$. Is there a finite subset $B$ containing $e$ such that
$$A\subset B^\#A\quad\text{and}\quad B\subset BA^\#$$
It is obvious there is no such $B$ when $... | https://mathoverflow.net/users/84700 | About the product of finite subsets of a torsion free group | If such $B$ exists then for some $n>0$ there must exist elements $a\_1,\dots ,a\_n\in A^\#$ such that $a\_1\dots a\_n=e$. (The argument is essentially the one you use to rule out the case $\lbrace e,x\rbrace$ when $G$ is torsionfree.)
Conversely, suppose that there exist $a\_1,\dots ,a\_n\in A^\#$ such that $a\_1\dot... | 5 | https://mathoverflow.net/users/6666 | 304402 | 133,193 |
https://mathoverflow.net/questions/304371 | 0 | Let $G$ be a finite group of order $2^7\cdot3^3\cdot5^2\cdot7$. Let $\mathrm{Irr}(G)$ be the set of all the irreducible $\mathbb{C}$-characters. Suppose that
(1) there is a character $\chi\in\mathrm{Irr}(G)$ such that $2^5\cdot7|\chi(1)$;
(2) there is a character $\theta\in\mathrm{Irr}(G)$ such that $5^2\cdot7|\t... | https://mathoverflow.net/users/99750 | Is $G$ non-solvable? | Such a group is not solvable. The existence of the irreducible characters given forces $F(G)$ to be a $2$-group (using Clifford's Theorem). If $G$ were solvable, that would imply that $G/O\_{2}(G)$ is isomorphic to a subgroup of ${\rm GL}(n,2)$ for some $n \leq 7.$ But no such ${\rm GL}(n,2)$ has order divisible by $25... | 5 | https://mathoverflow.net/users/14450 | 304403 | 133,194 |
https://mathoverflow.net/questions/304258 | 8 | Let $R$ be a (commutative) domain and let $Q$ be its fraction field.
Consider a morphism $f\colon R^n \to R^m$, i.e. a matrix $A \in M(m,n;R)$, and let $K= \operatorname{coker} f$.
Let $I\_k=(\det \operatorname{min}\_k(A))$ be the ideal of $R$ generated by all determinants of minors of $A$ of size $k$.
I wonder if
\... | https://mathoverflow.net/users/118707 | Generalized Smith Theorem for the torsion of cokernels | As Mohan already observed, the [Smith Norm Formal Theorem](https://en.wikipedia.org/wiki/Smith_normal_form) can be extended in your sense to the class of [Dedekind domains](https://en.wikipedia.org/wiki/Dedekind_domain). This is somehow the best we can get:
>
>
> >
> > **Claim.** Let $R$ be a Noetherian domain wh... | 3 | https://mathoverflow.net/users/84349 | 304406 | 133,195 |
https://mathoverflow.net/questions/304374 | 5 | I'm looking for a proof, in English, of the following theorem due to von Neumann (which apparently originates in the paper *Einige Sätze über Messbare Abbildungen*, Ann. of Math, 1932):
>
> Every automorphism of the Boolean algebra of (Lebesgue) measurable subsets of $[0,1]$ modulo null sets is realized by a Borel ... | https://mathoverflow.net/users/16107 | Von Neumann's theorem on realizing automorphisms of the measure algebra | Here is a hint for a proof, in English.
"Obviously, $[0,1]$ can be substituted with any standard finite measure space",
so let's substitute it with the cantor set $C=\{0,1\}^\mathbb{N}$ endowed with the standard measure $\mu=(1/2\delta\_0+1/2\delta\_1)^\mathbb{N}$
and let $T$ be an automorphism of the corresponding mea... | 4 | https://mathoverflow.net/users/89334 | 304408 | 133,196 |
https://mathoverflow.net/questions/304385 | 2 | Let $k$ be a finite extension of $\mathbb{Q}\_p$ very often we use the isomorphism that $Gal(\overline{k}/k)^{ab} \simeq \hat{(k^{\times})}$ given by local class field theory.
My question would be do we have (and if yes how can I prove it) $ \hat{O\_k^{\times}} \times \hat{\mathbb{Z}} \simeq \hat{(k^{\times})}$ ? and i... | https://mathoverflow.net/users/125529 | Decomposition of $\widehat{k^{\times}}$ occuring in local class field theory | Answering your questions: It is true that
$$\widehat{(k^{\times})} \simeq \widehat{O\_k^{\times} \times \mathbb{Z}} \simeq \widehat{O\_k^{\times}} \times \widehat{\mathbb{Z}} \simeq O\_k^{\times} \times \widehat{\mathbb{Z}}$$
as profinite groups, since
* $k^{\times}\simeq O\_k^{\times} \times \mathbb{Z}$ as groups by... | 2 | https://mathoverflow.net/users/24442 | 304415 | 133,199 |
https://mathoverflow.net/questions/304424 | 4 | Disclaimer : the following reasoning is a physicist's one and as such may not be suitable for this website. Still it may give rise to potentially interesting insights so I ask it anyway.
I stumbled upon an article whose abstract claims that for any sufficiently large $ x $ and any $ \delta>0.525 $, one has $\pi(x+x^{... | https://mathoverflow.net/users/13625 | Is there some numerical evidence that $ \pi(x+x^{1/e})-\pi(x)\geq 1 $ for any large enough $ x $? | Turning my comment into an answer: your conjecture is supported by numerical computation, but much stronger ones are also supported: for instance, [Cramér's conjecture](https://en.wikipedia.org/wiki/Cram%C3%A9r%27s_conjecture), based on models of the primes as a 'random set', suggests (with some slight modifications to... | 7 | https://mathoverflow.net/users/7092 | 304438 | 133,205 |
https://mathoverflow.net/questions/304422 | 19 | Let $E$ be an elliptic curve over $\mathbb Q$. Let's look at the group of points of this elliptic curve over $\mathbb Q(1^{1/\infty})$ which we get after adding all roots of unity to $\mathbb Q$. It is easy to prove that it is not finitely generated and the theorem of K.Ribet asserts that its torsion is finite. What el... | https://mathoverflow.net/users/88385 | Points of elliptic curves over cyclotomic extensions | $E (\mathbb Q^{ab})/\operatorname{tors}$ is a sum of countably many copies of $\mathbb Z$.
To prove this, take a countable basis $x\_1,x\_2,\dots$ of $E (\mathbb Q^{ab})/\operatorname{tors}\otimes \mathbb Q$. Suppose we show that $\oplus\_{i=1}^n x\_i \mathbb Q \cap E (\mathbb Q^{ab})/\operatorname{tors}$ is finitel... | 12 | https://mathoverflow.net/users/18060 | 304439 | 133,206 |
https://mathoverflow.net/questions/304434 | 11 | Consider the Hilbert space of functions $f \in L^2(\mathbb R)$ such that $x^2f \in L^2(\mathbb R) $ and $ f'' \in L^2(\mathbb R).$
I am wondering whether it is true that $xf'\in L^2(\mathbb R)$ as well?
It seems natural and perhaps some sort of interpolation should yield the claim but I fail to see how to show thi... | https://mathoverflow.net/users/126333 | $x f'$ bounded by $x^2f $ and $f''$? | By a cutoff function argument, it suffices to assume $f$ is compactly supported, so we can integrate by parts without picking up boundary terms.
Thus
$$\int (xf')^2 = \int (x^2f') f' = -\int 2xf'f - \int x^2 f'' f$$
Hence using Cauchy-Schwarz,
$$\|xf'\|\_2^2 \le \int |2xf f'| + \int |x^2 f f''| \le 2\|xf\|\_2 \|f'\|\... | 13 | https://mathoverflow.net/users/4832 | 304443 | 133,207 |
https://mathoverflow.net/questions/304456 | 2 | I am looking for a proof of the inequality as follows:
>
>
> >
> > Let $A\_1A\_2....A\_n$ be the regular polygon incribed in a circle $(O)$ with radius $R$. Let $B\_1B\_2....B\_n$ be a polygon incribed the circle $(O)$. We let $x\_{ij}=A\_iA\_j$ and $y\_{ij}=B\_{i}B\_{j}$. Let $f(x)=x^m$ (where $m=1, m=2$), I con... | https://mathoverflow.net/users/122662 | An inequality of a cyclic polygon | As for the sum of squares (m=2), denoting the vectors $\overline{OB\_i}=b\_i$ we get $\sum\_{i<j} y\_{ij}^2=\sum\_{i<j} (b\_i-b\_j)^2=n\sum\_i b\_i^2-(\sum b\_i)^2=n^2R^2-(\sum b\_i)^2$ that is maximal if and only if $\sum b\_i=0$ --- so, in particular, for a regular polygon.
| 2 | https://mathoverflow.net/users/4312 | 304461 | 133,214 |
https://mathoverflow.net/questions/304467 | 2 | If we have a second order quasilinear PDE of the form
$A\frac{\partial^2 u}{\partial x^2}+B\frac{\partial^2 u}{\partial y^2}+2C\frac{\partial^2 u}{\partial x\partial y}+ lower\,\, order \,\, terms=0$
where $A,B,C$ are functions of $x,y,u$,
then the equation is called elliptic if $det=\begin{vmatrix}A &C \\C & B\... | https://mathoverflow.net/users/126346 | Classification of a system of two second order PDEs with two dependent and two independent variables | Consider a determined linear system of differential order $k$ of the form $\sigma^{i\_1\cdots i\_k}\_{ab}(x) \partial\_{i\_1} \cdots \partial\_{i\_k} u^b(x) + l.o.t = 0$. The coefficients $\sigma^{i\_1\cdots i\_k}\_{ab}$, a square matrix in the $ab$ indices, constitute its *principal symbol*. By replacing $\partial\_i$... | 4 | https://mathoverflow.net/users/2622 | 304478 | 133,220 |
https://mathoverflow.net/questions/304481 | 4 | For computing homology of a simplicial complex, there is the well-known reduction algorithm.
How about for fundamental group of simplicial complexes? Is there any (implementable) algorithm to compute it? (By implementable I mean that it can be programmed on a computer and actually compute the fundamental group.)
I ... | https://mathoverflow.net/users/83274 | Algorithm for computing fundamental group of simplicial complexes | Depends on what you mean by "computing" and "algorithm". It is undecidable (even for a two-complex) whether the fundamental group is trivial, though computing a presentation is relatively easy.
| 3 | https://mathoverflow.net/users/11142 | 304484 | 133,223 |
https://mathoverflow.net/questions/304489 | 0 | I’m interested in the problem of simultaneous rational approximation to $k\geq 2$ numbers $\alpha\_1,…,\alpha\_k$ in the generic case where the $\alpha\_j$ are *transcendental* and algebraically independent. More precisely, what is the largest $m$ such that there exists an infinite sequence of positive integer denomina... | https://mathoverflow.net/users/22271 | Simultaneous rational approximation to transcendental and algebraically independent numbers | For any $m>1$ the set of such $k$-tuples $\alpha:=(\alpha\_1,\dots,\alpha\_k)$ have measure zero. For seeing this restrict onto $\alpha\in [0,1]^k$ and denote by $\Omega\_q$ the set of $\alpha$'s in $[0,1]^k$ for which this $q$ works. The measure of $\Omega\_q$ does not exceed, say, $2^k q^{k(1-m-1/k)}=2^k q^{-1-k(m-1)... | 5 | https://mathoverflow.net/users/4312 | 304491 | 133,226 |
https://mathoverflow.net/questions/304495 | 0 | Is it true that for any integer $k\geq 3$ there are $\aleph\_0$ many connected countably infinite, pairwise non-isomorphic $k$-regular graphs?
| https://mathoverflow.net/users/8628 | Infinite connected $k$-regular graphs | Take an $n$-cycle, add an infinite tree of the right degree at each vertex of the cycle (the vertex on the cycle having degree 2 less in the tree than the other vertices of the tree). This has only one cycle and it is of length $n$. So the graphs you get for two different values of $n$ are non-isomorphic.
| 4 | https://mathoverflow.net/users/18606 | 304496 | 133,227 |
https://mathoverflow.net/questions/304497 | 9 | Let $F = \mathbb{F}\_2$ be the field with two elements. I will denote the rings of polynomials and formal
power series over $F$ as $F[t]$ and $F[[t]]$ respectively. Suppose that $x \in F[[t]]$ is algebraic
over $F[t]$ (there exists a non-zero polynomial $P$ with coefficients in $F[t]$ such that $P(x) = 0$).
Is it true ... | https://mathoverflow.net/users/126017 | Algebraic power series over $\mathbb{F}_2$ as roots of polynomials of special form | Getting rid of the power series, your question boils down to showing that any polynomial $P(x)$ divides some polynomial $Q(x)$ which has the form $Q(x) = \sum\_{i} c\_i x^{2^i}$. Polynomials $Q$ which have this property are known as [additive polynomials](https://en.wikipedia.org/wiki/Additive_polynomial "additive poly... | 11 | https://mathoverflow.net/users/422 | 304501 | 133,228 |
https://mathoverflow.net/questions/304486 | 34 | The invention of the Jones polynomial led to hundreds of papers and a Fields medal. However, as far as I can tell it had few consequences in topology. After all, after Thurston’s work we already had algorithms to completely classify knots, so by itself a new invariant seems to be of limited value. Given all the excitem... | https://mathoverflow.net/users/126354 | Why should I care about the Jones polynomial? | Your question presupposes that people were excited about the Jones polynomial because it would help them to classify/distinguish knots. In fact, I suspect the interest came from the fact that this knot invariant was originally defined using operator algebras (rather than in the more combinatorial way people usually def... | 40 | https://mathoverflow.net/users/10839 | 304504 | 133,230 |
https://mathoverflow.net/questions/304475 | 3 | I'm looking for a proof for this problem on simplex which I think it is true
>
> **Question.** $A\_0A\_1...A\_n$ is a simplex in the Euclidean space $\Bbb E^n$. $G$ is its centroid and its center circumscribed sphere is $O.$ Let $O\_i$ be the centers circumscribed sphere of the simplex $GA\_{0}A\_{1}...A\_{i-1}A\_{... | https://mathoverflow.net/users/126268 | Centroid and center circumscribed spheres in simplex | Let $O$ be the origin, $\vec{OA\_i}=a\_i$, $\vec{OO\_i}=p\_i$, $\vec{OG}=a$.
Then $\|a\_i\|=1$ and $\sum a\_i/(n+1)=a$. We want to prove that $\sum p\_i=0$.
We know that for $i \neq j$:
\begin{align}
\|p\_i-a\|^2 &= \|p\_i-a\_j\|^2\\
(p\_i,a\_j) &= (p\_i,a)+\frac{\|a\_j\|^2-\|a\|^2}{2}.
\end{align}
Summing up over ... | 4 | https://mathoverflow.net/users/4312 | 304505 | 133,231 |
https://mathoverflow.net/questions/303871 | 0 | Let $X$ be a smooth projective curve of genus $g$. Consider the Ext space over $J\_{d\_1}\times J\_{d\_2}$ i.e., the vector space of isomorphism classes of extensions of line bundles of degree $d\_2$ by line bundles of degree $d\_1$. ($J\_i$ denotes the Jacobian of degree $i$).
There is an action of $Hom(L, Q)$ on Ex... | https://mathoverflow.net/users/nan | About universal Ext space | There is a natural action of $\operatorname{Hom}(L,Q)$ on $\operatorname{Ext}^1(L,Q)$, namely the trivial one.
There is a natural vector bundle stack over $J\_{d\_1}\times J\_{d\_2}$ whose fiber at every point $(L,Q)$ is the stack quotient $[\operatorname{Ext}^1(L,Q)/\operatorname{Hom}(L,Q)]$, that is a groupoid suc... | 1 | https://mathoverflow.net/users/84080 | 304507 | 133,233 |
https://mathoverflow.net/questions/304492 | -2 | Let $R$ be a reduced ring with all non-prime ideals finitely generated. Then is $R$ Noetherian ? If not, then is it true at least in the local case ?
Without reduced assumption, it is not true even in local case, although it is true when the ring is an integral domain; see [this previous question](https://mathoverflo... | https://mathoverflow.net/users/nan | Reduced ring with all non-prime ideals finitely generated | **Question: Let $R$ be a reduced ring with all
non-prime ideals finitely generated. Then is $R$ Noetherian?**
The answer is Yes.
To lessen my typing, let me use the abbreviation
**NFG** to mean not-finitely-generated.
The result proved here is
**Theorem.**
If $R$ is a commutative unital ring whose
NFG ideals are... | 11 | https://mathoverflow.net/users/75735 | 304519 | 133,238 |
https://mathoverflow.net/questions/304513 | 4 | I'm looking for a proof for the extremum problem on regular simplex.
>
> **Question.** Let $\mathcal{A}=A\_0A\_1...A\_n$ be a regular simplex in $\Bbb E^n$. $P$ is a point inside and on boundary of $\mathcal{A}$. Prove that the product $$\prod\_{i=0}^n PA\_i$$ attains maximum iff $P$ is one of the midpoints of edge... | https://mathoverflow.net/users/126268 | Extremum problem on regular simplex | It looks like for small $n$, your assertion is true. For large $n$, there are symmetrically placed pairs on the side achieving the maximum. Please see [this paper](https://link.springer.com/article/10.1007/s00022-006-1813-7) for details. (I do not have access to this paper yet.)
[Previuos comments]
The result is clea... | 2 | https://mathoverflow.net/users/104791 | 304527 | 133,239 |
https://mathoverflow.net/questions/304526 | 30 | For natural numbers $m, r$, consider the ratio of the number of subsets of size $m$ taken from a set of size $2(m+r)$ to the number of subsets of the same size taken from a set of size $m+r$:
$$R(m,r)=\frac{\binom{2(m+r)}{m}}{\binom{m+r}{m}}$$
For $r=0$ we have the central binomial coefficients, which of course are... | https://mathoverflow.net/users/23829 | When does doubling the size of a set multiply the number of subsets by an integer? | Put $n=m+r$, and then we can write $R(m,r)$ more conveniently as
$$
R(m,r) = \frac{(2n)!}{m! (n+r)!} \frac{m! r!}{n!} = \frac{\binom{2n}{n} }{\binom{n+r}{r}}.
$$
So the question essentially becomes one about which numbers $n+k$ for $k=1$, $\ldots$, $r$ divide the middle binomial coefficient $\binom{2n}{n}$. Obvious... | 33 | https://mathoverflow.net/users/38624 | 304529 | 133,241 |
https://mathoverflow.net/questions/304380 | 6 | Background
----------
Let $\sigma, \tau \in S\_n$. We will say that $\sigma$ and $\tau$ are *locally orthogonal* and write $\sigma \perp \tau$ if there exists $j \in \{1, 2, \ldots, n\}$ such that $\sigma(j) \neq \tau(j)$ but $\sigma^{-1}(j) = \tau^{-1}(j)$. (I will explain this terminology in the Motivation section ... | https://mathoverflow.net/users/396 | Can $S_n$ be partitioned into subsets containing an involution and satisfying $∀σ≠τ, ∃j$ s.t. $σ(j)≠τ(j),σ^{−1}(j)=τ^{−1}(j)$? | Yes. For each permutation $\sigma\in S\_n$, choose (arbitrarily) from each nontrivial cycle of $\sigma$ a point $x$ such that $x < \sigma(x)$, and let $f(\sigma)$ be the set of pairs $(x,\sigma(x))$ so obtained. Then $f$ is a map from $S\_n$ to the set of sets of disjoint ordered pairs $(i, j)$ with $i,j\in\{1,\dots,n\... | 3 | https://mathoverflow.net/users/20598 | 304530 | 133,242 |
https://mathoverflow.net/questions/304540 | 9 | Bott & Tu in *Differential forms in Algebraic Topology* write in Remark 5.17, pg.52
>
> The two Poincare duals of a compact orientated submanifold correspond to two homology theories - closed and compact homology. Closed homology has now fallen into disuse, while compact homology is known these days as the homology... | https://mathoverflow.net/users/35706 | What is closed homology? | I think that closed homology is another name for Borel-Moore homology, which certainly has the isomorphism you suggest: <https://en.m.wikipedia.org/wiki/Borel-Moore_homology>
It still is used quite frequently.
| 10 | https://mathoverflow.net/users/52918 | 304548 | 133,247 |
https://mathoverflow.net/questions/304543 | 4 | Is there a 2 dimensional Riemannian manifold $M$ whose curvature is not negative but its geodesic flow is an ergodic flow?
| https://mathoverflow.net/users/36688 | A kind of converse to the Hopf theorem on ergodicity of geodesic flow in negative curvature | It was proved by [Donnay](https://mathscinet.ams.org/mathscinet-getitem?mr=970551) that any compact orientable surface can be given a Riemannian metric for which the geodesic flow is ergodic. See [Theorem 1 of this article.](https://link.springer.com/chapter/10.1007%2FBFb0082827) On the other hand, there are no negativ... | 13 | https://mathoverflow.net/users/8588 | 304550 | 133,248 |
https://mathoverflow.net/questions/304549 | 1 | If $G$ is a group, we denote by $\text{Sub}(G)$ the [lattice](https://en.wikipedia.org/wiki/Lattice_(order)) of all subgroups of $G$, ordered by $\subseteq$. Given a cardinal $\kappa$, is there a group $G$ with $\text{Sub}(G) \cong {\cal P}(\kappa)$ where ${\cal P}(\kappa)$ denotes the power set lattice of all subsets ... | https://mathoverflow.net/users/8628 | Subgroup lattice isomorphic to the power set lattice | The only such groups are of the form $\bigoplus\_{p\in I}C\_p$ for some set $I$ of primes. So the only possible $\kappa$ are the countable ones.
Indeed, given such $G$, one can identify $\kappa$ to the set of minimal subgroups, and every nontrivial subgroup contains a minimal one (hence $G$ is torsion). Given two min... | 13 | https://mathoverflow.net/users/14094 | 304552 | 133,249 |
https://mathoverflow.net/questions/304557 | 19 | Let $g(x) = e^x + e^{-x}$. For $x\_1 < x\_2 < \dots < x\_n$ and $b\_1 < b\_2 < \dots < b\_n$, I'd like to show that the determinant of the following matrix is positive, regardless of $n$:
$\det \left (\begin{bmatrix}
\frac{1}{g(x\_1-b\_1)} & \frac{1}{g(x\_1-b\_2)} & \cdots & \frac{1}{g(x\_1-b\_n)}\\
\frac{1}{g(x\_2-... | https://mathoverflow.net/users/126373 | How to prove positivity of determinant for these matrices? | At first, we prove that the determinant is non-zero, in other words, the matrix is non-singular. Assume the contrary, then by the linear dependency of the columns there exist real numbers $\lambda\_1,\dots,\lambda\_n$, not all equal to 0, such that $F(x\_i):=\sum\_j \frac{\lambda\_j}{g(x\_i-b\_j)}=0$ for all $i=1,2,\do... | 33 | https://mathoverflow.net/users/4312 | 304559 | 133,251 |
https://mathoverflow.net/questions/304560 | 6 | Let $\Sigma\_{g}$ be a closed orientable surface of genus $g$. Let $d\_g$ denote the minimum dimension of a faithful representation of the mapping class group of $\Sigma\_g$. For $g=1$, the mapping class group is $SL(2, \mathbb{Z})$ so $d\_g=2$. For $g=2$, [this paper](https://arxiv.org/abs/math/0010310) proves that $d... | https://mathoverflow.net/users/nan | Minimum dimension of faithful representation of mapping class groups? | There has been no progress on this question for many years. In particular, the precise value of $d\_2$ is not known and it is not known if $d\_g$ is finite for any $g \geq 3$.
The only related paper is [this one](https://arxiv.org/abs/1104.4816) by Korkmaz, which proves that any representation of the genus $g$ mappin... | 7 | https://mathoverflow.net/users/317 | 304572 | 133,254 |
https://mathoverflow.net/questions/304573 | 7 | Let $f$ be a continuous, strictly increasing function from $[0,1]$ to itself with $f(0)=0, f(1)=1$. Let $\Gamma\_f$ denote its graph. What can be said about the Hausdorff dimension of $\Gamma\_f$? In particular, is it true that it is always 1?
If not, is there a link between $\dim\_H(\Gamma\_f)$ and $\dim\_H(\mu)$, w... | https://mathoverflow.net/users/8131 | Hausdorff dimension of the graph of an increasing function |
>
> **Theorem 1.** The Hausdorff dimension of the graph $\Gamma\_f$ of $f$ equals $1$.
>
>
>
**Proof.**
Take a partition of $[0,1]$ by intervals of length $1/n$. Since the function is increasing you can cover the graph by boxes of size $1/n\times (f((k+1)/n)-f(k/n))$, $k=0,1,\ldots,n-1$ located above the interva... | 12 | https://mathoverflow.net/users/121665 | 304574 | 133,255 |
https://mathoverflow.net/questions/304578 | 1 | Let $f:\mathbb R^n \rightarrow \mathbb R$ be a smooth function whose derivatives are all polynomially bounded and $f \in L^{\infty}.$
Such a function has the property that when multiplied with any Schwartz function $\varphi \in \mathcal S$ the product $f\varphi $ is again a Schwartz function.
Now consider the follo... | https://mathoverflow.net/users/126333 | $L^2$ function in Schwartz space? | This is quite *ad hoc*, but your integral equation can be reverse engineered to a differential equation (Maybe this is what you really want to solve?)
$$ \partial\_t\varphi=\alpha(t)f\varphi,\ \varphi(0)=\varphi\_0, $$
which has an explicit solution
$$ \varphi(x,t)=\varphi\_0(x)\exp\left( \int\_0^t \alpha(s)f(x)d... | 3 | https://mathoverflow.net/users/37103 | 304580 | 133,257 |
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.