parent_url stringlengths 37 41 | parent_score stringlengths 1 3 | parent_body stringlengths 19 30.2k | parent_user stringlengths 32 37 | parent_title stringlengths 15 248 | body stringlengths 8 29.9k | score stringlengths 1 3 | user stringlengths 32 37 | answer_id stringlengths 2 6 | __index_level_0__ int64 1 182k |
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https://mathoverflow.net/questions/417195 | 2 | Let $\{X\_i \}\_{i \in \mathbb{N}}$ be a sequence of i.i.d. random variables satisfying $\mathbb{E} X\_1 = 0$ and $\mathbb{E} X\_1 ^2 < \infty$. Assume that $\{S\_n \}\_{n \in \mathbb{N}}$ is a non-lattice random walk, where $S\_n = X\_1+...+ X\_n$. I am wondering whether there is a 'local large deviation theorem' run... | https://mathoverflow.net/users/41071 | Equivalent of a local limit theorem in the large deviation region and asymptotics of a convolution operator | For general classes of bounded pdf's of $X\_1$, including pdf's with exponential-like, super-exponential, and sub-exponential tails, your Theorem follows from the considerations in [Sections 2.1 and 2.2](https://www.sciencedirect.com/science/article/abs/pii/009630039490197X), with
$$\kappa\_n(x)=p\_{S\_n}(r)e^{-s\_0(x-... | 1 | https://mathoverflow.net/users/36721 | 417217 | 169,969 |
https://mathoverflow.net/questions/417214 | 9 | I'm reading Thurston's article "Shapes of polyhedra and triangulations of the sphere." In the introduction he claims the following:
>
> "${}^{(1)}$There are procedures to refine and modify any triangulation of a surface until every vertex has either 5, 6 or 7 triangles around it, or with more effort, ${}^{(2)}$so t... | https://mathoverflow.net/users/148805 | Refining a triangulation | Suppose that $S$ is a closed, connected surface with negative Euler characteristic. Suppose that $T$ is a triangulation of $S$.
Define "refine" to mean "replace each triangle by four triangles" (so that edge midpoints become vertices of valence six). This does not improve any of the vertices of "concentrated positive... | 7 | https://mathoverflow.net/users/1650 | 417219 | 169,970 |
https://mathoverflow.net/questions/417101 | 17 | Hurwitz's theorem says that the only division composition algebras over the real numbers $\mathbb{R}$ are the real numbers themselves $\mathbb{R}$, the complex numbers $\mathbb{C}$, the quaternions $\mathbb{H}$, and the octonions $\mathbb{O}$. However, in pure constructive mathematics without any weak axiom of choice, ... | https://mathoverflow.net/users/nan | Is Hurwitz's theorem true in constructive mathematics? | There is a weakening of Hurwitz's theorem that is true constructively, with essentially the same proof:
Let $A$ be a division composition algebra. Then any chain of proper subalgebras $\mathbb{R} = A\_0 \subsetneq A\_1 \subsetneq \cdots \subsetneq A\_n = A$ has length $n \leq 3$ (where "proper" means "contains an ele... | 3 | https://mathoverflow.net/users/100508 | 417226 | 169,972 |
https://mathoverflow.net/questions/413983 | 2 | It is well-known that an elliptic curve $E$ that has a point of order $2$ and is represented as $E=[0,a,0,b,0]$ has a *$2$-isogenous* curve $E^\prime=[0,-2a,0,a^2-4b,0]$, see e.g. p. 507 in
* A. Dujella, [*Number Theory*](https://web.math.pmf.unizg.hr/%7Eduje/numbertheorybook.html), University of Zagreb, Školska knji... | https://mathoverflow.net/users/95511 | $2$-isogenous to a curve in the Tate normal form | John Cremona has some explicit code for calculating the 2-torsion points of curves in the general Weierstrass [a1,a2,a3,a4,a6] format, and part of that formula is finding rational roots to the cubic equation
(1) $ P(x,[W]) = 4(x^3+e\_{a2}x^2+e\_{a4}x+e\_{a6})+(e\_{a1}x+e\_{a3})^2)$
where [a1,a2,a3,a4,a6] are from t... | 2 | https://mathoverflow.net/users/6046 | 417236 | 169,974 |
https://mathoverflow.net/questions/417246 | 2 | I am working on symplectic geometry and I have some questions about a degeneration of $\mathbb{P}^2$.
Question: Can we obtain the moment polytope (or the polytope associated with the anti-canonical divisor) of $\mathbb{P}(a^2,b^2,c^2)$ as a Newton-Okounkov body of $\mathbb{P}^2$? (Here $(a,b,c)$ is a Markov triple sa... | https://mathoverflow.net/users/11705 | Markov triples and Newton-Okounkov bodies of $\mathbb{P}^2$ | The polytopes you are interested in are related by sequences of combinatorial mutations, as described [here](https://arxiv.org/abs/1212.1785) and [here](https://arxiv.org/abs/1302.1152). If two polytopes $P\_1$ and $P\_2$ are related by a combinatorial mutation, then there is a construction due to Ilten ([here](https:/... | 3 | https://mathoverflow.net/users/104695 | 417255 | 169,981 |
https://mathoverflow.net/questions/417173 | 6 | I'm reading Milnor's [notes](http://www.math.stonybrook.edu/%7Ejack/DYNOTES/) on dynamical systems and in Lecture 3 he gives an example of an attractor with no natural measure, which he attributes to Mañé. I can find no other reference in which this example is discussed; no paper by Ricardo Mañé, no books or papers in ... | https://mathoverflow.net/users/477928 | Mañé's example of an attractor with no natural measure | **Q:** Has this or any other counterexample been published anywhere else?
**A:** In [A continuous Bowen-Mañé type phenomenon](https://www.aimsciences.org/journals/displayArticlesnew.jsp?paperID=3046) examples of vector fields without physical measure for certain parameters are discussed under the name "*Bowen-Mañé ty... | 3 | https://mathoverflow.net/users/11260 | 417269 | 169,985 |
https://mathoverflow.net/questions/417261 | 1 | Let $X$ be a vector space equipped with a norm $p$ and a seminorm $q$. Denote the completion of $X$ with respect to $p$ with $X\_p$ and with respect to $p+q$ by $X\_{p+q}$. Then the induced map $\iota : X\_{p+q} \to X\_p$ is well-defined and continuous but not necessarily injective as can be seen in analogy to [this an... | https://mathoverflow.net/users/18936 | When is a natural map between completions injective? | A simple sufficient condition for two norms $p\le r$ on $X$ to induce an *injective* continuous linear map $i:X\_r\to X\_p$ between the completions is that the unit ball $B\_r=\{x\in X: r(x)\le 1\}$ is $p$-closed.
Indeed, for $x\in X\_r$ with $i(x)=0$ choose a sequence $x\_n\in X$ with $x\_n\to x$ in $X\_r$. This seq... | 3 | https://mathoverflow.net/users/21051 | 417278 | 169,990 |
https://mathoverflow.net/questions/417276 | 5 | What's a good example of a simple algebra over a field of characteristic $0$ which has a non-inner derivation but also has the invariant basis number property (IBN)?
I'm under the impression that when an algebra is simple Artinian, all derivations are inner. If I'm missing something subtle about what can happen pleas... | https://mathoverflow.net/users/19965 | Finding non-inner derivations of simple $\mathbb Q$-algebras | Let $K$ be any field. I will give a simple $K$-algebra with IBN and a noninner derivation. My example will be a contracted monoid algebra. These have IBN by @PaceNielsen's nice answer [here](https://mathoverflow.net/questions/248345/can-the-trivial-module-be-stably-free-for-a-monoid-ring).
Let $X$ be an infinite set ... | 4 | https://mathoverflow.net/users/15934 | 417282 | 169,992 |
https://mathoverflow.net/questions/417283 | 3 | I came across the following inequality, which should hold for any integer $k\geq 1$:
$$\sum\_{j=0}^{k-1}\frac{(-1)^{j}2^{k-1-j}\binom{k}{j}(k-j)}{2k+1-j}\leq
\frac{1}{3}.$$
I have been struggling with this statement for a while. It looks valid for small $k$, but a formal proof seems out of reach with my tools. Any ... | https://mathoverflow.net/users/478035 | An inequality involving binomial coefficients and the powers of two | For $j=0,\dots,k-1$,
\begin{equation\*}
\frac1{2k+1-j}=\int\_0^1 x^{2k-j}\,dx.
\end{equation\*}
So,
\begin{equation\*}
\begin{aligned}
s:=&\sum\_{j=0}^{k-1}\frac{(-1)^{j}2^{k-1-j}\binom{k}{j}(k-j)}{2k+1-j} \\
&=\int\_0^1 dx\,\sum\_{j=0}^{k-1}(-1)^{j}2^{k-1-j}\binom{k}{j}(k-j)x^{2k-j} \\
&=\int\_0^1 dx\,kx^{k+1}(2... | 4 | https://mathoverflow.net/users/36721 | 417289 | 169,993 |
https://mathoverflow.net/questions/416648 | 17 | EDIT: immediately **after** bountying the question *(whoops ...)* I found, while looking for something else entirely, that Sauro Tulipani [gave an explicit algorithm](https://www.jstor.org/stable/2273991?refreqid=excelsior%3Abff6195c8445cbcfc1c488c4c2530272&seq=1#metadata_info_tab_contents) for producing a Horn sentenc... | https://mathoverflow.net/users/8133 | How hard is it to say "not exactly $p$" with a Horn sentence? | I will show how to improve Tulipani’s construction from $O(p^5)$ symbols to $O(p^3)$ symbols, or $O(p^3\log p)$ bits.
Recall that Tulipani’s sentence is
$$H\_p=\forall\vec x\,\exists\vec s\,\exists\vec u\,\exists y\:\Bigl(G\_p(\vec x,\vec s,\vec u)\land\let\ET\bigwedge\ET\_i(y=x\_i\land u\_p=x\_0\to x\_1=x\_0)\Bigr),... | 11 | https://mathoverflow.net/users/12705 | 417291 | 169,994 |
https://mathoverflow.net/questions/417253 | 2 | Suppose that $A,B$ are real analytic subsets of $\Omega\subseteq \mathbb{R}^n$ and $p\in A\cap B \neq \emptyset$. Does the intersection inequality from complex analysis still hold, i.e. does the following inequality hold:
$\mathrm{dim}^{\mathbb{R}}\_p(A\cap B) \geq \mathrm{dim}^{\mathbb{R}}\_p(A)+\mathrm{dim}^{\mathb... | https://mathoverflow.net/users/109193 | Dimension of intersection of real analytic sets | For a counterexample take a sphere in 3 space and a plane tangent to it .
| 5 | https://mathoverflow.net/users/4696 | 417293 | 169,996 |
https://mathoverflow.net/questions/417284 | 2 | Lets work with Harvey's [Friedman](https://cpb-us-w2.wpmucdn.com/u.osu.edu/dist/1/1952/files/2014/01/TalkAxiomSetThy-ts05xn.pdf) theory ${\sf K}(W)$, formulated in the language of set theory with a primitive constant symbol $W$ added, i.e. in ${\sf FOL}(\in,W)$
**Axioms:**
**Extensionality:** $$\forall Z \, (Z \in ... | https://mathoverflow.net/users/95347 | Is it consistent to add a generalization axiom on top of Ext.+Subworld Separation+Reduciblity? | Generalization holds in K(W). Suppose it did not, that ¬(W) holds and ∀infinite∈:.
Then ∃X((infiniteX)∧¬), since this is true when X is W. By Reducibility
∃X∈W((infiniteX)∧¬). But this contradicts the fact that ∀infinite∈:.
| 2 | https://mathoverflow.net/users/133981 | 417303 | 169,998 |
https://mathoverflow.net/questions/417304 | 3 | I have read in a paper about the following result:
Let $V$ be a separable Hilbert space and $(\Omega,A\_{\Omega},P)$ a probability space. Suppose that $Y\_1,Y\_2,...$ is a sequence of independent $V$-valued random variables. If $E\left(\Vert Y\_i\Vert\_{V}^m\right)\leq \frac{1}{2} m! B^2 L^{m-2}$ $\forall m\geq 2$, t... | https://mathoverflow.net/users/163533 | Concentration inequality for Hilbert space valued random variables | The condition $EY\_i=0$ cannot be dropped.
Indeed, if e.g. the $Y\_i$'s are iid with $\mu=EY\_i\ne0$ and $n\to\infty$, then, by the law of large numbers, the left-hand side of the inequality in question will go to $1$ for each $\epsilon\in(0,\|\mu\|\_V)$, whereas the right-hand side of the inequality will go to $0$.
... | 2 | https://mathoverflow.net/users/36721 | 417306 | 169,999 |
https://mathoverflow.net/questions/414893 | 2 | Let $G$ be a finite group and $\lambda \in \text{Irr}(G)$ an irreducible complex character of $G$.
Let $m(\lambda) := \min \{ \vert G : H \vert \mid H \leq G, \lambda\vert\_H \text{ has a linear component}\}.$
Is there a universal constant $c$, independent of $G$ and $\lambda$, such that
$$ \frac{m(\lambda)}{\lambda (1... | https://mathoverflow.net/users/173766 | Existence of universal bound related to characters | There is no such universal constant for solvable groups, at least. Essentially this is because $m(\chi)$ as defined in the question, is multiplicative. That is, if $G\_1$ and $G\_2$ have coprime orders, and if $\chi\_i\in\mathrm{Irr}(G\_i)$ for $i=1,2$ then $\chi\_1\times\chi\_2\in\mathrm{Irr}(G\_1\times G\_2)$ satisfi... | 1 | https://mathoverflow.net/users/313687 | 417310 | 170,000 |
https://mathoverflow.net/questions/403366 | 4 | Let $f : X \to Y$ be projective and smooth morphism of complex algebraic varieties. Here we care about the algebraic topology of $X$ and $Y$, so use classical topology for simplicity.
I can take the constant sheaf $\mathbb{Q}\_X$ and (derived) push it forward to get $f\_\* \mathbb{Q}\_X \in D^b\_c(Y,\mathbb{Q})$. The... | https://mathoverflow.net/users/919 | Decomposition of direct image of a smooth morphism, Deligne's theorem, motives | We can give many counterexamples to semisimplicity in positive characteristic using the observation that if the canonical morphism $k\_Y\rightarrow f\_\*k\_X$ in $D^b\_c(Y,k)$ is split, then the induced morphism in $k$ cohomology must be injective: $$f^\*:H^\*(Y,k)\rightarrow H^\*(X,k).$$
So we just need to cook up n... | 2 | https://mathoverflow.net/users/128502 | 417327 | 170,005 |
https://mathoverflow.net/questions/405604 | 1 | In [[1](https://www.jstor.org/stable/pdf/2157207.pdf?refreqid=excelsior%3Ac8cb5873955a58f5a2eba65be8b3f13c)], it is shown in theorem 1.2 that for symmetric $n \times n$ matrices $A$, $B$, we have
$$
\min\_{Y \in Y^\*} \text{tr}(Y^TAY) =
\text{tr}(X^TAX) =
\sum\_{i=1}^p \lambda\_i,
$$
with
$$
\text{
$X^TBX = I^p$
and ... | https://mathoverflow.net/users/401351 | Trace minimization for generalized eigenvalue problem | After some work, I figured out the proof, using -- indeed -- the Courant-Fischer theorem and parts of Cauchy's Interlacing and Poincaré's Separation theorems. I've carved out the part of my thesis proving this, which can be found [here](https://github.com/dnndbrkt/stratifyingautism/blob/6e4660b66c351cb83177736e963752e9... | 1 | https://mathoverflow.net/users/401351 | 417328 | 170,006 |
https://mathoverflow.net/questions/417324 | 5 | **Motivation**: The following is a theorem of Berrick-Hesselholt (essentially also due to Linnell, though not in this form):
>
> Let $G$ be a group. Suppose that for every subgroup of $G$ isomorphic to $\mathbb Q$, $G$ has a quotient in which the image of this subgroup is central and nontrivial. In this case the Ba... | https://mathoverflow.net/users/102343 | Potential counterexamples to Bass' trace conjecture | a) There are old results which directly imply the existence of such groups:
(1) Boone Higman 1972: every f.g. group with solvable word problem embeds into a simple subgroup of a finitely presented group.
(2) Every countable group with solvable word problem embeds into a f.g. group with solvable word problem (refere... | 9 | https://mathoverflow.net/users/14094 | 417334 | 170,009 |
https://mathoverflow.net/questions/417074 | 2 | Let us say a set of $n$ rectangles is *rectifiable* if all $n$ rectangles together form a big rectangle without gaps or overlaps.
**Question:** How hard computationally is the question of deciding whether a set of $n$ rectangles with all dimensions integers is rectifiable?
If we further constrain the question by di... | https://mathoverflow.net/users/142600 | On sets of rectangles that can all together form at least one big rectangle | The problem is NP-complete. There is a simple reduction from the bin packing problem [1].
Suppose we want to determine items $a\_1, \dots, a\_n$ fit in $k$ bins of size $h$, $kh = \sum\_{i=1}^n a\_i$.
Let $w \gt 2kh$. Each item of size $a\_i$ is represented as a rectangle of size $w \times 2 a\_i$. One rectangle of... | 4 | https://mathoverflow.net/users/476793 | 417337 | 170,010 |
https://mathoverflow.net/questions/417188 | 1 | I am studying a recent paper in which the author worked on the rectangular, flat 3 tori. It can be realized, the author explained, as $\mathbb{R}^3 \over (L\_1 \mathbb Z \times L\_2 \mathbb Z \times L\_3 \mathbb Z)$ with $L\_j \in (0, \infty),j=1,2,3.$ For notational convenience, we use the coordinates for the standard... | https://mathoverflow.net/users/471464 | The semigroup of Laplace-Beltrami operator on 3-flat torus | You are probably just overthinking it since this is basically just a multivariable calculus change of variables.
The transformation $\mathbb{R}^3 \to \mathbb{R}^3$ given by
$$ (x\_1, x\_2, x\_3) \mapsto (y\_1,y\_2,y\_3) = (L\_1 x\_1, L\_2, x\_2, L\_3 x\_3)$$
maps the torus $\mathbb{T}^3 = \mathbb{R}^3 / \mathbb{Z}^3$... | 2 | https://mathoverflow.net/users/3948 | 417345 | 170,011 |
https://mathoverflow.net/questions/417316 | 2 | Given an adjunction $F\dashv G:\mathcal{C}\rightleftarrows\mathcal{D}$ with unit $\eta$ and counit $\epsilon$, we naturally have a monad $(G\circ F,\eta,G\epsilon\_F)$ on $\mathcal{C}$ and a comparison functor $K:\mathcal{D}\to\mathcal{C}^{G\circ F}$ (where $\mathcal{C}^{G\circ F}$ is the [E-M category](https://ncatlab... | https://mathoverflow.net/users/92164 | Uniqueness of comparison functors | Your argument that any two functors that agree on arrows must agree on objects depends on assuming that the homsets of a category are disjoint, so that every arrow has exactly one domain and codomain. But even if the homsets of $\mathcal{C}$ are disjoint, the homsets of $\mathcal{C}^{G\circ F}$ won't generally be: a gi... | 5 | https://mathoverflow.net/users/49 | 417347 | 170,012 |
https://mathoverflow.net/questions/417287 | 4 | Consider the parabolic equation in $p: \mathbb R^2\to\mathbb R$
$$\partial\_t p + b(t)\partial\_x p + D(t,x)\partial^2\_{xx}p=0,$$
where $b$, $D$ are nice enough functions. I look for the continuity of the derivatives $\partial\_t p$, $\partial\_x p$ of the solution. It is known by Nash's paper ([Continuity of Solu... | https://mathoverflow.net/users/261243 | Reference request: continuity of the derivatives of the (fundamental) solution to a parabolic equation | The equation for $q = p\_x$ can be written in divergence form as
$$q\_t + b(t)q\_x + (D(t,\,x)q\_x)\_x = 0,$$
so Nash's theorem (which applies to divergence-form equations) implies that $p\_x$ is Holder continuous under mild hypotheses on the coefficients (boundedness and measurability).
If the coefficients are more ... | 4 | https://mathoverflow.net/users/16659 | 417352 | 170,013 |
https://mathoverflow.net/questions/417144 | 1 | The process $X(t)=\int\_0^t B(s) ds+B(t)$ is a centered continuous Gaussian process. Therefore it defines a Gaussian measure on $C[0,T]$. Therefore there is a Cameron-Martin space with Cameron-Martin norm. I can compute the covariance and get some complicated expression.
Is there a clean expression for the Cameron-Ma... | https://mathoverflow.net/users/341290 | What is the Cameron-Martin norm associated to $X(t)=\int_0^t B(s) ds+B(t)$? | I will show that for $x\in\mathcal C^1$, $x$ is in the Cameron-Martin space $\mathcal H$ and
$$ |x|\_\mathcal{H}^2 = \int\_0^T\left(x'(t)-\int\_0^t\mathbf e^{-(t-s)}x'(s)\mathrm ds\right)^2\mathrm dt. $$
Expanding the product and using Fubini's theorem, it can be rewritten as
$$ |x|\_\mathcal{H}^2 = \int\_0^T|x'(t)|^2\... | 1 | https://mathoverflow.net/users/129074 | 417357 | 170,014 |
https://mathoverflow.net/questions/417351 | 0 | Let $E,F$ be two holomorphic vector bundles on a compact Kahler manifold $X$. Denote by $\mathbb{P}(E), \mathbb{P}(F)$ the associated projective bundles and $L\_E=\mathcal{O}\_E(-1), L\_F=\mathcal{O}\_F(-1)$ the tautological line bundles. Let $f:E\to F$ a bundle map.
Is there a "canonical" induced map
$$\tilde{f}:\ti... | https://mathoverflow.net/users/102114 | Induced homomorphism on tautological line bundles $\mathcal{O}_E(1),\mathcal{O}_F(1)$ | Let $Z = \mathbb{P}(E) \times\_X \mathbb{P}(F)$. Consider the composition
$$
p\_1^\*L\_E \to p^\*E \to p^\*F \to p^\*F/p\_2^\*L\_F,
$$
where $p\_1 \colon Z \to \mathbb{P}(E)$, $p\_2 \colon Z \to \mathbb{P}(F)$, and $p \colon Z \to X$ are the natural projections,
and the middle arrow above is the pullback of $f$. Let
$$... | 2 | https://mathoverflow.net/users/4428 | 417358 | 170,015 |
https://mathoverflow.net/questions/417262 | 1 | Let $X$ be a smooth projective toric variety over $\mathbb{C}$. It is acted by the compact torus $T=(S^1)^n$.
The $T$-equivariant cohomology $H^\*\_T(X)$ (with coefficients in a field, say) is an algebra over the ring of the $T$-equivariant cohomology of the point $H^\*\_T(pt)$. The ideal $H^{>0}\_T(pt)\cdot H^\*\_T(... | https://mathoverflow.net/users/16183 | A connection between equivariant and non-equivariant cohomology of toric variety | Yes. The keyword here is equivariantly formal. More generally if $X$ is a (possibly singular) projective variety over $\mathbb{C}$ whose ordinary cohomology $H(X)$ vanishes in odd degrees, and if $X$ admits an algebraic action of a torus $T=(\mathbb{C}^\*)^n$ with compact torus $K=(S^1)^n$ then equivariant cohomology $... | 4 | https://mathoverflow.net/users/66536 | 417360 | 170,016 |
https://mathoverflow.net/questions/417344 | 4 | Let $ \Omega $ be a bounded domain with smooth boundary. Consider the Poisson equation
\begin{eqnarray}
-\Delta u&=&f\text{ in }\Omega\\
u&=&0\text{ on }\partial\Omega
\end{eqnarray}
where $ f\in C\_0^{\infty}(\Omega) $. By using the Lax-Milgram theorem, we can find the solution in $ H\_0^1(\Omega) $ and then enhance t... | https://mathoverflow.net/users/241460 | The behavior of $ \nabla u $ on the boundary for Poisson equations | The first observation is that the $u$ above satisfies $\nabla u=0$ on $\partial \Omega$ if and only if $f$ is orthogonal to all harmonic functions $v$ in $\Omega$, continuous up the the boundary. In fact, $\int\_{\Omega} fv=\int\_{\Omega} (\Delta u) v=\int\_{\Omega} u \Delta v=0$, by the boundary conditions. Conversely... | 5 | https://mathoverflow.net/users/150653 | 417361 | 170,017 |
https://mathoverflow.net/questions/417330 | 3 | Assume that $f\_0,f\_1,f\_2$ are polynomial functions of degree two in two variables. This means that the $f\_i$ are linear combinations with real coefficients of $x^2,xy,x,y^2,y,1$.
Consider the function $f = f\_1^2-af\_0f\_2:\mathbb{R}^2\rightarrow\mathbb{R}$ where $a\in \mathbb{R}\_{>0}$. Is it true that for a "ra... | https://mathoverflow.net/users/14514 | Positivity of real functions in two variables | $\newcommand\R{\mathbb R}\newcommand\c{\mathsf c}\newcommand\ep{\varepsilon}$The answer is no. Indeed, after clarifications given by the OP in comments and in the original post, the question can be stated as follows:
>
> For $i=0,1,2$, let
> $$f\_i(x,y)=a\_{i,0}+a\_{i,1}x+a\_{i,2}y+a\_{i,3}xy+a\_{i,4}x^2+a\_{i,5}y^... | 1 | https://mathoverflow.net/users/36721 | 417367 | 170,019 |
https://mathoverflow.net/questions/417364 | 0 | I am interested in the determinant of $W = X \* X'$, where $X \in \mathbb{R}^{k \times n}$ is a matrix with each row drawn IID from some sub-Gaussian distribution on $\mathbb{R}^{n}$. (I am aware of some universality results, so happy to also consider a "standard" Wishart matrix with parameter $k/n$). Edit: say e.g. th... | https://mathoverflow.net/users/134361 | LDP for Marchenko Pastur with k/n tending to 0 | For standard Gaussians, and with the matrix $W/n$,
the proof of the LDP given by Ben Arous-Guionnet adapts
to the Wishart setup. However, you will have different scalings and so the non-commutative entropy term (of exponential scaling $k^2$) will disappear, and the proof more or less trivializes.
If I did not make a st... | 1 | https://mathoverflow.net/users/35520 | 417370 | 170,020 |
https://mathoverflow.net/questions/417362 | 10 | Does there exists a triple $(G, X, \pi)$, where $G$ is a compact group, $X$ an infinite dimensional Banach space over $\mathbf{C}$, and $\pi : G \to B(X)$ a strongly continuous representation of $G$, such that if $Y$ is a non-zero closed invariant subspace of $X$ in the sense that $\pi(g)Y \subseteq Y$ for all $g \in G... | https://mathoverflow.net/users/40789 | Existence of a strongly continuous topologically irreducible representation of a compact group on an infinite dimensional Banach space? | For Banach spaces, the question is no : there is a form of the Peter-Weyl theorem, due to Shiga, which implies that in every Banach space representation of a compact group, the finite-dimensional sub-representations span a dense subspace. In particular, strongly continuous irreducible representations on a Banach space ... | 9 | https://mathoverflow.net/users/10265 | 417371 | 170,021 |
https://mathoverflow.net/questions/417372 | 4 | To simplify the notation, assume $V=L$. We have $\lvert V\_{\omega\_{1}} \rvert=\aleph\_{\omega\_{1}}$ and $\lvert H(\aleph\_{1})\rvert=\aleph\_{1}$, so in particular $V\_{\omega\_{1}} \models \exists x \forall \alpha\; x \not\in L(\alpha) $ since $L(\omega\_{1})=H(\aleph\_{1})$.
Using the Löwenheim–Skolem theorem we... | https://mathoverflow.net/users/472959 | A doubt about the Gödel condensation lemma | Your mistake is that taking the Mostowski collapse does **not** preserve elementarity.
We do have a countable transitive $A$ and a countable $M$ with $$A\cong M\preccurlyeq V\_{\omega\_1},$$ where $M$ comes from downward Lowenheim-Skolem applied to $V\_{\omega\_1}$ and $A$ is the collapse of $M$, but that does not im... | 13 | https://mathoverflow.net/users/8133 | 417377 | 170,023 |
https://mathoverflow.net/questions/416940 | 10 | $\DeclareMathOperator\Spec{Spec}\newcommand{\perf}{\mathrm{perf}}\DeclareMathOperator\SHC{SHC}$I have just finished reading the paper "The spectrum of prime ideals in tensor triangulated categories" in which Balmer proposes his notion of spectrum which nowadays is considered central in the understanding and classificat... | https://mathoverflow.net/users/131453 | Understanding Balmer spectra | I am not an expert in tt-geometry, but let me try to answer some of your questions.
(1) You are correct, the Balmer spectrum is typically not well-suited to study the "big" categories - this is because all definitions that appear only use "finitary" things : tensor products, cones/extensions, finite direct sums, retr... | 4 | https://mathoverflow.net/users/102343 | 417386 | 170,026 |
https://mathoverflow.net/questions/417393 | 1 | It is known that cobordism provides a complete classification of surfaces as: a surface is cobordant to either $S^2$ or $\mathbb{R}P^2$. I am looking for a reference with contains a proof of this fact.
| https://mathoverflow.net/users/475761 | Reference request: full classification of surfaces as being cobordant to $S^2$ or $\mathbb{R}P^2$ | The orientable closed surface with genus $g$ (denoted as $F\_g$) can be realized as the boundary of a central-symmetric closed body $K$ in $\Bbb{R}^3$ (for example, $F\_1 \cong \Bbb{T}^2$ can be realized as the boundary of a solid torus centered at the origin). Choose a large closed ball $B$ centered at the origin with... | 3 | https://mathoverflow.net/users/166298 | 417395 | 170,028 |
https://mathoverflow.net/questions/417390 | 3 | Let $A=[a\_{ij}]$ be a $3\times 3$ matrix, where $a\_{ii}$ is a real number, and $a\_{ji}=\overline{a\_{ij}}$ is the complex conjugate of $a\_{ij}$ for all $1\leq i,j\leq 3$, i.e $A^t=[\overline{a\_{ij}}]$. Let $\lambda\_1,\lambda\_2,\lambda\_3$ be eigenvalues of $A$ (not necessarily distinct) and $u\_i=(a\_i,b\_i,c\_i... | https://mathoverflow.net/users/36341 | An inequality relating to entries of eigenvectors | The inequality holds for any $n$ if all eigenvalues of the Hermitian matrix $A$ are distinct, so that the eigenvectors form a unitary matrix.$^\ast$
It does not hold if some eigenvalues are identical.
In that case the corresponding eigenvectors need not be orthogonal, you could choose them nearly parallel, say $u\... | 4 | https://mathoverflow.net/users/11260 | 417396 | 170,029 |
https://mathoverflow.net/questions/417392 | 1 | I have noticed experimentally that the following question has a positive answer.
Let $p>5$ and $H$ be a subgroup of $(\mathbb Z/p\mathbb Z) ^\*$, with $a\in H$ and $a>2$.
Is it true that $$(a-1)\; | \; p\sum\limits\_{h \in H} (-h/p \mod a) ?$$
| https://mathoverflow.net/users/110301 | Strange result of divisibility | I assume that in the calculations, you are identifying the elements of $\mathbb Z/p\mathbb Z$ with $\{0,\dots,p-1\}\subseteq\mathbb Z$, and likewise, that the $\bmod a$ operation takes values in $\{0,\dots,a-1\}$. Then the result follows from
>
> **Lemma:** If $a,b>0$ are coprime and $-a<h<b$, then $$a(ha^{-1}\bmod... | 4 | https://mathoverflow.net/users/12705 | 417398 | 170,031 |
https://mathoverflow.net/questions/417408 | 1 | I have an integral of the form
$$ \int\_R^{\infty} e^{-x} x^n \vert L\_m^{\alpha}(x) \vert^2 \ dx,$$
where $L\_m^{\alpha}$ is the generalized Laguerre polynomial and $n \ge 0.$
I would to get a nice explicit exponential bound on this integral in terms of $R$ (I intend to take $R$ large). However, most upper bounds on... | https://mathoverflow.net/users/457901 | Exponential decay bound on integral | Let $a:=\alpha$. The integral in question is
\begin{equation\*}
I(R):=\int\_R^\infty e^{-x} x^n L\_m^a(x)^2\, dx,
\end{equation\*}
[where](https://en.wikipedia.org/wiki/Laguerre_polynomials#Explicit_examples_and_properties_of_the_generalized_Laguerre_polynomials)
\begin{equation\*}
L\_m^a(x)=\sum\_{i=0}^m b\_i,\q... | 2 | https://mathoverflow.net/users/36721 | 417418 | 170,034 |
https://mathoverflow.net/questions/417414 | 1 | $T>0$ is a parameter.
Consider the linear Diophantine equation $ax+by=c$ where $a,b$ are coprime.
Suppose $a,b$ are of magnitude $T^{1+\epsilon}$ and $c$ is of magnitude $T^2$.
1. For how many such equations we can expect $x,y$ to be of magnitude $T^{1+\epsilon}$ for a fixed $c$ and we vary $a,b$ coprime of magni... | https://mathoverflow.net/users/10035 | Fundamental solutions to linear Diophantine equations and their existence and computation | Solve $ax'+by'=1$, then take $$x = x'c - b \left\lfloor \frac{x'c}{ b} \right\rfloor$$
$$y = y'c +a\left\lfloor \frac{x'c}{ b} \right\rfloor$$
then we have $ax+by= ax'c +by'c = c$ and (if $b>0$ for simplicity) $0 \leq x < b$ so $$|x| < b$$ and $$|y| = \left| \frac{c-ax}{b} \right| \leq \frac{|c|}{|b|} + \frac{|a||x|}... | 4 | https://mathoverflow.net/users/18060 | 417419 | 170,035 |
https://mathoverflow.net/questions/417404 | 3 | I have noticed experimentally that the following question has a positive answer.
Is it true that for all even and convex functions $f$, $g$:
$$\int\_0^1 f(\sin(1/x)) \times g(\cos(1/x)) dx \leq \int\_0^1 f(\sin(1/x)) dx \times \int\_0^1 g(\cos(1/x))dx? $$
| https://mathoverflow.net/users/110301 | $\int_0^1 f(\sin(1/x)) \times g(\cos(1/x)) dx \leq \int_0^1 f(\sin(1/x)) dx \times \int_0^1 g(\cos(1/x))dx? $ | $\newcommand\abs[1]{\lvert#1\rvert}$It already suffices that $f$ and $g$ be even and nondecreasing on $[0,1]$ (which of course is the case if $f$ and $g$ are even and convex). Indeed, then the identity $\abs\sin^2+\abs\cos^2=1$ implies that for all real $u$, $v$ we have
$$(\abs{\sin u}-\abs{\sin v})(\abs{\cos u}-\abs{\... | 7 | https://mathoverflow.net/users/36721 | 417421 | 170,036 |
https://mathoverflow.net/questions/417411 | 3 | $\DeclareMathOperator\Mod{Mod}$Let $S$ be a surface and $P=\{a\_1,...,a\_n\}$ be a pants decomposition of $S$. Denote by $\Mod(S)$ the mapping class group of $S$.
Define the stabilizer of $\Mod(S)$ on $P$ to be $$A=\{f\in \Mod(S), f(a\_i)=a\_i, i =1,...,n\}.$$
What is $A$?
I was once told that it is generated by Dehn t... | https://mathoverflow.net/users/nan | Stabilizer of the action of the mapping class group on a pants decomposition | Suppose that $S$ is closed (without boundary), connected, and oriented.
If $S$ has genus two then the stabiliser is generated by Dehn twists about the $a\_i$, the hyperelliptic, and a reflection.
If $S$ has genus greater than two, then there is no hyperelliptic symmetry, but there will still be a reflection symmetr... | 2 | https://mathoverflow.net/users/1650 | 417432 | 170,039 |
https://mathoverflow.net/questions/417424 | 2 | My question is similar to [The mean of points on a unit n-sphere $S^n$](https://mathoverflow.net/questions/231501/the-mean-of-points-on-a-unit-n-sphere-sn).
I have a unit $n$-sphere $S^n$ and a set $P$ of points lying on its surface.
I use geodesic distance metric $d(p,q)=\arccos(pq^T)$.
Additionally I have a guarant... | https://mathoverflow.net/users/163471 | The mean of positive points on a unit $n$-sphere $S^n$ | The answer is no. E.g., let $P$ be the set $\{1,e^{it},e^{i3t}\}$ of points on the unit circle in $\mathbb C=\mathbb R^2$, where $t$ is a small positive real number. Then the geodesic mean of $P$ is
$$e^{i(4/3)t}=1+\frac{4 i t}{3}-\frac{8 t^2}{9}-\frac{32 i t^3}{81}+O\left(t^4\right),$$
whereas the arithmetic mean of $... | 3 | https://mathoverflow.net/users/36721 | 417434 | 170,040 |
https://mathoverflow.net/questions/417439 | 2 | Let $p$ integer prime, $f$ a function of $A=\mathbb F\_p^n$ to $\mathbb F\_p$, with $n\geq p+1$.
Is it true that : for all $x\in A, \sum\limits\_{\sigma \in S\_n} s(\sigma) \times f(x\_\sigma) =0$?
$s$ the signature
$S\_n$ is the group of all bijection of $U\_n=\{1,...,n\}$ to $U\_n$.
If $x=(x\_1,...x\_n)$ then... | https://mathoverflow.net/users/110301 | A new convolution, on function of $\mathbb F_p^n$ to $\mathbb F_p$ still zero? | As long as $n \geq p+1$, two of the entries of $x$ must be the same by the pigeonhole principle. Let $\tau$ be a transposition fixing those two entries. Then $s(\sigma \circ \tau) = -s(\sigma)$ but $x\_{\sigma \circ \tau} = x\_\sigma$ so $f(x\_{\sigma \circ \tau} ) =f(x\_\sigma)$. Thus the terms in your sum for $\sigma... | 7 | https://mathoverflow.net/users/18060 | 417442 | 170,041 |
https://mathoverflow.net/questions/417444 | 6 | I have a question involving preservation of cofiltered limits. Ordinarily this would be a very boring question, but it comes up in condensed math in its analogue of the completeness concept.
The concept is that of "solid", which says that for each profinite set $S = \text{lim} S\_i$, maps of condensed abelian groups ... | https://mathoverflow.net/users/30211 | Condensed math and cofiltered limits | It should be noted that already in $\mathbf{Set}$, the free functor $\mathbf Z^{(-)} \colon \mathbf{Set} \to \mathbf{Ab}$ does not preserve cofiltered limits. For a cofiltered diagram $D \colon \mathcal I \to \mathbf{Set}$, write $S\_i$ for its value at $i \in \mathcal I$, write $S$ for its limit, and write $\pi\_i \co... | 12 | https://mathoverflow.net/users/82179 | 417452 | 170,043 |
https://mathoverflow.net/questions/417454 | 11 | Recently I learned a nice [constructive proof of the irrationality of $\sqrt{2}$](https://en.wikipedia.org/wiki/Square_root_of_2#Constructive_proof), which uses the 2-adic valuation of an integer: the count of how many times a number is divisible by 2. The valuation requires *some* induction to construct, and [this nic... | https://mathoverflow.net/users/4177 | How much induction does a p-adic valuation need? | If you want to stick to theories in the basic language of arithmetic $\langle0,1,+,\cdot,<\rangle$, the irrationality of $\sqrt2$ can be easily proved in the theory $IE\_1$ (i.e., using induction for bounded existential formulas), since it proves the $\gcd$ property; or even more directly, you can just prove
$$\forall ... | 9 | https://mathoverflow.net/users/12705 | 417460 | 170,045 |
https://mathoverflow.net/questions/333536 | 11 | $\DeclareMathOperator\PGL{PGL}\DeclareMathOperator\Conj{Conj}$Let $G$ be the symmetric group $S\_n$ or the projective general linear group $\PGL\_2(n)$.
Let $X$ be a cyclically reduced word in the abstract variables
$x\_1, x\_2, \ldots,x\_k$, i.e. $X$ is a product containing $x\_1, x\_2, \ldots,x\_k$ and their invers... | https://mathoverflow.net/users/125498 | Probability of words summing to $1$ in $S_n$ or $\mathrm{PGL}_2(n)$ | $\DeclareMathOperator\PGL{PGL}$I believe the best result in this direction is due to M. Larsen & A. Shalev (2012); see [this](https://www.researchgate.net/publication/256735756_Fibers_of_word_maps_and_some_applications) paper. I'll summarize their results here. This doesn't answer the questions whether the limit exists... | 5 | https://mathoverflow.net/users/474608 | 417466 | 170,047 |
https://mathoverflow.net/questions/417462 | 1 | I am looking for a copy of the paper "On some problems of Bellman and a theorem of Romanoff", P. Erdős, J. Chinese Math. Soc. 1951. Can someone help me by providing a link or copy of the paper?
| https://mathoverflow.net/users/160943 | Looking for a 1951 paper by Erdős titled "On some problems of Bellman and a theorem of Romanoff", published in J. Chinese Math. Society | I found a digital version of paper [1] in [the list of published papers of Paul Erdős](https://www.renyi.hu/%7Ep_erdos/Erdos.html): click on the title of the reference below and you'll see the same digital object.
**Reference**
[1] Paul Erdős, "[On some problems of Bellman and a theorem of Romanoff](https://www.ren... | 3 | https://mathoverflow.net/users/113756 | 417467 | 170,048 |
https://mathoverflow.net/questions/417238 | 0 | Let $\mu$ be a probability distribution on $\mathbb R^d$ with "sufficiently regular" density $p$. Let $f:\mathbb R^d \to \mathbb R$ be a "sufficiently regular" function. Finally, for every $t \ge 0$, define
$$
s\_f(t) := \mu(f^{-1}((-\infty,t])) = \int\_{f^{-1}((-\infty,t])}p(x)dx
$$
>
> **Question.** *What is the ... | https://mathoverflow.net/users/78539 | Time-derivative of integral over sub-level set $s(t) := \int_{f^{-1}((-\infty,t])}p(x)dx$ | This is a comment (but too long for that format) to suggest a simple and elementary approach: we can write
$$s\_f(t)=\int H(t-f(x))p(x)\,dx$$ and manipulate formally (but see below) to get
$$\frac{d}{dt}s\_f(t)=\int \delta(t-f(x))p(x)\,,dx$$ where $H$ and $\delta$ are the Heaviside function and the Dirac distribution r... | 1 | https://mathoverflow.net/users/317800 | 417471 | 170,049 |
https://mathoverflow.net/questions/417263 | 11 | It seems there are many subtly different notions of the shape of a topological space (and, more generally, toposes).
For instance, Lurie [*Higher topos theory*] defines this one:
**Definition 1.**
The shape of a topos $\mathcal{E}$ is the pro-object in $\mathcal{S}$ representing the endofunctor $p\_\* p^\* : \mathc... | https://mathoverflow.net/users/11640 | What is the connection between Lurie's definition of shape and Čech homotopy? | The plus construction has to be iterated, yes. The topological space from [this answer](https://mathoverflow.net/a/31386/20233) provides a simple counterexample. Let $X=\{a,b,c,d\}$ with opens $\{a\},\{b\},\{a,b\},\{a,b,c\},\{a,b,d\}$. Then the plus construction does not change the global sections of the constant presh... | 3 | https://mathoverflow.net/users/20233 | 417481 | 170,052 |
https://mathoverflow.net/questions/417477 | 9 | Let $\mathbb{F}\_2=\{0,1\}$ be the field with two elements, and let
$u:\mathbb{F}\_2^n\rightarrow \mathbb{F}\_2$. *Suppose that $n$ is odd.*
>
> Is it possible that
> $$
> \sum\_{x \in \mathbb{F}\_2^n}(-1)^{u(x)+u(x+a)}= 0,
> $$
> for every $a \neq 0$ in $\mathbb{F}\_2^n$?
>
>
>
I treat the sum here as natural... | https://mathoverflow.net/users/46290 | Are there functions $\mathbb{F}_2^n \to \mathbb{F}_2$ satisfying these special relations? | I think the answer is no. To see this, observe that
\begin{equation\*}
\left(\mathbb{E}\_{\mathbf{x}}[(-1)^{u(\mathbf{x})}]\right)^2=\mathbb{E}\_{\mathbf{x}}[(-1)^{u(\mathbf{x})}]\mathbb{E}\_{\mathbf{y}}[(-1)^{u(\mathbf{y})}]=\mathbb{E}\_{\mathbf{x},\mathbf{y}}[(-1)^{u(\mathbf{x})+u(\mathbf{y})}]=\mathbb{E}\_{\mathbf{x... | 12 | https://mathoverflow.net/users/170770 | 417489 | 170,054 |
https://mathoverflow.net/questions/417501 | 9 | What is an example of a Hopf algebra with a non-invertible antipode?
| https://mathoverflow.net/users/478224 | Hopf algebra with a non-invertible antipode | Theorem of Takeuchi (in *Free Hopf algebras generated by coalgebras, 1971*) asserts that free Hopf algebra $H(C)$ over a coalgebra $C$ has injective antipode, and it is bijective precisely (at least over alg. closed field) when $C$ is pointed.
On the other hand, in a paper *Faithful flatness over Hopf subalgebras - c... | 11 | https://mathoverflow.net/users/81055 | 417505 | 170,058 |
https://mathoverflow.net/questions/416734 | 4 | Let $(A,L)$ be a polarized abelian variety. I know that the degree of the polarization is the Euler characteristic of $L$, so that
$d = \chi(L) = \dim H^0(A,L)$
since $L$ is ample.
I've read in a lot of papers the sentence
>
> Let $(A,L)$ be a polarized abelian variety of dimension $g$ and of type $(d\_1, \dots, ... | https://mathoverflow.net/users/152522 | Type vs degree of a polarized abelian variety | **To answer your first question**, i.e., what is type on a (polarized) abelian variety $A$? It is already answered [here](https://mathoverflow.net/q/177246). Roughly speaking, for a polarized abelian variety $(A,L)$, there is an integral basis $\{dx\_i,dy\_i\}\_{i=1}^{g}$ of $H^1(A,\mathbb Z)$, and positive integers $d... | 1 | https://mathoverflow.net/users/74322 | 417512 | 170,060 |
https://mathoverflow.net/questions/417490 | 2 | Let $\mathfrak{so}(p,q)$ be the real definite/indefinite orthogonal Lie algebra, $p,q\ge0$, $p+q=n\in\mathbb{N}$, and $L\subset\mathfrak{so}(p,q)$ a Lie subalgebra with non-trivial centre, $\mathrm{Z}(L)\neq0$.
**Question:**
Is there an upped bound $c(p,q)$ on the dimension of $L$, $\dim L\le c(p,q)$, better than t... | https://mathoverflow.net/users/478218 | An upper bound on the dimension of a subalgebra of $\mathfrak{so}(p,q)$ with non-trivial centre | [Answer completely rewritten.]
Indeed we have:
>
> $\DeclareMathOperator\so{\mathfrak{so}}$Fix $n\ge 4$ and $p,q\ge 0$ with $p+q=n$; write $r=\min(p,q)$. Then the minimal codimension for the centralizer of a nonzero element in $\so(p,q)$ is $2n-6$ if $r\ge 2$ and $2n-4$ if $r\in\{0,1\}$.
>
>
>
(The cases $n=... | 3 | https://mathoverflow.net/users/14094 | 417519 | 170,062 |
https://mathoverflow.net/questions/417520 | 5 | Let $f = x^n + a\_{n-1}x^n + \cdots + a\_0$ be a monic polynomial of degree $n \geq 2$ with integer coefficients. By $\text{Gal}(f)$ we mean the Galois group over $\mathbb{Q}$ of the Galois closure of $f$. Define $H(f) = \max\{|a\_i|\}$ denote the naive or box height of $f$. Hilbert's irreducibility theorem asserts tha... | https://mathoverflow.net/users/160943 | Galois groups of specific classes of polynomials with one coefficient fixed | This is equivalent (by Hilbert) to asking whether the partially specialized polynomial still has symmetric Galois group (over the respective function field). This holds unless you specialized the constant coefficient to $0$.
According to *Cohen, S. D.*, [**The Galois group of a polynomial with two indeterminate coeff... | 9 | https://mathoverflow.net/users/127660 | 417527 | 170,064 |
https://mathoverflow.net/questions/417528 | 0 | This question is motivated by considerations on [conflict-free colorings](https://mathoverflow.net/questions/416793/conflict-free-coloring-of-mathbbr-with-the-euclidean-topology), which arose while studying assignment problems for frequencies in cellular networks.
A *[hypergraph](https://en.wikipedia.org/wiki/Hypergr... | https://mathoverflow.net/users/8628 | Conflict-free coloring of linear hypergraphs on $\omega$ | Let $V$ be the set of points in the affine plane $\mathbb Q^2$, and $E$ be the set of lines.
Assume that the coloring is possible. Say a line is *white* if it contains a unique black point, and *black* otherwise.
There exist two white points (on two parallel lines), hence a white line through them. There exists a w... | 1 | https://mathoverflow.net/users/17581 | 417532 | 170,067 |
https://mathoverflow.net/questions/417388 | 6 | Let $S$ be a monoid. On p. xvii of P.M. Cohn's *Free Ideal Rings and Localization in General Rings* (CUP, 2006), one reads that
* an element $u \in S$ is *regular* if (quote) "[...] it can be cancelled, i.e. $ua = ub$ or $au = bu$ implies $a = b$";
* $S$ is a *conical* monoid if (quote) "$ab = 1$ implies $a = 1$ (and... | https://mathoverflow.net/users/16537 | Problem 0.9.10 in Cohn's "Free Ideal Rings and Localization in General Rings" (CUP, 2006) | I mentioned this problem to George Bergman and he offered the following much simpler solution, which he has given permission for me to post. In his own words:
>
> Take any nonabelian group $G$ with a nontrivial homomorphism
> $f\colon G \to \mathbb{Z},$ and let $S$ be the submonoid of $G$ with
> underlying set $\{e... | 2 | https://mathoverflow.net/users/3199 | 417555 | 170,071 |
https://mathoverflow.net/questions/417550 | 4 | I need a series expansion to describe a general gaussian-like (bell shaped) function. I couldn't find a rigorous definition of "bell shaped" online but in essence the function should have the following:
1. $f(x)>0$ (positive)
2. $f(x)=f(-x)$ (symmetric)
3. $\int \_{-\infty}^{\infty}f(x)dx <\infty$
4. $f^{(n)}(x) = 0$... | https://mathoverflow.net/users/478260 | Series expansion for gaussian-like function | You could try a series of Hermite functions,
$$f(x)=\sum\_{n=0}^N a\_n \frac{d^{2n}}{dx^{2n}}e^{-x^2}.$$
The function $f$ satisfies your conditions 1,2,3,5 by construction, and if the $a\_n$'s decay rapidly with $n$ it will look "bell-shaped".
Actually, for unconstrained $a\_n$'s, and including also odd derivatives, ... | 2 | https://mathoverflow.net/users/11260 | 417569 | 170,076 |
https://mathoverflow.net/questions/417132 | 0 |
>
> If $f\in C^1(\mathbb R)$ satisfies $f'(x)>f(f(x))$ for all $x\in\mathbb R$, then $f(f(f(x)))\leq0$ for all $x\geq0$.
>
>
>
I have some trouble to prove this. I wonder if there's some relations between this problem and the ODE $ f'(x)=f(f(x)) $. Could anybody provide a solution or some hints on this problem?
... | https://mathoverflow.net/users/241460 | $f'(x)>f(f(x))$ implies $f(f(f(x)))\leq0$ for nonnegative $x$ | Well, this is Problem 4 of Day 1 of IMC 2012, proposed by Tomáš Bárta from Prague, see [here](https://www.imc-math.org.uk/imc2012/IMC2012-day1-solutions.pdf)
| 2 | https://mathoverflow.net/users/4312 | 417571 | 170,077 |
https://mathoverflow.net/questions/417493 | 1 | A copy of the Cantor set is a space homeomorphic to $2^{\omega}$.
Suppose that $X$ is a Hausdorff space that contains a copy $C^{\prime}$ of the Cantor set. Let $U$ be a nonempty subset open in $C^{\prime}$, also let $D$ be a countable set dense in $X$ such that $D\cap U$ is dense in $U$. Does anyone have any idea ho... | https://mathoverflow.net/users/475617 | Subsets of the Cantor set | To answer the question: every point of $U$ is an accumulation point of $D\cap U$, hence there are continuum many accumulation points. The ambient space $X$ plays no role here; everything takes place in the Cantor set.
| 3 | https://mathoverflow.net/users/5903 | 417579 | 170,080 |
https://mathoverflow.net/questions/417574 | 6 | Let $(X,d,m)$ be a metric measure space. We say that it is *doubling in the sense of metric spaces* if for every:
$x\in X$ and every $r>0$ there exists some **(metric) doubling constant** $C\_d\geq 0$ such that
$$
Ball(x,r) \mbox{ can be covered by at-most $C\_d$ balls of radius $r/2$}.
$$
There is a different, [rela... | https://mathoverflow.net/users/469470 | Relationship between doubling constant of a metric space and of a metric measure space | Apart from the obvious counterexample of the measure being $0$, if $(X,d,m)$ is doubling in the sense of metric measure spaces it will be doubling in the sense of metric spaces.
Consider a ball $B(x,r)$. If for some $n$, $B(x,r)$ cannot be covered by $n$ balls of radius $\frac{r}{2}$, then we can obtain by recursion ... | 10 | https://mathoverflow.net/users/172802 | 417580 | 170,081 |
https://mathoverflow.net/questions/417572 | 2 | On paper [A procedure for improving the upper bound
for the number of $n$-ominoes](https://www.cambridge.org/core/services/aop-cambridge-core/content/view/CA6D64DA91090FFAA49EBC83C098E3AF/S0008414X00050628a.pdf/a-procedure-for-improving-the-upper-bound-for-the-number-of-n-ominoes.pdf) by D. A. Klarner & R. L. Rivest, i... | https://mathoverflow.net/users/174530 | Number of polyominoes with area $n$ | Wikipedia suggests that no-one has improved on Klarner, D.A.; Rivest, R.L. (1973) [A procedure for improving the upper bound for the number of n-ominoes](https://www.cambridge.org/core/services/aop-cambridge-core/content/view/CA6D64DA91090FFAA49EBC83C098E3AF/S0008414X00050628a.pdf/a-procedure-for-improving-the-upper-bo... | 6 | https://mathoverflow.net/users/46140 | 417581 | 170,082 |
https://mathoverflow.net/questions/417256 | 6 | Motivation: for any ring $R$ there is the natural monomorphism $\mathrm{in} \colon R \to \mathrm{End}(R\_{add}): r \mapsto (x \mapsto rx)$, where $R\_{add}$ is an additive abelian group ( rings are assumed to be associative with identity, but not necessarily commutative). So a ring is exactly an abelian group with a di... | https://mathoverflow.net/users/148161 | Abelian groups such that $A \cong \mathrm{End}(A)$ and "complete rings" | The rings, you call ``complete'' are known as $E$-rings (as Ulrich Pennig mentioned in the comments).
Some comments on your questions
1. There are too many results on the $E$-rings to list them here and I'd rather direct you to the book by Göbel and Trlifaj *Approximations and endomorphism algebras of modules*. How... | 6 | https://mathoverflow.net/users/16678 | 417582 | 170,083 |
https://mathoverflow.net/questions/396372 | 2 | Let $K$ be a finite extension of $\mathbb{Q}\_p$. Let $\mathcal{O}$ be its ring of integers and $\mathfrak{m}$ the maximal ideal. Pick a uniformiser $\pi$. The construction using theory of Lubin--Tate formal groups yields an abelian Galois extension $K\_\pi$ of $K$ such that $\mathrm{Gal}(K\_\pi/K)\simeq\mathcal{O}^\ti... | https://mathoverflow.net/users/44005 | Field extension corresponding to a quotient of units of local fields | The isomorphism you write down involving the $\mathbb{Z}\_p$ is not canonical, so one cannot describe them without fixing additional data.
For $\mathbb{Q}\_p$ you have $a=1$ and this tower is the totally ramified (p-)cyclotomic tower.
| 3 | https://mathoverflow.net/users/478296 | 417585 | 170,084 |
https://mathoverflow.net/questions/417197 | 5 | Fix integers $1 \leq k \leq n$ and suppose $\mathbf{x} \in \mathbb{R}^n$ is such that $e\_j(x\_1,x\_2,\ldots,x\_n) \geq 0$ for all $1 \leq j \leq k$, where $e\_j$ is the $j$-th [elementary symmetric polynomial](https://en.wikipedia.org/wiki/Elementary_symmetric_polynomial).
>
> **Question 1:** Is it true that $x\_1... | https://mathoverflow.net/users/11236 | Dimension reduction for non-negativity of elementary symmetric polynomials | The answer to question 2, and therefore question 1, is "yes". We abbreviate $e\_k(x\_1, x\_2, \ldots, x\_{n-1})$ to $b\_k$ and $e\_k(x\_1, x\_2, \ldots, x\_{n-1}, x\_n)$ to $a\_k$, so $a\_k = x\_n b\_{k-1} + b\_k$.
We are trying to show that, if $a\_1$, $a\_2$, ..., $a\_k \geq 0$ then $b\_1$, $b\_2$, ..., $b\_{k-1} \... | 4 | https://mathoverflow.net/users/297 | 417597 | 170,087 |
https://mathoverflow.net/questions/417500 | 7 | Let $X$ be a compact Riemann surface of genus $g$, then $K^1\_{\mathrm{top}}(X)\cong\mathbb{Z}^{2g}$. Is there a explicit description of a set of basis of $K^1\_{\mathrm{top}}$? (e.g., For cohomology $H^1(X,\mathbb{Z})\cong\mathbb{Z}^{2g}$ we may take the 1-cochains ``around the holes'')
Furthermore, we define the Mu... | https://mathoverflow.net/users/nan | Topological K-theory of Riemann surface | Following @Kiran's suggestion in the comments, I'll outline why the map $U(1)\to U$ induces an isomorphism between cohomology and K-theory in this setting. At the end I'll also explain a different perspective that might be helpful.
The inclusion maps $U(n)\to U(n+1)$ are $(2n-1)$-connected, so the map $U(1)\to U$ is ... | 4 | https://mathoverflow.net/users/4042 | 417598 | 170,088 |
https://mathoverflow.net/questions/417602 | 2 | I'm reading a paper on the classical Gagliardo-Nirenberg interpolation inequality [arXiv link](https://arxiv.org/abs/1812.04281) and there is a inequality used
$$
|v-\overline{v}|\le \left\Vert v' \right\Vert\_{r,I} \ell^{1-\frac{1}{r}}, r\ge 1
$$
where $\overline{v}:=\frac{1}{\ell}\int\_I v(x)dx$, $I$ is an interval o... | https://mathoverflow.net/users/295572 | A simple 1-dimensional inequality, maybe Poincaré inequality or Hölder inequality? | By the mean value theorem, $\bar v=v(t)$ for some $t\in I$. So, for all $x\in I$,
$$|v(x)-\bar v|=|v(x)-v(t)|
=\Big|\int\_t^x v'\Big|
\le\int\_I|v'|\le\|v'\|\_r\, \ell^{1-1/r};$$
the latter inequality is an instance of Hölder's inequality.
| 3 | https://mathoverflow.net/users/36721 | 417606 | 170,091 |
https://mathoverflow.net/questions/416794 | 5 | I have several questions regarding proposition 2.3 in "Cherednik and Hecke algebras of varieties with a finite group action", by Pavel Etingof. Let $X$ be a complex affine algebraic variety and $g$ an automorphism of finite order. We define the $D(X)$-bimodule $D(X)g$ by $a(bg)c=abg(c)g$.
1. In the aforementioned pro... | https://mathoverflow.net/users/476832 | Two identities involving Ext functors in the context of D-modules | I asked Prof. Etingof and he answered the following:
1. A $D$-module on $X\times X$ is a module over $D(X)\otimes D(X)$, while initially, these are bimodules over $D(X)$, which are modules over $D(X)\otimes D(X)^{op}$. The equivalence I mentioned is a Morita equivalence between $D(X)$ and $D(X)^{op}$ which defines a ... | 2 | https://mathoverflow.net/users/476832 | 417615 | 170,093 |
https://mathoverflow.net/questions/417440 | 3 | $\DeclareMathOperator\Aut{Aut}\DeclareMathOperator\Gl{Gl}$Let $\mathcal{D}=\mathbb{C}[[t\_{1},\dotsc , t\_{n}]]$ be a formal polydisc over $\mathbb{C}$, and $G$ be a finite group. On Lemma 7.8 of [Etingof and Ma - Lecture notes on Cherednik algebras](https://arxiv.org/abs/1001.0432) it is stated that every action $\rho... | https://mathoverflow.net/users/476832 | On the linearizability of the action of a finite group on a formal polydisc | For $1 \to A \to B \to C \to 1$ an exact sequence of (not necessarily abelian) algebraic groups, we have a long exact sequence
$$ H^0(G, A) \to H^0(G, B) \to H^0(G, C) \to H^1(G, A) \to H^1(G, B) \to H^1(G, C) $$
It follows that if $H^1(G,A) = H^1(G, C) =0$ then $H^1(G,B)=0$.
We can now prove that for finite $G$,... | 2 | https://mathoverflow.net/users/18060 | 417617 | 170,094 |
https://mathoverflow.net/questions/417631 | 1 | This is a question about central limit theorems when the dimension is increasing. Suppose now I have a random vector $X\_N = (X\_{N1}, \cdots, X\_{Np})\in\mathbb{R}^p$. For all $c\_p\in\mathbb{R}^p$ with $\|c\_p\|\_2 = 1$, suppose we have $c\_p^\top X\_N \xrightarrow{d} N(0,1)$ as $N\to\infty, p\to\infty$. What can we ... | https://mathoverflow.net/users/465240 | Central limit theorem of random vectors when the dimension is increasing | In such generality, virtually nothing can be said about the asymptotic distribution of $V\_{Np}:=\sum\_{i=1}^p X\_{Ni}^2$ or even about the existence of such an asymptotic distribution. In particular, $V\_{Np}$ may have a non-normal asymptotic distribution or no asymptotic distribution at all.
Indeed, consider the fo... | 2 | https://mathoverflow.net/users/36721 | 417633 | 170,100 |
https://mathoverflow.net/questions/398509 | 7 | The definition of $L\_\infty$-algebra is by now pretty standard. I gather that the sign conventions given in Lada–Markl's paper *Strongly homotopy Lie algebras*, Communications in Algebra **23** Issue 6 (1995) (arXiv:[hep-th/9406095](https://arxiv.org/abs/hep-th/9406095)) are widely used, and I will keep to them here. ... | https://mathoverflow.net/users/4177 | What is the correct definition of weak map between 2-term $L_\infty$-algebras? | In the article *Classification of 2-term $L\_\infty$-algebras* (arXiv:[2109.10202](https://arxiv.org/abs/2109.10202)), Kevin van Helden gives the definition of a morphism of 2-term $L\_\infty$-algebras (Definition 2.3), and he was kind enough to share with me some private calculations that go through and checks the def... | 2 | https://mathoverflow.net/users/4177 | 417635 | 170,102 |
https://mathoverflow.net/questions/417640 | 10 | Given a finite field $\mathbb{F}\_q$ with $q=p^m$ where $p$ is the characteristic.
For any subset $S=\{a\_1,\dots,a\_n\}$ of $\mathbb{F}\_q$, if any partial sum (i.e. the sum of elements in a non-empty subset of $S$) is non-zero, then we may call $S$ a good subset.
My question is what's the maximal cardinality $f(q)$... | https://mathoverflow.net/users/471160 | The maximal subset of a finite field where the sum of any subset is non-zero | I could trace this question down to the paper of Erdős and Heilbronn "[On the addition of residue classes mod $p$](https://doi.org/10.4064/aa-9-2-149-159)" (Acta Arith. 17 (1970), 227–229), where it is shown that for a prime $p$, if $A$ is a subset of the $p$-element group with $\lvert A\rvert>6\sqrt{3p}$, then the sub... | 16 | https://mathoverflow.net/users/9924 | 417646 | 170,107 |
https://mathoverflow.net/questions/417621 | 8 | Let $K \subset S^3$ be a slice knot. Then it bounds a smooth embedded disk $D \subset B^4$. Let $S^3\_{p/q}(K)$ denote a $3$-manifold obtained by $p/q$-surgery on $K \subset S^3$.
The following theorem is due to Gordon:
>
> Gordon, C. M. (1975). Knots, homology spheres, and contractible 4-manifolds. Topology, 14(... | https://mathoverflow.net/users/475366 | Gordon's approach: slice knots and contractible $4$-manifolds | Yes, the generalisation is also true. This must be written somewhere, but I don't know where (any help from other users?), and finding such a statement is often hard.
So, here's the idea instead. Turn the surgery into an integral surgery, i.e. do 0-surgery on $K$ and $-n$-surgery on a meridian $L$ of $K$. 4-dimension... | 6 | https://mathoverflow.net/users/13119 | 417648 | 170,109 |
https://mathoverflow.net/questions/417110 | 1 | Consider the Cauchy problem
$$\left\{\hspace{5pt}\begin{aligned}
&-\dfrac{\partial u }{\partial t}
+a\dfrac{\partial^2 u}{\partial x^2}
+b \dfrac{\partial u }{\partial x}
+c u
= f(u) \leq 0& \hspace {10pt} &\text{for $(x,t) \in \mathbb{R} \times (0,T]$}
;\\
&u(x,T) = g(x)\geq 0 & \hspace{10pt} &\text{for $x \in \math... | https://mathoverflow.net/users/87922 | Local boundedness for Cauchy problem | I finally find a result in Theorem 11.17 in Chapter XI.6 in parabolic PDE book by Gary Lieberman. But it restricts on the case of one dimension space.
| 0 | https://mathoverflow.net/users/87922 | 417650 | 170,110 |
https://mathoverflow.net/questions/417187 | 2 | I encountered an example in a paper telling that $\underline{SM}(\mathbb{R}^{0|1},X)\cong \pi TX $, where $X$ is some fixed ordinary Riemannian manifold, $\pi TX $ is the supermanifold with base manifold $X$ and structural sheaf $\pi(\wedge^\*(TX)^\vee) $ according to my advisor, seen as a functor $\pi TX:=SM(-,\pi TX)... | https://mathoverflow.net/users/167862 | Parity reversed tangent bundle as a supermanifold | My advisor told me the answer; I was just one step away from it.
A morphism $\varphi:S\to \pi TX$ is determined by the natural transformation between the structural sheaves, which is locally determined, so it suffices to assume that $X$ has coordinates $x\_1,\cdots,x\_n$. Now $C^\infty(\pi TX)=\pi(\wedge^\*(TX)^\vee)... | 1 | https://mathoverflow.net/users/167862 | 417651 | 170,111 |
https://mathoverflow.net/questions/417587 | 2 | I need a (numerically) evaluable function for the number $N\_{n,k}$ of endofunctions $f: [n] \rightarrow [n]$ without fixed points that have exactly $k$ two-cycles, where $[n] := \{1,\dotsc,n\}$. In formal terms, what is
$$N\_{n,k} := \lvert \{ f:[n]\rightarrow [n] \mid \forall a: f(a) \neq a \land \lvert\{ a \in [n] \... | https://mathoverflow.net/users/119246 | Number of endofunctions in [n] without fixed points with exactly k two-cycles | First choose your 2-cycles, for a factor of $\binom{n}{2k}(2k-1)!!$. (Note that we require the convention that $(-1)!! = 1$). Then count functions $g: [n-2k] \to [n]$ with no fixed points or 2-cycles. There are $\binom{n-2k}{2j}(2j-1)!! (n-1)^{n-2k-2j}$ functions with no fixed points and at least $j$ 2-cycles, so an in... | 3 | https://mathoverflow.net/users/46140 | 417653 | 170,112 |
https://mathoverflow.net/questions/417655 | 5 | Apart from the abstract types of the crystallographic groups, are there any other abstract groups that admit a proper, co-compact, uniformly bilipschitz action on $\mathbb{R}^3$?
| https://mathoverflow.net/users/159356 | Infinite groups that admit a discrete, co-compact, bilipschitz action on $\mathbb{R}^3$ |
>
> Fix $k\ge 0$. Let $\Gamma$ be a discrete group. Then $\Gamma$ (a) has a proper cocompact, uniformly bilipschitz action on the Euclidean space $\mathbf{R}^k$ if and only if (b) it has an isometric one, if and only if (c) it has a finite index subgroup isomorphic to $\mathbf{Z}^k$.
>
>
>
Proof: That (c) implie... | 5 | https://mathoverflow.net/users/14094 | 417661 | 170,114 |
https://mathoverflow.net/questions/417570 | -1 | A simple, undirected graph $G=(V,E)$ is said to be *vertex-transitive* if for all $a,b\in V$ there is a graph isomorphism $\varphi:G\to G$ such that $\varphi(a) = b$.
If $G = (\omega, E)$ is vertex-transitive and connected, is there a bijection $p:\mathbb{Z}\to \omega$ such that $\{p(k), p(k+1)\} \in E$ for all $k\in... | https://mathoverflow.net/users/8628 | Hamiltonian $\mathbb{Z}$-paths in connected countably infinite vertex-transitive graphs | If $G$ is a regular tree of degree $d \geq 3$, there's clearly no such $p$.
| 2 | https://mathoverflow.net/users/66104 | 417664 | 170,115 |
https://mathoverflow.net/questions/417436 | 6 | Cross Posting this from MSE since it's been there for almost a month and it got a couple upvotes but no answers. MSE link [Is every finite subgroup the integer points of a linear algebraic group?](https://math.stackexchange.com/questions/4375025/is-every-finite-subgroup-the-integer-points-of-a-linear-algebraic-group)
... | https://mathoverflow.net/users/387190 | Is every finite subgroup the integer points of a linear algebraic group? | The answer to this question is no.
There exist finite groups $\Gamma \subset K$ of a compact connected Lie group $K$, such that for any algebraic $\mathbb Q$-group $G$ with $G(\mathbb R)=K$, the finite group $\Gamma $ cannot be a subgroup of $G(\mathbb Z)$:
We take $K=SU(2)$ and $\Gamma $ to be a Dihedral group of th... | 6 | https://mathoverflow.net/users/23291 | 417672 | 170,118 |
https://mathoverflow.net/questions/417669 | 4 | Are there $2^{\aleph\_0}$ pairwise non-isomorphic connected [vertex-transitive](https://en.wikipedia.org/wiki/Vertex-transitive_graph) graphs $G$ with $V(G) = \omega$?
| https://mathoverflow.net/users/8628 | Non-isomorphic connected vertex-transitive graphs on $\omega$ | Yes. In particular, there are continuously many pairwise non-quasi-isometric Cayley graphs with countably infinite vertex set. A proof of this is in the paper *Continuously many quasi-isometry classes of 2-generator groups* by Bowditch.
| 5 | https://mathoverflow.net/users/159356 | 417675 | 170,119 |
https://mathoverflow.net/questions/417676 | 3 | Let $R$ be a (not necessarily commutative) ring, $M$ a left $R$-module, and $N$ a right $R$-module. We say that a pairing
$$
\langle -,-\rangle:M \otimes\_R N \to R
$$
is non-degenerate if, for all $n \in N$ there exists an $m \in M$ such that $\langle m,n\rangle \neq 0$, \textbf{and} for all $m \in M$, there exists an... | https://mathoverflow.net/users/478224 | Nondegenerate pairings versus perfect pairings for finitely generated projective modules | No: for $R=\mathbb{Z}$ and even for $M,N$ both free (i.e. free Abelian groups), non-degenerate doesn't imply perfect. (You get finite index sublattices, so torsion quotients, unlike the case of vector spaces.)
| 5 | https://mathoverflow.net/users/13215 | 417677 | 170,120 |
https://mathoverflow.net/questions/417666 | 0 | Is there a method to make a rep-n rep-tile for any number n, using only triangles? And if there is no such method, what's the smallest number for which there's no example?
I'm only considering rep-tiles which are all of the same size, as they make up the larger congruent rep-tile.
I first asked the more general quest... | https://mathoverflow.net/users/478353 | Is there a method to make a rep-n rep-tile for any number n, using only triangles? | You can indeed do it for rectangles, by taking a ratio of sides equal to $\sqrt{n}$.
For triangles, M. Beeson (2012), [Triangle Tiling I: the tile is similar
to ABC or has a right angle](https://arxiv.org/pdf/1206.2231.pdf) (pre-print) gives the following results:
>
> When the tile is similar to $ABC$, we always ... | 3 | https://mathoverflow.net/users/46140 | 417681 | 170,122 |
https://mathoverflow.net/questions/417641 | 1 | Let $X$ be a reflexive Banach space, $z, x, v \in X$. If $f: X \times X \rightarrow \mathbb{R}$ is continuous regarding its first argument and locally Lipschitz regarding its second argument. $\{z\_i\}, \{x\_i\}$ and $\{v\_i\}$ are arbitrary sequences converging to $z, x$ and $v$, respectively.
**I wonder** if genera... | https://mathoverflow.net/users/478336 | Proof that Clarke generalized directional derivative is upper continuous | $\newcommand\R{\mathbb R}$The answer is no.
Indeed, apparently, in your post the Clarke derivative is meant with respect to the second argument of $f$:
$$f^0(z,x;v):=\limsup\_{\substack{y\to x\\ t\downarrow0}}
\frac{f(z,y+tv)-f(z,y)}t.$$
Let now $X:=\R$ and
$$f(z,x):=xe^{-x/|z|}\,1(z\ne0,x>0)$$
for real $z,x$ -- th... | 0 | https://mathoverflow.net/users/36721 | 417683 | 170,123 |
https://mathoverflow.net/questions/417684 | 11 | Does anyone know anything about M. Meyniel? [According to zbMath](https://www.zbmath.org/authors/?q=ai%3Ameyniel.m), he published precisely one mathematics paper, in which he gave a sufficient condition for hamiltonicity of digraphs:
>
> "[Une condition suffisante d'existence d'un circuit hamiltonien dans un graphe... | https://mathoverflow.net/users/2663 | Who is M. Meyniel? | You can be quite sure that M. Meyniel means "Monsieur Meyniel" (a common usage in French).
Here is what I think is definite proof that M. Meyniel is H. Meyniel: The acknowledgement of the 1973 paper by M. Meyniel thanks J.C. Bermond, so evidently Bermond knew the author. In the article [Cycles in digraphs - a survey]... | 29 | https://mathoverflow.net/users/11260 | 417685 | 170,124 |
https://mathoverflow.net/questions/417698 | 4 | Let $a$ be an element of $\mathbb{F}\_p$, which is not a quadratic residue.
Define $$f(x) = \frac{x + a}{x+1},$$ which is a rational function on $\mathbb{F}\_p$. In fact, if we set $f(-1)=\infty$ and $f(\infty)=1$, then $f:\mathbb{F}\_p\cup\{\infty\}\rightarrow \mathbb{F}\_p\cup\{\infty\}$ is a bijection.
What is t... | https://mathoverflow.net/users/166993 | Order of a rational function on $\mathbb{F}_p$ | As @DavidESpeyer [suggests](https://mathoverflow.net/questions/417698/order-of-a-rational-function-on-mathbbf-p#comment1072032_417698), I think you meant ${}+ a$ in place of ${}- a$. As @KevinCasto [says](https://mathoverflow.net/questions/417698/order-of-a-rational-function-on-mathbbf-p#comment1072031_417698), you are... | 7 | https://mathoverflow.net/users/2383 | 417700 | 170,131 |
https://mathoverflow.net/questions/417701 | 2 | Let $p > 2$ be a prime and $q = p^r$ for some $r \in \mathbb{Z}^+$. I will assume that all roots of unity lie in $\mathbb{C}\_p^{\times}$. Let $\zeta$ a primitive $p$-th root of unity. Let $Tr : \mathbb{F}\_q \to \mathbb{F}\_p$ be the trace. Also, denote $\pi$ to be the maximal prime in $\mathbb{Z}\_p[\zeta]$ such that... | https://mathoverflow.net/users/171396 | Value of the quadratic Gauss sum viewed in $\mathbb{C}_p$ | That $G(\psi,\zeta)^2 = \psi(-1)p$ is pure algebra, so it holds in $\mathbf C$ or $\mathbf C\_p$ or any other field not of characteristic $2$ that contains a nontrivial $p$th root of unity.
You could write down a $p$-adic formula for your quadratic Gauss sum using the Gross–Koblitz formula.
First let's normalize th... | 7 | https://mathoverflow.net/users/3272 | 417710 | 170,135 |
https://mathoverflow.net/questions/417539 | 1 | I recall two definitions from Banach space theory
>
> **Definition 1.** Let $E$ be a Banach space, then a basis $(e\_n)\_{n\in\mathbb{N}}$ of $E$ is called $1$-spreading if $$\left\|\sum\_{i=1}^k a\_i e\_{m\_i}\right\|\le\left\|\sum\_{i=1}^k a\_i e\_{n\_i}\right\|$$ whenever $k$ is a positive integer, $(a\_i)\_{i=0... | https://mathoverflow.net/users/141146 | Definition of $1$-spreading basis and spreading model | It might be more useful to read first the structure of classical $\ell\_p$ spaces before starting on spreading models. These questions are about those and has very little to do with spreading models (only exception is the pull back mentioned below).
1. The only Banach space with any one of these properties is the Hil... | 7 | https://mathoverflow.net/users/3675 | 417714 | 170,136 |
https://mathoverflow.net/questions/417502 | 4 | I'm not sure this question fully qualifies as a research-level math question, but from my (limited) past experience on stackexchanged I feared this question might not get an answer there.
**Setting**: the category $\cal{O}$ associated to a complex semisimple Lie algebra $\frak{g}=\frak{n}\oplus\frak{h}\oplus\frak{n}^... | https://mathoverflow.net/users/476521 | Spectral sequence from standard/Verma filtration/flag to compute Lie algebra cohomology of tensor product with respect to $\mathfrak{n}$ | I think the issue here is that the subquotients in the standard filtration have weights (your $\lambda\_i$'s) which are ordered the other way. To be clear, if the weights of $\nu\_0$, ..., $\nu\_n$ of $L$ are ordered so that $\nu\_i \le \nu\_j$ implies $i\le j$, then one obtains a standard filtration for $M\_\lambda \o... | 4 | https://mathoverflow.net/users/138150 | 417716 | 170,137 |
https://mathoverflow.net/questions/417690 | 148 | Teaching group theory this semester, I found myself laboring through a proof that the sign of a permutation is a well-defined homomorphism $\operatorname{sgn} : \Sigma\_n \to \Sigma\_2$. An insightful student has pressed me for a more illuminating proof, and I'm realizing that this is a great question, and I don't know... | https://mathoverflow.net/users/2362 | Conceptual reason why the sign of a permutation is well-defined? | (This is a variant of Cartier's argument [mentioned](https://mathoverflow.net/a/417727) by Dan Ramras.)
Let $X$ be a finite set of size at least $2$. Let $E$ be the set of edges of the complete graph on $X$. The set $D$ of ways of directing those edges is a torsor under $\{\pm1\}^E$. Let $G$ be the kernel of the prod... | 95 | https://mathoverflow.net/users/2757 | 417732 | 170,145 |
https://mathoverflow.net/questions/417691 | 0 | Let $X$ be a reflexive Banach space, $z, x, v \in X$. $\{z\_i\}, \{x\_i\}$ and $\{v\_i\}$ are arbitrary sequences converging to $z, x$ and $v$, respectively.
**I would like to know** under which **conditions of the function $f: X \times X \rightarrow \mathbb{R}$**, generalized directional derivative $f^{0}(z, x; v)$ ... | https://mathoverflow.net/users/478336 | The conditions used to prove upper semicontinuous of generalized directional derivative (in Clarke sense) | $\newcommand\R{\mathbb R}$The conditions that you can accept are not enough for the conclusion that you desire.
Indeed, as in the [previous answer](https://mathoverflow.net/a/417683/36721), let $X:=\R$ and
$$f(z,x):=xe^{-x/|z|}\,1(z\ne0,x>0)$$
for real $z,x$ -- that is, $f(z,x)=xe^{-x/|z|}$ if $z\ne0$ and $x>0$, and ... | 0 | https://mathoverflow.net/users/36721 | 417734 | 170,147 |
https://mathoverflow.net/questions/417740 | -1 | What is an example of a simple graph $G = (\{1,\ldots,n\}, E)$, where $n\in\mathbb{N}$ is a positive integer, with the following properties?
1. There is a path in $G$ of length $n$,
2. every vertex has at least $2$ neighbors, and
3. $G$ does *not* have a [Hamiltonian cycle](https://en.wikipedia.org/wiki/Hamiltonian_p... | https://mathoverflow.net/users/8628 | Path of length $n$ but no Hamilton cycle | Assuming you mean a graph with a Hamiltonian path but no Hamiltonian cycle,
* The Petersen graph is a standard example.
* Any pendant-free graph with a Hamiltonian path and a bridge is an easy example; e.g. take two pendant-free graphs $G\_1$ and $G\_2$ with Hamiltonian paths (and optionally with Hamiltonian cycles).... | 3 | https://mathoverflow.net/users/46140 | 417749 | 170,151 |
https://mathoverflow.net/questions/417744 | 1 | Lets say I take an arbitrary closed and smooth $d$-manifolds $\mathcal{M}$. Now, it is a well-known fact that whenever I take two (sufficiently nice embedded) closed $d$-balls $B\_{1}$ and $B\_{2}$ in $\mathcal{M}$, then the manifold obtained by cutting out the interior of $B\_{1}$ is homeomorphic to the manifold obtai... | https://mathoverflow.net/users/259525 | Annulus theorem for pseudomanifolds | The homeomorphism type of the space left after deleting the interior of the ball can depend on whether the ball intersects the singular locus (necessarily in the boundary of the ball) or not. If your ball intersects the singular locus, then once you remove the ball interior such points need not have neighborhoods homeo... | 3 | https://mathoverflow.net/users/6646 | 417751 | 170,152 |
https://mathoverflow.net/questions/417747 | 4 | Let $K=\mathbb{Q}(\sqrt{6})$. I am looking to determine all $K$-rational points on the curve
$$C: y^{2}=3x^6-24x^5+24x^4-54x^3+24x^2-24x+3.$$
More precisely, $C$ is a twist of the modular curve $X\_{0}(26)$. I know that Bruin and Najman (<https://arxiv.org/abs/1406.0655>) have determined all quadratic points on $X\_{0}... | https://mathoverflow.net/users/409515 | Finding the $K=\mathbb{Q}(\sqrt{6})$-rational points on the twist of $X_{0}(26)$ | I used Magma to point search on $C/K$ up to a height of $1000$ and it appears that $C(K) = \emptyset$. If that's true, then one can probably use the Mordell-Weil sieve to prove it. Here's a bit more detail.
The curve $C$ has four automorphisms defined over $\mathbb{Q}$ and for one of these (the map $(x,y) \mapsto (1/... | 5 | https://mathoverflow.net/users/48142 | 417763 | 170,154 |
https://mathoverflow.net/questions/417759 | -1 | Given two compact oriented surfaces that have the same number of genus and boundary components. How to construct a homeomorphism that sends one to another?
| https://mathoverflow.net/users/nan | Construct a homeomorphism between two surfaces | A closed surface of genus g can be cut along 2g closed curves (all at one base point) to obtain a 4g-gon. Do this for both surfaces. Then choose a homeomorphism between the 4g-gons which matches the corresponding pairs of curves, so you get a well-defined homeomorphism of surfaces. (A 4g-gon is a cone over some interio... | 2 | https://mathoverflow.net/users/39082 | 417764 | 170,155 |
https://mathoverflow.net/questions/417755 | 2 | Let $K(x)\_{n\times n}$ be a positive definite matrix defined on $x\in D$ and $K\_{i,j}(x)\in C^2(D)$ (or generally $C^k$) for any $1\le i,j\le n$. Of course for any $x$, there exists a invertable matrix $A(x)$ such that $A(x)A(x)^t=K(x)$, and $A(x)$ is not unique. What I want to ask is, whether $K(x)$ has $C^2$ (or $C... | https://mathoverflow.net/users/176547 | Decomposition of a positive definite matrix | The [Cholesky decomposition](https://en.wikipedia.org/wiki/Cholesky_decomposition) does what you want. It depends smoothly on the input matrix, because every step in the algorithm is a smooth function. It's all just basic arithmetic and square roots.
| 3 | https://mathoverflow.net/users/3041 | 417771 | 170,159 |
https://mathoverflow.net/questions/417558 | 2 | It is known that the $T\_0$ and $T\_2$ axioms are not preserved under open, closed and continuous maps (for instance, see here: [An example of open closed continuous image of $T\_0$-space that is not $T\_0$](https://math.stackexchange.com/questions/1411751/an-example-of-open-closed-continuous-image-of-t-0-space-that-is... | https://mathoverflow.net/users/146942 | Preservation of separation axioms under perfect functions | Yes, it seems to be true.
First, towards a counterexample. Let $f\colon X \to Y$ be a perfect map from a $T\_0$ space. If $Y$ is not $T\_0$, it contains a two-point indiscrete space $B$, and $f\colon f^{-1}[B] \to B$ is also a perfect map. So if there is a counterexaple at all, there is a counterexample realized by a... | 2 | https://mathoverflow.net/users/112373 | 417779 | 170,162 |
https://mathoverflow.net/questions/417484 | 6 | *This is a minor variation of a question originally [asked on MSE](https://math.stackexchange.com/questions/4349128/partial-consistency-of-peano-arithmetic) by user779130 and bountied by me, without success. Throughout, "length" refers to the number of symbols, not lines, in a proof.*
For $T$ an "appropriate" theory,... | https://mathoverflow.net/users/8133 | Proving short consistency: can we do better than brute force search? | Since the concept involves two theories, but the notation in the question only indicates one of them, I will instead write $p\_{S,T}(n)$ for the shortest $S$-proof of $\DeclareMathOperator\con{Con}\con\_T(\def\ob{\overline}\ob n)$, expressing that there is no $T$-proof of contradiction of length $<n$.
Here, each of $... | 3 | https://mathoverflow.net/users/12705 | 417783 | 170,163 |
https://mathoverflow.net/questions/417780 | 8 | By Deligne's theorem, each coherent topos has enough points. What would be an example of a Grothendieck topos with enough points which is *not* coherent?
| https://mathoverflow.net/users/478438 | Topos with enough points but not coherent | Here are some examples :
1. For any topological space $X$, the topos of sheaf $\operatorname{Sh}(X)$ has enough points. In most cases this is not a coherent topos. If I remember correctly (for $X$ sober), this is only coherent if $X$ is a [spectral space](https://en.wikipedia.org/wiki/Spectral_space). In any case, sp... | 17 | https://mathoverflow.net/users/22131 | 417793 | 170,166 |
https://mathoverflow.net/questions/417782 | 0 | Let S be a compact orientable surface. Let A and B be two subsurfaces of S that have the same signature. How to check if there is a homeomorphism of S that sends A to B and if so, find one?
Here a subsurface of $S$ is a component of S \ simple closed curves.
I saw this question [Mapping-Class Groups of Subsurfaces ... | https://mathoverflow.net/users/nan | Construct a homeomorphism of a surface that sends a subsurface to another subsurface | We are assuming that $A$ and $B$ are both connected and contain their boundary points. (If we do not assume $A$ and $B$ are connected then the problem becomes much harder.) Let $(A\_i)$ be the closures of the components of $S - A$. Let $(B\_j)$ be closures of the components of $S - B$. Define $(g\_i, b\_i, s\_i)$ to be... | 1 | https://mathoverflow.net/users/1650 | 417796 | 170,168 |
https://mathoverflow.net/questions/370283 | 17 | A set $X\subseteq\omega^\omega$ is **unravelable** iff there is a possibly larger set $A$ and a clopen set $Y\subseteq A^\omega$ (with respect to the product topology coming from the discrete topology on $A$) such that winning strategies for the game on $A$ with payoff set $Y$ can be converted to winning strategies for... | https://mathoverflow.net/users/8133 | What sets can be unraveled? | I emailed Itay Neeman, and he told me the following:
>
> As far as I know it's open. I don't think anything is known about
> unraveling beyond what you can get from my methods. These give the
> Suslin operation on $\Pi^1\_1$ sets, and various iterations of that. In
> terms of the large cardinal hierarchy it's still... | 6 | https://mathoverflow.net/users/8133 | 417802 | 170,169 |
https://mathoverflow.net/questions/417797 | 4 | I am interested in the following Poincaré-type inequality,
$$ \int\_{S(r)} \lvert u-\bar{u}\rvert^2 d\sigma \leq C(N) \int\_{S(r)} |u\_{\theta}|^2 d\sigma$$
where $\bar{u} = \frac{1}{\lvert S(r)\rvert}\int\_{S(r)} u d\sigma$ and $u\_{\theta}$ denotes the tangential derivative of $u$. The domain $S(r)$ is just the $N$-d... | https://mathoverflow.net/users/100801 | Best constant for Poincaré inequality on spheres | The best constant is just the multiplicative inverse of the smallest positive eigenvalue of the Laplacian on the sphere. On $\mathbb{S}^N$ this the smallest eigenvalue is $N$, so $C(N)$ in that case equals $1/N$.
You can figure out the appropriate $r$ scaling yourself.
| 8 | https://mathoverflow.net/users/3948 | 417807 | 170,171 |
https://mathoverflow.net/questions/417804 | 9 | Do there exist integers $x$ and $y$ such that $\frac{x^3-x^2-2 x+1}{y^2-4}$ is an integer?
In other words, can any integer representable as $x^3-x^2-2 x+1$ have any divisor representable as $y^2-4$?
This is the simplest non-trivial example of my earlier question [Integer points of rational function in 2 variables](... | https://mathoverflow.net/users/89064 | Can $y^2-4$ be a divisor of $x^3-x^2-2 x+1$? | No. The roots of $x^3 - x^2 - 2x + 1$ are $-(\zeta + \zeta^{-1})$ where $\zeta$ is a 7th root of unity; this soon implies [*see below*] that any prime factor is either $7$ or $\pm 1 \bmod 7$, and thus that all factors of $x^3 - x^2 - 2x + 1$ are congruent to $0$ or $\pm 1 \bmod 7$. In particular it is not possible for ... | 30 | https://mathoverflow.net/users/14830 | 417811 | 170,173 |
https://mathoverflow.net/questions/417787 | 2 | I'll begin with a broad question: if $M$ is a smooth manifold and $E \to M$ is a stably trivial bundle, can one determine lower bounds on the rank $k$ of the trivial bundle needed such that $E \oplus \underline{\mathbb{R}}^k$ is trivial?
An obvious example is for the tangent bundle of spheres: $TS^n \to S^n$. Here, $... | https://mathoverflow.net/users/121144 | Quantitative results for stabilizing tangent bundles of homology spheres | If $E \to X$ is a rank $r$ real vector bundle, then it is classified by a map $X \to BO(r)$. The existence of an isomorphism $E \cong E\_0\oplus\underline{\mathbb{R}}$ (equivalently, the existence of a nowhere-zero section of $E$), corresponds to lifting $X \to BO(r)$ through the map $BO(r-1) \to BO(r)$ induced by the ... | 4 | https://mathoverflow.net/users/21564 | 417820 | 170,176 |
https://mathoverflow.net/questions/417790 | 3 | This question is about the argument for Lemma 3.7 in [Forcing axioms and stationary sets](https://doi.org/10.1016/0001-8708(92)90038-M) ([MSN](https://mathscinet.ams.org/mathscinet-getitem?mr=MR1174395)) by Boban Veličković.
He defines a game $G\_\alpha$ between two players, playing objects in $H\_\kappa$, depending ... | https://mathoverflow.net/users/11145 | Veličković's model game | What constitutes a partial play doesn't depend on $\alpha$. And $\kappa$ is assumed to have cofinality strictly larger than $\aleph\_1$. So he means: define $\sigma$ applied to a partial play to just be some ordinal below $\kappa$ that is above all the outputs of the $\aleph\_1$-many strategies $\sigma\_\alpha$.
(And... | 4 | https://mathoverflow.net/users/26319 | 417838 | 170,180 |
https://mathoverflow.net/questions/417800 | 25 | I am working on a paper which will extend a result in my thesis and have boiled one problem down to the following: show that the symmetric matrix $M\_p$, whose definition follows, is invertible for all odd primes $p$. Letting $p>3$ be prime and $\ell = \frac{p-1}{2}$, we define
$$M\_p = \begin{pmatrix} 2ij - p - 2p\lef... | https://mathoverflow.net/users/477847 | Show that these matrices are invertible for all $p>3$ | Experimentally, we have the following formula for $p$ prime:
$$\det(M\_p)=(-1)^{(p^2-1)/8}(2p)^{(p-3)/2}h\_p^-\;,$$
where $h\_p^-$ is the minus part of the class number of the $p$-th cyclotomic
field, itself essentially equal to a product of $\chi$-Bernoulli numbers.
I have not tried to prove this, but since there are ... | 24 | https://mathoverflow.net/users/81776 | 417854 | 170,183 |
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