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https://mathoverflow.net/questions/433120 | 6 | In the German Wikipedia entry for $L^p$-Raum it is stated ([Link](https://de.wikipedia.org/wiki/Lp-Raum))
>
> Das $L$ in der Bezeichnung geht auf den französischen Mathematiker Henri Léon Lebesgue zurück, da diese Räume über das Lebesgue-Integral
> definiert werden.
>
>
>
>
>
> "The $L$ in the name goes b... | https://mathoverflow.net/users/142414 | Origin of $L$ in $L^1$ and $L^2$ norms | On page 453 of his 1910 paper [Untersuchungen über Systeme integrierbarer Funktionen](https://eudml.org/doc/158473) Riesz writes (in a footnote)
>
> \*) H. Lebesgue [Sur les intégrales singulières](https://eudml.org/doc/72816) (1909) appeared after the completion of this work. That paper touches on many of the poin... | 5 | https://mathoverflow.net/users/11260 | 433128 | 175,216 |
https://mathoverflow.net/questions/433125 | 9 | There is some work which generalises the usual [Wilson loop](https://en.wikipedia.org/wiki/Wilson_loop) in QFT to higher dimensions and constructs non-abelian [Wilson surface functionals](https://www.sciencedirect.com/science/article/pii/S0001870810003804) in the context of [non-abelian gerbes](https://www.sciencedirec... | https://mathoverflow.net/users/119114 | Physics application of Wilson surface observables | Wilson surfaces have been used to describe non-Abelian quasiparticles in topological states of matter, see
* [Framed Wilson Operators,
Fermionic Strings, and Gravitational Anomaly in 4d](https://arxiv.org/abs/1404.4385)
* [Bosonic Topological
Insulators and Paramagnets: a view from cobordisms](https://arxiv.org/abs/1... | 7 | https://mathoverflow.net/users/11260 | 433129 | 175,217 |
https://mathoverflow.net/questions/433130 | 3 | A fact about triangulated categories is that (exact) localisation functors and so-called colocalisation functors come in pairs, making an exact localisation triangle. I've tried to come up with less traditional examples.
Let $A$ be a commutative ring, and $I \subset A$ an ideal. Then tensoring by $A / I$ is idempoten... | https://mathoverflow.net/users/488857 | Is there an elementary reason that this colocalisation map of complexes is a quasi-isomorphism? | The derived tensor product with $A/I$ is typically not idempotent (because ${\rm Tor}\_{\*}^A(A/I, A/I)$ is nontrivial), so this won't give you a localization sequence.
One case where it is idempotent is when $I = eA$ for $e \in A$ an idempotent. In this case, tensoring with $A/I$ is the same as localizing at $1-e$, ... | 4 | https://mathoverflow.net/users/131945 | 433134 | 175,219 |
https://mathoverflow.net/questions/433138 | 3 | Let $\mathcal{O}$ be a compact good orbifold, where we understand a *good orbifold* to be an orbifold obtained as a global quotient $M/G$, where $M$ is a manifold and $G$ is a discrete group. Are there a compact manifold $\widetilde{M}$ and a discrete group $\widetilde{G}$ acting on $\widetilde{M}$ such that $\mathcal{... | https://mathoverflow.net/users/151395 | Can a compact good orbifold be realized as a global quotient of a compact manifold? | This is true in dimension two (work of Fox from the 1950’s) and in dimension three (the so-called orbifold theorem). I don’t know the status in dimension four (or higher).
| 3 | https://mathoverflow.net/users/1650 | 433141 | 175,223 |
https://mathoverflow.net/questions/433135 | 2 | We are given a $d$-dimensional convex shape $S$ inscribed in the hypercube $[-1,1]^d$. We want find an approximation of its volume based only on a set of curves given by the intersection of the $S$ boundary and a finite number of $2$-planes.
We denote by $\gamma\_{i,j}$ the curve given by the intersection of the $S$ ... | https://mathoverflow.net/users/115803 | Estimating the volume of a convex shape in higher dimensions based only on normal sections | Those constants don't exist for any $d\geq4$, here is an idea of why.
For each $\varepsilon>0$ let $A\_\varepsilon=\{(x\_1,\dots,x\_d)\in[-1,1]^d;\lvert (d-1)x\_d-\sum\_{i=1}^{d-1} x\_i\rvert\leq\varepsilon\text{ and }\lvert\frac{(d-1)(d-2)}{2}x\_d-\sum\_{i=1}^{d-1}ix\_i\rvert\leq\varepsilon\}$. Note that there is a ... | 2 | https://mathoverflow.net/users/172802 | 433158 | 175,228 |
https://mathoverflow.net/questions/432148 | 11 | $\DeclareMathOperator\PSL{PSL}\DeclareMathOperator\SL{SL}\DeclareMathOperator\SO{SO}\DeclareMathOperator\SU{SU}$Let $ p $ be a prime for which $ \PSL(2,p) $ is simple (so $ p \neq 2,3 $).
Is the minimal irrep of $ \PSL(2,p) $ defined over a quadratic extension? In particular I wish to ask:
for $ p $ congruent to $ 1 ... | https://mathoverflow.net/users/387190 | Minimal irrep of $\mathrm{PSL}(2,p) $ | $\def\QQ{\mathbb{Q}}\def\FF{\mathbb{F}}\def\Sp{\text{Sp}}\def\SL{\text{SL}}\def\GL{\text{GL}}$Okay, time to do both cases using the Weil representation. This is going to rely on some Key Facts which I can check from my [explicit matrices for the Weil representation](https://mathoverflow.net/questions/432407) and which ... | 3 | https://mathoverflow.net/users/297 | 433161 | 175,229 |
https://mathoverflow.net/questions/433117 | 3 | Let $G$ be a regular $n$-vertex graph which is edge transitive. How large can the degrees of $G$ be if it is not a complete $r$-partite graph?
The best I can do is about $n/2$ by considering two disjoint cliques of the same size. A relatively easy argument shows that any $n$-vertex edge transitive graph with degree $... | https://mathoverflow.net/users/106377 | Super dense edge-transitive graphs | Take $G=\overline{K\_m\square K\_m}$, the complement of the Cartesian product of two copies of the complete graph on $m$ vertices. It has $m^2$ vertices, valency $(m-1)^2$ and is edge-transitive. (In fact, we have $\mathrm{Aut}(G)=S\_m\wr S\_2= (S\_m\times S\_m)\rtimes S\_2$.)
| 5 | https://mathoverflow.net/users/22377 | 433163 | 175,230 |
https://mathoverflow.net/questions/432834 | 0 | Given $n$ balls, all of which are initially in the first of $n$ numbered boxes, $a(n)$ is the number of steps required to get one ball in each box when a step consists of moving to the next box every second ball from the highest-numbered box that has more than one ball.
Not so long ago I proposed this sequence to the... | https://mathoverflow.net/users/231922 | Recurrence for the number of steps required to get one ball in each box | Generalise $a$: $a(n, k)$ is the number of steps to perform this process with $n+k$ boxes and balls starting with $n \ge 1$ balls in the first box and one ball each in the next $k$ boxes. Then the original $a(n)$ is $a(n, 0)$.
Clearly $a(1, k) = 0$. If $n > 1$ then we propagate $\lfloor \tfrac n2 \rfloor$ balls right... | 1 | https://mathoverflow.net/users/46140 | 433195 | 175,238 |
https://mathoverflow.net/questions/433153 | 6 | I found myself trying to prove the following, but I had to compute everything explicitly.
It is well known that if $u:\mathbb{R}^n\to\mathbb{R}$ is an harmonic function on $\mathbb{R}^n$, then the so-called Kelvin transform of $u$
$$(Ku)(x):=\frac{1}{|x|^{n-2}}u\biggl(\frac{x}{|x|^2}\biggr)$$
is harmonic in $\mat... | https://mathoverflow.net/users/170893 | Why is $\frac{1}{|x|^{n-2}}u(\frac{x}{|x|^2})$ harmonic if $u$ is harmonic? | An explanation can be the following. Take a harmonic function of the form $u(x)=r^\alpha P(\omega)$, with $r=|x|$, $|\omega|=1$ and $P$ a polynomial. Then $0=\Delta u=r^{\alpha-2}\left (\alpha(N-2+\alpha)P(\omega)+\Delta\_S P(\omega)\right)$ with $\Delta\_S$ the Laplace-Beltrami on the unit sphere $S$. Then $\Delta\_S ... | 7 | https://mathoverflow.net/users/150653 | 433212 | 175,246 |
https://mathoverflow.net/questions/125878 | 14 | I'm sure this is a fairly basic question, but I can't seem to find a solid answer:
My primary question is: Is there a reasonably nice subsystem of second-order arithmetic corresponding essentially to "primitive recursive comprehension?" I'm interested only in $\omega$-models - that is, I don't care about how much ind... | https://mathoverflow.net/users/8133 | Reverse mathematics below RCA | It looks like there is now a paper explicitly treating such a system! [Bazhenov/Fiori-Carones/Liu/Melnikov, *Primitive Recursive Reverse Mathematics*](https://arxiv.org/pdf/2210.13080.pdf) just appeared on the arxiv. Here are a couple key points:
* Models of their base theory, $\mathsf{PRA}^2$, are precisely pairs $(... | 2 | https://mathoverflow.net/users/8133 | 433216 | 175,248 |
https://mathoverflow.net/questions/433081 | 3 | For $s\in(0,1],$ consider the following non-local fractional laplacian:
$$(-\Delta)^sv= f ~~\text{on } \mathbb{R}^n.$$
Then how to use "the standard elliptic estimate" to obtain:
* for $p\in[1, \frac{n}{n-2s}),$
$$\|v\|\_{L^p(B\_2\setminus B\_1)} \lesssim \|f\|\_{L^1(B\_3\setminus B\_{\frac{1}{2}})}+\|v\|\_{L^1(B\_4\... | https://mathoverflow.net/users/166368 | Endpoint Calderon-Zygmund inequality of nonlocal fractional laplacian | I think the estimate is false, at least for $n=1$ and $0 < s < 1/2$, due to the non-locality you mention (I imagine similar arguments would work in other non-local cases). If it held, then one would have
$$ \|v\|\_{L^p([1,2])} \leq C \| v \|\_{L^1([-4,4])} \quad (1)$$
for some fixed constant $C$ whenever $(-\Delta)^s... | 9 | https://mathoverflow.net/users/766 | 433221 | 175,250 |
https://mathoverflow.net/questions/433217 | 3 | Suppose $X$ is a Banach space with the following property: For any $x\in X$ there exists a two dimensional subspace $E$ *isometric* with $l\_2^2$ such that $x\in E$. Does this property characterize a (separable) Hilbert space?
What about the stronger property: For any $x$ and any $n\in\mathbb{N}$ there exists a $n$-d... | https://mathoverflow.net/users/69275 | Do these properties characterize Hilbert spaces? | For any Banach space $X$ you can consider $X\oplus l^2$, with norm $||(x,y)||:=(||x||^2+||y||^2)^\frac{1}{2}$. Then for each $x\in X$, span$(x)\oplus l^2$ is isometric to $l^2$, so $X\oplus l^2$ is covered by isometric copies of $l^2$
| 6 | https://mathoverflow.net/users/172802 | 433225 | 175,251 |
https://mathoverflow.net/questions/433181 | 8 | Let $X$ and $Y$ be metric space, $X$ be compact, $C(X,Y)$ denote the set of continuous functions from $X$ to $Y$ with uniform convergence on compacts topology, and $\operatorname{Lip}(X,Y)$ denote the subspace of Lipschitz functions from $X$ to $Y$. Under what conditions is $\operatorname{Lip}(X,Y)$ a dense subset of $... | https://mathoverflow.net/users/491352 | Uniform density of Lipschitz maps is space of continuous function — for general metric spaces | Let $(X,\rho)$ be a compact metric space, and let $(Y,d)$ be a separable metric space. Then I claim that one can endow $X$ with a compatible metric $d$ such that every continuous $f:X\rightarrow Y$ can be uniformly approximated by a Lipschitz function.
Let $(f\_n)\_{n\geq 0}$ be a sequence of continuous functions suc... | 14 | https://mathoverflow.net/users/22277 | 433227 | 175,252 |
https://mathoverflow.net/questions/433226 | 7 | Are the categories of sets, abelian groups, and commutative rings unique? Independence results like the independence of the generalized continuum hypothesis, the Whitehead problem, and the global dimension of $\prod\_{n = 1}^\infty \mathbb{F}\_2$ from ZFC seem to indicate no. And yet we don't say "Let $\mathbf{Set}$ be... | https://mathoverflow.net/users/17218 | Are the categories of sets, abelian groups, and commutative rings unique? | *Introduction to pluralism*
A version of this question lies at the heart of the ongoing dispute on pluralism in the philosophy of mathematics. Is there at bottom just one mathematical reality? Does every mathematical question, whether about arithmetic, about the real continuum, or about set theory, have a definite ma... | 23 | https://mathoverflow.net/users/1946 | 433235 | 175,256 |
https://mathoverflow.net/questions/433239 | 5 | Leap years are determined by a scheme in which every $4$th year is a leap year, but every $4\cdot 25$th year is exempted, but every $4\cdot 25 \cdot 4$th year is reinstated $\ldots $ and there we stop, because that's good enough in practice to approximate the actual length of a year in terms of days. But what if we wan... | https://mathoverflow.net/users/134299 | Leap year formula to arbitrary precision | The greedy algorithm always produces a suitable expansion. The proof follows.
**Lemma**: If $0<x<1$, then $x$ may be written as $\frac 1n(1-y)$ for some $n$ with $0\le y<\frac 1{n+1}$.
Proof: Let $n$ be such that $\frac 1{n+1}<x\le \frac 1n$. Then $0\le \frac 1n-x< \frac 1{n(n+1)}$. In particular, writing $y=n(\fra... | 8 | https://mathoverflow.net/users/11054 | 433243 | 175,258 |
https://mathoverflow.net/questions/433142 | 5 | Cross-posted from [MSE](https://math.stackexchange.com/questions/4539025/can-solvable-lie-groups-have-maximal-subgroups).
Many interesting manifolds can be expressed as $ G/H $ for $ G $ a connected Lie group and $ H $ a maximal closed subgroup. Examples include the projective spaces $ \mathbb{C}P^n \cong \operatorna... | https://mathoverflow.net/users/387190 | Can solvable connected Lie groups have maximal subgroups? | $\newcommand{\g}{\mathfrak{g}}$We can obtain the characterization by classifying just non-nilpotent connected Lie groups.
>
> Let $G$ be a solvable, non-nilpotent connected Lie group. Suppose that every quotient Lie group of $G$ of dimension $<\dim(G)$ is nilpotent. Then $G$ has the form $H\ltimes V$ with both $H,V... | 3 | https://mathoverflow.net/users/14094 | 433250 | 175,261 |
https://mathoverflow.net/questions/433066 | 7 | A beautiful and surprising (to me at least) result around the axiom of choice is that, while full $\mathsf{AC}$ is preserved by forcing, a model of $\mathsf{ZF}$ + "There are no amorphous sets" may have a (set-)generic extension in which there do exist amorphous sets. This was proved by Monro, *[On generic extensions w... | https://mathoverflow.net/users/8133 | How hard is it to get "absolutely" no amorphous sets? | Turning my comment into an answer, an $X$ which is the universe of any finitely axiomatized theory with an infinite model must be orderable, and there must be a bijection between between $X$ and $X^2.$ Both of these follow from $X$ satisfying a large finite fragment of PA. In fact this is a characterization of such $X$... | 6 | https://mathoverflow.net/users/109573 | 433255 | 175,264 |
https://mathoverflow.net/questions/433254 | 7 | Let $K$ be a global field (ie either a number field or the function field of a curve over a finite field). Let $A,B$ be abelian varieties over $K$ and let $\phi:A\to B$ be an isogeny. Associated with $\phi$ is the well-known Selmer group ${\rm Sel}(\phi)\subseteq H^1(K,{\rm ker}(\phi))$, which contains $B(K)/\phi(A(K))... | https://mathoverflow.net/users/17308 | Reference request. Finiteness of the Selmer group | The paper is Milne, J. S. Elements of order p in the Tate-Šafarevič group. Bull. London Math. Soc. 2 (1970), 293–296. He deduces his statement about the Tate-Shafarevich group from a statement about the Selmer group. He also notes that if you omit any places in the definition of these groups, then you may get infinitel... | 8 | https://mathoverflow.net/users/492821 | 433263 | 175,267 |
https://mathoverflow.net/questions/433264 | 3 | I am interested in the following congruence
$$\binom{ap^n}{bp^n}\equiv \binom{a}{b}\pmod{p^n}$$
I am aware that by some reference in a book the above it should actually hold modulo $p^{3n}$; the reference in question is Zieve (1999) but could not find any trace of the paper.
I am also aware of the various generalizatio... | https://mathoverflow.net/users/41010 | Binomial coefficient congruence modulo $p^n$ | This is false in general.
To show this, I will use the congruence
$$ \binom{p^k a}{p^k b} \equiv \binom{p^{k-1} a}{p^{k-1}b} \bmod p^{3k}$$
for $p\ge 5$ and $k \ge 1$.
This is originally due to Ljunggren and Jacobsthal in "On the divisibility of the difference between two binomial coefficients" in Skand. Mat.-Kongr... | 7 | https://mathoverflow.net/users/31469 | 433268 | 175,269 |
https://mathoverflow.net/questions/433237 | 0 | Let $e\_{1}$ and $e\_{2}$ be involutions in the algebraic group $G=\operatorname{PGL}\_{n}(\mathbb{C})$. Do we have $$C\_{G}(\langle e\_{1},e\_{2}\rangle)^{\circ} = C\_{G}(e\_{1})^{\circ}\cap C\_{G}(e\_{2})^{\circ}.$$ It's obvious $C\_{G}(\langle e\_{1},e\_{2}\rangle)^{\circ} \leqslant C\_{G}(e\_{1})^{\circ}\cap C\_{G}... | https://mathoverflow.net/users/488802 | Intersection of identity components | $\DeclareMathOperator\Cent{C}\newcommand\oG{\overline G}\newcommand\oe{\overline e}$Put $\oG = \operatorname{PGL}\_n(\mathbb C)$. I hope you will permit me to denote the involutions by $\oe\_i$ instead of $e\_i$.
At first, it matters only that we are dealing with semisimple elements (which is implied by being an invo... | 1 | https://mathoverflow.net/users/2383 | 433275 | 175,274 |
https://mathoverflow.net/questions/433274 | 4 | Suppose $p$ is a prime, that $F$ is a finite extension of the field $\mathbb{Q}\_p$, $D$ is the division quaternion algebra over $F$ and $\mathcal{O}\_D$ is the valuation ring of $D$. What is the abelianisation of the group of units $\mathcal{O}\_D^\times$? I'd also appreciate a reference.
Apologies that this may loo... | https://mathoverflow.net/users/345 | Abelianization of unit quaternions over a p-adic field | The answer is yes: it is a result of Riehm, Corollary to Theorem 21 in The norm 1 group of $\mathfrak{p}$-adic division algebras *Amer. J. Math.* **92** 2 (1970), 499--523, see also Theorem 1.9 p.33 and the following Remark in Platonov and Rapinchuk, Algebraic groups and number theory *Pure and Applied Math.* **139** (... | 5 | https://mathoverflow.net/users/40821 | 433291 | 175,280 |
https://mathoverflow.net/questions/322023 | 3 | An algebra $(X,\*)$ is said to be self-distributive if it satisfies the identity $x\*(y\*z)=(x\*y)\*(x\*z)$ for all $x,y,z\in X$. If $(X,\*)$ is an algebra, then a subset $L\subseteq X$ is said to be a left-ideal if $x\*y\in L$ whenever $y\in L$. An element $x$ is said to be a left-identity if $x\*y=y$ for $y\in X$. Le... | https://mathoverflow.net/users/22277 | Can Laver tables go extinct? | **Multigenic Laver tables can go extinct.** Examples of multigenic Laver tables that go extinct with 2 generators are rare; one is not likely to find a multigenic Laver table that goes extinct unless one is looking for a multigenic Laver table that goes extinct, and one needs to use computer calculations to produce suc... | 2 | https://mathoverflow.net/users/22277 | 433293 | 175,281 |
https://mathoverflow.net/questions/433290 | 1 | Given two random variables $X,Y$ which are both $\mathbb{N}$-valued and have the same expected value (which is some fixed positive constant), and denote their probability mass functions by ${\bf p} = (p\_0,p\_1,\ldots)$ and ${\bf q} = (q\_0,q\_1,\ldots)$, respectively. We also assume that $q\_n > 0$ for all $n \in \mat... | https://mathoverflow.net/users/163454 | Upper bound Wasserstein distance by $\chi^2$ distance | Such a real-valued function $f$ does not exist.
Indeed, for any natural $N$, let
$$(p\_N,p\_{2N},p\_{3N})=\tfrac18(1,4,2),\ (q\_N,q\_{2N},q\_{3N})=\tfrac18(2,2,3),$$
so that $p\_N+p\_{2N}+p\_{3N}=q\_N+q\_{2N}+q\_{3N}=\frac78$ and $Np\_N+2Np\_{2N}+3Np\_{3N}=Nq\_N+2Nq\_{2N}+3Nq\_{3N}$. Next,
for $n\in J:=\{0,1,\dots\}\... | 1 | https://mathoverflow.net/users/36721 | 433297 | 175,284 |
https://mathoverflow.net/questions/433292 | 26 | The Riemann hypothesis for finite fields can be stated as follows: take a smooth projective variety X of finite type over the finite field $\mathbb{F}\_q$ for some $q=p^n$. Then the eigenvalues $\alpha\_j$ of the action of the Frobenius automorphism on the $i$th $\ell$-adic étale cohomology (are algebraic numbers and) ... | https://mathoverflow.net/users/158123 | Why do we care about the eigenvalues of the Frobenius map? | The Riemann hypothesis is very important for the relationship between the cohomology and combinatorics of the variety.
First, the Riemann hypothesis lets us read off the Betti numbers from the point counts over finite fields, i.e. the $i$'th Betti number is the number of zeroes/poles of $$e^{ \sum\_j \# X(\mathbb F\_... | 23 | https://mathoverflow.net/users/18060 | 433299 | 175,285 |
https://mathoverflow.net/questions/433308 | 4 | Let $f:x\to z$ and $g:y\to z$ be morphisms in an $\infty$-category $\mathcal C$. It seems that the square
$$\require{AMScd}
\begin{CD}
\operatorname{Map}\_{\mathcal C\_{/z}}(f,g) @>>> \operatorname{Map}\_{\mathcal C^{\Delta^1}}(f,g)\\
@VVV @VVV \\
\Delta^0 @>{\operatorname{id}\_z}>> \operatorname{Map}\_{\mathcal C}(z,z... | https://mathoverflow.net/users/37110 | Maps in the slice category vs. maps in the arrow category | Let us use the fat slice $\mathcal{C}^{z/}$ (See HTT, $\S$4.2.1) and the model $\operatorname{Hom}\_{\mathcal{C}}(x,y)=\operatorname{Fun}(\Delta^1,\mathcal{C})\times \_{\mathcal{C}\times \mathcal{C}}\{(x,y)\}$ of the mapping space. By computation, we can check that the square
$$\require{AMScd}
\begin{CD}
\operatornam... | 6 | https://mathoverflow.net/users/144250 | 433312 | 175,289 |
https://mathoverflow.net/questions/359754 | 4 | Let $f= f(t,x) : \mathbb{R}\_+ \times \mathbb{R}^d \to \mathbb{R}$ be a Lipschitz function such that
$$
\partial\_t f - |\nabla f|^2 = 0 \qquad \text{almost everywhere in } \mathbb{R}\_+ \times \mathbb{R}^d.
$$
As is well-known, this condition does not determine the function $f$ uniquely in terms of the initial conditi... | https://mathoverflow.net/users/56892 | Uniqueness condition for Hamilton-Jacobi equation? | Proposition A.2 of <https://arxiv.org/abs/2104.05360> shows that the answer is "yes". In fact, this is also valid for a general nonlinearity in the equation in place of the squared norm here.
| 0 | https://mathoverflow.net/users/56892 | 433317 | 175,290 |
https://mathoverflow.net/questions/433160 | 4 | I preface this by saying that I am fairly new to the enveloping von Neumann algebra scene, so there may be some gaps in my understanding.
Given a $C^\*$-algebra $A$ and a state $\phi$ on $A$, one may consider $\phi$ as a normal state on the universal enveloping von Neumann algebra $A''$ of $A$. In this case, there sh... | https://mathoverflow.net/users/480800 | Support projection vs closed support projection of a normal state in enveloping von Neumann algebra | I am not quite sure what the question is, so let me try to understand what your unease is. In the example we take $A=C([0,1])$ and for the state $\phi$ take normalised Lebesgue measure. What is the support projection in $A''$? When you write the following:
>
> ... which is smaller than the operator of multiplicatio... | 3 | https://mathoverflow.net/users/406 | 433321 | 175,292 |
https://mathoverflow.net/questions/433320 | 3 | This might seem like a silly question considering my relatively elementary knowledge of representation theory.
The question is regarding Eugen Hellman 's paper titled ["On the derived category of the Iwahori-Hecke algebra"](https://arxiv.org/abs/2006.03013). Specifically on the paragraph above Lemma 2.18. First, let ... | https://mathoverflow.net/users/157428 | Algebraic representations and vector bundles | The definition of algebraic representation should be exactly what you suggested in 1).
For what concerns 2), you can construct a $\operatorname{Gl}\_n$-torsor over $G/B$ given a representation $B \to \operatorname{Gl}\_n$ as you suggested. The reasoning should be the following: from the homomorphism $B \to \operatorn... | 4 | https://mathoverflow.net/users/146464 | 433322 | 175,293 |
https://mathoverflow.net/questions/433325 | 2 | I started trying to learn about Gromov-Witten invariants by reading the book "$J$-holomorphic curves and Symplectic Topology" and I have a doubt in an example the authors provide. It's example $7.1.3$ where we consider the homology class $A=0$.
So when we are focusing in the case where $k=3$ and we want to compute th... | https://mathoverflow.net/users/155363 | Question on Gromov-Witten invariants when $A=0$ | In this case, the J-holomorphic curves are all constant, so the
evaluation pseudocycle is the tridiagonal $\{(x,x,x) : x\in
M\}$. You take cycles $A\_1,A\_2,A\_3$ Poincare dual to
$a\_1,a\_2,a\_3$ respectively and intersect the tridiagonal with
$A\_1\times A\_2 \times A\_3$. This gives you precisely
$\{(x,x,x) : x\in A... | 4 | https://mathoverflow.net/users/10839 | 433329 | 175,297 |
https://mathoverflow.net/questions/433335 | 8 | Let $u \in C^{\infty}(\mathbb R^3)$ be harmonic. Suppose that $u$ has no critical points outside the unit ball but that it has at least one critical point inside the unit ball.
Does it follow that $u$ is a polynomial?
| https://mathoverflow.net/users/50438 | On critical points of harmonic functions | This is not true in dimension 2.
Function $f(z)=ze^z$ is entire, and $f'(z)=0$ at one point,
$z=-1$. It follows that the function
$$u(x,y)=\mathrm{Re}f(x+iy)=e^x(x\cos y-y\sin y)$$
is harmonic, has one critical point $(x,y)=(-1,0)$, and evidently not a polynomial.
Now one can construct a similar example in even dimen... | 10 | https://mathoverflow.net/users/25510 | 433339 | 175,301 |
https://mathoverflow.net/questions/433218 | 2 | My question may be simple to an expert, but I'm not:
Let's consider $u \in C^{s}(\mathbb{R}^d)$ be a Hölder function sor some $s\in [0,1/2)$ which we may take very close to $0$.
Of course, $u^2 \in C^{s}(\mathbb{R}^d)$ so that $(u^2)' \in B^{s-1}\_{\infty,\infty}$, is a well-defined distribution, with some explicit... | https://mathoverflow.net/users/94414 | Microlocal approach to definition of product of distributions | Too long for a comment. For $u$ in $C^s$, $s\in (0,1)$, you can indeed define $u^2$ and then the distribution-derivative of $u^2$, which belongs to $B^{s-1}\_{\infty,\infty}$. Now that does not define the product $uu'$, unless you decide to define that product as $\frac12(u^2)'$. Assuming for instance that $u$ is also ... | 2 | https://mathoverflow.net/users/21907 | 433345 | 175,303 |
https://mathoverflow.net/questions/433337 | 2 | Assume that $C$ is a projective curve and $X$ is an elliptic fibration over $C$.
>
> What is the picard group of $X$? can we say something about it?
>
>
>
I think it should be generated by multisections and (components of ) fibres. How can I see it formally?
| https://mathoverflow.net/users/nan | Picard group of an elliptic fibration is generated by multisection and fibres | Do you know about the [Shioda-Tate formula](https://planetmath.org/shiodatateformula). (I'm surprised there isn't a Wikipedia article about it.) Anyway, it says that if the elliptic fibration does not split as a product and if you have at least one section, then the Neron-Severi group is generated by the following divi... | 6 | https://mathoverflow.net/users/11926 | 433346 | 175,304 |
https://mathoverflow.net/questions/430142 | 4 | I'm not sure if this is completely relevant to MO, let me know if this would be better on MSE.
I have been told today by a professor of mine that the following is a classic result of Cartan. Suppose $M$ is a closed parallelizable smooth $n$-manifold and $X\_1, \ldots, X\_n$ are everywhere linearly independent vector ... | https://mathoverflow.net/users/143629 | On a result of Cartan for homogeneous manifolds arising from a quotient of discrete subgroups | The result that you are looking for is not in Élie Cartan's 1936 book *La topologie des groupes de Lie* because it was not known to be true at the time the book was written. Indeed, as Cartan remarks in the book (which is the lecture notes from an October 1935 conference where he spoke), it was not even known at that t... | 5 | https://mathoverflow.net/users/13972 | 433360 | 175,306 |
https://mathoverflow.net/questions/433375 | 7 | I asked this on Math Stack Exchange, but apparently no one paid attention to it. So, I am asking it again, filling in the background necessary to understand it.
Consider a countably infinite set $P$ of propositional atoms, indexed by the positive integers like so: $p\_1,p\_2,p\_3,p\_4,...$. We also have the connectiv... | https://mathoverflow.net/users/43439 | Are no infinite subsets of the set of all propositional atoms definable in this structure, even with parameters? | It's a nice question. This Boolean algebra, known as the [*Lindenbaum algebra*](https://en.wikipedia.org/wiki/Lindenbaum%E2%80%93Tarski_algebra), is a countable atomless Boolean algebra — it is atomless because we can always take the conjunction of any formula with a new atom or its negation — and all such Boolean alge... | 11 | https://mathoverflow.net/users/1946 | 433377 | 175,313 |
https://mathoverflow.net/questions/433378 | 10 | Consider a polynomial in one variable $p(x)$ with $p(0)>0$, and that is not a polynomial in $x^m$ for any $m>1$ (that is, the $gcd$ of the exponents appearing in $p(x)$ is 1). I would like to find necessary and sufficient conditions so that a power of $p$ has no negative coefficients. I know of a sufficient condition: ... | https://mathoverflow.net/users/66323 | Conditions for a power of a polynomial to have no negative coefficients | Marcus Michelen, Julian Sahasrabudhe, A characterization of polynomials whose high powers have non-negative coefficients,
<https://arxiv.org/abs/1910.06890>
| 11 | https://mathoverflow.net/users/25510 | 433383 | 175,314 |
https://mathoverflow.net/questions/433364 | 3 | This is a generalisation of an older [question inspired by a football tournament](https://mathoverflow.net/questions/432988/on-a-combinatorial-design-inspired-by-a-football-soccer-tournament) (which does not have an answer yet).
Let $\frak P$ be a partition of $\omega$ into blocks, that is, pairwise disjoint sets, su... | https://mathoverflow.net/users/8628 | Cycling through a general combinatorial design on $\omega$ | If there are infinitely many blocks, then indeed you can find the permuations $\pi\_n$ as desired. Indeed, there are computable such $\pi\_n$. Furthermore, you don't need that every block has size at least two, but rather only that there is at least one block of size at least two.
We build the sequence of permuations... | 4 | https://mathoverflow.net/users/1946 | 433387 | 175,315 |
https://mathoverflow.net/questions/433261 | 4 | Suppose that the closed, piecewise $C^1$-curve $f(\mathbb T)$ has exactly $n$ points that are run through twice, all other points are run through once. Is it true that the compact set $f(\mathbb T)$ has exactly $n+1$ holes? For $n=1$ this is true and is based on the well-known fact that a crosscut in a Jordan domain de... | https://mathoverflow.net/users/61993 | Curves in the plane and their number of holes | Here's a slightly more hands-on point of view, using just the Jordan curve theorem (although of course in the end it does come down to the same thing as Euler's formula somehow, as described by Alex.)
Take a closed curve $\gamma\colon [0,1]\to S^2$, and assume that there are only finitely many pairs of points in $[0,... | 1 | https://mathoverflow.net/users/3651 | 433393 | 175,318 |
https://mathoverflow.net/questions/433366 | 0 | Let $G:(0,1)\to(0,1)$ be the Gauss map, i.e., $G(x)=\left\{\frac1{x}\right\}$, which is known to act as the shift on the space of continued fraction expansions.
**Question.** Is there an explicit expression for the natural extension of $G$ in $\mathbb R^2$? What about its density?
| https://mathoverflow.net/users/8131 | Natural extension of the Gauss map | As it was already mentioned by Christophe Leuridan for classical continued fraction
we have extended Gauss measure
$$d\overline{\nu}(x,y)=\frac{1}{\log2}\cdot\frac{dx\,dy}{(1+xy)^2}=\frac{1}{\log2}\cdot\begin{vmatrix} 1 & \mp x\\
\pm y & 1
\end{vmatrix}^{-2} dx\,dy,\qquad (x,y)\in[0,1]^2,$$
which is invariant under the... | 3 | https://mathoverflow.net/users/5712 | 433400 | 175,320 |
https://mathoverflow.net/questions/433200 | 4 | Let $M$ be a compact manifold and $\varphi:M\rightarrow M$ a diffeomorphism. The invariant differential forms
$$
\Omega^{k}\_{inv}(M)=\{\alpha\in\Omega^{k}(M):\varphi^{\*}\alpha=\alpha\}
$$
form a subcomplex $(\Omega\_{inv}(M),d)$ of the de Rham complex. Denote by $H\_{inv}(M)$ its cohomology.
**Question:** Is there ... | https://mathoverflow.net/users/150945 | Cohomology of invariant differential forms | There is no such thing: any $\phi$-invariant exact
1-form is a differential of $\phi$-invariant function.
Indeed, let $\alpha$ be an exact $\phi$-invariant
form, $\alpha=df$, where $f$ is not $\phi$-invariant.
Then $d(\phi^\* f -f)=0$, hence
$\phi^\* f = f + C$, where $C$ is a constant.
This gives $\sup f = \sup \phi... | 4 | https://mathoverflow.net/users/3377 | 433405 | 175,322 |
https://mathoverflow.net/questions/433294 | 2 | For the sake of simplicity, assume $f$ is a non-cm eigenform of weight $k$ on the group $\mathrm{SL}(2, \mathbb{Z})$. Are there any known results or conjectures regarding any special values of the associated $L$-function $L(f, n)$ for any integers $n$ for weight $k > 2$? If so, what can be said about the rational facto... | https://mathoverflow.net/users/138669 | Special values of non-cm $L$-functions | Since the OP is interested in particular by $L(\Delta,11)$, the relevant
theorems come into the framework of Deligne's theory of special points and
special values, while Bloch-Kato would be for $s>11$. Here the theorems are
due to Shimura and especially Manin, which for $\Delta$ (and for general eigenforms similarly) s... | 3 | https://mathoverflow.net/users/81776 | 433407 | 175,323 |
https://mathoverflow.net/questions/433406 | 2 | I am working on a problem that involves an iterative application of a function I think might be a trapdoor function.
Formally, I have a function $f:X \to X$ that can be described as
$$
[x\_{1,N+1}, ..., x\_{s,N+1}]=f([x\_{1,N},...,x\_{s,N}])\\
\forall\_{i<s-1}x\_{i,N}=x\_{i+1,N+1}
$$
and $x\_{s, N+1}$ is probabili... | https://mathoverflow.net/users/44865 | Proving that a function is a trapdoor function | First, there is a site crypto.stackexchange.com that is typically better for questions like this.
Second,
>
> Coming from a CS background, if I wanted to prove a problem was NP-hard, I would try to prove an equivalence to a known NP-hard problem. Could a similar approach work for trapdoor functions?
>
>
>
Th... | 5 | https://mathoverflow.net/users/101207 | 433411 | 175,324 |
https://mathoverflow.net/questions/278822 | 9 | Natural theories extending EFA (exponential/elementary function arithmetic) are well-ordered by $Π^0\_1$ provability, and we would like a formal definition of the well-ordering that is robust yet as fine as possible. Here is the definition I came up with.
Fix an elementary ordinal notation system such that the predec... | https://mathoverflow.net/users/113213 | $Π^0_1$ Proof Ordinals | Modulo the fact that Beklemishev [1] considers consistency with cuts as the basic consistency notion, his $\mathsf{Con}(\mathsf{EA}\_\alpha)$ are equivalent to your's $\mathsf{Con}\_{\alpha}$. It is quite easy to account for the effects of the switch between cut-free and full consistency using the result of Visser [2] ... | 3 | https://mathoverflow.net/users/36385 | 433414 | 175,325 |
https://mathoverflow.net/questions/433422 | 1 | Does there exist a simple smooth closed curve $\gamma:S^1\to \mathbb C$ such that
$$ \int\_{0}^{2\pi} e^{\gamma(e^{it})} \, |\gamma'(e^{it} )|\,dt =0?$$
| https://mathoverflow.net/users/50438 | On a property of complex exponentials | After the change of the variable $w=\gamma(t)$ your integral
becomes
$$\int\_\gamma e^wds,$$
where $ds$ is the length element, $ds=\sqrt{dx^2+dy^2}$, and $\gamma$ is a Jordan curve of zero index about the origin. Now notice that on every vertical segment of length $2\pi k$, where $k$ is an integer,
this integral is zer... | 4 | https://mathoverflow.net/users/25510 | 433434 | 175,330 |
https://mathoverflow.net/questions/433435 | 0 | In Chapter 3 of the textbook: [An Introduction to Random Matrices](https://www.google.com.hk/url?sa=t&rct=j&q=&esrc=s&source=web&cd=&cad=rja&uact=8&ved=2ahUKEwiC8aL504P7AhW_k2oFHTv9BK8QFnoECBUQAQ&url=http%3A%2F%2Fwww.wisdom.weizmann.ac.il%2F%7Ezeitouni%2Fcupbook.pdf&usg=AOvVaw0wbMPwbj788zuq0QFfT4Cp&cshid=16669853834661... | https://mathoverflow.net/users/168083 | Do we have the universal property of the edge of the spectrum for the Wigner matrix? | There is an extensive literature on the universality of the Tracy-Widom distribution. Here are some pointers:
* [Quantitative Tracy-Widom
laws for the largest eigenvalue of generalized Wigner matrices](https://arxiv.org/abs/2207.00546)
studies the general case of a Wigner matrix.
* [A necessary and
sufficient conditi... | 1 | https://mathoverflow.net/users/11260 | 433437 | 175,331 |
https://mathoverflow.net/questions/433425 | 3 | I am reading a paper where they refer to a certain algebra as a **PBW algebra**. What does this mean exactly? I would infer from the $U(\frak{g})$ setting that this means the existence of an ordered generating set $\{x\_1, \dots, x\_n\}$ such that
$$
x\_1^{d\_1} \cdots x\_n^{d\_n}, ~~~~~~ d\_k \in \mathbb{N}\_0.
$$
for... | https://mathoverflow.net/users/491434 | What is a PBW algebra? (I.e., an algebra generalising properties of $U(\frak{g})$) | This concept, and the name Poincare Birkhoff Witt algebra, is due to Stewart Priddy in his paper [Koszul resolutions, Trans. Amer. Math. Soc. 152 (1970), 39–60] that also first defined Koszul algebras. See section 5 of his paper for the PBW criterion, which is a bit more flexible than what you wrote. (The basis might n... | 6 | https://mathoverflow.net/users/102519 | 433442 | 175,332 |
https://mathoverflow.net/questions/433278 | 73 | It seems like the article ["The Twin Primes Conjecture is True in the Standard Model of Peano Arithmetic: Applications of Rasiowa–Sikorski Lemma in Arithmetic (I)"](https://link.springer.com/article/10.1007/s11225-022-10017-2) by Janusz Czelakowski published in *Studia Logica* yesterday, claims to have proven that the ... | https://mathoverflow.net/users/156061 | Czelakowski's claimed proof of the Twin Prime Conjecture | The error in the paper is in the proof of Theorem 7.2. The proof of Theorem 7.2 is immediately suspicious because of how vague it is in places and because of how lofty the expository text before and after it is. In the proof, the author claims that because we can identify the set of variables $(v\_i)\_{i \in \mathbb{N}... | 118 | https://mathoverflow.net/users/83901 | 433444 | 175,333 |
https://mathoverflow.net/questions/432969 | 1 | Let $R$ be a commutative Noetherian ring, and let $\text{mod } R$ denote the abelian category of finitely generated $R$-module. Consider the bounded derived category $D^b(\text{mod } R) $ which is a triangulated category. Every object $X \in \text{mod } R$ can be naturally identified in $D^b(\text{mod } R) $. For an ob... | https://mathoverflow.net/users/493381 | Finitely generated module, which is a virtually small complex, embeds into a module of finite projective dimension? | For every $M$, $M\oplus R$ is virtually small, so your question is equivalent to the question: Does every finitely generated $R$-module embed in a finitely generated module of finite projective dimension?
The answer is no. For example, let $R$ be a local commutative finite dimensional algebra over a field $k$, that i... | 2 | https://mathoverflow.net/users/22989 | 433460 | 175,335 |
https://mathoverflow.net/questions/433315 | 1 | Let $\mathbf{C}\_S \subset \mathbf{R}^{2n}$ be a Simons cone, where the dimension is large enough that it is area-minimizing: $n \geq 4$. Let $T$ be a leaf of the Hardt–Simon foliation with $\operatorname{dist}(T,0) = 1$.
For all $R > 1$, $T$ intersects $\partial B\_R$ transversely. Thus $T \cap \partial B\_R$ is a s... | https://mathoverflow.net/users/103792 | A paradox based on Simons cones | The issue here is that (I suspect) your intuition is failing you as your first bullet point is incorrect.
A good first visualization is the Clifford torus in $\mathbf{S}^3$ all the parallel surfaces are tori, until a critical distance is reached and one gets two circles on each side.
Another, way to think about the... | 2 | https://mathoverflow.net/users/127803 | 433464 | 175,337 |
https://mathoverflow.net/questions/433475 | 2 | The categories $\mathbf{Top}$ of topological spaces, $\mathbf{sSet}$ of simplicial sets and $\mathbf{Cat}$ of small categories are all equipped with a functor $\pi\_0$ into the category $\mathbf{Set}$ of sets, which is a left adjoint and measures the number of connected components. There are also plenty of functors bet... | https://mathoverflow.net/users/479945 | Which functors preserve the number of connected components? | Ignoring issues with what TOP really should be, let me focus on the question whether $\pi\_0(Sd^\infty(X))=\pi\_0(X)$ for a simplicial set $X$. If we look at $X=\Delta^1$, we should get a counterexample. Here $\pi\_0$ should denote the equivalence class of vertices, where two vertices are equivalent, if there is a fini... | 8 | https://mathoverflow.net/users/3969 | 433485 | 175,343 |
https://mathoverflow.net/questions/433487 | 5 | A propeller is a chiral structure. Propellers can have low or large pitch. Pitch differences can be seen as distinguishing propellers with low chirality from those with large chirality.
Is there a general way, given a closed 3d shape, to quantify its chirality? Is there some sort of integral over the shape that yield... | https://mathoverflow.net/users/493794 | Is there a way to quantify the chirality of a 3d shape? | This is a topic of some research, summarized in [On quantifying chirality — Obstacles and problems towards unification](https://link.springer.com/article/10.1007/BF01277559). One metric is the Hausdorff chirality, which quantifies the chirality of a geometric representation of an object by measuring the degree of coinc... | 7 | https://mathoverflow.net/users/11260 | 433488 | 175,344 |
https://mathoverflow.net/questions/433489 | 12 | **Context**: I am currently working on a rather important paper for my career, in the sense that it is a culmination of the past 5 years of (post Ph.D.) research. I started this particular article with 3 other members but only 1 other member (senior) has contributed. The other two likely cannot even explain our contrib... | https://mathoverflow.net/users/36886 | How to indicate when another author has done nothing significant | It seems clear to me that you need to talk to the senior author who actually did contribute and see if they can talk to the other two. You wrote that "one of the authors said that they were experts who are so great (bla bla)" so it sounds like it wasn't really your choice to add them in the first place.
Let's call th... | 31 | https://mathoverflow.net/users/11540 | 433492 | 175,345 |
https://mathoverflow.net/questions/433448 | 2 | Suppose that $\mathcal{D}$ is a Johnson-Lindenstrauss (JL) distribution on $\mathbb{R}^{r\times n}$ ($1 \le r \le n$), meaning that there exist constants $\epsilon, \delta \in(0,1)$ such that
$$
\mathbb{P}\_{A \sim \mathcal{D}}((1 - \epsilon)\|x\| \le \|Ax\|\le (1 + \epsilon)\|x\|) \ge \delta
\quad \text{for all}\quad... | https://mathoverflow.net/users/170208 | Distribution of scaled Johnson-Lindenstrauss transforms | $\newcommand\ep\epsilon\newcommand{\de}{\delta}\newcommand{\R}{\mathbb R}$We have
\begin{equation\*}
P((1-\ep)\|x\|\le\|Ax\|\le(1+\ep)\|x\|)\ge\de \tag{1}\label{1}
\end{equation\*}
for some $\ep,\de$ in $(0,1)$ and all $x\in\R^n$.
The OP asks if then
\begin{equation\*}
P((1-\ep')\|x\|\le\|S(A)x\|\le(1+\ep')\|x\|)\g... | 2 | https://mathoverflow.net/users/36721 | 433499 | 175,347 |
https://mathoverflow.net/questions/433470 | 1 | I know that the wave equation doesn't satisfy a maximum principle but I have also heard that hyperbolic equations do not satisfy any maximum principle. But I don't know any reference or proof regarding that. It would be really helpful if I get a help or resource regarding the same.
| https://mathoverflow.net/users/493046 | Maximum principle for hyperbolic PDEs | Fundamentally, the classical maximum principles for second order elliptic PDEs are based on the simple facts that
1. The Hessian of a function at a local maximum is positive semidefinite.
2. The full contraction (Hilbert-Schmidt product) between two (symmetric) positive semidefinite matrices is non-negative.
In poi... | 4 | https://mathoverflow.net/users/3948 | 433500 | 175,348 |
https://mathoverflow.net/questions/433508 | 5 | I get the follow equation in a paper. Let $A \in \mathbb{R}^{2 \times 2}$, then $M = A^TA$ is a positive semi-definite matrix, the nuclear norm of $A$ is:
$$ \Vert A \Vert\_\* = \sqrt{\operatorname{tr}(M) + 2\sqrt{\det(M)}}.$$
Are there any analytical form of nuclear norm for $n \times n$ matrix like the above form... | https://mathoverflow.net/users/493804 | Analytical form for the nuclear norm of an $n \times n$ matrix | **As Gro-Tsen pointed out, I had not computed the Galois group of the governing polynomial, and, in fact, my original answer was wrong. I believe that the following answer is correct, though.**
If by 'like the above form' you mean an expression in terms of radicals of the traces of the powers of $M$, the answer is 'n... | 6 | https://mathoverflow.net/users/13972 | 433517 | 175,355 |
https://mathoverflow.net/questions/432732 | 4 | Let $X$ be a subset of a group $G$. We say that $X$ is *left amenable* with respect to $G$ if there is a function $\mu:\mathcal P(G)\to [0,\infty]$ with the following three properties.
1. $\mu(A\cup B)=\mu(A)+\mu(B)$ for every pair of disjoint subsets $A, B$ of $G$.
2. $\mu(gA)=\mu(A)$ for all $g\in G$ and $A\in \mat... | https://mathoverflow.net/users/132356 | Amenable subsets of groups | Here is a slightly modified version of an example that was communicated to the author of the question by Nicolas Monod.
Let $k>1$ be an integer and $G$ the Baumslag-Solitar group $\mathrm{BS}(1,k)$; that is,
$$G:=\langle a, b\ :\ bab^{-1}=a^k\rangle.$$
Let $A$ and $B$ be the infinite cyclic subgroups generated by $a$... | 4 | https://mathoverflow.net/users/132356 | 433527 | 175,357 |
https://mathoverflow.net/questions/433521 | 3 | Assume that $\Omega$ is a bounded connected domain and $\partial \Omega \in C^{\infty}$. Denote $\Gamma\_1,\Gamma\_2,\cdots,\Gamma\_n$ are $n$ connected components of $\partial\Omega$. This notation leads to $\partial \Omega=\cup^n\_{i=1}\Gamma\_i$. Consider the following problem.
\begin{cases}
\Delta \phi&=0\\
\phi|\_... | https://mathoverflow.net/users/176379 | How to use comparison principle to prove the following inequality about Laplace equation? | Let $\psi$ be harmonic in $\Omega$, with $\psi=\phi$ on $\cup \_{i \in S} \Gamma\_i$, $\psi=m$ on $\cup\_{i \not \in S}\Gamma\_i$, where $m=\max\_{i \not \in S} \max\_{\Gamma\_i} \phi$. By comparison, $\psi \geq \phi$ and then $\nabla \psi \cdot n \leq \nabla \phi \cdot n$ on $\cup \_{i \in S} \Gamma\_i$. Then
$$
0=\in... | 5 | https://mathoverflow.net/users/150653 | 433536 | 175,361 |
https://mathoverflow.net/questions/433535 | 0 | I have been thinking about this for the last few days but I was not able to produce a definitive answer.
Take an integrable function $g$ that maps in $\mathbb{R}$ and with domain contained in $[0,M]$ (not necessarily equal to this interval). Consider now a family of probability measures $\mu\_{\theta}$ such that $\mu... | https://mathoverflow.net/users/493038 | Deduce that a function is zero on interval $[0,M]$ | As you say, it is not difficult to prove that $g=0$ when there are finitely many fluctuations in sign.
More precisely, suppose we have points $0=a\_0,a\_1,\dots,a\_n=M$ such that $g\geq0$ or $g\leq0$ in each interval $[a\_i,a\_{i+1}]$. Then, if we have measures $\mu\_\theta$ as you say and $\int\_0^\theta g(z) \, d \... | 1 | https://mathoverflow.net/users/172802 | 433545 | 175,364 |
https://mathoverflow.net/questions/433547 | 14 | For a little more context: I'm currently an undergrad (sophomore) at a small liberal arts college with a (from my experience so far) solid math program. So far, I've taken Calc I, II, III, linear algebra, and an introduction to logic class. This semester I'm taking graph theory and differential equations, an intro to s... | https://mathoverflow.net/users/493850 | How do I, as an undergraduate, find interesting, accessible questions to work on to see if I'd be interested in research mathematics? | Welcome to MathOverflow! I am a professor at a liberal arts college. The best place to start is by talking to your favorite math professor, or your advisor in the department. The graph theory course will have plenty of accessible problems that will give you a taste of research mathematics. Abstract algebra does, too, a... | 18 | https://mathoverflow.net/users/11540 | 433550 | 175,366 |
https://mathoverflow.net/questions/433533 | 4 | Consider the left-regular representation $\lambda : G \to B(L^2(G))$, $\lambda\_g f(h) = f(g^{-1}h)$, for a locally compact group.
It is well-known that this is a unitary faithful and strongly-continuous representation, but is it also a homeomorphism onto its image $\lambda(G)$ (equipped with the strong-operator topo... | https://mathoverflow.net/users/485160 | Is the left-regular representation of a locally compact group a homeomorphism onto its image? | Yes. It's more generally true for every faithful $C^0$ unitary representation $\pi$ of $G$. (Recall that a unitary representation $\pi$ is $C^0$ if for all $v,w$ in the Hilbert space, one has $\langle \pi(g)v,w\rangle\to 0$ when $g$ leaves compact subsets of $G$.)
Indeed if this is not a homeomorphism onto its image,... | 7 | https://mathoverflow.net/users/14094 | 433555 | 175,369 |
https://mathoverflow.net/questions/433556 | 2 | Let $\mu$ be the Haar measure defined on the space of unimodular lattices, identified with $\text{SL}(d,\mathbb R)/\text{SL}(d,\mathbb Z)$.
The classical Siegel's formula in geometry of numbers states that for $f\in L^1(\mathbb R^d)$, let $\Lambda$ be a unimodular lattice, and let
$$\hat{f}(\Lambda):= \sum\_{v\in \... | https://mathoverflow.net/users/489992 | Proof of generalized Siegel's mean value formula in geometry of numbers | Such a generalization (roughly) exists, known as *Rodger's Integration Formula*.
See [Section 1.2 of Seungki Kim's Dissertation](https://stacks.stanford.edu/file/druid:ms598tw2644/thesis%20%281%29-augmented.pdf) for a reference.
Theorems 1.2 and 1.3 are of interest.
>
> **Theorem 1.2**: (Rogers). Let $k < n$, and $... | 4 | https://mathoverflow.net/users/101207 | 433557 | 175,370 |
https://mathoverflow.net/questions/433539 | 4 | Given any linear space $L$ over an ordered field $F$, consider the *equiproportion relation* $${\sim}=\{((x,y,z),(a,b,c))\in L^3\times L^3: \exists t\in[0,1]\subseteq F\;(y{-}x=t(z{-}x)\wedge b{-}a=t(c{-}a))\},$$called the *standard equiproportion relation* on $L$.
This relation seems to describe the affine geometry of... | https://mathoverflow.net/users/61536 | Is the affine geometry a geometry of proportions? | The midpoint of $a$ and $b$ can be defined from equiproportion as the unique $m$ for which $(a,m,b)$ and $(b,m,a)$ are equiproportional.
Similarly, the collinearity of $a,b,c$ can be defined as one of $(a,b,c)$, $(b,c,a)$ or $(c,a,b)$ being equiproportional to itself.
Using these, $\mathbb{R}^n$ should be character... | 1 | https://mathoverflow.net/users/nan | 433562 | 175,373 |
https://mathoverflow.net/questions/433551 | 3 | I don't know much (anything) about sieves, but as I read the section on the Selberg upper bound sieve from Greaves's *Sieves in Number Theory*, there is a theorem 4 which says that
If $Y\ge X \ge 2$, then
\begin{equation}
\pi(Y)-\pi(Y-X) \leq \frac{2X}{\log X} + O\left(\frac{X}{\log^2 X}\right).
\end{equation}
1. I... | https://mathoverflow.net/users/392272 | Best available bounds for $\pi(Y)-\pi(Y-X)$? | If $Y\geq X\geq 2$, then as Daniel Johnston wrote, [Montgomery and Vaughan](https://deepblue.lib.umich.edu/bitstream/handle/2027.42/152543/mtks0025579300004708.pdf) proved that
$$\pi(Y)-\pi(Y-X)\leq \frac{2X}{\log X}.$$
Whether this constitutes a "best bound" requires a definition of what you consider to be "best".... | 6 | https://mathoverflow.net/users/111215 | 433570 | 175,377 |
https://mathoverflow.net/questions/433581 | 12 | I would like to read Pincus' article [**Adding dependent choice**](https://doi.org/10.1016/0003-4843(77)90011-0), where he proves, among other things, the consistency of the theory $\mathsf{ZF+DC+O+\neg AC}$, where $\mathsf{DC}$ stands for Dependent Choice and $\mathsf{O}$ is the linear ordering principle, i.e. the sta... | https://mathoverflow.net/users/141146 | Is there a more modern account of the main results of "Adding Dependent Choice" by D. Pincus? | No.
Most of the work of Pincus that involved these sort of iterated constructions, where one "pushes the counterexamples out" has never been reworked into modern terms.
In my Ph.D. one of the reasons to develop the notion of an iteration of symmetric extensions was to help and simplify these proofs (also the works ... | 15 | https://mathoverflow.net/users/7206 | 433582 | 175,381 |
https://mathoverflow.net/questions/433572 | 4 | Let $h=[\Bbb CP^1]\in H\_2(\Bbb CP^2;\Bbb Z)$. By a theorem of Kronheimer and Mrowka (Theorem 1 of this paper: [https://people.math.harvard.edu/~kronheim/thomconj.pdf](https://people.math.harvard.edu/%7Ekronheim/thomconj.pdf)), a class $nh \in H\_2(\Bbb CP^2;\Bbb Z)$ can be represented by a smoothly embedded 2-sphere i... | https://mathoverflow.net/users/164671 | Homology classes in connected sum of $\Bbb CP^2$'s that can be represented by smoothly embedded spheres | As Mike says in his comment, the answer is not known in full generality, not even for $n=2$, so for classes in $\mathbb{CP}^2\#\mathbb{CP}^2$. I think the paper that he links (<https://arxiv.org/abs/2210.12486>, by Marengon, Miller, Ray, and Stipsicz) describes the state of the art for $n=2$.
Some results are known f... | 6 | https://mathoverflow.net/users/13119 | 433588 | 175,382 |
https://mathoverflow.net/questions/433554 | 31 | I've often seen Lurie's *Higher Topos Theory* praised as the next "great" mathematical book. As someone who isn't particularly up-to-date on the state of modern homotopy theory, the book seems like a lot of abstract nonsense and the initial developments unmotivated. I'm interested in what the tools developed concretely... | https://mathoverflow.net/users/99902 | Higher Topos Theory- what's the moral? | It seems there are really two questions here:
1. Why higher category theory? What questions can you pose without the language of higher category theory which are best answered using higher category theory?
2. Why does Lurie's work specifically set the standard for the foundations of higher category theory?
These ar... | 33 | https://mathoverflow.net/users/2362 | 433590 | 175,383 |
https://mathoverflow.net/questions/433579 | 0 | Let $X=(X\_t:t\ge 0)$ be a stochastic process (martingale in general) starting at $X\_0=0$. For $T>0$ and $a<b$, let $U^T\_{a,b}(X)$ be the number of upcrossings of $X$ across the interval $[a,b]$ over $[0,T]$, i.e. $U^T\_{a,b}(X)$ is the supremum of the nonnegative integers $n\in\mathbb N$ such that there exist times ... | https://mathoverflow.net/users/493556 | Is the number of uncrossing invariant under time-change? | We have $U\_{a,b}^T(Y) = U\_{a,b}^{h(T)}(X)$.
By construction, the function $h$ maps $[0,T]$ onto $[0,h(T)]$. More precisely, each $t' \in [0,h(T)]$ can be written $h(h^\leftarrow(t'))$,
where $h^\leftarrow(t') := \inf\{t \in [0,T] : h(t) \ge t'\} \in [0,T]$.
The function $h^\leftarrow$ thus defined is strictly incre... | 0 | https://mathoverflow.net/users/169474 | 433592 | 175,384 |
https://mathoverflow.net/questions/433597 | 1 | From the paper of [Ambrosio-Crippa](https://www.cambridge.org/core/journals/proceedings-of-the-royal-society-of-edinburgh-section-a-mathematics/article/abs/continuity-equations-and-ode-flows-with-nonsmooth-velocity/DD2A02C300757F63F7F0A96FAD8A70C5), it is known that if $\beta:\mathbb R^d\times[0, T[\longrightarrow\math... | https://mathoverflow.net/users/160186 | Continuity equation for a density of a measure | I am not so sure to understand the problem, maybe I am missing something. You should not use $(\triangle)$ but instead go back to the equation satisfied by the measure. Indeed, since $\mu$ is solution of your conservative transport equation, you have for any test function $\varphi\in\mathscr{D}(\mathbf{R}\_+\times\math... | 2 | https://mathoverflow.net/users/27767 | 433602 | 175,385 |
https://mathoverflow.net/questions/433402 | 5 | I have a monoidal (not symmetric) triangulated category $(A,\otimes, 1)$ with unit 1.
Define $C$ the localizing subcategory of $A$ generated by the unit 1.
* is $(C, \otimes, 1) $ a symmetric monoidal triangulated category?
* Do we have a natural isomorphism $c\otimes a\cong a\otimes c$ for any $a\in A$ and any $c\in... | https://mathoverflow.net/users/141114 | Monoidal triangulated categories | The answer to question 2, even with the assumption that $a\in C$, is no in general. It follows that the answer to question 1 is also no.
For a counterexample, one can consider for example a (derived) category of bimodules. Let $k$ be a base commutative ring, and $R$ a (say) flat $k$-algebra. Consider the derived cate... | 5 | https://mathoverflow.net/users/102343 | 433606 | 175,386 |
https://mathoverflow.net/questions/433589 | 1 |
>
> Define $\mathbb{W} = H\_{0}^{1}(-1,1) \times H\_{0}^{1}(-1,1)$, where $u \in H\_{0}^{1}(-1,1)$ if $u,u^{\prime} \in L^{2}(-1,1)$ and $u(-1) = u(1) = 0$. Consider the sesquilinear form $a: \mathbb{W} \times \mathbb{W} \to \mathbb{C}$ given by
> $$
> a((u,v),(w,z)) = \int\_{-1}^{1}u^{\prime}\overline{w}^{\prime}dx ... | https://mathoverflow.net/users/481556 | Proof that sesquilinear form in is coercive | The first eigenvalue of the second derivative with Dirichlet b.c. on $(-1,1)$ is $\pi^2/4$ (with eigenfunction $\cos \frac{\pi x}{2}$) and then Poincare' inequality with optimal constant is $\|u\|\_2^2 \leq \frac{4}{\pi^2}\|u'\|\_2^2$. It follows that
$$2| Re \int\_{-1}^1 u\bar v | \leq 2\|u\|\_2\|v\|\_2 \leq \|u\|\... | 1 | https://mathoverflow.net/users/150653 | 433610 | 175,388 |
https://mathoverflow.net/questions/433604 | 2 | Setting and definitions
-----------------------
Let $X = \{X(t), t \in T \}$, $T \subset \mathbb{Z}$, be an infinite-variance associated stochastic process, i.e.
$$
\text{Cov}(f(X(I)), g(X(J))) \geq 0
$$
for all finite disjoint subsets $I, J \subset T$ and bounded, coordinate-wise increasing Borel functions $f: \math... | https://mathoverflow.net/users/302666 | Infinite-variance associated processes are (BL, $\theta$)-dependent | The answer is no. E.g., Let $X(t)=Z$ for all $t$, where $Z$ is any random variable with infinite variance. Then the process $(X(t)\colon t\in\mathbb Z)$ is positively associated. On the other hand, for $f\_n(x):=\min(n,\max(-n,x))$ and each natural $j$ we have
$$Cov(f\_n(X\_0),f\_n(X\_r))\to Var\,Z=\infty$$
as $n\to\in... | 2 | https://mathoverflow.net/users/36721 | 433615 | 175,389 |
https://mathoverflow.net/questions/433598 | 0 | I have one more question about the Example (I.5.1) on page 7 from
Rick Miranda's the basic theory of elliptic surfaces:
Let $C\_1$ be a smooth cubic curve in $\mathbb{P^2}$ and let $C\_2$
be any other cubic. Let $F\_1, F\_2 \in \mathbb{C}[X,Y,Z]$ the homogeneous cubic polynomials
generating the vanishing ideals of $C... | https://mathoverflow.net/users/108274 | Calculate blowup of a pencil of cubics "by hand" | If the two cubics $C\_1=V(F\_1)$ and $C\_2=V(F\_2)$ do not share a common component, then the ideal $I = (F\_1,F\_2)$ defines a $0$-dimensional subscheme $V(I)\subset \mathbb{P}^2$ length 9, which is a complete intersection of codimension 2.
In the fancy language above, the graded ring $\bigoplus\_{n\geq0}I^n$ is gen... | 2 | https://mathoverflow.net/users/104695 | 433616 | 175,390 |
https://mathoverflow.net/questions/433603 | 4 | Stallings' celebrated Fibration Theorem states that if a closed irreducible $3$-manifold $M$ admits a short exact sequence
\begin{equation}
1 \to N \to \pi\_1(M) \to \mathbb{Z} \to 1,
\end{equation}
where $N$ is finitely generated, then $M$ fibers over $S^1$.
My question is that whether it is possible to give the exa... | https://mathoverflow.net/users/493880 | Stallings' fibration theorem - Explicit description | Your question ("is $M$ homeomorphic to $T\_f$?") is answered in the affirmative by Theorem 2 of Stallings' paper \*On fibering certain 3-manifolds". You will also need his Theorem 1. Here are the statements (slightly simplified).
**Theorem 1**: Suppose that $M$ is a compact connected three-manifold. Suppose that $\Ga... | 3 | https://mathoverflow.net/users/1650 | 433619 | 175,391 |
https://mathoverflow.net/questions/432563 | 1 | I am coming from [this question](https://mathoverflow.net/questions/336149/about-the-cartans-moving-frame-method?noredirect=1&lq=1), which has not being completely answered but I think is very interesting.
In several works ([Chern], [Griffiths] and [Clelland]) the Maurer-Cartan form for $E(n)$ is worked out in the fo... | https://mathoverflow.net/users/129995 | Maurer-Cartan form and Levi-Civita connection | I have been working in question 2 and I think I have a good explanation. At the end is a triviality, but that's what (almost) always happens in math when you understand something.
I am going to discuss here the Euclidean plane from two different perspectives.
Euclidean plane
===============
From the point of view... | 0 | https://mathoverflow.net/users/129995 | 433620 | 175,392 |
https://mathoverflow.net/questions/433611 | 3 | Some years ago, I found a paper with all the formulas for the balls into bins problem when the "areas" (i.e., probabilities to capture a ball) of the bins are all different. However, the formulas looked quite involved and cumbersome in the general case. Now, I am instead trying to solve an elementary version of the bal... | https://mathoverflow.net/users/115803 | A linearly distributed version of the balls into bins problem | From the referenced paper, I am writing in terms of their variables, $k$ is the number of bins or type of coupons:
Let $n\_1$ be the time where the last of the missing events is observed. Let $n\_2$ be the time where the second last of the missing events is observed, etc. until $n\_k$:
Define
$$
S\_kf(p\_1,\ldots,p... | 4 | https://mathoverflow.net/users/17773 | 433621 | 175,393 |
https://mathoverflow.net/questions/433418 | 7 | Let $X$ be a compact complex manifold, $L$ be a holomorphic line bundle on $X$, then the exponential exact sequence $0\to \mathbb Z\hookrightarrow \mathcal O\to \mathcal O^\*\to 0$ induces the map $c:H^1(X,\mathcal O^\*)\to H^2(X,\mathbb Z)$, it is well-known that the line bundle $L$ can be seen as an element in $H^1(X... | https://mathoverflow.net/users/99826 | When Atiyah class and Chern class coincide? | $\def\ZZ{\mathbb{Z}}\def\CC{\mathbb{C}}\def\cO{\mathcal{O}}$To spell out my comment a little more, let $Z^1$ be the sheaf of $\partial$-closed holomorphic $(1,0)$-forms. Since "holomorphic" means $\overline{\partial}$-closed, this can also be described as the space of closed $(1,0)$forms. Then we have a commutative dia... | 4 | https://mathoverflow.net/users/297 | 433622 | 175,394 |
https://mathoverflow.net/questions/433486 | 10 | For the purposes of this question, a *$T$-interpretation with arity $n$* will be a tuple $\Phi=(\delta,\eta,F)$ where
* $\delta$ and $\eta$ are individual formulas of arity $n$ and $2n$ respectively,
* $T$ proves that $\eta$ defines an equivalence relation on the set picked out by $\delta$, and
* $F$ is a (possibly i... | https://mathoverflow.net/users/8133 | Can we always "sharpen" interpretations? | If you allow to add to an interpretation infinitely many predicate symbols (infinite $G\setminus F$) then any interpretation in $\mathsf{ZFC}$ could be extended to a sharp interpretation in $\mathsf{ZFC}+L=V$. So you original question has a positive answer.
However, if you restrict your attention to extensions by fin... | 5 | https://mathoverflow.net/users/36385 | 433625 | 175,395 |
https://mathoverflow.net/questions/433516 | 6 | Let $D$ be a small category. Does the category of diagrams $\mathsf{Top}^{D^{\text{op}}}$ have a classifier of (strong?) subobjects? I tried following the "sieve construction" for the category of presheaves, but I don't see what topology to put on the set of sieves on an object in $D$ (or perhaps this won't work anyway... | https://mathoverflow.net/users/78650 | Subobject classifier in $\mathsf{Top}^{D^{\text{op}}}$? | If any of these categories had a subobject classifier, every monomorphism would be regular, so that's not happening.
The indiscrete two-point space is a strong subobject classifier in $\mathsf{Top}.$ Similarly, the subobject classifier for $\mathsf{Set}^{D^{\mathrm{op}}},$ equipped with the indiscrete topology on its... | 5 | https://mathoverflow.net/users/43000 | 433634 | 175,399 |
https://mathoverflow.net/questions/433652 | 0 | Consider a product of projective varieties $X\times Y$ and two cohernet sheaves $\mathcal{F},\mathcal{G}$ such that
$$\mathcal{F}|\_{x\times Y}\cong\mathcal{G}|\_{x\times Y}$$
for any $x\in X$. Thanks to user 40297, we know that in general we do not have $\mathcal{F}\cong q^\*\mathcal{M}\otimes\mathcal{G}$ for some lin... | https://mathoverflow.net/users/nan | Does $\mathcal{F}|_{x\times Y}\cong\mathcal{G}|_{x\times Y}\Rightarrow q^*\mathcal{M}\otimes\mathcal{F}\cong q^*\mathcal{N}\otimes\mathcal{G}$ | No. For instance, take $Y$ to be a point, $X = \mathbb{P^1}$, $F = \mathcal{O} \oplus \mathcal{O}(1)$, and $G = \mathcal{O} \oplus \mathcal{O}$.
| 2 | https://mathoverflow.net/users/4428 | 433657 | 175,406 |
https://mathoverflow.net/questions/429284 | 4 | Let $V$ be an $n$-dimensional vector space over a finite field $F$ (of order $q$). Denote by $\mathrm{AGL}(V)$ the group of invertible affine transformations of $V$; so $\mathrm{AGL}(V)$ consists of all mappings $x\mapsto Tx+v$ where $T\in GL(V)$ and $v\in V$.
Clearly, $\mathrm{AGL}(V)$ acts on the poset of all affin... | https://mathoverflow.net/users/15934 | Representation of $\mathrm{AGL}(V)$ on the homology of the poset of affine subspaces of $V$ | I suspect that
Solomon, Louis The affine group. I. Bruhat decomposition.
proves what you are looking for.
Let $A\_n(q)$ denote the poset of proper affine subspaces of $\mathbf{F}\_q^n$. The only non-vanishing reduced homology group
of $A\_n(q)$ is in degree $n-1$ and it is free abelian of rank $|E\_n(q)|$ where $... | 2 | https://mathoverflow.net/users/50509 | 433663 | 175,408 |
https://mathoverflow.net/questions/433671 | 0 | So, if we have system of differential equations obtained from Lagrange function, by means of Noether theoerem (if we know some one-parameter symmetry group), we can derive conserved quantity.
But how does it work with arbitrary system (not derived from Euler-Lagrange equations).
Does it mean that knowledge of symmetr... | https://mathoverflow.net/users/493428 | Conserved quantities | If the dynamics does not have a variational formulation a conservation law is not necessarily related to a symmetry. A more general approach than starting from the Euler-Lagrange equations is described in [Construction of conservation laws: how the direct method generalizes Noether’s theorem](https://personal.math.ubc.... | 1 | https://mathoverflow.net/users/11260 | 433673 | 175,411 |
https://mathoverflow.net/questions/433594 | 4 | Let $K$ be a field, $K\_s$ its separable closure, $K$ $\subseteq$ $F$ $\subseteq$ $K\_s$ an extension with $[F:K]$ $=$ $n$, $R$ $\subseteq$ $K$ a Dedekind domain with quotient field $K$, $S$ the integral closure of $R$ in $F$, and $\mathfrak{p}$ a maximal ideal of $R$.
> Can we find a $d>n$ with $(d,n)=1$ and an ext... | https://mathoverflow.net/users/31923 | Finding field extensions in which a given prime is inert | Thanks to Arno Fehm's observation above, the answer in the general case is **NO**.
In the number field case, $R$ $=$ $\mathfrak{O}\_K$ and $S$ $=$ $\mathfrak{O}\_F$ (or overrings thereof in $K$ resp. $F$), and we can argue as follows. Let $p\mathbb{Z}$ = $\mathfrak{p}\cap\mathbb{Z}$. Let $q\_1,\cdots,q\_s$ be the rat... | 0 | https://mathoverflow.net/users/31923 | 433674 | 175,412 |
https://mathoverflow.net/questions/433584 | 13 | Let $K$ be an algebraically closed field, $A$ and $B$ two finite type $K$-algebras which are assumed to be UFD. Is $A \otimes\_K B$ again a UFD?
This question has been already asked [here](https://math.stackexchange.com/questions/3578801/is-the-tensor-product-of-ufds-again-ufd) and [here](https://math.stackexchange.c... | https://mathoverflow.net/users/37214 | Tensor product of finite type UFD algebras over an algebraically closed field is again UFD? | My [previous answer](https://mathoverflow.net/a/433662/82179) gives a partial result under some additional hypotheses, but these are not needed.
**Theorem** (Boissière–Gabber–Serman). *If $X$ and $Y$ are locally factorial varieties over an algebraically closed field $k$, then so is $X \times Y$. If $\operatorname{Cl}... | 7 | https://mathoverflow.net/users/82179 | 433678 | 175,414 |
https://mathoverflow.net/questions/433680 | 5 | In an abelian category $\mathcal{A}$, for a system $\{F\_i,\phi\_{ij}\}$ we have an exact sequence
$0\to \lim F\_i\to \prod F\_i \to \prod F\_i$
where the second map is given by $id-\prod\phi\_{ij}$. Is there a version of this for stable $\infty$-categories? Meaning, if $\mathcal{C}$ is a stable $\infty$-category a... | https://mathoverflow.net/users/197402 | limits and products stable $\infty$-category | In the case of an $\mathbb N^{op}$-indexed system specifically, the answer is yes (note that this is implicit in the Stacks project link you gave); in fact if you replace "fiber sequence" by "equalizer", this holds in an arbitrary $\infty$-category with the appropriate limits (namely products and equalizers). The descr... | 7 | https://mathoverflow.net/users/102343 | 433681 | 175,416 |
https://mathoverflow.net/questions/433675 | 13 | I've seen various fast algorithms for computing the first few, or directly the $n$-th, digits of $\pi$.
However, it seems to me that all these algorithms assume (see [last sentence here](https://en.wikipedia.org/wiki/Bailey%E2%80%93Borwein%E2%80%93Plouffe_formula#BBP_digit-extraction_algorithm_for_%CF%80)) that there a... | https://mathoverflow.net/users/955 | Can we compute the first $n$ digits of $\pi$ in $F(n)$ time? | Mahler proved(1) that for any $p,q$, $\left | \pi -\frac{p}{q} \right| > \frac{1}{q^{42}}$. It follows that if one can compute $2^{ 42n} \pi$ to within an error of at most $1$, one can compute the first $n$ digits of $\pi$, since the only way the knowledge of $2^{42n}\pi$ to within an error of $1$ would not be determin... | 18 | https://mathoverflow.net/users/18060 | 433683 | 175,418 |
https://mathoverflow.net/questions/433623 | 3 | The *total variation distance* between (say discrete) probability distributions, represented as vectors over their support, is defined to be
$$\Delta(\vec p,\vec q) = \frac{1}{2}\lVert \vec p-\vec q\rVert\_1.$$
The (squared) Hellinger distance is then defined to be
$$H^2(\vec p, \vec q) = \frac{1}{2}\lVert \sqrt{... | https://mathoverflow.net/users/101207 | When are the total variation distance and Hellinger distance comparable? | $\newcommand{\R}{\mathbb R}\newcommand{\al}{\alpha}\newcommand{\be}{\beta}
\newcommand{\De}{\Delta}$What you want is impossible for any reasonable class of probability distributions, including the class defined by your condition that $0<c<p\_i/q\_i<C$ for some real $c,C$ and all $i$.
Indeed, for simplicity of writing... | 2 | https://mathoverflow.net/users/36721 | 433684 | 175,419 |
https://mathoverflow.net/questions/433686 | 40 | Do editors for top math journals ever read a submitted paper, agree that there are no mistakes and the result is new, yet still reject it on the basis that this is a top math journal and someone could've done that before but chose not to? Maybe some arrogant mathematician goes "I could've proven that in a day or week b... | https://mathoverflow.net/users/127521 | Math papers where the only issue is that someone else could've done it but didn't | As Sam Hopkins comments, the short answer to the stated question is "yes, all the time." You'd be hard-pressed to find a professional mathematician who *hasn't* received a referee report that basically boils down your first paragraph. Often, the referee or editor can't find anything mathematically wrong with the result... | 91 | https://mathoverflow.net/users/11540 | 433691 | 175,422 |
https://mathoverflow.net/questions/432871 | 2 | Let $M\_t$ be a continuous time martingale, and assume its quadratic variation is identically zero with some positive probability less than $1$.
To be more precise, assume there exists some event $E$ with $0 < \mathbb P(E) < 1$ such that for almost every $\omega \in E$, $\langle M, M\rangle\_t (\omega) = 0$ for all $... | https://mathoverflow.net/users/173490 | Is a martingale constant on the event that its quadratic variation is zero? | To answer your question, the following Lengart's inequality is useful: (Please refer to
S. W. He *et al., Semimartingale Theory and Stochastic Calculus*, Sci. Press and CRC(1992), p.239, Theorem 9.23.)
**Theorem** Let $ X $ be an adapted cadlag process, dominated by an predictable process $ A $. Then for arbitrary co... | 2 | https://mathoverflow.net/users/103256 | 433723 | 175,430 |
https://mathoverflow.net/questions/433709 | 8 | The following question was [asked years ago on MSE](https://math.stackexchange.com/q/54761/53976), but let me recap it:
>
> **Question**: Is there anything currently known about the exact consistency strength of "$\mathsf{ZF}$ + both $\omega\_1$ and $\omega\_2$ are singular?" Or could we find a better bound?
>
>
... | https://mathoverflow.net/users/48041 | Some relevant questions about the consistency strength of singularity of $\omega_1$ and $\omega_2$ | We know little to nothing about the exact strength. Here are the facts, as you already know them.
1. To get $\omega\_1$ and $\omega\_2$ both singular we need to have *at least* a Woodin cardinal.
2. If there are infinitely many Woodin cardinals, then it is consistent that $\omega\_1$ and $\omega\_2$ are singular (go ... | 8 | https://mathoverflow.net/users/7206 | 433724 | 175,431 |
https://mathoverflow.net/questions/433740 | 3 | What is an example of a Frobenius algebra that is not Koszul? Are there reasonable requirements for a Frobenius to be Koszul?
| https://mathoverflow.net/users/491434 | What is an example of a Frobenius algebra that is not Koszul? | If an algebra A is Koszul with Koszul daul B, one has the formula gldim A=Loewy Length B-1.
Thus when a Frobenius algebra is Koszul (and not semisimple), the quadratic dual must be infinite dimensional.
For example for preprojective algebras of Dynkin type, the quadratic dual is finite dimensional and thus they can... | 5 | https://mathoverflow.net/users/61949 | 433741 | 175,438 |
https://mathoverflow.net/questions/433752 | 11 | I wanted to compute $\mathit{KO}^{-1}(\mathbb{R}P^3)$ and regrettably I could only think of using the Atiyah Hirzebruch spectral sequence, which seemed like a big overkill but looking at similar computations of $K^0(\mathbb{R}P^n)$ done in Atiyah's book do not look particularly simpler as he uses equivariant $\mathit{K... | https://mathoverflow.net/users/494010 | Computing KO^-1 of RP^3 without AHSS | The $KO$-theory of all truncated real projective spaces (including the calculation you want) was carried out very systematically by Frank Adams in his famous paper on the vector fields on spheres: see section 7 of [J.F.Adams, Ann.Math. (1962)]. He also invented/constructed `Adams operations' (not his term!) in this pap... | 17 | https://mathoverflow.net/users/102519 | 433753 | 175,439 |
https://mathoverflow.net/questions/433725 | 9 | Let $A$ be a selfadjoint operator on some Hilbert space $H$, let $U(t)=e^{itA}$ be the corresponding continuous group, and let $f\in H$ be orthogonal to all eigenvectors of $A$. Are there examples such that $U(t)f$ does not converge weakly to 0 as $t\to+\infty$?
(From the RAGE Theorem, if $K$ is a compact operator on... | https://mathoverflow.net/users/7294 | Counterexamples to weak dispersion for the Schrödinger group | The answer is yes.
A measure-preserving invertible shift $T: X \to X$ on a probability space $(X,\mu)$ is said to be [weakly mixing](https://en.wikipedia.org/wiki/Mixing_(mathematics)) if $\lim\_{N \to \infty} \frac{1}{N} \sum\_{n=1}^N |\langle f \circ T^{-n}, g \rangle|^2 = 0$ for all $f,g$ in the Hilbert space $L^2... | 10 | https://mathoverflow.net/users/766 | 433756 | 175,442 |
https://mathoverflow.net/questions/433396 | 4 | Let $f: \mathbb R \to \mathbb R$ be a $C^1$ convex function, satisfying the growth conditions
$$\lim\_{x \to -\infty} \nabla f(x) = -\infty, \lim\_{x \to \infty} \nabla f(x) = \infty.$$
and let $\gamma\_t: [0, \infty) \to \mathbb R$ be a deterministic, Borel measurable process satisfying the following conditions:
... | https://mathoverflow.net/users/173490 | Convergence of a continuous time stochastic gradient descent algorithm | here "[An SDE perspective on stochastic convex optimization](https://arxiv.org/abs/2207.02750#:%7E:text=We%20analyze%20the%20global%20and,problems%20with%20noisy%20gradient%20input.)" they study these type of SDEs (Stochastic gradient descent) and basically ask bounded and an L2 condition for gamma\_t in theorem 3.1.
... | 1 | https://mathoverflow.net/users/99863 | 433758 | 175,443 |
https://mathoverflow.net/questions/433685 | 18 | The question was cross-posted from Math.SE: <https://math.stackexchange.com/questions/4566017/strengthening-ax-grothendieck>
---
The question is simple. The Ax-Grothendieck theorem says a polynomial map $p\colon\mathbb C^n\to\mathbb C^n$ that is injective is also surjective.
>
> Is assuming $p$ has finite fib... | https://mathoverflow.net/users/123673 | Strengthening Ax-Grothendieck | A counterexample for $n = 2$ is the map $(x\_1, x\_2) \mapsto (x\_1x\_2 - 1, x\_2(x\_1x\_2 - 1) + x\_1)$. This example, which I learned from a paper of Zbigniew Jelonek, was mentioned in an [earlier answer of mine](https://mathoverflow.net/a/343994/1508).
| 17 | https://mathoverflow.net/users/1508 | 433760 | 175,445 |
https://mathoverflow.net/questions/433763 | 3 | Circulant matrices are very useful in digital image processing.
I found the general formula for determinant of circulant matrix.
But I think it is not suitable for block-circulant matrices.
For example, consider the formula for $\det(K)$,
where $$K = \left(\begin{array}{cccc} A & B & C & D \\
D & A & B & C \\
C &... | https://mathoverflow.net/users/369335 | One question on block-circulant matrices | The formula for the specific case is
$$\det K=\det(A+B+C+D)\det(A-B+C-D)\det(A+iB-C-iD)\det(A-iB-C+iD).$$
More generally, for a block-circulant matrix with $n$ square blocks $A\_0,\ldots,A\_{n-1}$, the formula is
$$\det K=\prod\_{\omega^n=1}\det(A\_0+\omega A\_1+\cdots+\omega^{n-1}A\_{n-1}).$$
To see this, observe that... | 6 | https://mathoverflow.net/users/8799 | 433765 | 175,446 |
https://mathoverflow.net/questions/433757 | 1 | Let $d,n\ge 1$ be fixed integers. Given some compact subset $E\subset \mathbb R^d$, consider the function $f: E^n\ni (x\_1,\ldots, x\_n) \longrightarrow f(x\_1,\ldots, x\_n)\in \mathbb R$ defined by
$$f(x\_1,\ldots, x\_n):= \max\_{(c\_1,\ldots,c\_n)\in\mathbb R^n}\left\{\int\_E \left(\min\_{1\le i\le n}|y-x\_i|^2-c\_... | https://mathoverflow.net/users/493556 | Differentiability of some function defined as the maximum | Suppose that there is an open subset $U$ of $E$ such that the Lebesgue measure of $E\setminus U$ is $0$. Since $E$ is compact, the function $E^n\ni(x\_1,\dots,x\_n)\mapsto|y-x\_i|^2$ is $L$-Lipschitz for some real $L>0$ and each $i\in\{1,\dots,n\}$ and each $y\in E$. Therefore and because the $\max$, $\min$, and integr... | 1 | https://mathoverflow.net/users/36721 | 433767 | 175,448 |
https://mathoverflow.net/questions/433764 | 8 | Let $ X\_N = \text{span} \{\cos(2\pi lx): l=0, \cdots, N-1 \} $ with $ x \in [0, 1] $ and $ Y\_N = \{v =(v\_0, \cdots, v\_{N-1}): v\_j \in \mathbb{C}\} = \mathbb{C}^N $. Then $ X\_N $ is the space of trigonometric polynomials. We equip $ X\_N $ with the usual $ L^p (1 \leq p \leq \infty ) $ norm and equip $ Y\_N $ with... | https://mathoverflow.net/users/484187 | Bounding the discrete $l^p$ norm by the continuous $L^p$ norm for trigonometric polynomials | Yes, this goes back to the work of
*Plancherel, M.; Pólya, George*, [**Fonctieres entières et intégrales de Fourier multiples**](http://dx.doi.org/10.1007/BF01258191), Comment. Math. Helv. 9, 224-248 (1937). [ZBL0016.36004](https://zbmath.org/?q=an:0016.36004).
(see for instance Theoreme III). Nowadays one would us... | 12 | https://mathoverflow.net/users/766 | 433774 | 175,449 |
https://mathoverflow.net/questions/433583 | 2 | Let $D \subset \mathbb{R}^d$ be a bounded $C^1$ domain. We consider a reflected Brownian motion $X=(\{X\_t\}\_{t \ge 0},\{P\_x\}\_{x \in \overline{D}})$ on $\overline{D}$. Let $\{p\_t\}\_{t>0}$ denote the semigroup of $X$ (in other words, $\{p\_t\}\_{t>0}$ is the Neumann semigroup). It is known that $\{p\_t\}\_{t>0}$ i... | https://mathoverflow.net/users/68463 | On a core for Neumann Laplacian on $C(\overline{D})$ | I think $\mathcal C$ is not even dense in the space of continuous functions!
To be specific: consider a 2-D domain $D$ lying above the graph of a $C^1$ function $\phi : \mathbb R \to \mathbb R$, and assume that $\phi'$ is continuous, but nowhere differentiable — say, a generic sample path of the Wiener process.
Let... | 3 | https://mathoverflow.net/users/108637 | 433776 | 175,450 |
https://mathoverflow.net/questions/433637 | 3 | Are there any connective $E\_1$ rings $R$ over $\mathbb{F}\_p$ satisfying the following?
1. $\pi\_\*(R)$ is a finite dimensional $\mathbb{F}\_p$ vector space
2. $R$ is compact as a module over $R \otimes R^{op}$
3. $R$ is not concentrated in degree 0
Motivation: I am looking for a smooth proper category with negati... | https://mathoverflow.net/users/136287 | Are there strictly connective smooth proper algebras over $\mathbb{F}_p$? | Here's an example: consider the $\infty$-category $Fun(\Delta^1,Perf(\mathbb F\_p))$. It has two canonical generators $A= \mathbb F\_p\to \mathbb F\_p$ and $B=\mathbb F\_p\to 0$; and I claim that $R= End(A\oplus \Sigma B)$ is an example of what you're looking for (derived endomorphisms - more generally, everything here... | 3 | https://mathoverflow.net/users/102343 | 433806 | 175,455 |
https://mathoverflow.net/questions/433782 | 4 | Let $\mathcal{F}\_N$ be the set of all strictly increasing sequences of positive integers. For every two $F\_1, F\_2\in\mathcal{F}\_N$, if we use $\delta(F\_1,F\_2)$ to denote the first $n$-th coordinate where $F\_1(n)\neq F\_2(n)$, then $d(F\_1, F\_2) = \exp[-\delta(F\_1, F\_2)]$ defines a metric on the space $\mathca... | https://mathoverflow.net/users/151332 | Could the range of $\sum_{k\geq 1}r^{n(k)}$ for $r\in \big(\frac{1}{2}, 1\big)$ be continuous? | For (1), I'm not sure you're going to find a better condition than "$r$ is an algebraic number which is the root of some polynomial with coefficients $0$ and $1$"; I certainly don't think there's an intrinsic necessary and sufficient condition there.
For (2), the answer is yes, the function is always surjective. You ... | 5 | https://mathoverflow.net/users/116357 | 433820 | 175,461 |
https://mathoverflow.net/questions/433779 | 2 | I'm trying to check that certain examples of Young functions in the harmonic analysis literature are actually Young functions, and in doing so need to prove the following convexity-like inequality for $p > 1, \delta > 0$ and $0 \leq a \leq 1 < b$:
\begin{equation}\displaystyle \left( \frac{a+b}{2} \right)^p\left[ 1 +... | https://mathoverflow.net/users/15280 | Elementary convexity example | Here is how to remove the assumption that $p-2+\delta\ge0$.
Let
\begin{equation\*}
s:=p-1+\delta.
\end{equation\*}
The conditions $p>1$ and $\delta>0$ imply $s\ge0$. No other conditions on $p$ and $s$ will be used or needed in what follows.
The inequality in question will follow from the inequality
\begin{equatio... | 2 | https://mathoverflow.net/users/36721 | 433821 | 175,462 |
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