Trending questions
159,041 questions
2
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Leray spectral sequence for étale homology
Let $X$, $Y$, $Z$ be quasi-projective varieties over an algebraically closed field $k$, $f: X \to Y$ and $g: Z \to X$ proper (even projective) maps with $f$ smooth, and $h: Z \to Y$ their composite. ...
- 658
8
votes
1
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868
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Brauer–Siegel's Theorem and application
$\newcommand\underto[1]{\xrightarrow[#1]{}}$Brauer–Siegel's Theorem says that if we consider an infinite sequence of number fields $K_i/\mathbb{Q}$ such that $$\frac{[K_i:\mathbb{Q}]}{\log{|D_i|}}\...
- 103
7
votes
2
answers
323
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Formula for compositions of Steenrod squares that produce a form in the top degree
On a smooth $d$ dimensional compact connected manifold $M$, for an $\mathbb{Z}_2$-valued $(d-j)$-cocycle $x_{d-j}$ we have the formula ${\text{Sq}}^{j} (x_{d-j}) = u_{j} \cup x_{d-j}$. Here $u_j \in ...
- 283
7
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330
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Is the partial order of all equations in the signature of magmas a lattice?
$\newcommand\Eq{\mathrm{Eq}}$I asked this question on math stack exchange, here, but there were no comments or answers. So, I am asking it here on mathoverflow. Consider the signature of a single ...
- 2,023
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vote
1
answer
132
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Packing problems where parts of objects are allowed to intersect
I'm interested in packing problems where the objects are allowed to intersect.
For a simple example, consider stacking 1×2 tiles on a nxn chessboard. Each 1×2 tiles consists of part X and Y (both 1×1)....
- 11
4
votes
2
answers
287
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Nash equilibria of a "minority game"
An odd number $N \geq 3$ of players are playing a game - they bet on the outcome of a biased coin that comes up heads $p > \frac{1}{2}$ of the time, where $p$ is known to all of the players in ...
- 6,323
4
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390
views
Gaussian mixtures are dense in total variation?
Let $M_{TV}(\mathbb{R}^d)$ denote the set of probability measures on $\mathbb{R}^d$ with finite total variation norm which are absolutely continuous with respect to the Lebesgue measure.
By a Gaussian ...
- 5,405
12
votes
0
answers
271
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What is known about G. A. Croes
G. A. Croes is the author of the first description of the 2-opt moves heuristic for improving non-optimal traveling salesman tours:
Croes, G. A. “A Method for Solving Traveling-Salesman Problems.” ...
- 13.2k
1
vote
2
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127
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Homeomorphism and boundary of a complementary component
Let $X\subset \mathbb R^2$ be compact and connected. My question is whether homeomorphisms of $X$ preserve boundaries of complementary components.
More precisely, let $h:X\to X$ be a homeomorphism.
...
- 3,317
1
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118
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Are most true statements about Math unprovable (undecidable)? [duplicate]
In an essay titled: All Questions Answered, Donald Knuth states that “In fact, we now know that in some sense almost all correct statements about mathematics are unprovable.” How do we know that?
I’m ...
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135
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Reconstructing a scheme from its quotient stack
Let $X/S$ be a scheme over a base $S$, with a group action by a nice group $G/S$ (typically $G/S$ affine smooth).
Can we reconstruct $X$ from its quotient stack $[X/G]$?
It seems that we can expect $X$...
- 645
43
votes
4
answers
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On what basis does a paper get accepted into a top journal?
Since the initial question was closed, but seems to be attracting a lot of discussion even after the fact, some of the comments qualifying as full answers IMO, I believe it is a reasonable question to ...
1
vote
2
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118
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If $f\in C([0,\infty))$, does $\delta>0$ and $g\in C^1((0,\delta))\cap C([0,\delta])$ s.t. $g\geq f$ on $[0,\delta]$ and $g(0)=f(0)$ exist?
The question is the following:
Suppose $f : [0,\infty) \rightarrow \mathbb{R}$ is a continuous function. Can I find $\delta \in (0,\infty)$ and a function $g : [0,\delta] \rightarrow \mathbb{R}$ such ...
- 309
3
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76
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What exactly does it mean for the moduli space of stable sheaves to have a universal family étale locally?
It is known that in general, the moduli space $M$ of (Gieseker) stable sheaves do not have a universal family. Therefore, a map $f \colon S \to M$ does not necessarily induce a family of sheaves.
On ...
3
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123
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Reference request: ray class group as quotient of finite ideles
Let $K$ be a number field, and write $\mathbb{A}_{K,f}^\times$ for the group of finite ideles of $K$. That is
$$
\mathbb{A}_{K,f}^\times = \{(u_v)_v \in \prod_{v \nmid \infty} K_v^\times : v(u_v) = 0 \...
- 411
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93
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Relative algebraic $K$ theory of Galois extensions
Let $R$ be a domain, e.g. $\mathbb{Z}$, and let $R\to R[x]/I$ be an integral extension, e.g $\mathbb{Z}[i]$.
Is anything known about the relative algebraic $K$ groups, $K_*(R',R)$? For example, when $...
- 2,907
1
vote
1
answer
63
views
Need bound for absolute value of complex-valued special functions (Taylor coefficients of Faddeeva's w(z))
To guarantee accuracy for code [1] that computes Faddeeva's w(z) [2] using Taylor expansions around different centers, I would need upper bounds for the absolute values $|w_n(z)|$ of the coefficients
$...
- 111
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80
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Reference for computing cohomology of $Q_nS^0$ (the $n$-th component of infinite loop space)?
$Q_nS^0=\varinjlim_k(\Omega^kS^k)_n$, where $(\Omega^kS^k)_n$ refers to homotopy classes of maps $(S^k,\ast)\to (S^k,\ast)$ of degree $n$. I already know the case of $n=0$.
Is there any reference ...
8
votes
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answer
635
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Availability of a copy the first volume of Segre's "Forme differenziali e loro integrali"
I am precisely referring to the following, first volume of the textbook/lecture notes/monograph written by Beniamino Segre in the fifties of the twentieth century (I own a copy of the second volume)
...
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5
votes
1
answer
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How many well-orders of reals are there?
It's commonly known that the cardinality of the set of all well-orders on $\aleph_0$ is the continuum (correct me if I'm wrong plz). What about that of all well-orders on $\mathbb{R}$? Is there a ...
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votes
1
answer
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Is there a ternary Cayley graph on 27 vertices that is a non-complete core?
Is there a non-complete ternary Cayley graph that is a core with $3^3 = 27$ vertices?
By a ternary Cayley graph, I mean a (simple, undirected) graph whose vertex set is $\mathbb{Z}_3^n := \bigoplus_{i ...
- 331
7
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224
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Finitistic interpretation of Nelson's internal set theory
What does “standard” in internal set theory really mean?
Is it secretly a way of reconciling conventional mathematics with (ultra)finitism?
Until recently I thought “standard” was just a way of ...
- 15.9k
4
votes
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answers
286
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Does $\mathbf R\text{Hom}_R(k, -)\otimes_R^{\mathbf L} k$ commute with co-products?
Let $(R, \mathfrak m, k)$ be a commutative Noetherian local ring. Then, is it true that $\mathbf R\text{Hom}_R(k, -)\otimes_R^{\mathbf L} k$ commutes with arbitrary co-products?
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189
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Can ZFC be interpreted in this infinitary logic theory?
Working in language $\mathcal L_{\Omega^+,\Omega^+}$ where $\Omega$ is the first strongly inaccessible cardinal. If we add a primitive partial $\Omega$-ary function $F$ and a primitive constant $\...
- 11.3k
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votes
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92
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Chern class of roots of the canonical bundle
How can one compute the Chern class of line bundles that are roots of the canonical bundle? For example the theta characteristic is K^{1/2} and in general we can have line bundles that look like K^{m/...
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1
vote
1
answer
203
views
Is $\mathcal H/\Gamma$ defined over a number field when $\Gamma$ is not congruent?
Let $\mathcal H$ denote the upper half plane of complex numbers, and $\Gamma$ an arithmetic subgroup of $\operatorname{SL}_2(\mathbb Z )$ (not necessarily congruent). I wonder whether $\mathcal H/\...
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3
votes
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answers
192
views
What smoothing to use for PNT-like results?
Consider a Dirichlet series $\sum_n a_n n^{-s}$ with desirable analytic properties (e.g., analytic extension to $\Re s>0$); one example would be $a_n=\mu(n)$. Say we want to estimate $\sum_{n\leq x}...
- 20.2k
-1
votes
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answers
94
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Why define Schwartz by supremum rather than limit? [migrated]
The Schwartz space is defined as the set of all indefinitely differentiable functions such that the supremum over the free variable of any (order) derivative times any (order) power is finite.
However,...
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3
votes
0
answers
64
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Congruences regarding $4n$-dimensional lattices
A sequence of integers $(a_n)_{n\geq 1}$ satisfies Gauss congruence if
$$\sum_{d\mid n}\mu(d)a_{n/d}\equiv 0\pmod{n}$$
for every $n\geq 1$. Such sequences are also called Dold sequences, Newton ...
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3
votes
1
answer
246
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Model structure on simply-connected topological spaces in which the weak equivalences are the rational homotopy equivalences
I recently started learning rational homotopy theory, and found the claim on page 7 of this survey that there is a suitable model category structure in which the weak equivalences are the rational ...
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2
votes
1
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146
views
How to integrate this differential equation?
I am trying to solve the following differential equation:
\[
\frac{1}{\sqrt{1+y}} \frac{dx}{dy} - \frac{2\sqrt{1+y}}{x} = 2(x+5).
\]
After performing the substitution:
\[
p = \sqrt{1+y}, \quad y = p^2 ...
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4
votes
1
answer
92
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Does an indexed functor $C \rightarrow \mathbb{B}$ extend to $\operatorname{Psh}(C) \rightarrow \mathbb{B}$?
This question concerns indexed categories and functors, as well as internal categories and functors.
$\newcommand{\Psh}{{\operatorname{Psh}}}$
$\newcommand{\Id}{{\operatorname{Id}}}$
$\newcommand{\...
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answers
127
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Solving the quintic using the eta quotients $\frac{\eta(\tau)}{\eta(2\tau)},\,\frac{\eta(\tau)}{\eta(3\tau)},\,\frac{\eta(\tau)}{\eta(4\tau)},$ etc?
I. Reduced quintics
The general quintic can be reduced to the one-parameter forms,
$$x^5+5x+\alpha=0\\[5pt]
x^5+5\alpha x^2-\alpha=0$$
for some generic alpha. The first is the Bring form and there are ...
- 12.6k
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votes
0
answers
85
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Taking hyperplane section remains dominant
I was reading Kollar's book "Rational curves on algebraic varieties". This is from Chapter IV, proposition $1.3$. I don't understand the proof from $1.3.3$ to $1.3.1$, Page- 182. Suppose I ...
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3
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307
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Approximate square root of Dirac delta function on $\mathrm{SL}_2(\mathbb{R})$
$\DeclareMathOperator\SL{SL}\DeclareMathOperator\AdS{AdS}$I hope to find a sequence of complex-valued functions $\{f_i(g)\}$ on the group element $g$ of a locally compact group $\SL(2,\mathbb{R})$ so ...
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103
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Prove or disprove that $|(1/\zeta)^{(n)}(x)| \leq \frac{n!}{(x-\frac{1}{2})}$ for all real $x>1$
$|(1/\zeta)^{(n)}(x)| \leq \frac{n!}{(x-\frac{1}{2})}$ for all real $x>1$.
I had this conjecture for a long time. I tried various methods and techniques but they all failed. It might also be wrong ...
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3
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0
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155
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Colimits in commutative Banach algebras?
Let $K$ be a complete non-Archimedean field. It is known that the category $\mathrm{Ban}_K$ of $K$-Banach spaces with bounded linear maps does not have infinite colimits. The usual argument for $\...
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votes
1
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177
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Mellin transform at $0$
Let $f$ be a smooth complex valued function with support on $[0,2]$ and with Mellin transform $\tilde{f}$. Recall that the Mellin transform is defined as $$\tilde{f}(w)=\int_0^{\infty}f(x)x^{w-1}dx.$$ ...
user543269
3
votes
1
answer
103
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Designing an SDE satisfied by $\frac{B(t)}{1+t}$
Let $B$ be the Brownian motion. I want to find a stochastic differential equation satisfied by the process $$X(t) = \frac{B(t)}{1+t}.$$ I am trying to use Itô's lemma for $f(x,t) = \frac{x}{1+t}$ but ...
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1
vote
1
answer
91
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Sobolev inequality with weight in the case $1<n\leq p$
Assume that $1<n\leq p$. Does there exist a (non-negative) measure $\mu$ (preferably with some positive density function with respect to the Lebesue measure $dx$) and $q>p$ so that for all $f\in ...
- 459
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vote
0
answers
81
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Is every homogeneous line bundle pulled back from the quotient stack?
Let $G= \mathbb{G}_m^k$ act on a variety $X$.
Let $\mathcal{L}$ be a line bundle on $X$ and assume that for each $g \in G$ the pullback $g^\star \mathcal{L}$ is isomorphic to $\mathcal{L}$.
Does it ...
- 323
3
votes
1
answer
158
views
Three congruences for a Perrin-like sequence and pseudoprimes
Let $ V(n) $ be defined by the recurrence relation:
$$
V(n) = 3\,V(n-2) + V(n-3)
$$
with the initial conditions:
$$
V(0) = 3, \quad V(1) = 0, \quad \text{and} \quad V(2) = 6.
$$
If $ n $ is an odd ...
0
votes
1
answer
169
views
Existence of a "universal" measure-preserving transformation on the unit interval
Let $I = [0,1]$ be the unit interval equipped with the Lebesgue measure $\lambda$. Let $\mathcal{M}$ be the set of all Lebesgue measure-preserving transformations $T: I \to I$. We say a transformation ...
user545331
1
vote
0
answers
149
views
Inequalities in the classic proof of perfect matching in Erdős–Rényi graph
I am checking the classic paper by Erdős and Rényi, "On the existence of a factor of degree one of a connected random graph" with the link here. I am curious about the computation of the ...
- 97
5
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0
answers
204
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A proof for an $L^p$-$L^p$ inequality
This is a transfer of the question
https://math.stackexchange.com/questions/4996853/an-lp-lp-inequality
Let $a\in (0,1)$ and $1<p<\infty$ and use $L^{p}$ to denote the space $L^{p}([0,\infty))$ ...
- 852
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votes
0
answers
70
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Is the hypothesis "uniformly equivalent" needed?
I am reading S. Shimorin's paper titled Complete Nevanlinna-Pick property of Dirichlet-type spaces. My question concerns Lemma 2.3. which is as follows:
Assume $\mathscr{H}$ is a Hilbert space of ...
- 151
1
vote
0
answers
58
views
'Invert' perturbed vorticity equation to forced Euler system
Given the vorticity form of the Euler equations in $2D$ with stream function $\psi$
\begin{align}
\omega_t + \nabla^\perp \psi \cdot \nabla\omega &= 0 \\
\Delta \psi = \omega
\end{align}
we know ...
- 255
0
votes
0
answers
37
views
Maximise norm over the boundary of a convex set
Let $K\subset \mathbb R^2$ be compact, convex and connected. What is the know numerical scheme to find the extremal points of $K$?
Denote by $\partial K$ the collection of all extremal points of $K$. ...
- 1,399
8
votes
1
answer
199
views
Topological property of the space of probability measures
Suppose that $\mathbb{P}$ is the metric space of Borel probability measures on the interval $[0,1]$ equipped with the topology of $w^*$ convergence.
Consider also $\mathbb{P}_{ac}, \mathbb{P}_{s}$ the ...
1
vote
2
answers
268
views
Is there any counterexample of the statement that the residue field extension of a separable extension is also separable?
Recently I'm reading Serre's Local Fields, and in page 22 (English version) he says that
The residue field extension $\overline L/\overline{K}$ is separable in each of the following cases (which ...
- 515