Newest Questions
159,095 questions
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For unit sphere bundle over sphere there exist a real vector bundle equipped with inner product structure? [closed]
For unit sphere bundle over sphere there exists a real vector bundle equipped with an inner product structure?
I did't get any results relative to this extension till now. Is there any result ...
- 11
3
votes
1
answer
358
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A question about Bockstein homomorphisms
For $r\geq 1$, we have the following short exact sequence
$$0\rightarrow \mathbb{Z}/2^r\mathbb{Z} \xrightarrow{\cdot 2^r}\mathbb{Z}/2^{2r}\mathbb{Z} \xrightarrow{\bmod2^r}\mathbb{Z}/2^r\mathbb{Z}\...
- 545
11
votes
0
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374
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Example of abelian variety over finite field which doesn't lift
What is an example of an abelian variety over a finite field $\mathbb{F}_p$ which doesn't lift to $\mathbb{Z}_p$? This question seems to hint that they should exist, but no example is given.
Note that ...
- 21.4k
2
votes
1
answer
235
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What is the homotopy type of the smash power of Moore spectra $(S/2)^{\otimes n}$?
Let $S/2$ be the mod $2$ Moore spectrum, and let $n \in \mathbb N$.
Question:
What is the homotopy type of the $n$th smash power $(S/2)^{\otimes n}$?
Notes:
When $p$ is odd, we have $S/p \otimes S/p =...
- 64k
5
votes
1
answer
310
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Lee-Parker Yamabe problem proposition 4.6
I believe there may be a gap towards the end of the proof of proposition 4.6 in the Bulletin of the AMS paper The Yamabe Problem by Lee and Parker : https://projecteuclid.org/journals/bulletin-of-the-...
- 457
1
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0
answers
85
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Smooth proper local model of a smooth projective curve
Say I have a curve $C/K$, where $K$ is a number field. Let $v$ be a place of $K$, and denote by $K_v$ the $v$-adic completion of $K$. Further assume $C$ is smooth and proper over $K$. Denote by $C_v$ ...
- 2,907
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votes
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274
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Comparison of model-theoretic and axiomatic approaches to NSA
This question is motivated by the discussion in the comments to this
post. The question concerns
a comparison of model-theoretic (extension) approaches to nonstandard
analysis, and axiomatic (...
- 16.6k
4
votes
0
answers
88
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Diameters of permutation groups with transitive generators
Suppose that for a permutation puzzle on $n$ elements we have a subroutine to place any $m>3$ elements in arbitrary positions (possibly scrambling the rest). Can we solve the puzzle (if it is ...
- 7,545
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votes
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99
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Reflections of an apartment in building — Weyl groups
I will give the definition of what I mean by reflection which is in Suzuki's group theory I.
Let $\Sigma $ be an apartment of a building that contains adjacent chambers $C$ and $C'$. Then there are ...
- 139
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1
answer
99
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Recovering the openness of a map from the openness of its scalar projections
Good morning. I have been thinking about the following question for a while without much success, therefore I'm starting to doubt its validity, although I don't have a clear counterexample in mind.
...
- 311
0
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0
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95
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Prove that $\forall x,y \in \mathbb{R}^d , P_x\{y\in B\mathopen]0,1]\}=0$
I'm folowing the proof of corollary 1.8 page 5 of Mörters - Sample path properties of Brownian motion.
I want to show that $$\forall x,y \in \mathbb{R}^d , P_x\{y\in B\mathopen]0,1]\}=0$$ where $B$ is ...
- 11
0
votes
2
answers
131
views
Reshaping data vector into a matrix for deconvolution using a circulant matrix
Suppose we have a circulant matrix S made from pseudorandom binary sequence of length $N$ consisting of $0$'s or/and $1$'s. $1$ means that we can inject something for chemical analysis and $0$ means ...
- 879
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0
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78
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Could the patterns in the roots of the Riemann-Liouville differintegral of $(s-1)\,L(s, \chi_{q,j})$ be explained?
This question aims to extend this question to (automorphic) Dirichlet L-functions.
Take: $$f(s,q,j) = (s-1)\,L(s, \chi_{q,j})$$
with $q$ the modulus and $j$ the index of a character $\chi$. A fast way ...
- 4,169
4
votes
1
answer
269
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Submodule lattices of preprojective algebras
Let $A$ be a preprojective algebra of Dynkin type.
Question 1: Let $P$ be an indecomposable projective $A$-module. It is known that the submodule lattice of $P$ is finite in Dynkin type $A_n$. Does ...
- 26.5k
1
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0
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130
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Resolving singularities in numerical integration
I am now trying to compute numerically the following integral.
$$
\begin{split}
L_1^s(\hat{\phi}_s)(r,\zeta,\theta_\zeta) &=\frac{1}{\sqrt{2}\pi}
\int\limits_{0}^{2\pi} d\varphi
\int\limits_0^{\pi}...
- 11
7
votes
1
answer
453
views
Strong limits of nilpotent operators
Let $H$ be an infinite-dimensional Hilbert space.
Is it possible that the Identity $H\to H$ is a strong limit of nilpotent compact operators?
user473423
1
vote
0
answers
127
views
Facility location on a graph
Given a connected graph $G$ with weighted edges (weights can be assumed as edge lengths and are non-negative). We need to identify the vertex such that the sum of the distances from other vertices to ...
- 5,979
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82
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Name of the power of the exponent of a $p$-group
Is there a name for the power of the exponent of a $p$-group? So, if $\mathrm{exp}(G):=\max\lbrace o(g)|g\in G\rbrace=p^k$ for some $k\in\mathbb{N}$, is there a name for the $k$? Additionally, is ...
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answers
124
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Does this construct Platonic solids?
Consider $\mathrm{O}(3)\curvearrowright\mathbb{R}^3$. Let $\Gamma\subseteq\mathrm{O}(3)$ be a finite group. Let $x\in \mathbb{R}^3\setminus\{0\}$ be a point such that $\mathrm{Stab}_\Gamma(x)$ has ...
- 930
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154
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Hochschild cohomology of path algebra as a cohomology of simplicial complex
M. Gerstenhaber and S. D. Schack have shown that a cohomology of simplicial complex can be expressed as a Hochschild cohomology of path algebra constructed from this complex (link).
Is the opposite ...
- 91
20
votes
3
answers
3k
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What is the simplest proof that the density of coprime pairs does not go to zero?
By density of coprime pairs, I mean the proportion of pairs integers between $1$ and $x$ which are coprime.
This is known to be asymptotically $1/\zeta(2)$.
I want something much weaker, namely that ...
- 19.1k
9
votes
4
answers
475
views
Minimum number of common edges of triangulations
Let $S$ and $T$ be two triangulations.
We define
$c(S,T)$ as the number of edges shared by $S$ and $T$.
With this, we can define
$f(n) = \min_{P} \min_{S,T} c(S,T)$.
Here the first minimum goes over ...
- 479
1
vote
0
answers
103
views
Is there a bound on the number of $p$-adic semisimple representations?
Faltings proved the following:
Fix integers $w, d \geqslant 0$, and fix a number field $K$ and a finite set $S$ of primes of $\mathcal{O}_K$. There are, up to conjugation, only finitely many ...
- 785
1
vote
0
answers
74
views
Idempotent completeness
We say a category $\mathcal{N}$ is exact if it is additive and is endowed with an exact structure. In brief, it is an additive category with a predetermined class of short exact sequences in its ...
- 167
7
votes
4
answers
1k
views
Amending flawed "proof" that homology groups are zero
I am trying to prove a certain statement that seems true based on computational data, and there is a nice argument that proves it, assuming all cycles are the simplest ones (e.g., when the only 1-...
- 807
1
vote
1
answer
236
views
Derived McKay correspondence between a weighted projective plane and a Hirzebruch surface
Let $k$ be an algebraically closed field of $\text{ch}(k) =0$.
Let $\mathbb{P}(1,1,2)$ be the weighted projective plane of weight $(1,1,2)$ as a stack.
Let $\mathbf{P}(1,1,2)$ be the weighted ...
- 13
0
votes
1
answer
113
views
Where does this coupling result use independence when bounding total variational distance?
I am reading this paper, which gives the following coupling result:
Throughout this, I'll assume the dimension $k$ is clear. Let $e_i$ be the $i$-th basis in the $k$ dimensional standard basis.
A $k$ ...
- 662
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votes
0
answers
58
views
Role of basins of attraction in the Morse decomposition
Let $M$ be a differentiable manifold and $F \in \mathcal{X}(M)$. We define a DS by
$$\dot{x}=F(x(t))$$
An ordered collection $\mathcal{M}=\left\{M_{1}, \ldots, M_{l}\right\}$ of compact subsets of ...
- 247
2
votes
1
answer
155
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Is the vector field associated with an element of the boundary at infinity on a Hadamard manifold smooth?
A Hadamard manifold $M$ (complete, simply connected, non-positive sectional curvature) has a so-called boundary at infinity $\partial M$ whose elements are equivalence classes of unit-speed geodesic ...
- 23
3
votes
1
answer
293
views
How can I verify this family of values for hypergeometric functions?
This Wolfram MathWorld page on hypergeometric functions states that
An infinite family of rational values for well-poised hypergeometric functions with rational arguments is given by $$_kF_{k-1}\left[...
- 33
1
vote
1
answer
110
views
Gorenstein property from initial ideal
My question is:
If $I$ is a homogenous ideal of $S=K[x_1,\dots,x_n]$ and $in_{<}(I)$ is the initial ideal of $I$, with respect to a term order $<$ on $S$, then $S/I$ is Gorenstein if and only if ...
- 115
2
votes
1
answer
200
views
An inequality related to matrix trace
$$Tr(A|A^T U \Sigma U^T|) \leq Tr(AA^T U \Sigma U^T)$$ where $A \in \mathbb{R}^{m\times n}$ is a real rectangular matrix, $U \in \mathbb{R}^{n \times n}$ is a orthonormal matrix and $\Sigma$ is a ...
- 21
0
votes
0
answers
232
views
Lower bounds for sub-Gaussians?
For a random variable $X$, define
$$\lVert X\rVert_{\psi_2} =\inf \{k>0\mid \mathbb{E}[\exp((X/k)^2)]\leq 2\}$$
and for a random vector $\vec X$, define
$$\lVert \vec X\rVert_{\psi_2} = \sup_{\...
- 2,183
3
votes
2
answers
253
views
On algebraic topology of coset complexes without geometry
I'm interested in understanding the algebraic topology of "coset complexes" from a "combinatorial" perspective (i.e., without relying on geometric realizations of the complexes). ...
- 173
1
vote
0
answers
165
views
Rotation number for homeomorphisms of a Lie group other than $S^1$
Let $G$ be a Lie group whose Lie algebra is $\mathfrak{g}$ with exponential map $\exp:\mathfrak{g}\to G$.
For what kind of Lie group $G$ the standard process of definition of rotation number ...
- 366
1
vote
0
answers
178
views
Is the Poincare Birkhoff theorem valid if we change the volume form of the annulus region?
Is the Poincare-Birkhoff theorem valid if we change the volume form of the annulus region?
Note: A possible approach could be the following: Is it true to say that the answer is affirmative ...
- 366
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1
answer
160
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Embedding of symmetric square in Jacobian
Let $C$ be a projective curve defined over a field $K$, and let $C^{(2)}$ and $J$ be its symmetric square and Jacobian, respectively.
There is a natural map $C^{(2)}\hookrightarrow J$, defined as ...
- 2,907
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0
answers
58
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Given a proper submersion $f: X \setminus F_0 \to D \setminus 0$ which extends to $X \to D$, are cycles on $X$ which run around $0$ boundaries in $X$?
I have a proper map of complex manifolds
$$f: X \to D,$$
where $D \subset \mathbb C$ is the unit disc. By assumption, $f$ has connected fibers, is smooth over $D \setminus 0$, and a smooth fiber $F$ ...
- 1,286
3
votes
0
answers
132
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How does the number of connected components of the Néron model change in a family of abelian varieties?
Given an elliptic curve $E/\mathbb{Q}_p$, it is known that the component group of the Néron model of $E$ is cyclic of order $-v(j(E))$ when $E$ has split multiplicative reduction, and in all other ...
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votes
1
answer
1k
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If some powers of polynomials are linearly independent, does it imply higher powers are also independent?
Let $P_1,\dotsc,P_k$ be polynomials. Assume they are pairwise non-proportional (i.e., any two of them are linearly independent). Suppose $N$ is a power such that $P_1^N,\dotsc,P_k^N$ are linearly ...
- 6,237
3
votes
1
answer
122
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Number of categories of product of fusion rings
Given two fusion rings $A$, $B$, and let $n_A$, $n_B$ be the number of gauge-inequivalent fusion categories belonging to $A$ and $B$. When one looks at the number of categories $n_{A\otimes B}$ of the ...
- 303
1
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0
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126
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Canonical basis of cycles of Riemann surfaces
Let $\Gamma$ be the compact Riemann surface defiend by the algebraic curve
$$
f(x, y) = y^n + a_1(x)y^{n-1} + a_2(x) y^{n-2} + \dots + a_{n-1}(x)y + a_n(x) = 0,
$$
where $a_1(x), \dots, a_n(x)$ are ...
- 89
5
votes
0
answers
128
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Ratio of theta functions as roots of polynomials
I already asked the same question here, but received no answer. I did some little progress and so I'm asking again.
I was playing with the theta functions with argument $ z = 0 $
$ \vartheta_2(q) =\...
- 387
3
votes
0
answers
786
views
Is this set a manifold?
Take a general spacetime that is not strongly causal.
Call this spacetime $(M, g) $ where $M$ is a connected time-oriented manifold and $g$ is the Lorentzian metric that satisfies the Einstein's Field ...
- 612
2
votes
0
answers
70
views
Reference request : A SPDE model
Let $\Omega_0\subset\mathbb R^d$ be open and bounded with sufficiently smooth boundary $\partial\Omega_0$. Let $O\subset \Omega_0$ be a random open subset. Set $\Omega:=\Omega_0\setminus O$. Consider ...
- 1,409
2
votes
1
answer
184
views
Gorenstein projective module over commutative local algebras
Let $A$ be a local commutative finite dimensional algebra over a field $K$.
An $A$-module $M$ is called Gorenstein projective if $M$ is reflexive, $Ext_A^i(M,A)=0=Ext_A^i(M^{*},A)$ for all $i>0$ ...
- 26.5k
1
vote
0
answers
106
views
Benjamini-Schramm convergence: convergence on metric balls implies weak convergence?
In the famous paper by Benjamini-Schramm 2001, they consider the space of rooted graphs with uniformly bounded degrees, this space modulo rooted graph isomorphism is denoted by $\mathcal X$.
This ...
- 502
1
vote
0
answers
133
views
Find the number of tuples $(x_0,x_1,\dots)$ such that $n=\sum\limits_{j=0}^\infty x_j2^{j}$ and $m=\sum\limits_{j=0}^\infty x_j(2^{j}-1)$
While studying the homology mod-2 of a certain family of groups $\lbrace G_n\rbrace$, where $n$ is the number of generators of the group, I was able to prove the following:
The homology group $H_m(G_n,...
- 911
2
votes
1
answer
245
views
Markov property for groups?
My question again refers to the following article:
Koji Fujiwara, Zlil Sela, The rates of growth in a hyperbolic group, Invent. math. 233 (2023) pp 1427–1470, doi:10.1007/s00222-023-01200-w, arXiv:...
- 309
3
votes
0
answers
147
views
Derived categories and minimal model program
I read Kawamata's note, and the goal of this note is to construct an equivalence
$$ \Phi:D^{b}(X)\to D^{b}(Y)$$
for a flop $X\to Z\leftarrow Y$, and I wonder to the moduli approach and it seems to me ...
- 392