Newest Questions
159,101 questions
5
votes
1
answer
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Between BV and Baire 2
My question is about functions of bounded variation (BV) on the reals.
On one hand, Helly's selection theorem provides (fairly restrictive) conditions under which a sequence of BV-functions has a sub-...
- 4,359
1
vote
0
answers
150
views
Invariant polynomials under a non-standard group action
There is a whole theory of finding the invariant polynomials for matrix groups $\Gamma$ acting on the polynomial ring $\mathbb{C}[x_1,\ldots,x_n]$. I would be interested in finding invariant ...
3
votes
1
answer
288
views
For any integer $n>6$, does there always exist a prime $p>n+1$ such that $p\mid 2^n-1$?
For any integer $n>6$, does there always exist a prime $p>n+1$ such that $p\mid 2^n-1$?
It's true for $6<n<100$. But for $n>100$?
- 103
2
votes
1
answer
99
views
Definite negative functions and length functions
$\DeclareMathOperator\ND{ND}$I am reading E. Bedos paper on heat properties for groups.
Let's denote, for a group G, $$\ND^+_0(G) := \{d : G \to [0,+\infty[\; : \;d \text{ is negative definite and }d(...
- 21
3
votes
0
answers
105
views
When can we lift transitivity of an action from geometric points to a flat cover?
Let $G$ a nice group scheme (say, over $S$), $X$ a smooth $G$-scheme over $S$, that is, $\pi : X \to S$ a smooth, $G$-invariant morphism. Assume that the action is transitive on algebraically closed ...
- 605
1
vote
1
answer
134
views
If $(f,g)$ and $(f,h)$ are maximal ideals, then $ag+bh=P(f)$ for some $a,b \in k, P(t) \in k[t]$?
Let $k$ be an algebraically closed field of characteristic zero, for example $k=\mathbb{C}$.
Let $f,g,h \in k[x,y]$, $g \neq h$, satisfy the following two conditions:
(1) $(f,g)$ is a maximal ideal of ...
- 2,837
0
votes
0
answers
139
views
Integration on algebraic curves
Consider the plane algebraic curve
$$f(x, y) = y^4 - (2x - 1)y^2 - (4x - 1) y + x^2 + x + 1 = 0.\tag{1}$$
Its compactification results in a Riemann surface $C_1$ of genus $1$.
Hence, it can be ...
- 89
10
votes
2
answers
436
views
The additive structure of clusters of nonstandard models of arithmetic
Given $\frak M$ a countable nonstandard model of $\sf PA$ and let $a\in M$ be a nonstandard element. A "cluster around $a$" is the set of successors and predecessors of $a$, a cluster is a ...
- 1,676
7
votes
1
answer
579
views
Progress on determining which partial orders embed into the rationals
The following result is relatively well-known: (for example in Math StackExchange answer #37161)
For every countable linear order $(R,\prec)$, there is an $X\subseteq\mathbb Q$ such that $(R,\prec)$ ...
- 2,031
0
votes
0
answers
75
views
Goldstine theorem in quasi-Banach spaces
A classical theorem of Goldstine is the following: Let $X$ be a Banach space and $J \colon X \to X''$ the natural inclusion. Then $J(B_X)$ is $\sigma(X'', X')$-dense in $B_{X''}$, where $B_Y$ is the ...
0
votes
2
answers
371
views
How can I derive functional properties of (the solutions of) this simple functional differential equation?
I've not yet finished a course in functional analysis so I'm unsure how to go about this, but I've always been fascinated by a simple functional differential equation I concocted for almost no reason.
...
1
vote
0
answers
140
views
Kernel of restriction map in Galois cohomology
Let $S$ be the algebraic group $SL_2/\mathbb{Q}_p$ with a $G=G_{\mathbb{Q}}$ action, (acts by conjugation with a representation $\rho: G\longrightarrow GL_2$.)
Let $G_p$ be the decomposition group at ...
- 2,907
2
votes
0
answers
100
views
Deformations of invertible sheaves admitting global sections
We follow Sernesi's treatment of algebraic deformations, working over the complex numbers.
Given a pair $(X,L)$ consisting of a compact complex manifold $X$ and an invertible sheaf $L$ on $X$, we ...
- 518
8
votes
1
answer
380
views
Given an embedded disk in $\mathbb{R}^n$, is there always another disk which intersects it nontrivially in a disk?
We call an open subset $D\subset X$ of a manifold $X$ an embedded disk, if there exists a homeomorphism $D\cong \mathbb{R}^n$.
The precise formulation of the question in the title is as follows:
Let $...
- 725
1
vote
0
answers
145
views
The free products of finitely many finitely generated groups are hyperbolic relative to the factors
Are there any references how to show that:the free products of finitely many finitely generated groups are hyperbolic relative to the free factors. More precisely, how to show that
$G = A \ast B $ is ...
- 81
2
votes
1
answer
536
views
Bianchi's identity in a principal bundle
Let us consider a principal bundle $P$, with a Lie-algebra-valued connection one-form $\omega\in\mathfrak{g}\otimes\Omega^1(P)$ and a Lie-algebra-valued curvature two-form $\Omega\in\mathfrak{g}\...
- 41
8
votes
2
answers
649
views
Analogous results in geometric group theory and Riemannian geometry?
As you can see from my other question I concern mmyself with the following article at the moment:
Koji Fujiwara, Zlil Sela, The rates of growth in a hyperbolic group, Invent. math. 233 (2023) pp 1427–...
- 309
3
votes
1
answer
157
views
Embedding of half open half closed $n$-set in $n$-space
Let $n\geq 2$. Set $\Sigma= \{x\in \mathbb{R}^n: 1\leq |x|<2\}$. Assume $h:\Sigma
\rightarrow \mathbb{R}^n$ is continuous and injective.
Question: Must $h$ also be an embedding?
Some thoughts:
$h|...
2
votes
1
answer
149
views
Functors of tilting modules
Let $k$ be a field and $G$ be a reductive group over $k$. Inside the category $Rep_k(G)$ of $k$-linear representations of $G$, there's the subcategory $Tilt_k(G)$ of tilting modules.
Suppose I have a ...
- 125
4
votes
1
answer
299
views
Does Kalai's $3^d$ conjecture hold for simplicial spheres?
Kalai's $3^d$ conjecture asserts that every centrally symmetric $d$-polytope has at least $3^d$ non-empty faces. This is open in general, but has been proven for simplicial polytopes.
Question: Does ...
- 13.6k
2
votes
2
answers
538
views
Tensor functor between rigid tensor categories preserves $\text{Hom}$-objects
I was looking at these notes on Tannakian categories. Let me briefly recall the notion of tensor functors:
Let $(\mathcal{C},\otimes)$ and $(\mathcal{C'},\otimes')\DeclareMathOperator{\uphom}{\...
- 737
6
votes
1
answer
363
views
Morita equivalences and centers of some algebras
Let $k $ is an algebraically closed field of $\text{ch}(k)=0$.
Let $$E := k \langle x_0, x_1, x_2 ,x_3 \rangle/(x_ix_j-q_{ij}x_jx_i )_{0 \leq i,j \leq 3},$$ where $$(\text{deg}(x_0), \text{deg}(x_1), \...
- 889
5
votes
0
answers
135
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Geometric interpretation of pairing between bordism and cobordism
In page 448 of these notes, a pairing between bordism and cobordism
$$\langle \ ,\ \rangle: U^m(X)\otimes U_n(X)\rightarrow \Omega^U_{n-m}$$
is defined as follows. Assume $x\in U^m$ is represented by $...
- 171
4
votes
1
answer
259
views
For any integer $n>0$, does there always exist a prime $p>n$ such that $p\mid 2^n-1$?
For any integer $n>0$, does there always exist a prime $p>n$ such that $p\mid 2^n-1$?
It's easy to verify this result for $1<n<100$ by computer.
But for any integer $n>0$, is it always ...
- 103
15
votes
2
answers
913
views
Sequences that don't count algebraic structures on finite sets
People count $n$-element groups, $n$-element monoids, $n$-element commutative monoids, etcetera - always up to isomorphism. The algebraic structures I've listed, and many more, are studied ...
- 22.3k
4
votes
2
answers
406
views
Is a positive degree self map on a Riemann surface homotopic to a holomorphic self map?
Let $S$ be a compact Riemann surface and $f:S\to S$ be a continuous self map of positive degree. Is $f$ homotopic to a holomorphic map on $S$?
Motivation: I had intention to consider this question ...
- 366
0
votes
0
answers
104
views
Automorphism group of symmetric square
Say I have a hyperelliptic curve without any automorphism beyond the hyperelliptic involution.
Is it possible for its symmetric square to obtain new automorphisms beyond the one induced by the ...
- 2,907
0
votes
0
answers
46
views
Prove lower collinearity on the tails of Gaussian blob
Let us consider a $n$-dimensional Gaussian blob, i.e. a set of $N$ random vectors $\{\boldsymbol{X}^{(j)}\}_{j=1}^N$, with $n$ independent components, $X_i^{(j)}$, and such that $X_i^{(j)} \sim \...
- 305
2
votes
1
answer
155
views
Variation of concept of a Lusin space
Citing from Wikipedia,
A Hausdorff topological space is a Lusin space if some stronger topology makes it into a Polish space.
Is there a (previously studied) analogous concept of a Hausdorff (...
- 661
8
votes
3
answers
509
views
Free probability: A unitary group heuristic for the relationship between additive free convolution and free compression
From one perspective, free probability is the study of how the eigenvalues of large random matrices interact under the basic matrix operations. The free probability operations of free additive ...
- 484
0
votes
0
answers
145
views
Pullback of a translation map of a divisor in Birkenhake-Lange's book "Complex Abelian Varieties"
I'm currently studying the book 'Complex Abelian Varieties' by Birkenhake and Lange. On page 74, after lemma 1.5, the authors make the following statement:
'Another observation, which will prove ...
5
votes
2
answers
790
views
What is lost in General Relativity without Hahn-Banach axiom in the ZF+HB set theory?
In the same spirit of this question:
How much of mathematical General Relativity depends on the Axiom of Choice?
I want to go radically further ahead and ask for what remains of mathematical general ...
- 612
1
vote
0
answers
112
views
Is this a correct description of the BPS monopole of charge $1$?
I am reading the book "The Geometry and Dynamics of Magnetic Monopoles", by M.F. Atiyah and N.J. Hitchin, and I got to this part:
"... let $H$ be the Hopf line bundle over $S^2$ and ...
- 5,215
3
votes
0
answers
118
views
Divide Euclidean space by surfaces
It is well known that $n$ hyperplanes in $\mathbb{R}^k$ can divide $\mathbb{R}^k$ into at most $p$ regions where $p$ is
\begin{equation}
1 + n + C^2_n + \cdots + C^k_n
\end{equation}
Is there similar ...
- 185
4
votes
0
answers
247
views
Cantor-Bernstein phenomena for structures (and a "moderate zigzag" property)
My favorite proof of the Cantor-Bernstein theorem is the one that argues by "histories" - given injections $f:A\rightarrow B$ and $g:B\rightarrow A$, we identify each element of $A$ as ...
- 20.7k
8
votes
1
answer
433
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Is "do-almost-nothing" ever winning on large CHOMP boards?
This is a special case of a question asked but unanswered at MSE:
Consider the combinatorial game CHOMP (presented as in the linked notes so that the "poison" square is bottom-left). In any $...
- 20.7k
1
vote
0
answers
169
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On two formulas involving the $k$-fold divisor function $d_k$ and the function $r_k$
I have a puzzle which needs some help form the experts here. Let $d(n)$ be the divisor function, and $d_k(n)$ the $k$-fold divisor function.
I) It is known that, for any positive integer $h$,
$$d(n+h)...
- 563
2
votes
2
answers
239
views
Sign-reversing involution for $q$-binomial coefficient at $q=-1$
Consider the q-binomial coefficient $\binom{n}{k}_q$.
One combinatorial way to define it is as follows. Let $W_{n,k}$ be the set of binary words of length $n$ with $(n-k)$ 0's and $k$ 1's. An ...
- 24.2k
4
votes
1
answer
230
views
Conditions for distinct nonzero eigenvalues in product DAD for symmetric matrix A with repeated nonzero eigenvalues and diagonal matrix D
Let $A\in\mathbb{R}^{n\times n}$ be an real symmetric matrix with eigenvalues $\lambda_1, \lambda_2, \cdots, \lambda_n$ with some of which be nonzero and repeated, i.e., there exist $\lambda_i \ne 0$ ...
- 187
5
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0
answers
124
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Tits indices over $\mathbb{Q}$
Does every Tits index belong to some semisimple algebraic group defined over the field of rational numbers?
- 2,773
11
votes
0
answers
121
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Description of the canonical equivalence between Adelman's free abelian category and Freyd's free abelian category on an additive category?
Let $\mathcal A$ be an additive category. Then there is a free abelian category $F(\mathcal A)$ on $\mathcal A$. I'm aware of two constructions in the literature, and I'd like to relate them.
The ...
- 64k
2
votes
0
answers
124
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Uniqueness in interpolation of Hilbert spaces
I am wondering under what condition it is true that for Hilbert spaces $H$ and $H_0$ such that $H \hookrightarrow H_0$ there is uniqueness in the existence of a Hilbert space $H_1 \hookrightarrow H$ ...
- 21
10
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2
answers
473
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Is the set of permissible numbers of models of various cardinalities computable?
This question arose in the comments to this question.
Let $X$ be the set of pairs $(m,k)$ such that there is some (consistent complete countable first-order) theory $T$ with exactly $m$ models of size ...
- 20.7k
5
votes
1
answer
208
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Sphere packing and modular forms in known dimensions (maybe 2)
Viazovska constructed magic functions via integral transforms of (quasi-)modular forms that gives a tight bound for linear programming bounds in 8 and 24 dimensions (with other mathematicians after ...
- 2,215
53
votes
7
answers
7k
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Are there any undecidability results that are not known to have a diagonal argument proof?
Is there a problem which is known to be undecidable (in the algorithmic sense), but for which the only known proofs of undecidability do not use some form of the Cantor diagonal argument in any ...
- 114k
5
votes
1
answer
206
views
Bias of DS literature to polynomial ODEs
In the literature on continuous time dynamical system, we generally deal with an open set $U \subset \mathbb{R}^n$ and a vector field $F: U \rightarrow \mathbb{R}^n$ and define a DS by the ODE
$$\...
- 247
1
vote
0
answers
201
views
Exceptional locus of proper birational morphism from smooth variety to normal variety
Let $f:Y\rightarrow X$ be a proper birational morphism. Suppose that $X$ is normal and $Y$ is smooth.
Let us write the largest open subset U of X such that $f^{-1}$ can be defined.
I want to show that ...
- 11
2
votes
2
answers
135
views
Directed sets of positive elements in noncommutative $\mathrm L^p$ spaces
Let $\tau$ be a faithful normal semifinite trace on a von Neumann algebra $\mathcal M$.
If $1<p<\infty$ and $E$ is a nonempty subset of $\mathrm L^p(\mathcal M,\tau)_+$ such that
for every $x\...
- 407
8
votes
0
answers
231
views
Lattice point counts on the determinantal variety
I recently came across the following result of Katznelson [1]. It says that for some $C>0$, the following lattice point count holds for $n> m\geq k$.
$\#\{A \in M_{m \times n}(\mathbb{Z}) \mid \...
- 885
4
votes
0
answers
179
views
Is $H^*($vanishing cycles$)$ computed by the twisted de Rham complex?
In notes by Sabbah (Theorem 3), it is stated that the cohomology
$$\text{H}^*(X,\varphi_f)$$
of the vanishing cycle sheaf of a function $f:X\to \mathbf{A}^1$ for certain $X$ is expected to be the same ...
- 5,711