Trending questions
159,051 questions
3
votes
1
answer
300
views
Motives and birational invariance
One can construct non-isomorphic smooth projective varieties which define the same motive by blowing up $\mathbb{P}^2$ at five points. I think I learned this here at MathOverflow. But these examples ...
- 205
1
vote
0
answers
182
views
"Reflex field" for $\mathbb H/\Gamma$ for $\Gamma$ non-congruence
Suppose $\Gamma$ is a non-congruence arithmetic subgroup of $PGL_2(\mathbb Z)$, and $\mathbb H$ is the upper half plane of $\mathbb C$. Then by Belyi's theorem we know $\mathbb H/\Gamma$ is an ...
- 785
1
vote
1
answer
67
views
Does convergence in probability of iid samples imply convergence in measure of the sampled functions?
Let $g_i: [0, 1] \to \mathbb R$ be $L^1$ functions, equibounded in $L^1$ norm. Let $X_i$ a sequence of iid uniform random variables on $[0, 1]$. Suppose that
$$\frac{1}{n} \sum_{i = 1}^n g_i (X_i) \to ...
- 6,323
20
votes
1
answer
620
views
Conjecture on the number of roots of $z^n + P(z)$ within the unit disk
Some other people and I have noticed that the following seems to be true.
Fix an integer polynomial $P \in \mathbb{Z}[x]$. Let $a_n$ be the number of roots of $z^n + P(z) = 0$ that lie in the unit ...
- 303
9
votes
0
answers
284
views
Meaning of the Ehrhart polynomial at $-1/2$?
I am studying a large collection of lattice polytopes, all of them being simple and empty. The dimension can be any integer. The dilatation by $2$ gives non-empty polytopes.
For many of these ...
- 3,597
7
votes
2
answers
244
views
Prove that $ n \leq d+1 $ under ordering constraints in $\mathbb{R}^d$
Let $x_1, \dotsc, x_n \in \mathbb{R}^d$ and $\theta_1, \dotsc, \theta_n \in \mathbb{R}^d$ be vectors such that for every $k \in [n]$, the following inequality holds:
$$
\langle x_k, \theta_k \rangle &...
4
votes
1
answer
343
views
Minimum eigenvalue of a symmetric matrix
I was solving a problem and got stuck on the following:
Let $[p] = \{1, \ldots, p\}$ where $p \in \mathbb{N}$. Let $P(n, r)$ denote the set of all injective functions from $[r]$ to $[n]$ and write a ...
- 41
0
votes
1
answer
188
views
Does the second Bourgain–Delbaen space belong to C_p?
The second Bourgain–Delbaen space $Y$ is a separable $\mathcal{L}_\infty$ space such that $Y$ contains no copy of $c_0$ or $\ell_1$. The dual space is isomorphic to $\ell_1$.
An operator $T:X\to Y$ ...
1
vote
1
answer
74
views
Positivity of caloric measure density on a cylinder
Let $u$ be a solution to the heat equation $u_t = \Delta u$ in the unit cylinder $B_1\times(-1,0) \subset \mathbb R^{n+1}$.
Then, it is well known (see for instance Chapter 2 in "Watson - ...
1
vote
2
answers
225
views
Bounds of zeta function near $\Re(s)=1$
Richert proved in
https://link.springer.com/article/10.1007/BF01399533
that $$ \zeta(s) =O\left( |\Im(s)|^{100(1-\Re(s))^{3/2}} (\log |\Im(s)|)^{2/3}\right)$$ uniformly in the region $\Re(s)\in [1/2,1]...
- 3,062
2
votes
1
answer
337
views
Application of the adjoint functor theorem to get the right adjoint of the forgetful functor from $\delta$-rings to rings (the Witt vectors)
I am studying $\delta$-rings and Witt vectors from [K] (the definition of $\delta$-ring is [K, 2.1.1]), and I am having trouble verifying that everything in Kedlaya's definition for the Witt vectors ...
6
votes
1
answer
133
views
Number of semistandard tableaux of all possible shapes fitting within some rectangle
Suppose $n$ and $k$ are two integers. Then I am interested in having a closed form for the sum
$$\sum_{\lambda \subset k \times n} S_\lambda (\mathbb{C}^n),$$
where $S_\lambda$ denotes the Schur ...
- 553
-3
votes
0
answers
137
views
Approximation on Dirichlet's arithmetic progression by means of central limit theorem
In this video lecture on
Number theory over function fields taught by Will Sawin
is presented a 'conceptional' reason for error estimation
$\#\{p \in \Bbb P: p =a \ \text{mod} \ N, p <x \}
=\frac{1}...
- 619
5
votes
0
answers
181
views
Deformations of cotangent bundles
Let $M$ be a smooth variety of even dimension over $\mathbb C$. I am interested in necessary or sufficient conditions such that $X$ is a deformation of a family of cotangent bundles.
In other words, ...
- 6,622
3
votes
0
answers
94
views
What axioms are needed to show that the range of a finitely additive diffuse measure on $\mathbb N$ is not closed?
The other day I learned of a small error in the book Theory of Charges: A Study of Finitely Additive Measures. Example 11.4.1 goes as follows.
Let $\mu_0$ be a finitely additive probability measure ...
- 869
1
vote
1
answer
80
views
$p$-torsion related to algebraic groups
Definition $14.14$ from "Linear algebraic groups and finite groups of Lie type" by Malle and Testerman:
A prime $p$ is a torsion prime for a linear algebraic group $G$ if the fundamental ...
- 217
9
votes
2
answers
803
views
Explanation for Lurie's SAG Remark 25.1.3.7
I am trying to understand the theory of simplicial commutative rings or animated rings. I just find a remark in Lurie's book Spectral Algebraic Geometry:
Remark 25.3.1.7. Let $f : R[x_1,\ldots ,x_n]\...
- 93
16
votes
2
answers
602
views
$\mathrm{GL}_n(\mathbb{Z}_2)=\mathrm{Out}(F_n)/\langle\langle \epsilon_1,\dots,\epsilon_n\rangle\rangle$
In a paper I found the following result:
$$\mathrm{GL}_n(\mathbb{Z}_2)=\mathrm{Out}(F_n)/\langle\langle \epsilon_1,\dots,\epsilon_n\rangle\rangle$$
However, they got the result as a corollary of a ...
- 911
4
votes
2
answers
205
views
Lower bounding a sumset quantity
Given $A,B \subset[0,...,d]^n$ such that $A \cap B = \phi$. Can we show
$$ |(2A \cup 2B) \triangle (A + B)| \geq \Omega_d({\rm poly}(|A|,|B|))$$
where $2A = A+A, 2B = B+B$ and we are taking the ...
- 343
7
votes
2
answers
398
views
Tangent space to infinite dimensional manifolds
In finite dimensional geometry, there is a single invariant of a vector space - its dimension. This characterizes finite dimensional manifolds as being glued from Euclidean balls.
This situation is ...
- 593
4
votes
0
answers
48
views
Resolution of constant sheaf by $L^2$ function sheaves
Let $X$ be a compact Hausdorff space equipped with a Radon measure of full support.
Then $U\mapsto L^2(U)$ is a fine sheaf, hence can be taken for a first step in an acyclic resolution of the constant ...
- 492
0
votes
0
answers
87
views
How to prove the following equation (involving multiple binomial coefficients sum)?
I encountered the equation below, encountered a problem that has been bothering me for a long time
Does anyone have an idea how to prove it? I would be extremely grateful to you if you come up with an ...
- 41
0
votes
0
answers
47
views
max eigenvalue and schatten-1 norm of depolarizing channel acting on trace-0 Hermitian matrix
Denote $\mathcal{H}^n$ as the $n-$dimension Hermitian matrices. Depolarizing channel $\Delta_p:\mathcal{H}^2\to\mathcal{H}^2$ is defined as $\Delta_p(A)=p\text{ tr }(A)~I/2+(1-p)A$ where $A\in \...
- 91
-4
votes
1
answer
173
views
To which arithmetic\set theory this theory is bi-interpretable?
$\textbf{Logic:}$ Mono-sorted first order logic with equality.
$\textbf{Extralogical Primitives: } <, \in$
$ \textbf{Axioms:}$
$ \textbf{Order:} \ x < y < z \to x < z $
$ \textbf{...
- 11.3k
1
vote
1
answer
158
views
Comparison of special metrics on Riemann surfaces with the hyperbolic one
Let $X$ be a complex algebraic curve, or equivalently a compact Riemann surface, of genus $a\ge2$. Denote $\omega_1, \dots , \omega_a$ a basis of holomorphic differentials, and consider the Riemaniann ...
- 31
6
votes
0
answers
132
views
Do there exist strictly contracting eikonal functions on $\mathbb R^n$?
A function $f: \mathbb R^n \to \mathbb R$ is said to be a strict contraction if
$$|f(x) - f(y)| < |x - y|$$
for all $x \neq y$.
A function $f$ is said to be eikonal if it is differentiable ...
- 6,323
1
vote
0
answers
84
views
Does sets of positive capacity rule out constant functions?
Let $U\subset \Bbb R^d$ be bounded with Lipschitz boundary $K\subset \bar{U}$ be compact. The capacity of $K$ in $U$ is defined by
\begin{align*}
\text{Cap}_{p}(K, U) :=
\inf \left\{
\int_U |\...
- 1,101
1
vote
2
answers
202
views
Spaces $X$ with every compactification $0$-dimensional with $\beta X\setminus X$ not locally compact
Previously, in this post I've shown the following characterization of spaces with only zero-dimensional compactifications:
Theorem. Let $X$ be strongly zero-dimensional and $\beta X\setminus X$ ...
- 1,211
17
votes
3
answers
691
views
Large "internal" categories and "finite" products
The question is basically "do we really have a good way to talk about large categories internally in an elementary topos?"
An internal small category in a topos $E$ is just a category object ...
- 42.4k
6
votes
0
answers
102
views
Computer program for free restricted Lie polynomial
I am conducting numerical experiments involving the Gröbner–Shirshov Basis for restricted Lie algebras. At each step of the computation, I need to work with restricted Lie polynomials. Specifically, I ...
- 1,013
5
votes
1
answer
109
views
Duals and direct summands in an abelian monoidal category
This question may be seen as a continuation of Duals and sub-objects in a monoidal category.
In an abelian monoidal category, i.e. an abelian category with biadditive monoidal product, if $X \oplus Y$ ...
- 1,484
2
votes
0
answers
102
views
Finite groups of Lie type
Table $22.1$ Finite groups of Lie type from "Linear algebraic groups and finite groups of Lie type" by Malle and Testerman:
For the type $A$: $G_{sc}^{F}=\operatorname{SL}_{n}(q)$ and $G_{ad}...
- 217
0
votes
0
answers
30
views
Question on the closed proper convex functions
I'm confused with the definition of the closed proper convex functions when reading the paper
https://people.orie.cornell.edu/aslewis/publications/00-dykstras.pdf
It appears that, when a function $f$ ...
- 1,399
2
votes
1
answer
169
views
Ratio of inscribed/circumscribed ellipsoids: geometrical proof?
Let $K$ be a convex subset of ${\mathbb R}^n$, with non-void interior. The Löwner-John theorem states that there are a minimal volume ellipsoid $\cal E$ containing $K$, a maximal one $\cal F$ ...
- 52.4k
11
votes
2
answers
387
views
Bounds for the difference in the number of ones in $M$ and $M^{-1}$
If $M$ is a full rank $n$ by $n$ binary matrix over $\mathbb{F}_2$, how much larger or smaller can the number of $1$s in $M^{-1}$ be, compared to the number of $1$s in $M$?
Clearly the identity matrix ...
- 3,377
0
votes
1
answer
71
views
Limit distribution of this discrete time Markov chain is standard normal?
Consider a discrete time, uncountable state space Markov chain with one-step transition density
$$
p^{(1)}(z_0,z)=\mathbf 1_{\{z\leq z_0-\theta\lor z\geq z_0+\theta\}}\phi(z)
+\delta(z-z_0)\left(\Phi(...
3
votes
0
answers
81
views
Combinatorial/probabilistic interpretation of a quantity of union closed family
Let $\mathcal{F}\subseteq2^{[n]}$ be a union-closed family of sets. For a set $S\in[n]$ (not necessary belong to $\mathcal{F}$), define $w_{\mathcal{F}}(S)$ to be the number of subset of $S$ which ...
- 1,462
3
votes
0
answers
76
views
Natural transformation and Hochschild cohomology
I am reading the lecture note by Căldăraru:https://arxiv.org/pdf/math/0501094, in the last chapter of this note, he said that we should consider dg category instead of the derived category of coherent ...
- 395
4
votes
0
answers
107
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Partial uniformization under AD
Under ZF + AD, and especially $\text{AD}^+$, even if uniformization fails for reals, in some ways it must almost hold.
For a notion of small, we say that uniformization holds on a co-small set of ...
- 7,545
2
votes
2
answers
427
views
Questions about some parallel between polynomial and differential equation
Do the relations between Galois groups and solutions to polynomial equations with one variable have a counterpart between Lie groups and solutions to differential equations ?
Do the relations between ...
- 3,104
12
votes
1
answer
323
views
Does every mapping class group embed into some $\mathrm{Out}(F_n)$?
The title is pretty much the whole question. Let $S_g$ be a closed, oriented surface of genus $g$. Does there exist $n$ such that the mapping class group $\mathrm{Mod}(S_g)$ embeds as a subgroup of $\...
- 3,740
4
votes
1
answer
238
views
When does a cofibrantly generated model category have this factorization property?
Let $\mathcal{C}$ be a cofibrantly generated model category, which is generated by $I$ and $J$. According to the small object argument (Hovey Theorem 2.1.14) of cofibrantly generated model categories, ...
- 143
4
votes
1
answer
195
views
Asymptotic spectrum of a complex Sturm-Liouville differential operator
Let $\varepsilon > 0$ and consider the (complex) Sturm–Liouville differential operator on $[0,1]$ given by
$$
\mathcal{L}_\varepsilon f(x) = \varepsilon^2 f''(x) + i V(x) f(x),
$$
with Neumann ...
- 333
19
votes
4
answers
1k
views
Generalization of a mind-boggling box-opening puzzle
Motivation. Suppose we are given $6$ boxes, arranged in the following manner:
$$\left[\begin{array}{ccc} 1 & 2 & 3 \\ 4 & 5 & 6 \end{array}\right]$$
Two of these boxes contain a ...
- 48.9k
45
votes
6
answers
8k
views
Situation with Artemov's paper?
Artemov's paper on Goedel's theorem has been on the arxiv since 2019. There was a (less than fully friendly) discussion of this on FoM. At stackexchange, I found only a brief mention at this MSE ...
- 16.6k
9
votes
3
answers
1k
views
Examples of combinatorial problems where the only known solutions, or most "natural" solutions, use representation theory?
In Solution of two difficult combinatorial problems with linear algebra, Robert Proctor presents two simply stated combinatorial problems, and gives solutions to them using a linear algebraic approach ...
7
votes
1
answer
357
views
Is this true in the ordered Mostowski model?
The ordered Mostowski model $M$ is the permutation model used by Mostowski to show that the axiom of choice is independent from the ordering principle. The set $A$ of atoms of this model is isomorphic ...
- 1,792
1
vote
1
answer
173
views
Are all integers up to $x$ but possibly $O_{\varepsilon}(x^{\varepsilon})$ the sum of $a$ squares and $b$ primes with $a+b\leq 3$?
This question is related to https://math.stackexchange.com/questions/3710032/conjecture-all-but-21-non-square-integers-are-the-sum-of-a-square-and-a-prime.
We know since Lagrange that every natural ...
- 6,982
1
vote
0
answers
32
views
Balanced cocircuit cover
Are there studies on matroids which can be covered by $r$ cocircuits ($r$ is the rank of the matroid), so each element is covered by a small number of times?
For example, it is known graphic matroids ...
- 613
3
votes
1
answer
168
views
Express fundamental group of $\mathcal H/\Gamma$ by $\Gamma$
Suppose $\mathcal H$ is the upper half plane, and $\Gamma$ is an arithmetic subgroup of $\operatorname{PSL}_2(\mathbb Z)$, I want to ask can we interpret the fundamental group of $\mathcal H/\Gamma$ ...
- 785