Trending questions
159,054 questions
10
votes
1
answer
501
views
What is the least $\alpha$ such that $L_\alpha$ contains a non-measurable set
What is the least level of the constructable hierarchy that contains a non-measurable (Lebesgue) subset of $2^\omega$. If it makes a difference assume we are working inside L (V=L).
I'm pretty sure it ...
- 3,029
3
votes
1
answer
139
views
$L^\infty$-bound on Laplace-eigenfunctions
Suppose we are on a closed Riemannian manifold $M$. Any function $f\in C^\infty(M)$ may be decomposed as $$f = \sum_{j = 0}^\infty f_j\phi_j,$$ where $\phi_j\in C^\infty(M)$ are the Laplace ...
2
votes
0
answers
109
views
Can a function and its Fourier transform both vanish on a convex cone?
It's all in the title :)
Theorem A' of Shapiro (see below) yields that if a tempered function $f$ on $\mathbb R^d$ has a spectral gap (i.e. $\hat f$ in the distributional sense vanishes on a non-empty ...
- 1,352
0
votes
0
answers
43
views
Equivalent conditions for $z$-embeddability
I am looking for where this specific theorem of Blair is originally located:
Theorem. Let $S\subseteq X$, the following are equivalent:
$S$ is $z$-embedded
If $A, B\subseteq S$ are disjoint zero-...
- 1,211
2
votes
1
answer
87
views
Explicit construction of integers with prescribed digit sum and residue class conditions
Let $q\geq 2$ be an integer, and $p,m\in \mathbb{N}$. Let $S_q$ be the function sum of digits in base $q$. If $\gcd(q-1,m)=1$, I was wondering if there is simple way to construct $k\in \mathbb{N}$ ...
- 23
0
votes
1
answer
131
views
Poset definition of dimension
Let $\mathsf{k}$ be an algebraically closed field and $X$ an abstract variety (an integral separated scheme of finite type over $\mathsf{k}$).
Is there any way to define the usual dimension of $X$ ...
- 3,318
2
votes
2
answers
136
views
Non-linear recursion relation with fractional exponent
I'm trying to solve the following non-linear recursion relation:
$a_{n+1} = a_n + c\cdot a_n^b, \quad n \geq 1$,
where $a_1 > 0$, and $c > 0$ and $0 < b < 1$ are constants. Mostly I'd like ...
1
vote
0
answers
44
views
Differential system of equations I would like to simplify
I have 2 functions of time $f(t),g(t)$ and a condition for the time-derivative of a third function $h(t)$, say $$\dot{h}(t)=\dot{g}(t)\cos{f(t)},$$ so $h$ is defined provided a value for $h(0)$ (as $h(...
- 11
2
votes
1
answer
138
views
small monodromy for polarized Abelian families over a torus?
In Lawrence-Venkatesh, they proved in Prop 5.3 a
beautiful finiteness property for a locus, successfully avoiding the using of Tate conjecture. Note that the introduction of $size_v$ and friendly ...
- 131
3
votes
0
answers
246
views
Fundamental group of degree 4 del Pezzo surface minus 16 (-1)-curves [Reference request]
Let $S$ be a degree $4$ del Pezzo surface (over $\mathbb{C}$).
That is, $5$ points blow-up of $\mathbb{P}^2$, or $4$ points blow-up of $\mathbb{P}^1 \times \mathbb{P}^1$.
The classical fact is that $...
- 111
5
votes
1
answer
633
views
Consistency of ZFC with inaccessible cardinals but no measurable cardinals
Let $S$ be a set and k a infinite field. The injection $S \to k\mathrm{Alg}(k^S, k)$ (sending a point to the evaluation in it) is a bijection if and only if $S$ is a non-measurable cardinal (see for ...
- 6,962
0
votes
0
answers
78
views
Factoring totient of a prime
Is it any easy to factor $p-1$ when $p$ is a prime compared to general factorization problem?
What about when $2p+1$ is also a prime?
- 13.9k
6
votes
1
answer
303
views
A Simplification of the computation of local heights in Gross-Zagier
At the end of the following document, https://public.websites.umich.edu/~asnowden/seminar/2014/gz/L07.pdf , it was stated that to prove the formula of Gross and Zagier, it is not necessary to compute ...
- 213
-3
votes
1
answer
74
views
Non-Ramsey function $f:[\omega]^{<\omega}\to\{0,1\}$ [closed]
Let $\newcommand{\o}{\omega}\o$ be the set of non-negative integers, and for any set $X$, let $\newcommand{\oo}{[\o]^{<\o}}X^{<\o}$ denote the collection of all finite subsets of $X$.
What is an ...
- 48.9k
4
votes
1
answer
133
views
Restrict sigma algebra in measure-preserving system
Consider a measure space $(X,\mathcal{A},\mu)$ and a measure-preserving transformation $\phi \colon X\rightarrow X$, that is, $\phi$ is measurable and $\phi_*\mu = \mu$.
My intuition tells me that we ...
- 257
1
vote
1
answer
184
views
Metric currents on singular measures in $\mathbb R^d$
Unless I am misunderstanding a lot of works, it is my understanding that a finite and non negative measure $\mu=g\mathcal{H}^\alpha$, where $\mathcal{H}^\alpha$ is the $\alpha$-Haudorff measure, ...
- 391
5
votes
1
answer
884
views
Is this ring isomorphic to a quotient of a group algebra?
Consider the quotient of the free algebra $\mathbb{Q}\langle \alpha, \beta, \gamma, \delta, \varepsilon, \zeta \rangle$ by the two-sided ideal $I$ subject to the relations $$ \alpha\delta=\delta\alpha=...
- 1,093
5
votes
1
answer
179
views
Semisimplicity of algebras in fusion categories
Let $\mathcal{C}$ be a fusion category and $A \in \mathcal{C}$ be an algebra object. We say that $A$ is semisimple if its category of (right) modules $\mathsf{mod}_A(\mathcal{C})$ is a semisimple ...
- 61
7
votes
2
answers
298
views
Compactly generated and paracompact $\Rightarrow$ Hausdorff?
In A Concise Course in Algebraic Topology by May, a proposition is stated that any open cover of a paracompact space has a numerable refinement, where the space is assumed to be compactly generated ...
6
votes
1
answer
568
views
Can one show that $|\zeta'(x) / \zeta^2(x)| \leq 1/(x-.5)$ for $x\in\mathbb{R}\cap [1,\infty)$?
I have found that $\left|\frac{\zeta'(x)}{\zeta^2(x)}\right|\leq \frac{1}{x-\frac{1}{2}}$ for all real $x$ such that $x>1$ seems to be true. I have plotted the inequality and got this inequality ...
- 178
4
votes
1
answer
349
views
Do Frobenius subalgebras form a lattice?
A finite-dimensional, unital, associative algebra $A$ over a field $k$ is termed a Frobenius algebra if it is endowed with a nondegenerate bilinear form $\sigma : A \times A \to k$ satisfying the ...
12
votes
2
answers
622
views
Countable set meeting uncountable family of positive measure sets
Suppose $\mu:\mathcal{P}([0, 1]) \to [0.1]$ is a probability measure and $\{A_i: i < \omega_1\}$ is a family of subsets of $[0, 1]$ such that $\mu(A_i) \geq 1/2$ for every $i < \omega_1$. Can we ...
- 129
0
votes
0
answers
98
views
A question about the sheaf supported on the zero section
Let $X$ be the total space of the cotangent sheaf on $\mathbb{P}^{2}$ and $i \colon \mathbb{P}^{2} \hookrightarrow X$ be thezero section. Suppose that $E$ is a coherent sheaf on $X$ which is set-...
4
votes
1
answer
214
views
Asymptotic behavior of weighted sums involving the fractional part function
Currently, I am studying the asymptotic behavior of sums of the form
\begin{equation}\label{eq1}\tag{1}
\sum_{k=1}^{n-1} f(n-k) \left\{ \frac{n}{k} \right\}
\end{equation}
In this context, based on ...
- 611
3
votes
0
answers
87
views
Is every finite plane with a characteristic Desarguesian?
By a projective plane I understand a mathematical structure $(X,\mathcal L)$ consisting of a set $X$ of points and a family $\mathcal L$ of subsets of $X$, called lines such that the following four ...
- 42k
2
votes
0
answers
92
views
Geometric interpretation of flags and the role of the rook monoid and Kazhdan–Lusztig theory in $M_n(\mathbb{C})$
Let $G = GL_n(\mathbb{C})$, $B$ be its Borel subgroup, and $P$ a parabolic subgroup. The space $G/B$ corresponds to complete flags in $ \mathbb{C}^n$, and $G/P$ corresponds to partial flags. The ...
- 141
15
votes
6
answers
2k
views
What does keep you "doing what you do"? [closed]
I am towards the end of my Phd (with some difficultues to overcome, I can say I am really satisfied about it) and I was wondering about what to do next. There are basically two paths: academia or ...
- 835
3
votes
1
answer
190
views
Irreducibility under etale ring map
Let $A\rightarrow B$ be a etale ring map between finite type algebra over algebraically closed field $k$.
If $A$ is one dimensional integral domain, is $B$ direct product of finite type integral ...
- 328
3
votes
1
answer
243
views
Metrics of constant Gauss curvature on 2-cylinder
Let $C=S^1\times[0,1]$ be a compact cylinder. Given positive numbers $l,\lambda>0$. Is it possible to construct a smooth Riemannian metric on $C$ of constant Gauss curvature -1 such that one ...
- 21.8k
13
votes
0
answers
332
views
Lie theory for quantum groups?
$\DeclareMathOperator\SU{SU}$I know about quantum groups from two perspectives:
Compact quantum groups in the sense of Woronowicz.
Deformation of the universal enveloping algebra of a Lie algebra in ...
- 357
161
votes
37
answers
17k
views
Conceptual reason why the sign of a permutation is well-defined?
Teaching group theory this semester, I found myself laboring through a proof that the sign of a permutation is a well-defined homomorphism $\operatorname{sgn} : \Sigma_n \to \Sigma_2$. An insightful ...
- 64k
16
votes
2
answers
1k
views
CH in non-set theoretic foundations
I asked this question on stack exchange and got little attention, barring a nice example I intend to look into. The original post can be found here: https://math.stackexchange.com/q/4941233/1053681
I ...
- 261
0
votes
0
answers
146
views
On the pointwise limit of a sequence of analytic functions
I have been confused with this problem for weeks now. Suppose I have Banach spaces $E$ and $F$ and a sequence of functions $f_{n}: U \subset E \to F$, where $U$ is open and nonempty. Let $x \in U$ be ...
1
vote
0
answers
100
views
Cohen-Macaulayness of the homogeneous coordinate ring of projective monomial curves
Let $A = \{a_0, a_1, \ldots, a_{n-1}\} \subset \mathbb{N}$ be a set of non-negative integers where we assume that $a_0 < \cdots < a_{n-1}$ and set $d := a_{n-1}$. For every $s \in \mathbb{N}$, ...
-1
votes
0
answers
74
views
Why is there in theory no morphism/isogenies when enlarging a prime field sharing a common suborder/subgroup? [closed]
Simple question : I have a prime field having modulus $p$ where $p−1$ contains $O$ as prime factor, and I have a larger prime field $q$ also having $O$ as its suborder/subgroup. Why are there no ...
- 87
5
votes
1
answer
163
views
I believe that all facets of a Voronoi-cell of a lattice are centerally symmetric. Is my argument correct? Is this true?
So let $L$ be a full dimensional lattice in $\mathbb{R}^{n}$. Then the Voronoi-cell of the lattice are precisely the points in $\mathbb{R}^{n}$ that are at least as close to the origin, as to any ...
7
votes
1
answer
309
views
Proper Forcing Axiom for $|\mathbb{P}| \leq \mathfrak{c}$
Let $\mathsf{PFA}(\mathfrak{c})$ denote the Proper Forcing Axiom (PFA) restricted to posets $|\mathbb{P}| \leq \mathfrak{c}$. I think $\mathsf{PFA}(\mathfrak{c}) \implies \mathfrak{c} = \aleph_2$, but ...
- 1,442
1
vote
0
answers
48
views
Is the finite/countable union of topological Markov shifts a topological Markov shift?
A topological Markov shift (TMS), or countable state shift of finite type (SFT), is a shift space $X$ over a countably infinite alphabet $\mathcal{A}$ defined by a transition matrix $T=(t_{ij})_{\...
- 119
1
vote
1
answer
109
views
Orbit spaces of n-tuples of square matrices under simultaneous conjugation
Let $n, p, \geq 1$ be integers. Denote the set of ordered partitions of $p$ by $\Pi$: each $\pi \in \Pi$ is an ordered $k$-tuple $(p_1,p_2, \dotsc, p_k)$ where $p_1+\dotsb+p_k = p$. Write $\pi \leq \...
- 185
4
votes
1
answer
167
views
Pressureless explicit solutions to incompressible Euler
What are some examples of (semi-)explicit solutions of the incompressible Euler equations which satisfy the following
they are pressureless
they are periodic in space
they have nontrivial time ...
- 94
2
votes
0
answers
182
views
Algorithm for $\frac{1}{1-x} = \sum\limits_{n=0}^{\infty}a(n)x^n\prod\limits_{k=1}^{n}\frac{1-kx}{1+kx}$
Let $a(n)$ be A208832. Here
$$
\frac{1}{1-x} = \sum\limits_{n=0}^{\infty}a(n)x^n\prod\limits_{k=1}^{n}\frac{1-kx}{1+kx}.
$$
Start with vector $\nu$ of fixed length $m$ with elements $\nu_i = 1$ (that ...
- 4,938
2
votes
0
answers
96
views
Free, easy-to-use program for noncommutative algebra over finite fields
I am looking for a computer program that can handle computations in noncommutative algebra over a finite field of prime order $p$.
My requirements are:
The program should be free, as I do not have ...
- 1,013
6
votes
3
answers
491
views
Boolean rings with many automorphisms
Does there exist an infinite Boolean ring $R$ (not assume unital, only associative) with the property that for any nonzero $x,y\in R$, there is a ring automorphism $\varphi\colon R\to R$ such that $\...
- 131
16
votes
2
answers
1k
views
Identifying short introductory book on non-commutative geometry I read c.2008
I’m trying to identify a book I remember reading and enjoying in grad school around 2008–9; I’ve forgotten its author and title, and haven’t been able to find a book matching my memories in half an ...
- 21.3k
13
votes
2
answers
802
views
For which rationals is this exponential sum bounded?
Given $x \in [0, 1]$, we denote by $e(x)$ the complex number $e^{2 \pi i x}$.
Can we characterise the set of rationals $x$ for which the sum
$$A_N(x)\, :=\, \sum_{n = 0}^N e(2^n x)$$
remains bounded ...
- 6,313
175
votes
39
answers
31k
views
Short exact sequences every mathematician should know
I'd like to have a big-list of "great" short exact sequences that capture some vital phenomena. I'm learning module theory, so I'd like to get a good stock of examples to think about. An ...
4
votes
1
answer
327
views
Is the pushforward of a closed immersion ever fully-faithful at the level of Derived Categories?
Let $i: Z \rightarrow X$ be a closed immersion of schemes. Then, for any $\mathcal{O}_{Z}$-module $\mathcal{G}$, the counit of adjunction $i^{*}i_{*}\mathcal{G} \rightarrow \mathcal{G}$ is an ...
- 137
1
vote
0
answers
146
views
Can we find curves with many rational points using linear algebra?
Probably this is impossible, but let us try.
Working over $\mathbb{Q}[x_1,...,x_n]$.
Let $T_i$ be $n$ sets of rationals with cardinality $B$.
Assume we are given $n-2$ linear equations $f_i$ which are ...
- 25.4k
12
votes
2
answers
868
views
Sets that project to zero measure on all lines except one
It is a (difficult) exercise to show that there exists a measurable set $E \subset [0,1]^2$ (necessarily with zero 2-dimensional Lebesgue measure) such that the projection on every line passing ...
- 355
7
votes
3
answers
708
views
Properties of $P_{n}(x)={e}^{-x}\sum_{k=0}^{\infty}\frac{a_{k,n}{x}^{k}}{k!}$
I know this will sound like a general question, but given the structure $$P_{n}(x)={e}^{-x}\sum_{k=0}^{\infty}\frac{a_{k,n}{x}^{k}}{k!}$$ where $$a_{k,n} = \frac{1}{\prod_{i=1}^{n} (k+2i) }, $$ what ...
