All Questions
152,872
questions
2
votes
1
answer
204
views
Finite flat pullback of the diagonal
Let $X, Y$ be smooth projective connected complex varieties of the same pure dimension $d$ and $f : X\to Y$ a finite flat surjective morphism.
Let $\Delta_X$ be the closed subscheme of $X\times X$ ...
5
votes
1
answer
507
views
Is the singular simplicial complex functor $\operatorname{Sing}_\bullet:\operatorname{Top} \to \operatorname{sSets}$ fully faithful for nice spaces?
$\DeclareMathOperator{\Hom}{Hom} \DeclareMathOperator{\Sing}{Sing} \DeclareMathOperator{\Top}{Top}
\DeclareMathOperator\sSets{sSets}$Is there a category of “sufficiently nice” topological spaces such ...
0
votes
0
answers
83
views
Does it help for graph isomorphism to know power of the permutation matrix?
Here all matrices are square $n \times n$ with integer entries.
If you prefer, all entries are $0-1$.
Observation: the discrete logarithm for permutation matrices is
polynomial in $n$, since the ...
1
vote
0
answers
68
views
A ZFC example of a star-$K$-Menger space which is not star-$K$-Hurewicz
An open cover $\mathcal U$ of a space $X$ is said to be $\gamma$-cover if $\mathcal U$ is infinite and for each $x\in X$, the set $\{U\in\mathcal U : x\notin U\}$ is finite.
A space $X$ is said to be ...
1
vote
0
answers
94
views
Does the center of any finitely generated associative algebra over a field have finite type?
Consider the monoid algebra $R:=K\langle x_1,\dots,x_n\rangle$ generated by $n$ letters $x_1,\dots,x_n$ for $n>1$ over field $K$. Equivalently, $R$ is the tensor algebra $T(V)$ on the $n$-...
3
votes
1
answer
401
views
Positive definiteness of a matrix-valued function
This question is a repost from math.se, where I didn't receive an answer.
Are there simple conditions on an $d \times d$ matrix B under which
$$
f(t, s)
=
\begin{cases}
\exp(-B |t - s|^\alpha), &...
6
votes
1
answer
314
views
Deformations of the 4-sphere with 8-dimensional isometry groups
I am looking for deformations of the 4-sphere with 8-dimensional isometry group, like a 4-dimensional Berger sphere.
14
votes
3
answers
864
views
(An introduction to) deformation theory (written) for differential geometers
Question is as mentioned in the title:
Are there any introductory notes on deformation theory that are easier to read for differential geometers?
I am learning about differential graded Lie algebras (...
1
vote
1
answer
137
views
Dense properties of weighted Sobolev space define on $\mathbb{R}^n$
Consider the usual Sobolev space $H^1(\mathbb{R}^n)$ and $H^1_0(\mathbb{R}^n)$, where $H^1_0(\mathbb{R}^n)$ is the closure of $C_0^\infty(\mathbb{R}^n)$ with respect to the norm of $H^1(\mathbb{R}^n)$....
1
vote
1
answer
147
views
Trivial convergent sequences in $\beta X$
Let $X$ be a Tychonoff space and denote by $\beta X$ its Stone-Čech compactification. We know, for example, that if $X$ is an $F$-space then $\beta X$ is an $F$-space and, therefore, in $\beta X$, the ...
0
votes
1
answer
266
views
Estimating a sum involving the von Mangoldt function
I'd like to know the estimate of the following sum
$$\sum_{n\leq x}\sum_{d|n}\Lambda(d)\frac{\phi(d)}{d} $$
where $\Lambda(d)$ is the von mangoldt function and $\phi(d)$ is the Euler totient function. ...
5
votes
3
answers
493
views
If the Fourier coefficient $\hat{f}(k)$ of $f\in C^1(\mathbb T)$ is zero for all $|k|<N$, then $\|f\|_{L^\infty}\leq \frac CN \|f'\|_{L^1}$?
Let $f\in C^1(\mathbb T)=C^1(\mathbb R/\mathbb Z)$ be a function such that
$$\hat f(k):=\int_{\mathbb T}f(x)e^{-2\pi ikx}\,dx=0,\qquad \forall k\in\{-N+1,\cdots,-1,0,1,\cdots, N-1\}.$$
Do we have $\|f\...
1
vote
0
answers
75
views
Integration over a finite-dimensional subspace of Hilbert space
Let $H$ be a separable Hilbert space with inner product $\langle,\rangle$, let $\{e_k\}_{k=1}^\infty$ be an orthonormal basis of $H$, and let $A: H\to H$ be a symmetric, positive definite and ...
8
votes
1
answer
278
views
What are twisted Verma modules? Basic properties?
Let $\mathfrak{g}$ be a simple Lie algebra over $\mathbf{C}$, $\lambda$ be a weight in the nonnegative Weyl chamber and $w_1,w_2\in W$. Then the twisted Verma modules are defined e.g. in Andersen and ...
6
votes
0
answers
295
views
Looking for analogs of the rational, $p$-adic, and real numbers in homotopy theory via the passage $\mathbb{N}\to\mathbb{Z}\to\mathbb{Q}\to\mathbb{R}$
We can view the construction of the real or $p$-adic numbers as the end result of a series of constructions each of which adds more and more structure to the object at hand:
First, we start with the ...
6
votes
1
answer
229
views
Is there a general method for computing finitely generated normalizers?
I'm looking to compute normalizers of finite subgroups of $\mathrm{GL}(n, \mathbb{Z})$ and its possible that they are infinite but they are always finitely presented. For $\mathrm{GL}(n, \mathbb{Z})$ ...
0
votes
0
answers
170
views
Geometric interpretation of normalization inside a finite extension of function field
$\DeclareMathOperator\Spec{Spec}$Suppose $X = \Spec A$ is a smooth affine variety over $\mathbb C$ and suppose $L/K$ is a finite extension of its function field. Let $Y = \Spec B$, where $B$ is the ...
2
votes
0
answers
90
views
References (and a question) on the "fine" topology of powersets
Recently I've been trying to understand powerset topologies better, and came upon the following reference:
Frank Wattenberg, Topologies on the set of closed subsets. Pacific J. Math. 68(2): 537-551 (...
2
votes
2
answers
151
views
expectation and variance of the norm of a random matrix
Suppose $X \in \mathbb{R}^{n \times d}$ is a random matrix where $n > d$. Given a matrix $A \in \mathbb{R}^{n \times n}$ such that $AX$ is a zero matrix in expectation, i.e., $\mathbb{E}_{X}[AX] = ...
3
votes
0
answers
140
views
Extension of work by Novelli and Thibon on noncommutative symmetric functions and Lagrange inversion
(Edit May 12, 2023: I just put up a brief summary of some of my notes on the partition polynomials described below in my WordPress mini-arXiv at "As Above, So Below". It contains multinomial ...
5
votes
1
answer
325
views
About an asymptotic behavior in number theory
Where can I read about the asymptotic behavior (with $N$ tending to infinity) of the sum of the fractional parts obtained from dividing $N$ by all prime numbers up to $N$ divided by the number of ...
3
votes
0
answers
170
views
Are "very conservative" connectives already definable?
I'm sadly an outsider to nonclassical propositional logics. All terminology below comes from Humberstone's book The Connectives, specifically section 4.2.
A new connective - a bit more precisely, a ...
2
votes
0
answers
156
views
Weil's "Sur la formule de Siegel dans la théorie des groupes classiques"
I am struggling to understand the following passage on Weil's text "Sur la formule de Siegel dans la théorie des groupes classiques" on page 16.
If I understood correctly, in the second ...
2
votes
0
answers
96
views
Hamiltonian action as a group homomorphism
It is sometimes demanded that a Hamiltonian group action $G \times M \to M$ allow for a Lie algebra homomorphism from $\mathfrak{g}$ to $C^\infty(M)$ with the Poisson bracket.
Is there a natural ...
2
votes
0
answers
83
views
Proving that $u_0$ is a canonical solution
I've asked this question on stack exchange before but no one could help me so I wish I can get some help here.
Let's first start with the definition of the canonical solution:
Consider $\frac{\...
16
votes
1
answer
740
views
The second stable homotopy group
I am interested in the low-degree stable homotopy group $\pi_2^{s}(X)$ of a path-connected space $X$. Using the Atiyah-Hirzebruch spectral sequence, we have the short exact sequence $0\to H_1(X,\...
1
vote
0
answers
155
views
Interpretation of Tate conjecture using motivic homotopy
For a smooth projective variety $X$ over a field $k$ the Tate conjecture says that the cycle class maps
$$CH^i(X)\otimes \mathbb{Q}_l \to H^{2i}(X_{\bar{k}},\mathbb{Q}_l(i))^{G_k}$$
are surjective. To ...
15
votes
1
answer
2k
views
How did Fermi calculate this integral?
In his 1926 paper Fermi states without further explanation that it follows from the Thomas-Fermi equation
$$\frac{d^2\psi(x)}{dx^2}=\frac{\psi(x)^{3/2}}{\sqrt{x}},\label{1} \tag{1}$$
and boundary ...
5
votes
1
answer
196
views
Step in the Bruhat decomposition for reductive Lie groups
Err, not research but if anyone has read this part of Knapp's book recently, I'd be obliged if they could help me out. Also posted on MSE.
I'm stuck on a line in the proof of Theorem 7.40 in Knapp's '...
6
votes
1
answer
251
views
A ZFC example of a Menger space which is not Scheepers
$\Omega$: The collection of all $\omega$-covers of a space $X$. An open cover $\mathcal U$ of $X$ is said to be $\omega$-cover if $X\notin\mathcal U$ and for each finite $F\subseteq X$ there exists a $...
2
votes
1
answer
77
views
Request for literature recommendations on isotonic mappings
An isotonic mapping is a function between two partially ordered sets that preserves the ordering between the elements. Specifically, given two partially ordered sets $(X,\le)$ and $(Y,\le)$, a ...
2
votes
0
answers
50
views
Convergence of minimiser of empirical risk to minimiser of population risk
Let $X_1, \dots, X_n \sim \mu$ be some random elements of a space $\mathcal{X}$. Let $H$ be a Hilbert space of functions $f: S \to \Re$ with norm $\|\cdot\|_H$.
Let $\|f^*\|_{L_2(\mu)} < \infty$ ...
3
votes
1
answer
231
views
Closed subset of unit ball with peculiar connected components
Let $n\geq 2$ and denote by $B\subset \mathbb{R}^n$ the closed unit ball.
Does there exist a closed subset $A\subset B$ containing $0\in \mathbb{R}^n$ with the following properties i,ii,iii?
i) $\{0\}$...
1
vote
0
answers
98
views
Schrödinger equation approximation – continuity of eigenvalues with respect to potential
The question has been crossposted from Stackexchange after receiving no answers.
Setup: the time-independent Schrödinger equation (eigenvalue problem):
$(-\frac{\hbar^2}{2m}\Delta +V)\psi = E\psi$
(On ...
3
votes
1
answer
193
views
Does positive set theory prove the existence of a set of all ordinals and itself?
Does positive set theory $\sf GPK^+_\infty$ prove the existence of a set $K= \{x \mid x \text { is von Neumann ordinal } \lor x=K\}$
1
vote
0
answers
64
views
Inside-out dissections of polygons - a generalization
Definitions (Inside-out polygonal dissections): a polygon P has an inside out dissection into P' if P′ is congruent to P and the perimeter of P becomes interior to P' and so the perimeter of P' is ...
3
votes
0
answers
226
views
Grothendieck schemes and the Sheffer differential op calculus (Rota, Roman, et al. finite operator calculus)
In "Left differential operators on non-commutative algebras" on p. 4, Michiel Hazewinkel displays "precisely the right definition of differential operator" as
$$D\; X^n = F(\tfrac{...
1
vote
0
answers
57
views
Probability that a Lévy process "closely" follows a predefined trajectory
For a Brownian motion $(B_t)_{t\geq 0}$ it is well-known [Thm 38, David Freedman, Brownian motion and diffusion], that if $f:[0,1] \to \Bbb R$ is a continuous function with $f(0)=0$ then for $\...
3
votes
0
answers
215
views
An attempt to extend polynomial rings
Suppose $\mathbb{K}$ is a field of characteristic $0$. Let $S_n=\mathbb{K}[[x_1,\dots,x_n]]$ be the ring of formal power series in $n$ variables, $L_n$ be its fraction field and $W_n=\mathbb{K}[x_1,\...
5
votes
1
answer
394
views
What is the correct definition of semisimple linear category?
I have been using the notion of "semisimple linear category" for a while now, but I never bothered to write down a definition. I've always just waved my hands and said "Schur's lemma ...
11
votes
0
answers
434
views
What sequence maximizes the final distance?
This problem was created by professor Ronaldo Garcia from Universidade Federal de Goiás (UFG) and he showed it to me at an event in my university. This problem has a lot of history and he told me he ...
4
votes
1
answer
133
views
Kernels of actions on truncated polynomial algebra
Let $p$ be an odd prime, and let $k=\mathbb{F}_p$ be the field with $p$ elements. Let $G=\text{GL}_n(k)$. The group $G$ acts on the truncated polynomial algebra $A:=k[x_1,\ldots, x_n]/(x_1^p,\ldots, ...
3
votes
1
answer
153
views
Fastest algorithm for calculating optimal tours in weighted $K_5$
Weighted $K_5$ have the unique property that their edge set can be interpreted as the disjoint union of their shortest and their longest Hamilton cycle.
That makes $K_5$ attractive for designing new ...
0
votes
0
answers
66
views
Computing and isotopy of curves in $\mathbb{R}^3$
Imagine a piece of string in the ocean moving gently with the currents; the string bends but does not change its length.
The (stationary) string can be modelled by a unit speed curve:
$$[0,1] \...
0
votes
0
answers
78
views
Describing a time-varying process with a manifold
I am a beginner in topology and I am trying to define a model for some computations. My questions are speculative:
I am wondering what is the proper way to add time in a manifold so as to describe a ...
2
votes
1
answer
184
views
Dispersion of a "random" subset of $[-1,1]^2$
Let $C$ be the square $[-1,1]^2$. Let $a_1,\dots,a_m$ be points chosen independently and uniformly at random from $C$. Let $d_m$ (dispersion) be the random variable $\max_{x \in C}{\min_{j \in [m]}{\|...
1
vote
1
answer
152
views
Quantitative version of Lebesgue points theorem
Let $A \subset [0,1]^n$ with $A$ measurable and such that $\mathcal{L}^n (A)= \delta >0$, and consider a partition of $[0,1]^n$ in $\epsilon$-cubes (i.e. cubes of side $\epsilon)$. For $\epsilon \...
1
vote
2
answers
206
views
How much "room" in inequality $\displaystyle \int_a^b \varphi' ov ~\mathrm{d}x \leq 0$
Let $[a, b]$ be a nonempty interval, $o \in C^1([a, b])$ be such that $o>0$ and $o'<0$ and assume we found some $v \in L^\infty(\mathbb{R})$ such that
\begin{equation}\tag{1}\label{1}
\int_a^b \...
4
votes
0
answers
97
views
Systems of parabolic equations -- Petrovskii's condition
Consider the flat torus $\mathbf{T}^d:=\mathbf{R}^d/\mathbf{Z}^d$ and define the corresponding periodic-parabolic cylinder $Q_T:=(0,T)\times\mathbf{T}^d$.
Given a matrix field $A:Q_T\rightarrow\text{M}...
0
votes
1
answer
150
views
Littlewood-Paley characterisation of Hölder regularity
I am going through Terence Tao's "Nonlinear Dispersive Equations (Local & Global Analysis)" and trying to work through some of his exercises. However, I find myself being stumped by ...