All Questions
15,509 questions
5
votes
0
answers
180
views
Left Adjoint From the Category of Topological Groups to the Category of Condensed Groups
In Scholze's Lecture Notes on Condensed Sets, the author states that (Remark 1.8) the functor that takes a topological group $G$ to its condensation $\underline{G}$ has a left-adjoint, but we do not ...
- 241
0
votes
0
answers
52
views
Reference request for the determinant of a matrix constructed from Pascal's triangle
One can prove by induction that the matrix $M^{(n)}$ given by
$$ \begin{pmatrix}
1 & 1 & 1 & 1 & \dots & \binom{n}{0} \\
1 & 2 & 3 & 4 & \dots & \binom{n+1}{1} \...
- 5,546
13
votes
0
answers
333
views
Lie theory for quantum groups?
$\DeclareMathOperator\SU{SU}$I know about quantum groups from two perspectives:
Compact quantum groups in the sense of Woronowicz.
Deformation of the universal enveloping algebra of a Lie algebra in ...
- 357
2
votes
0
answers
26
views
Reference for the biequivalence between the bicategory of distributors and the bicategory of two-sided discrete fibrations
It is well known that a distributor/profunctor $A \not\rightarrow B$, i.e. a functor $B^{\text{op}} \times A \to \mathrm{Set}$, is equivalent to a two-sided discrete fibration from $A$ to $B$. ...
- 10.7k
1
vote
1
answer
76
views
Determinant formula for a certain parametrized M-matrix
Let $P_{ij}$ be variables, and let $A \in \mathbb{R}^{n\times n}$ be the matrix defined by
$$
A_{ij} = \begin{cases}
-P_{ij} & i \neq j,\\
P_{i1} + P_{i2} + \dots + P_{in} & i=j.
\end{cases}
$$...
- 20.2k
0
votes
0
answers
80
views
Relation between Chow groups and K theory
I am reading about Chow groups and algebraic K-theory of schemes. I get to know that for smooth schemes the re is a strongly convergent spectral sequence
$$E_2^{p,q} = CH^{-q}(X,-p-q) \implies K_{-p-q}...
- 629
4
votes
0
answers
180
views
Subgroups that conjugate-cover the ambient group
Let $G$ be a finite group, and suppose that a set of proper subgroups $H_1,\dotsc,H_n$ satisfy $G=\bigcup_{g\in G}\bigcup_{i=1}^nH_i^g$, where $H_i^g$ is the conjugate of $H_i$ by $g$. In this case, ...
2
votes
0
answers
152
views
What are alternative mathematical definitions of observers beyond Bennett and Hoffman's framework?
Motivation:
This question is inspired by a talk from Avi Wigderson given on Randomness, where the idea that the randomness is in the eye of the observer is suggested.
In the study of information ...
- 3,144
4
votes
1
answer
271
views
Examples of discrete-space continuous-time dynamical systems
Something that I see occur repeatedly in my work is the need for formal notions of discrete-space continuous-time dynamics — these are generally realized as digital oscillators that are interact using ...
- 41
2
votes
0
answers
230
views
Any rigorous construction of $\phi^4$ theories without the mass term in the Lagrangian? (revised)
There are various papers on rigorous construction of massive $\phi^4$ theories in $2$ or $3$ Euclidean dimensions.
In 2D, there are in fact more general results such as this one by Glimm, Jaffe and ...
- 3,507
4
votes
0
answers
133
views
Ring theoretical aspects of the DAHA
The double affine Hecke algebras (DAHA) were introduced by Cherednik in his study of Macdonald's inner product conjectures (which were solved affirmatively).
Nowdays there are many variations of the ...
- 3,318
11
votes
1
answer
885
views
Which paper is the "Taubes trick" from?
In symplectic geometry, the "Taubes trick" is an argument used to show that a moduli space $\mathcal{M}(J)$, depending on a parameter $J \in \mathcal{J}$, is cut out transversely for generic ...
- 243
0
votes
2
answers
224
views
Reference to get quickly to modern discrete probability theory
I've had some formal training in Analysis - Functional Analysis, Basic Operator Algebra - and I've started working on probability - specifically Combinatorial Statistical Mechanics and Spin-Glasses. ...
8
votes
0
answers
245
views
Strengthening of Frankl's union-closed sets conjecture: An algebraic approach
Let $\mathcal F$ be a union-closed family of subsets of $[n]=\{1,2,...n\}$ and $n$ real numbers $x_1,x_2,...,x_n\geq 1$.
Conjecture: There exists $k\in [n]$ such that:
$$\sum_{k\in A,A\in \mathcal F}\...
- 1,462
2
votes
0
answers
76
views
Does a matrix ring over a ring satisfy the Koethe conjecture if the coefficient ring itself satisfies the Koethe conjecture?
I just want to know whether the following statement is true or false.
If $R$ is a ring satisfying the Koethe conjecture, then the matrix ring over $R$ also satisfies the Koethe conjecture.
Or is it ...
3
votes
0
answers
101
views
Tuple rearrangement: a combinatoric problem emerging from the Hurwitz action on Coxeter groups
I am working on Artin Groups, so called Dual Artin groups and the conjecture that they are isomorphic. Tuples of $n$ group elements can be acted on by the braid group $B_n$ in a particular way called ...
- 31
4
votes
0
answers
128
views
Errata for "Foliations and Geometric Structures" by Aurel Bejancu and Hani Reda Farran
I'm reading "Foliations and Geometric Structures" (2006) by Aurel Bejancu and Hani Reda Farran and have been looking for an errata sheet. Unfortunately Prof. Bejancu has passed away. I ...
2
votes
1
answer
300
views
G-equivariant homotopy between G-spaces
I apologize for asking too many questions in a single post. I am not very conversant with equivariant homotopy theory. While discussing with some faculty I was told that certain fact is true. All ...
- 139
1
vote
1
answer
141
views
Understanding quadrature rule of a function multiplied by another $C^{\infty}$ function
Define a function $f \in C^m[-1,1],m \in \mathbb{N}$ and $g\in C^{\infty}[-1,1]$. Also define a quadrature rule $Q$ for approximating the integral $\int_{-1}^1 h(x)dx$ for some function integrable ...
- 69
9
votes
1
answer
304
views
About the normal subgroups of Burnside groups
I was reading "On periodic groups of odd period $n\ge 1003$" of V. S. Atabekyan. He found that the Burnside group $B_n$ with $n\ge 1003$ has uncountably many normal subgroups. However, I was ...
- 91
2
votes
1
answer
93
views
Reference needed: estimate of the second order derivatives
In $\mathbb{R}^d$ there is estimate (see 1.3, Chapter III of E.M.Stein' book Singular Integrals and Differentiability Properties of Functions)
$$\left\|\frac{\partial^2 f}{\partial x_i \partial x_j} \...
11
votes
1
answer
500
views
Uncountable families of measurable sets with pairwise positive intersections
Let $(X,\mathcal{B},\mu)$ be an arbitrary finitely additive probability measure space, let $a>0$ and let $(A_i)_{i\in I}$ be an uncountable family of subsets with measure $\geq a$.
Is there an ...
- 10.6k
3
votes
0
answers
74
views
Reference for PDEs from system of SDEs
I'm working with a system of SDEs
\begin{align*}
dX_t &= b(X_t, t) + \sigma dB_t\\
dY_t &= c(X_t, Y_t, t) + \sigma dB_t.
\end{align*}
Here, the Brownian motion is the same.
I know that ...
5
votes
2
answers
218
views
Smooth toric variety which is a cube is a bott tower (reference request)
According to Lee, Masuda and Park (page 3), the following result is "well-known in toric topology". I've found a proof, but I would like a published reference.
Let $X$ be a toric variety. ...
- 156k
0
votes
0
answers
106
views
Generalizing the property of linear independent set in infinite dimensional TVS
Given a infinite dimensional Hilbert space $H$, and a countable set of vectors $\{v_{i}\}_{i=1}^{\infty}$. I want to study the following property of $\{v_{i}\}_{i=1}^{\infty}$:
There exists sequences $...
- 523
2
votes
1
answer
105
views
Looking for review of delay differential equations involving $f(x)$ and $f(x/k)$
A research problem unexpectedly leads me to a delay differential equation of the form
$$
f(x)=\alpha(f(x),f(x/2))\,f'(x)+\beta(f(x),f(x/2))\,f'(x/2)+\gamma(f(x),f(x/2))
$$
For special cases of $\alpha,...
- 3,065
3
votes
0
answers
140
views
Polynomial from degrees of Weyl group
Let $d_1, \dotsc d_n$ be the degrees (of fundamental invariants) of the Weyl group $W$ of a simple Lie group, (in the reflection representation; see table given on the Wikipedia page for their ...
3
votes
0
answers
267
views
Cohomology for quantum groups
I'm interested in quantum groups for two perspectives:
Compact quantum groups in the sense of Woronowicz.
Deformation of the universal enveloping algebra of a Lie algebra in the sense of Drinfeld &...
- 357
0
votes
0
answers
33
views
Non-positive definite solution for differential Riccati equation
Consider the matrix-valued differential Riccati equation (DRE):
$$
\dot P_t +PA+A^\top P+Q-(B^\top P+S)^\top (B^\top P+S)=0, \quad t\in [0,T];\quad P(T)=G,
$$
where all coefficients are continuous.
...
- 503
8
votes
1
answer
635
views
Availability of a copy the first volume of Segre's "Forme differenziali e loro integrali"
I am precisely referring to the following, first volume of the textbook/lecture notes/monograph written by Beniamino Segre in the fifties of the twentieth century (I own a copy of the second volume)
...
- 6,377
5
votes
0
answers
114
views
Finitely generated projective modules over Noetherian endomorphism ring
Let $\mathcal A$ be a locally Noetherian Grothendieck abelian category and $M\in \mathcal A$ be a Noetherian object. Set $B:=\text{End}_{\mathcal A}(M)$. Let $B$-mod be the category of finitely ...
- 413
11
votes
3
answers
672
views
Merging single-sorted and multi-sorted theories
The general theory of single-sorted (say, algebraic) theories is very similar to the general theory of multi-sorted (algebraic) theories. Each variable gets a sort, but apart from that nothing really ...
- 63.1k
5
votes
1
answer
202
views
Independent stationary increment process but with finite propagation speed
Intuitively, standard Brownian motion has infinite propagation speed, as it has a non-zero probability of reaching any point in any arbitrarily short time. This is due to the fact that the probability ...
- 807
7
votes
1
answer
416
views
Is there a “Closure-of-Range Theorem” for Banach spaces?
The classic Closed Range theorem states that for a linear bounded operator $T:X\to Y$ between Banach spaces, and its transpose $T^*:Y^*\to X^*$, the four conditions:
$T(X)$ is $s$-closed; $T(X)$ is $...
- 60.6k
10
votes
0
answers
288
views
Coefficients of polynomials vs trigonometric product
Let's consider the family of sequences of coefficients in the expansion
$$\prod_{i=0}^{n-1}(1+x^{3^i}+x^{3^{i+1}})=\sum_{k\geq0}a_n(k)\, x^k.$$
Remark. Evidently, the RHS is a finite sum.
Here is a ...
- 43.2k
8
votes
0
answers
150
views
What is the forcing $\bf U$ from Bartoszyński-Judah?
In Set Theory - on the structure of the Real Line by Bartoszyński & Judah, a forcing notion $\bf U$ is mentioned on page 339, allegedly corresponding to $\rm{cof}(\cal N)$ as it has several ...
- 403
4
votes
2
answers
383
views
Gibbs measure as stationary distribution of SDEs
I have been trying to understand how one can mathematically explain some of the results from statistical mechanics, especially regarding certain distributions like the Gibbs distribution. It would be ...
- 807
1
vote
1
answer
292
views
General algebraic definition of mirror symmetry
I'm trying to understand the following statement of Hori-Vafa from the algebraic perspective:
The mirror of the Hirzebruch surface $\mathbb{F}_{n}$ is the Landau-Ginzburg model $x+y+\frac{a}{x}+\frac{...
- 305
6
votes
1
answer
407
views
Good reduction for the universal elliptic curve
Let $X$ be a modular curve, i.e. a quotient of the upper half plane $\mathbb{H}$ by a congruence subgroup $\Gamma$. When $\Gamma=\Gamma_0(N)$, it is known that $X$ has a smooth model denoted $\mathcal{...
- 2,907
1
vote
0
answers
52
views
Stability of Euler discretization
I am looking at the discretization of an ODE:
$$x_{n+1} = x_n + \alpha f(x_n),$$
where $x_n\in R^d$ and $f$ is continuously differentiable and such that $f(0)=0$ and $f'(0)$ is Hurwitz (i.e., the real ...
- 562
5
votes
1
answer
214
views
Stability of ODEs with polynomial nonlinearity
Consider the following ODE system:
$$
x′=f(x)\iff
\begin{pmatrix}
x_1^\prime \\
\vdots\\
x_k^\prime\\
\vdots\\
x_n ^\prime
\end{pmatrix} =
\begin{pmatrix}
f_1(x) \\
\vdots\\
f_k(x)\\
\vdots\\
f_n(x)
\...
- 807
2
votes
0
answers
168
views
Centralizer of PSL in PGL and of SL in GL: reference request
$\DeclareMathOperator\GL{GL}\DeclareMathOperator\SL{SL}\DeclareMathOperator\PGL{PGL}\DeclareMathOperator\PSL{PSL}$Consider the general linear group $\GL(n,q)$ over a finite field with $q$ elements and ...
- 361
3
votes
0
answers
130
views
A Talagrand inequality for the supremum of partial sums over function classes under dependence. (Reference request)
As a consequence to the Talagrand concentration inequality, it is well known that for a measurable space $(S,\mathcal{S})$ and an i.i.d. sample $X_1,...,X_n$ of $S$-valued random variables, if $\...
- 141
2
votes
1
answer
77
views
Locating elementary argument for leaf volume bound of codimension one compact foliations
Where in Reeb's thesis (or other expository reference) is the fact claimed by the following two blurbs proved? These both refer to Reeb's thesis.
Edwards-Millett-Sullivan's Foliations with all leaves ...
1
vote
0
answers
86
views
Unitary representations of the symmetric group over finite fields
I am interested in understanding the unitary representations of the symmetric group over $\mathbb{F}_{q^2}$. In general, some comments here are relevant
Unitary representations of finite groups over ...
- 571
9
votes
1
answer
1k
views
Is the number of varieties of groups still unknown?
A variety of groups is a class of groups satisfying a specified set of equations. Equivalently, it is a class of groups that is closed under homomorphic images, subgroups, and direct products. A ...
- 63.1k
1
vote
0
answers
82
views
Behavior of translation functors in characteristic $p$
Let $G$ be a semisimple and simply connected algebraic group over an algebraically closed field of characteristic $p>0$, and let $\mathfrak g$ be the Lie algebra of $G$. Let $U_\chi(\mathfrak g)$ ...
- 2,974
4
votes
0
answers
97
views
Characterization of Vilenkin group
It is shown in [1, Section 1] by C.W. Onneweer that every infinite compact, metrizable, zero-dimensional commutative group is a Vilenkin group. My question is does this implication also hold if we ...
- 85
1
vote
0
answers
69
views
Unique continuation of Laplace eigenforms
Let $M$ be a compact Riemannian manifold and $\Delta = d\delta + \delta d$ denote the (positive definite) Hodge Laplacian acting on differential forms. Call a smooth differential form $\omega$ a ...
- 1,407
1
vote
0
answers
101
views
Locating volume 2 of certain conference proceedings in analytic number theory
Does anyone know where one might locate "Analytic Number Theory: Proceedings of a Conference in Honor of Heini Halberstam, Volume 2"? There exists Volume 1 here: https://link.springer.com/...
- 1,890