All Questions
15,613 questions
3
votes
1
answer
135
views
Concentration of sample median for iid Gaussians
Let $X_1, \dots, X_n$ be iid according to $\mathcal{N}(0, 1)$, and let $M_n$ be the median of the $X_1, \dots, X_n$. I recall reading a concentration inequality for $M_n$ that was (roughly) as follows:...
- 75
0
votes
0
answers
54
views
Number of indecomposable modules over representation-finite hereditary algebras
Let $A$ be a finite dimensional $K$-algebra over a field $K$ that is hereditary and of finite representation type.
It is well known that they are classified by Dynkin diagrams.
For algebraically ...
- 26.5k
1
vote
0
answers
104
views
Reference about the semiabelian variety associated to a stable curve
If a stable curve $C$ of genus $g$ is such that its dual graph has Betti number $1$, then the Jacobian of $C$ is a semiabelian variety of torus rank $1$. It is described by Mumford that the data of a ...
8
votes
1
answer
849
views
Representations of groups with the same derived group, how much control do we have over the central character?
Let $G_1 \subset G$ be the rational points of $p$-adic reductive groups sharing the same derived group. There are some well known results relating representations of $G_1$ to representations of $G$, ...
CommunityBot
- 1
11
votes
4
answers
950
views
Is there a name for finite unions of intervals?
Finite unions of intervals are simple sets that are used quite often, e.g. in measure theory. (The construction of the Cantor set is a noble example). I realised that I do not have a name for them. Is ...
- 6,367
11
votes
1
answer
340
views
Number of odd-dimensional irreducible representations of $S_n$
In this paper the structure of odd-dimensional irreducible representations of the symmetric group is described, but what is the asymptotic behaviour of the number of such representations? (Or, if it ...
- 12.9k
2
votes
0
answers
83
views
Random time change and ergodicity
I guess it is a standard question in ergodic theory but I failed to find any reference to similar problems and I have no clue on how to tackle it.
Let $(B_{t})_{t\in \mathbb{R}}$ be a standard ...
- 3,065
2
votes
1
answer
91
views
References for Green's functions right focal boundary-value problem
Could you please give me some references for the computation of a Green's function for a second-order right focal difference equation?
For this problem:
\begin{gather*}
\Delta^2 u(t)=f(t), \; t\in\{0,...
CommunityBot
- 1
8
votes
0
answers
115
views
optimal regularity for the Neumann heat equation on Lipschitz domains
$\newcommand{\R}{\mathbb R}$Let $\Omega\subset\R^d$ be a bounded Lipschitz domain, possibly non-convex (but not too nasty either, whatever that means). I am looking for well-posedness and optimal ...
- 5,380
3
votes
1
answer
90
views
Original references for Cordes-Nirenberg estimates
Cordes-Nirenberg estimates look like:
Let $u \in H^1(B_1)$ a weak solution of
\begin{equation}
- \operatorname{div}(a_{ij}(x)\nabla u(x)) = 0 \quad \text{in} \quad B_1
\end{equation}
Then, for any $0&...
- 7,817
7
votes
1
answer
170
views
Topological rigidity of cartesian product with $\mathbb{R}$
It seems that the following is true : if $V$ and $W$ are compact differentiable manifold of the same dimension, and $\mathbb{R} \times V$ is diffeomorphic to $\mathbb{R} \times W$, then $V$ and $W$ ...
- 12.3k
6
votes
1
answer
489
views
What inequalities for convex sets are known since the work of Scott and Awyong?
In 2000, Paul R. Scott and Poh Way Awyong published the paper Inequalities for Convex Sets, which nicely collates the known results relating various natural geometric functionals (diameter, area, etc.)...
CommunityBot
- 1
15
votes
1
answer
2k
views
Sublattices of Young's Lattice
Young's Lattice is the lattice of inclusions of Young tableaux, or integer partitions.
In R. Stanley's Enumerative Combinatorics Vol. 1, Proposition 1.4.4., there is a generalization of integer ...
- 2,183
2
votes
1
answer
243
views
Uniform Lipschitz function approximation by shallow neural networks
Fix $d\in \mathbb{N}$. Let $F_1$ be the set of all 1-Lipschitz functions mapping $[0, 1]^d$ to $\mathbb{R}$.
For $\varphi: \mathbb{R} \rightarrow \mathbb{R}$ and $m \in \mathbb{N}$, let $N_\varphi^m$ ...
- 3,599
17
votes
3
answers
2k
views
Theoretical results on neural networks
With this question I'd like to have a recollection of theoretical rigorous results on neural networks.
I'd like to have results that have been settled, as opposed to hypothesis. As an example, this ...
- 3,599
3
votes
0
answers
120
views
Convergence of gradient descent to critical point
Does there exist a generalization of this theorem by Yurii Nesterov in Introductory Lectures on Convex Optimization (2004) which relaxes the convexity assumption and shows that gradient descent ...
- 5,405
3
votes
3
answers
382
views
On subfields of the cyclotomic field $\mathbb{Q}(\zeta_p)$
Let $p$ be an odd prime. Let $\zeta_p=e^{2\pi{\bf i}/p}$ and let $1\le k\le p-1$ be a divisor of $p-1$. Recently, when I learnt algebraic number theory, I met the following problem.
If we let
$$U_k=\{...
- 50.6k
2
votes
1
answer
66
views
How many non-trivial solutions can a semilinear elliptic equation have on a smooth star-shaped bounded domain with 0-Dirichlet boundary conditions?
I am not an expert in elliptic partial differential equations, but while studying the attractor structure of evolutionary PDEs, I frequently encounter problems related to elliptic equations. ...
- 52.3k
7
votes
1
answer
441
views
Road map and references for combinatorial Hodge theory
I'm a PhD student. I'm familiar with graduate level algebraic geometry and toric varieties.
I wanted to know a road map for getting into combinatorial Hodge theory and other prerequisites that I'll ...
- 1,046
3
votes
0
answers
97
views
Notion of a finite generator in an abelian category
Let us say that an abelian category admits a generator $g$, if for every object $x$ in the category there is an epimorphism $g^{\oplus I} \to x$ for some index set $I$. I am interested in weakening ...
- 1,474
10
votes
3
answers
541
views
Curvature of the boundary vs. normal derivative of the first eigenfunction
Disclaimer. I posted this question in Math.SE, but it haven't received enough attention.
Let $\varphi_1$ be the first eigenfunction of the zero Dirichlet Laplacian in a planar bounded domain $\Omega$....
6
votes
1
answer
348
views
p cohomological dimension of a profinite group
I would like to know what is the $p$-cohomological dimension of $\textrm{Gal}(\mathbb{Q}_S/\mathbb{Q}_{cyc})$. Here $S$ is a finite set of primes containing $p$ and the Archimedean primes and $\mathbb{...
3
votes
1
answer
267
views
Volume of a double class of a parahoric subgroup
Let $F$ be a non-archimedean local field with residue field $F_q$. Let $G$ be the group of $F$-rational points of a connected reductive group defined and split over $F$. Fix a maximal split torus $T$ ...
- 171
5
votes
0
answers
99
views
Differential equations analogue of fundamental theorem of symmetric functions
In Gian-Carlo Rota's article "Ten Lessons I Wish I Had Learned Before I Started Teaching Differential Equations", at the end of the third lesson he states a theorem:
"Every differential ...
- 869
3
votes
0
answers
47
views
Lax morphism classifiers via lax-idempotentification
Let $T$ be a 2-monad on a nice 2-category $\mathcal K$, so that the inclusion $T\text{-}\mathbf{Alg}_s \to T\text{-}\mathbf{Alg}_l$ of the 2-category of (strict) $T$-algebras and strict $T$-algebra ...
- 10.6k
7
votes
2
answers
292
views
Quotient topoi as quotient objects
In Lawvere's Open problems in topos theory; quotient topoi are treated as connected geometric morphisms of Grothendieck topoi.
Is there a good reference for where these come from? Is there any sense ...
- 1,347
1
vote
2
answers
237
views
Calderón–Zygmund/$L^p$ estimates for the linear heat equation
Let $C_r$ denote the open cylinder
$$
C_r = \{(x,t) \in \mathbb R^{n+1} : |x| < r, -r^2 < t < 0\}
$$
and consider a classical $C^{2,1}_{x,t}(C_1)$-solution to the linear heat equation
$$
\...
- 12.9k
11
votes
4
answers
707
views
Deriving an asymptotic for $\pi(x)$ directly from $\log \zeta(s)$?
Denote by $\pi(x)$ the number of primes $p\leq x$. We generally give approximations for $\pi(x)$ by first approximating $\psi(x) = \sum_{n\leq x} \Lambda(n)$. Part of the reason is presumably that, if ...
- 14.6k
5
votes
2
answers
540
views
Boundary regularity for elliptic PDE in Lipschitz domains
In section 2.6 of Fernandez-Real and Ros-Oton's book "Regularity theory for elliptic PDEs" it is stated that solutions of the Dirichlet problem with smooth data for the Laplacian are
$C^{1-...
5
votes
0
answers
261
views
Primes generated by cyclotomic polynomials
Let $p$ be an odd prime, and let $f=\Phi_p$ be the $p$-th cyclotomic polynomial. Denote by $S_p$ the set of primes $q$ such that there exists a sequence of primes $p_1,\dots, p_g$ such that $p_1=f(1)=...
CommunityBot
- 1
4
votes
1
answer
175
views
Looking for J.-C. Deville technical report from 2000
Yves Tillé's book Sampling Algorithms mentions several times a technical report by J.-C. Deville:
J.-C. Deville (2000), Note sur l’algorithme de Chen, Dempster et Liu, Tech.
rept. CREST-ENSAI, Rennes....
- 82.7k
8
votes
1
answer
464
views
Smooth numbers in short intervals
Let $\psi(x,y)$ be the number of positive integers up to $x$ which are $y$-smooth, that is, integers whose prime factors are at most of size $y$. There has been, for a few decades now, a lot of ...
- 14.6k
1
vote
1
answer
56
views
How to study the convergence of the sample mode for arbitrary probability spaces
(This is not the problem I actually care about, but an analogy with similar issues to the problem I'm actually considering.)
Consider a probability space with i.i.d. random variables $X_i$ producing ...
- 128k
1
vote
0
answers
25
views
Characterising rank-$2$ lattices $\Lambda$ and conjugate-linear translate $g \sigma(\Lambda)$, given elementary divisors
Let $E/F$ be a quadratic unramified extension of local fields with $\operatorname{char} F = 0$. Let $\Lambda \subseteq E^2$ be an $O_E$-lattice of rank $2$. Let $g \sigma \in \operatorname{GL}_2(E)$ ...
- 111
3
votes
0
answers
89
views
Ordering the elements of a semigroup by $a \le b$ iff $a=b$ or $b=ab=ba$
Let $S$ be a semigroup, written multiplicatively. The binary relation $\le$ on (the underlying set of) $S$, whose graph consists of all pairs $(a,b) \in S \times S$ such that $a = b$ or $b = ab = ba$, ...
- 10.5k
2
votes
0
answers
205
views
When should the empirical measure of an infinite sequence be defined?
Let $(x_n)_{n \in \mathbb{N}}$ be a (deterministic) sequence of nonnegative reals, possibly even with $x_n \in \mathbb{N}$ if you prefer. Then we'd like to define the empirical measure of such a ...
- 6,406
11
votes
3
answers
779
views
Is every recursively axiomatizable and consistent theory interpretable in the true arithmetic (TA)?
I am looking for a scholarly text that discusses this issue in detail.
- 47.1k
4
votes
1
answer
193
views
Canonical decomposition as wedge sum up to homotopy equivalence
I am curious: is there a canonical way to decompose a finite simplicial complex into a wedge sum up to homotopy equivalence? More formally:
Let $X$ be a finite simplicial complex. Is $X$ homotopy ...
- 16.2k
16
votes
0
answers
188
views
Representation theory of Pin groups
I am (still) thinking about branching rules from $\mathfrak{so}(n+m)$ to $\mathfrak{so}(n) \oplus \mathfrak{so}(m)$, using Proctor's paper as the starting point.
Proctor describes this rule for $m = 2$...
- 1,574
1
vote
0
answers
95
views
References on the claims of moduli spaces of additive compactifications by Hassett-Tschinkel
I am considering the classification problem of Artinian local $\mathbb{C}$-algebras, and notice the paper
Hassett, Brendan; Tschinkel, Yuri, Geometry of equivariant compactifications of (\mathbb{G}^...
- 930
27
votes
4
answers
4k
views
Who introduced the terms "equivalence relation" and "equivalence class"?
Consider that the question does not concern the origin of the ideas of equivalence relation and equivalence class. It exactly concerns the origin of the terms "equivalence relation" and "equivalence ...
- 61
0
votes
1
answer
73
views
Computing spectrum of very simple Schrödinger operator
I asked this question recently on a thread in math stack exchange, but with no real answers suggested. I think this is a relatively simple variation on the classical free Laplacian spectrum, so I ...
- 17.2k
2
votes
0
answers
103
views
Variant on ring objects in the category of complete lattices
Let $L$ be a complete lattice and denote its top and bottom elements by $0$ and $\infty$ respectively. Consider two binary operations $+$ and $\times$ defined on $L$ such that $(L^{op},+,0)$ is a ...
- 4,219
3
votes
1
answer
205
views
Inertia Action on Kummer Sheaves
In 7.0.2 of Katz's book "Gauss Sums, Kloosterman Sums, and Monodromy Groups", Katz states the following (when $x=0$).
Let $\chi:\mathbb{F}_q^\times\to\mathbb{Q}_\ell^\times$ be a ...
- 148k
11
votes
2
answers
430
views
On the convex cone of convex functions
$\newcommand\R{\Bbb R}$Let $F$ be the set of all functions of the form $\max(a,b,c)$, where $a,b,c$ are affine functions from $\R^2$ to $\R$ and the maximum is taken pointwise. Let $G$ be the set of ...
- 91.8k
2
votes
1
answer
315
views
A reductive group is the complexification of a compact subgroup even if not connected?
The definition of a linear algebraic complex reductive group is sometimes using the connectedness hypothesis for the complex algebraic group sometimes not.
Here I use the following definition : a ...
- 12.9k
4
votes
0
answers
87
views
Statistics of random Voronoi S-tessellations
Given a locally finite set of points $\{x_1,x_2,\dots\}\subset\mathbb{R}^d$, the Voronoi cell of a point $x_{i}$, denoted by $C(x_{i})$, consists of all the points in $\mathbb{R}^d$ that are closer to ...
2
votes
0
answers
121
views
Singular cohomology as a sheaf of $\infty$-categories
In several expositions of $\infty$-categories, I read that singular cohomology of a topological space with integral coefficients is a sheaf valued in $D(\mathbb{Z})$, if we consider Top and $D(\mathbb{...
3
votes
1
answer
368
views
Lot sizing problem: how to add these cuts efficiently
Consider the set of constraints of the uncapacitated lot sizing problem:
$$
\{(x,s,y)\in \mathbb{R}^n_+ \times \mathbb{R}^n_+ \times \mathbb{B}^n \;|\;s_{t-1}+x_t = d_t+s_t,\; x_t \le My_t,\; t=1,\...
CommunityBot
- 1
7
votes
2
answers
517
views
Tensor product of a DGA and an $A_\infty$ algebra
In general there seems no way to naturally define the tensor product of two $A_\infty$ algebras $A$ and $B$. But, if $(A, m^A_1,m^A_2)$ is only a DGA(differential graded algebra) and $(B, m^B_k, k\ge ...
