All Questions
15,614 questions
1
vote
0
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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 ...
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 ...
- 109k
2
votes
1
answer
215
views
Number of binary matroids of rank $r$ on a ground set with $n$ elements
How many simple binary matroids are there, up to isomorphism, of rank $r$ on an $n$-element ground set, where $r \le n < 2^r$? Write this number as $a_r(n)$. Is there somewhere where I can get this ...
- 331
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
2
votes
0
answers
83
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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 ...
- 21
55
votes
2
answers
3k
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Is it known? A sum over lattice parallelograms of area one is equal to $\pi$
I recently discovered a formula, my proof is really a high school proof in three lines.
$$4\sum_{x, \, y \, \in \, \mathbb Z_{\geq 0}^2, \, \det(x \ \ y) = 1} \frac{1}{\lVert x\rVert^2\cdot\lVert y\...
- 5,053
3
votes
1
answer
76
views
Polarities for intuitionistic linear logic formulas inside classical linear logic (without linear implication)
In the article on intuitionistic linear logic on the LLWiki, it is stated that a polarization of formulas in classical linear logic is enough to make it equivalent to intuitionistic linear logic, ...
- 343
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$ ...
3
votes
0
answers
119
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 ...
- 51
2
votes
1
answer
66
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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. ...
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
3
votes
3
answers
379
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=\{...
- 87
3
votes
1
answer
88
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&...
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 ...
- 226
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
3
votes
1
answer
162
views
Ind-completion commutes with category product
$\def\A{\mathcal{A}}
\def\C{\mathcal{C}}
\def\D{\mathcal{D}}
\def\ind{\operatorname{Ind}}
\def\op{\mathrm{op}}
\def\Hom{\operatorname{Hom}}$I am trying to understand the following result from ...
2
votes
1
answer
79
views
What is weak convergence of random permutons?
In various papers on permutons you can find statements similar to this (see Maazoun's thesis)
For any $n$ let $\sigma_n$ be a random permutation of size $n$. TFAE:
$(\mu_{\sigma_n})_n$ converges in ...
- 677
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
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
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 ...
- 277
13
votes
3
answers
670
views
How algebraic can the dual of a topological category be?
(I'm going to try to use definitions from Abstract and Concrete Categories: The Joy of Cats by Adámek, Herrlich, and Strecker, since both of the adjectives in the title of my question seem to have at ...
- 12.4k
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)=...
2
votes
1
answer
186
views
On local Galois deformation rings
Let $p,\ell$ be two different primes. Let $K$ be a finite field extension of $\mathbb{Q}_{\ell}$ and $ \bar{\rho}:G_{K}\to {\rm GL}_{n}(\mathbb{F}_p) $ be a continuous mod $p$ representation of the ...
- 667
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 ...
- 8,401
2
votes
0
answers
204
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
0
votes
1
answer
73
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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 ...
- 633
3
votes
1
answer
204
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 ...
- 65
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
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 ...
- 128k
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{...
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
1
vote
1
answer
64
views
What is $\left[ \begin{array}{c} K_i;0\\ \ell\\ \end{array} \right] _{\varepsilon _i}$ in the restricted specialization in QUE algebras?
I have a question about the book A Guide to Quantum Groups written by Vyjayanthi Chari and Andrew Pressley. It comes from section $9.3$ on page $300$ of this book
In Section 9.1, the authors define ...
- 137
9
votes
0
answers
274
views
Has a computer search for inconsistency of large cardinals been carried out before?
In discussion about the consistency of large cardinals, a justification which occasionally appears is that despite years of work, no inconsistency has currently been discovered. For example, here are ...
- 2,031
4
votes
0
answers
167
views
What textbooks/papers should I read to try to make this rigorous?
Consider a surface of revolution $S$ and an embedding $e:S \hookrightarrow X^3$ for $X^3=[0,1]^3$ with cone points $p,q$ elements of $\partial X^3$ where $\partial X^3=X^3-(0,1)^3$ for $\mathrm {sup}~ ...
- 383
4
votes
1
answer
271
views
Is there a non-semistable simple sheaf?
Let $C$ be a smooth projective (irreducible) curve over an algebraically closed field $k$.
A sheaf is said to be simple if its endomorphism algebra is isomorphic to $k$.
It is known that a stable ...
- 405
0
votes
0
answers
52
views
References on a variant of Geometric Calculus
Geometric algebra and (standard) calculus, when synthesized, give rise to geometric calculus, a very powerful formalism.
I have read a bit about fractional calculus and time-scale calculus, both very ...
user537376
5
votes
1
answer
151
views
Dimension from Hilbert series with variable-weighted grading?
Suppose $R = F[x_1,...,x_n]/I$ is a finitely generated ring over the field $F$, and suppose there is a function $d:[n] \to \mathbb{Z}$ such that defining the degree of a monomial $x_1^{e_1} ... x_n^{...
- 3,230
2
votes
0
answers
54
views
Distance between a Hölder function and a Sobolev ball
Let $\Omega$ denote $[0, 1]^n$ and let $\|\cdot\|_{k, p}$ and $|\cdot|_{m, \alpha}$ denote norms of Sobolev space $W^{k,p}(\Omega)$ and Holder space $C^{m, \alpha}(\Omega)$, respectively.
My question ...
- 460
8
votes
1
answer
685
views
The state of the art on topological rings - the Jacobson topology
I was recently studying the Jacobson density theorem and I found it quite interesting.
Most textbooks I've seen, including Jacobson's own Basic Algebra, only spend a few lines about the reason why it ...
- 183
1
vote
0
answers
46
views
Reference on polynomial attached to permutation group
Let $G$ be a permutation group acting on some set. Let $C(g)$ be the set of associated cycles of an element $g\in G$, and define $l(c)$ to be the length of a cycle $c$. Now set
$$T(G) = \sum_{g\in G}\...
0
votes
0
answers
28
views
Analyzing simple DDE with simple characteristic test
I'm wondering if anyone can comment on the stability of delay DE given that we can analyze its characteristic equation.
For instance, let's say we have the DDE $\frac{d}{dt}x(t) = x(t-a),$ where $a$ ...
- 11
0
votes
0
answers
85
views
Uniqueness of compatible cycle decomposition for Eulerian trail
Fleischner mentions in his article Uniqueness of maximal dominating cycles in 3-regular graphs and of hamiltonian cycles in 4-regular graphs about the uniqueness of compatible cycle decomposition that ...
3
votes
0
answers
87
views
cubic twists of Mordell curve and their rank
Let $a$ be a non-zero integer. Consider the elliptic curve $E_a/\mathbb{Q}$ given by the equation
$$
E_a: y^2 = x^3 + a.
$$
For a cube-free integer $D$, define the elliptic curves $E_{aD^2}/\mathbb{Q}$...
- 1,283
2
votes
1
answer
191
views
Sums of multiplicative functions over residue classes
It was stated in this Shiu, P. work, page 169, Theorem 2, that $$\sum_{\substack{n\le x\\ n\equiv a\pmod k}}d_r^{\ell}(n)\ll\frac{x}{k}\left(\frac{\phi(k)}{k}\log x\right)^{r^{\ell}-1}.$$
Here, $d_r(n)...
user536681
0
votes
0
answers
72
views
Regularity estimates of Double Layer potential
Let $\Omega$ is a bounded open subset of $\mathbb{R}^n,n\ge 2$ with $C^{\infty}$ boundary. Define $$I\left[ \phi \right](x) := -\frac{1}{\omega_n}\int_{\partial \Omega} \frac{(x-y)\cdot \nu_y}{|x-y|^n}...
- 69
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....
- 82.6k
1
vote
0
answers
159
views
Where can I find Stefan Wewers' doctoral thesis "Construction of Hurwitz Spaces"?
I am looking for a copy of Stefan Wewers' doctoral thesis titled "Construction of Hurwitz Spaces," which was defended at the University of Essen in 1998. I have tried searching through ...
- 560
1
vote
0
answers
66
views
Sum of k vectors with largest possible norm
Suppose I have a family of $n$ vectors in $\mathbb{R}^d$:
$v_1,\dots,v_n.$ Is there a poly-time algorithm that computes a subset $S\subset [n]:=\{1,\dots,n\}$ of size $1\leq k\leq n$ for which the (...
5
votes
1
answer
367
views
Reference request: locally erasable delta-functor is universal
It is well-documented that an erasable delta-functor $(F^n,\delta)$ is universal. However, in 'small' abelian categories (in the technical sense or otherwise), there aren't always enough objects to ...
- 28.7k
4
votes
1
answer
253
views
Reference to formal approach to homotopy analysis method
I'm currently reading a book about the Homotopy Analysis Method (HAM), but it isn't very rigorous (it explains most things with a single example), which is bothering me.
I'm searching for papers where ...
- 151