All Questions
15,509 questions
-2
votes
1
answer
210
views
Reference request on dynamics and hyperbolic dynamics (hyperbolicity in absence of periodic orbits)
I would appreciate if you introduce me a reference (paper or book) who address the concept of hyperbolic dynamics but with emphasis on absence of periodic orbits. a possible ...
- 366
8
votes
1
answer
333
views
Standard reference for double category theory
Is there a ‘standard reference’ for double category theory?
Ideally something along the lines of CWM for $1$-category theory or Johnson and Yau’s book for $2$-category theory; some reference that ...
- 10.1k
0
votes
0
answers
108
views
looking for reference for two elliptic curves with equal formal group
I am looking for a reference.
In this post, @Chris Wurthrich made the following comment:
If the formal group laws (probably upon particular choice of coordinates) of two elliptic curves over any ring ...
- 195
2
votes
1
answer
101
views
Bound on number of extensions of Q unramified outside a fixed prime
Are there any known asymptotic bounds on the number of degree $d$ extensions of $\mathbb{Q}$ unramified outside a fixed prime $p$?
- 2,907
1
vote
1
answer
70
views
Classifying lisse conformal vertex algebras using singularities of associated varieties
For the sake of keeping terminologies consistent, let me say that a conformal vertex algebra is a vertex algebra (VA) with a specified conformal vector, and a vertex operator algebra (VOA) is a ...
- 1,516
2
votes
1
answer
118
views
Reference request for a proof of the fact that every congruence-permutable variety is semidegenerate
Given an algebra $\mathbf{A}$, a pair of congruences
$ \alpha ,\beta \in Con(\mathbf {A})$ are said to permute when
$ \alpha \circ \beta =\beta \circ \alpha$, and an algebra
$\mathbf{A}$ is called ...
- 21
6
votes
1
answer
154
views
Derivations and central extensions of some infinite dimensional simple Lie algebras in characteristic zero
Let $F$ be a field of characteristic zero and consider the polynomial algebra $A=F[x_1,x_2,\ldots,x_n]$ in $n$ indeterminates over $F$. I recall that the derivations of $A$ form a simple Lie algebra $...
- 99
2
votes
0
answers
145
views
$L$ and $\epsilon$ factors for local Langlands correspondence
I am looking for an English reference for the construction of $L$ and $\epsilon$ factors in the local Langlands correspondence.
At the moment I am reading "The local Langlands correspondence for ...
- 367
2
votes
0
answers
97
views
Centralizer bound for irreducible representations of $\operatorname{SU}_n(\mathbb{C})$
Let $G$ be a finite group, $χ$ be the character of an irreducible representation $V$ of $G$, and $g ∈ G$. Then a classical bound on the trace of $g$ is given by: $|χ(g)|² ≤ |C_G(g)|$, where $C_G(g)$ ...
- 2,907
2
votes
1
answer
233
views
Reference for surjectivity of the canonical map $R^{G_1} \otimes R^{G_2} \to R^{G_1 \cap G_2}$
Let $R$ be a commutative ring, $G$ a finite group with an action over $R$. Let $G_1, G_2 \subset G$ be two subgroups. Then the canonical map $R^{G_1} \otimes R^{G_2} \to R^{G_1 \cap G_2}$ is ...
- 23
3
votes
1
answer
232
views
Non-degeneracy in hyperplane intersections of canonical curves
Let $C$ be a smooth projective non-hyperelliptic curve over $\mathbb{C}$ of genus $g = 4$. The canonical bundle $\omega_C$ induces a canonical embedding $C \longrightarrow \mathbb{CP}^3 $ such that $C$...
- 343
1
vote
0
answers
67
views
Upsampling parameters in the Takahashi-Alexander model
Let me start by begging your forebearance; this question might at first glance appear to belong more on a forum for economics, but I hope by the end to convince you that there is mathematical content ...
- 721
9
votes
2
answers
418
views
Reference request: Parabolic Equations
I am a PhD student working mainly on Elliptic Equations. With the other PhDs of my department, we organised a reading group, meaning that we agreed on a book we were all interested in, we meet weekly ...
- 452
5
votes
1
answer
210
views
Reference for homotopy groups of filtered homotopy colimits
It seems to be well known that for a filtered category $I$
and a functor to the category of pointed spaces $X:I \to \mathcal{S}_*$
the homotopy groups of the filtered homotopy colimit are colimits of ...
- 1,484
6
votes
1
answer
393
views
Decimal expansion definition of real numbers, constructively
The two most common definitions of $\mathbb{R}$ are as Dedekind cuts or Cauchy sequences of rational numbers.
A real analysis student of mine is working out of the book Real Analysis and Applications ...
- 10.1k
2
votes
0
answers
160
views
Centre of centralisers in connected reductive groups
Let $G$ be a connected reductive group over an algebraically closed field. Let $T$ be a maximal torus and $x\in T$. Let $G_x$ denote the centraliser of $x$ in $G$.
Question: What is an explicit ...
- 2,751
4
votes
1
answer
165
views
Reference Request: on an explicit formula for class-1 Whittaker functions on split reductive groups over p-adic fields
The 1978 preprint by S.Kato 'On an explicit formula for class-1 Whittaker functions on split reductive groups over p-adic fields' is cited by papers involving unramified computations of local ...
- 43
4
votes
1
answer
170
views
About $CW(512,16^2)$
Definitions: A weighing matrix $W = W(n,k)$ with weight $k$ is a square matrix of order $n$ and entries $w_{ij}$ in $\{0, \pm 1\}$ such that $WW^T=kI$,
where $I$ is the identity matrix. A circulant ...
- 716
3
votes
0
answers
161
views
Generalized dimension property for rings
My question is very basic, I am looking for a characterization (and name) of rings $R$ satisfying the following property $\star$.
For any $V, W$ two finitely generated $R$-modules such that $V\oplus W\...
- 223
2
votes
1
answer
155
views
What conditions on the rate matrix $Q$ ensure unique convergence in continuous-time Markov chains?
In the study of discrete-time Markov chains, the conditions under which all initial distributions converge to a unique stationary distribution are well-understood. Specifically, if the transition ...
- 807
2
votes
1
answer
217
views
Number of distinct higher dimensional integer partitions
By a distinct partition, I mean a partition into distinct parts, i.e., $10 = 5+4+1$ is one, but $10=6+2+2$ is not. The number of distinct partitions of $k$ all whose parts are at most $n$ is given by ...
- 281
1
vote
0
answers
88
views
Hausdorff distance and Hausdorff measure of symmetric difference
Let $X_n$ be a sequence of $k$-dimensional piecewise smooth submanifolds of $\mathbb{R}^m$, converging in Hausdorff distance to a $k$-dimensional piecewise smooth submanifold $Y \subset \mathbb{R}^m$, ...
- 11
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 ...
- 41
2
votes
0
answers
60
views
Relative Dolbeault cohomology using currents
I need to compute the cohomology groups of some relative holomorphic $i$-forms $H^\bullet(X, \Omega^i_{X/Y})$ for a fibration of complex manifolds $X\to Y$, using a kind of distributional de Rham ...
- 2,054
9
votes
1
answer
435
views
On the origin of a fundamental theorem of additive number theory
Given $a, b \in \mathbb Z$, set $[\![a,b]\!] := \{x \in \mathbb Z: a \le x \le b\}$. A basic result in additive number theory goes as follows:
If $A$ is a finite subset of $\mathbb N$ with $0 \in A$ ...
- 10.5k
7
votes
0
answers
141
views
Frenkel-Kac's vertex operator realisation of the basic representation of an untwisted affine Kac-Moody algebra
Let $\mathfrak{g}$ be a finite-dimensional simple Lie algebra over $\mathbb{C}$ and let $\hat{\mathfrak{g}} := (\mathfrak{g}[t^{\pm 1}] \oplus \mathbb{C} c) \rtimes \mathbb{C} D$ be the corresponding ...
- 1,516
12
votes
4
answers
994
views
Reference for the proof that Möbius transformations extend to isometries of hyperbolic 3-space
Consider the group $\operatorname{PSL}(2,\mathbb C)$ acting by Möbius transformations of the Riemann sphere. It is known that this action can be extended to an action on the unit ball which is ...
- 481
1
vote
2
answers
239
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
$$
\...
- 233
4
votes
0
answers
168
views
Explicit bounds on gaps between zeros of $\zeta^\prime(s)$
In $\S$9.1 of "Theory of the Riemann Zeta Function", Titchmarsh uses Borel-Carathéodory and Hadamard Three Circles to show that every circle of radius 6 and center $3+iT$ contains a zeros of ...
- 11.1k
5
votes
1
answer
177
views
Orthogonal projection onto cones in inner product spaces
Let $H_n$ denote the space of $n\times n$ Hermitian matrices. For every $A\in H_n$, using the spectral decompostion of $A$,
$$A=\sum_i \lambda_i x_ix_i^*,$$
one can define the positive and negative ...
- 4,484
11
votes
1
answer
1k
views
Had this attempt to salvage naïve comprehension been studied before?
Is the following a possible way to overcome inconsistency with naive comprehension:
We add an $\in_n$ symbol for each natural $n$ to the signature of this theory, which is a first order theory with ...
- 11.3k
7
votes
0
answers
144
views
Bhargava's "Higher composition laws V" - where I can find it?
TSIA. There are four papers by Bhargava on higher composition laws that are publicly available:
I: A new view on Gauss composition, and quadratic generalizations
II: On cubic analogues of Gauss ...
- 2,215
0
votes
2
answers
216
views
Papers related to a diophantine equations about Magic square of squares for $n=3$
The open problem of magic squares of squares explained here. Consider the following magic square of squares:
$$
\begin{aligned}
&a^2&b^2&&c^2\\\\
&d^2&e^2&&f^2\\\\
&...
4
votes
0
answers
151
views
Computable subsets of non-standard models of arithmetic
By Tennenbaum's theorem, there exists no computable non-standard model of $\mathsf{PA}$. That is, for any nonstandard model $M$, we cannot define an encoding of the integers $x\in M$ such that $+_M$, $...
- 291
7
votes
0
answers
349
views
An open set which is not the union of a closed set and a countable set
The following fact is probably a known result:
Fact. Let $X$ be an uncountable Polish space. Then there exists an open subset of $X$ which is not the union of a closed set and a countable set.
Proof:...
- 1,511
1
vote
0
answers
114
views
An urn model with weighted objects and replacement
Consider the following game:
In an urn, there are $K$ balls, $x_0$ of them are blue and light (mass $m_0$), $x_1$ are blue and heavy ($m_1$), $x_2$ are red and light ($m_2$), the rest $x_3$ are red ...
- 31
3
votes
0
answers
76
views
Bow lemma with angles
First, let me recall the statement of the bow lemma.
Let $\gamma_1: [a,b] \to \mathbb{R}^2$ and $\gamma_2: [a,b] \to \mathbb{R}^2$ be two smooth unit-speed curves.
Assume $\gamma_1$ and its chord ...
- 45k
6
votes
1
answer
199
views
$L(\mathbb{R})$-absoluteness from a proper class of Woodins: source?
For a paper I'm writing I need to use (as a blackbox) the following theorem: if there is a proper class of Woodin cardinals and $G$ is set-generic, then $L(\mathbb{R})$ and $L(\mathbb{R})^{V[G]}$ are ...
- 20.7k
15
votes
1
answer
603
views
Topological spaces in which countable intersections of dense open sets have dense interior
In certain topological spaces, known as Baire spaces (e.g., completely metrizable spaces), a countable intersection of dense open sets is dense.
Now consider the following strengthening of the Baire ...
- 32.5k
2
votes
0
answers
90
views
What's known about the Steinerian map's indeterminacy locus?
Sorry if this question is asked already, a quick google search didn't yield any answers.
In Classical algebraic geometry, §1.1.6, Dolgachev defines the Steinerian hypersurface $\operatorname{St}(X)$ ...
- 305
3
votes
2
answers
237
views
Lengths of closed geodesics and geodesic segments
Let $M$ be a closed Riemannian manifold of dimension $n \geq 2$. I am looking at sufficient conditions for the following two properties:
existence of closed geodesics of arbitrarily long length on $M$...
- 31
3
votes
1
answer
203
views
Cohomology of the complex of differential forms with Schwartz coefficients
Let $U$ be an open manifold (say an open subset of $\mathbb{R}^n$ for simplicity). Denote by $\mathscr{S}(U)$ the space of Schwartz functions on $U$. Schwartz functions are defined as usual to be ...
- 1,374
0
votes
0
answers
34
views
Locally compact groupoid with range map restricted to isotropy groupoid is open
Suppose the action groupoid 𝐺=𝐻⋉𝑋, where 𝐻 is a locally compact group and 𝑋
a locally compact space is such that isotropy subgroups of H are isomorphic to each other.
Can this be an example of a ...
2
votes
0
answers
179
views
A Brun-Titchmarsh type result for divisor sums; asymptotic/improved bound
In Shiu's work ('A Brun-Titchmarsh theorem for multiplicative functions') he proved that if $r\le x$ is a natural number, we have $$\sum_{r<n\le x}d(n)d(n-r)\ll x\log^2x\sum_{d|r}\frac{1}{d}.$$
I ...
user344548
4
votes
1
answer
195
views
When is one dynamical system an approximation of another?
I've been thinking about the question of when a discrete time dynamical system $f : X \to X$ (or possibly other objects) can be said to approximately model another dynamical system. So far I've mostly ...
- 141
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
1
answer
143
views
Algorithm for computing isogeny class of elliptic curve
Is there an algorithm for computing the entire isogeny class of a given elliptic curve $E/\mathbb{Q}$?
References/ideas are welcome. Thanks!
- 2,907
3
votes
0
answers
68
views
Reference request: inverse image in singular homology as in Chow groups
I come from algebraic geometry and I have trouble finding a reference to check the construction of the inverse image in singular homology, analogous to that of the Chow groups. Let me be more precise:
...
- 2,871
3
votes
0
answers
59
views
Reference request: History of the fact that the family of probability distributions of Cauchy type is closed under l.f.t.s of random variables
Two probability distributions on (Borel subsets of) $\mathbb R$ are of the same "type" if for any random variable $X$ having one of those distributions, the other distribution is that of $\...
5
votes
1
answer
302
views
Higher homological mirror symmetry?
The bounded derived category $D^b\mathrm{Coh}(X)$ is the homotopy category of a stable $\infty$-category $\mathbb{D}^b\mathrm{Coh}(X)$. Apparently there are reasons, such as "nonfunctoriality of ...
- 355