Newest Questions
159,031 questions
2
votes
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144
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Examples of counting holomorphic curves in cylindrical reformulation of Heegaard Floer
In 2005, Robert Lipshitz reformulated Heegaard Floer in a "cylindrical setting" by counting holomorphic curves in $\Sigma \times [0,1] \times \mathbb{R}$ where $\Sigma$ is a Heegaard surface ...
- 21
9
votes
0
answers
180
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Are there fast rank and unrank algorithms for integer vectors under the action of a permutation group?
We are distributing $m$ indistinguishable balls in $k$ numbered boxes $S=\{1,2,\ldots,k\}$. A distribution is a tuple of nonnegative integers $a=(a_1,\ldots,a_k)$ whose sum is $m$. We also have a ...
- 4,219
1
vote
0
answers
135
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Conformal laplacian on asymptotically flat manifolds with boundary
Let $g$ be an asymptotically flat metric on $M = \mathbb{R}^3 \setminus B_1$ where $B_1$ is the unit ball.
Suppose $X$ is a smooth vector field on $M$ that is decaying exponentially and satisfies
$$\...
- 969
1
vote
1
answer
192
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Lie group framing and framed bordism
What is the definition of Lie group framing, in simple terms?
Is the Lie group framing of spheres a particular type of Lie group framing? (How special is the Lie group framing of spheres differed ...
- 447
3
votes
1
answer
173
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What are the internal adjunctions in the bicategory $\mathsf{Span}$?
Recently I've been trying to understand spans better, in particular how they relate to relations, as both may be thought of as "multivalued functions between sets" (see Bruni and Gadducci - ...
- 11.8k
4
votes
1
answer
230
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Pontryagin product on the homology of cyclic groups
Consider the cyclic group $C_{p^N}$ of order $p^N$, and let $k$ be a field of characteristic $p$. I would like to know what the algebra structure on the homology $H_*(C_{p^N};k)$ induced by the ...
- 103
2
votes
0
answers
76
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Framed bordism and string bordism in 3-dimensions vs topological modular form
In simple colloquial terms, how are the framed bordism and string bordism in 3-dimensions related to the study of the theory of topological modular form TMF? I want to know some simple derivable ...
- 10.5k
2
votes
2
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212
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Isometric embeddings of metric $K_{n+1}$ in $\mathbb{R}^n$
Question:
is it always possible to embed a complete, symmetric and metric graph $G$ with $n+1$ vertices isometrically in $\mathbb{R}^n$?
I'm convinced it must be true, but can't remember having seen ...
- 13.2k
1
vote
0
answers
98
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Adjunction correspondence for Blow up of double point
Let $C$ a curve over an algebr closed field $k$ with a singular double point singularity at $x$ and $\pi: C' \to C$ the blowup in $x$ and let $x_1,x_2 \in C'$ be the two points over $x$.
Why holds for ...
- 6,038
10
votes
6
answers
880
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Countable chain condition in topology
A topological space $X$ is said to have the countable chain condition (ccc) if every collection of open and disjoint subsets of $X$ is at most countable. This definition can be found in L. Steen, J. ...
- 467
6
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0
answers
179
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Elementary equivalence for rings
Let $\mathcal{L}$ be a first-order language, and $M$ and $N$ be two $\mathcal{L}$-structures. We say that $M$ and $N$ are elementarily equivalent (write $M \approx N$) if they satisfy the same first-...
- 3,318
2
votes
0
answers
77
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How to write the division values of $\operatorname{sn}(u;k)$ as rational functions of theta functions with zero argument?
Define the "thetanulls" (theta functions (https://dlmf.nist.gov/20) with one argument equal to zero) as follows:
$$\vartheta_{00}(w) = \prod_{n = 1}^{\infty} (1-w^{2n})(1+w^{2n-1})^2,$$
$$\...
- 317
2
votes
0
answers
210
views
Calculate genus of reducible nodal curve
Let $C$ be be a connected reducible nodal curve over alg closed field $k$, such that all (finitely many) irred components $C_i$ of $C$ are smooth and intersections between different components are ...
- 6,038
36
votes
5
answers
4k
views
When has the scaffolding been more important than the completed building?
Niels Abel once said(1) of Gauss, "He is like the fox, who effaces his tracks in the sand with his tail." to which Gauss replied, "No self-respecting architect leaves the scaffolding in ...
8
votes
3
answers
696
views
Is "the purely probabilistic version of Freiling's axiom of symmetry" disprovable in ZFC?
I'm trying to pinpoint the "intuitive argument" for Freiling's Axiom of Symmetry. It's meant to be a "probabilistic" argument, so thinking about what seems to me to be the ...
- 708
2
votes
0
answers
163
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The existence of a positive Green function for the Laplacian on $\mathbb R$
One can show explicitly and easily that the function $G(x,y) = \frac 1 2 |x-y|$ is a positive Green function for the Laplacian $\frac {\mathrm d ^2} {\mathrm d x ^2}$ on $\mathbb R$ (endowed with the ...
- 5,407
0
votes
1
answer
185
views
Can we approximate a Hölder pdf by higher-order Hölder pdf's?
$\newcommand{\RR}{\mathbb R}\newcommand{\NN}{\mathbb N}$
Let $\alpha \in (0, 1)$ and $j \in \NN$. We denote by $H^{j + \alpha} := H^{j + \alpha} ({\RR}^d)$ the space of real-valued functions $f$ on $\...
- 825
2
votes
0
answers
135
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Does every faithful action on a scheme act freely on a dense open subset?
Disclaimer: I have asked this question on math exchange a week ago (here), but sadly to no avail. So I decided to escalate my question:
Let $G$ be a finite group acting faithfully on a smooth quasi-...
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1
vote
1
answer
184
views
Average distance between points of lower dimensional simplices in $\mathbb R^n$
Notation: By a simplex, we mean the convex hull of a finite set of distinct points in $\mathbb R^n$, which are called the vertices of the simplex. $\mathcal H^n$ will denote the $n$-dimensional ...
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1
vote
0
answers
210
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How to show that every Von Neumann algebra is unital?
I was reading the book on operator algebra by Kehe Zhu. The proof of theorem 17.7 (page 107) goes like this :
He first considered the set of all non-empty finite subsets of the set of all projections ...
- 173
5
votes
1
answer
590
views
On the Riemannian integrability of the bounded derivative
Let $f:[a,b]\to\mathbb R$ be a differentiable function with $f'$ bounded. According to this post, $f'$ is not necessarily Riemann integrable on $[a,b]$, see also Volterra's function.
I wonder, if $f'$...
- 299
1
vote
0
answers
116
views
Higman's lemma and well-quasi-ordering theory [closed]
Higman's Lemma is basic to well-quasi-ordering (WQO) theory, but has many specific forms, for example: the Cartesian product of two WQOs is a WQO. Any new extensions?
Usually proved by minimal bad ...
3
votes
1
answer
332
views
Sparse representation for continuous function?
I recently came across the field of "Sparse representation".
A talk is given here : https://www.youtube.com/watch?v=2bW4TkfTk-M.
The goal of sparse representation is taking a signal and ...
- 115
4
votes
0
answers
70
views
A question about the existence of surjective contractions
A few years ago I was doing some research in origami, and was motivated to as the following questions:
Consider $\mathbb{R}^2$ with the Euclidean metric and Lebesgue measure. Does there exist a ...
- 384
1
vote
1
answer
296
views
An example of non-invertible operator $F$ such that $P_nF$ is invertible on $\operatorname{Im}P_n$ or proving that It is impossible
Given:
$X$ - any Banach space
$F : X \to X$ (linear bounded and non-invertible)
$P_n$, which is projector that strongly converges to the identity operator $I$ as $n \to\infty$
Can you help me come ...
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4
votes
0
answers
222
views
Bochner theorem for (non-abelian) discrete groups
I am interested in Pontryagin duality-like theories for discrete groups, more particularly, whether an analogue to Bochner's theorem for abelian groups exists in the discrete non-finite and non-...
- 303
3
votes
1
answer
153
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Posets of equational theories of "bad quotients"
This is a follow-up to an older question of mine:
Suppose $\mathfrak{A}=(A;...)$ is an algebra (in the sense of universal algebra) and $E$ is an equivalence relation - not necessarily a congruence - ...
- 20.5k
2
votes
1
answer
335
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Hahn-Banach theorem and ultrafilter lemma
I'm unable to understand a remark in "Two application of the method of construction by ultrapowers to analysis" by Luxemburg, which uses the ultrafilter lemma to prove the Hahn-Banach ...
- 95
3
votes
1
answer
747
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Is $1 = \sum_{n=1}^{\infty} \frac{\pi(p_n^2)-n+2}{p_n^3-p_n}$ , where $\pi$ denotes the prime counting function and $p_n$ denotes the $n$-th prime?
Is
$$1 = \sum_{n=1}^{\infty} \frac{\pi(p_n^2)-n+2}{p_n^3-p_n},$$
where $\pi$ denotes the prime counting function and $p_n$ denotes the $n$-th prime?
Context:
This question came out as a result in ...
- 3,074
-1
votes
1
answer
294
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Derived algebraic geometry and Bridgeland stability conditions [closed]
In the context of derived algebraic geometry can someone elucidate the intricacies of Bridgeland stability conditions on derived categories of coherent sheaves? Where can one find examples involving ...
anon
0
votes
0
answers
155
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Embedding of the first Hirzebruch surface in $\mathbb{P}^4$ as a cubic surface
The first Hirzebruch surface (the blow-up of $\mathbb{P}^2$ at one point) is a projective toric surface that naturally embeds into $\mathbb{P}^4$ as a cubic surface (sometimes called the cubic scroll)....
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-1
votes
1
answer
235
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Is $\text{Sym}(\omega)/\text{(fin)}$ embeddable in $\text{Sym}(\omega)$? [duplicate]
Let $\omega$ denote the set of natural numbers, let $\text{Sym}(\omega)$ be the collection of bijections $\psi:\omega\to\omega$, and let $\text{(fin)}$ be the set of members of $\text{Sym}(\omega)$ ...
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6
votes
1
answer
392
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How to show that $\log 2(1/2\log 2\log 4 + 1/3\log 3\log 6 + \dotsb) + 1/2\log 2 - 1/3\log 3 + 1/4\log 4 - \dotsb = 1/\log 2$ [closed]
I've been studying Ramanujan's work and I stumbled upon this question in the book: Collected Papers of Srinivasa Ramanujan. In there I found question number 769 which is about an infinite sum with ...
2
votes
1
answer
138
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Solvability of derivation Lie algebras of local finite-dimensional commutative algebras
Let $A$ be a finite-dimensional local commutative algebra (with one) over a characteristic zero field $k$. Is it true that the Lie algebra $\operatorname{Der}_k(A)$ of $k$-derivations of $A$ is ...
- 18.4k
4
votes
1
answer
301
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Perturbation of Wasserstein distance: looking for references
I am doing readings related to Optimal transport which is new to me and I often encounter the following statement regarding a sort of derivative of the Wasserstein distance: $u$ and $v$ be two ...
- 1,101
4
votes
1
answer
192
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Gonality of specific Riemann surfaces $y^k=\tfrac{z^k-1}{z^k+1}$
The gonality of a compact Riemann surface $\Sigma$ is defined to be the lowest degree $d$ of a non-constant holomorphic map $f\colon \Sigma\to\mathbb CP^1.$ This means the gonality is 1 only for $\...
- 6,825
3
votes
0
answers
76
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Anisotropic kernel of groups of type A
I'm studying the results of classification of reductive groups using Tits index and anisotropic kernel.
It is known that simple groups with Tits index $^1 A_{n,r}^{(d)}$ are of the form $SL_{r+1}(D)$, ...
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10
votes
2
answers
548
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Wick rotation for Laplace and wave equations
I have seen Wick rotation used to describe the relationship between the heat and Schrodinger equations. That is, if $u(t,x)$ solves the heat equation then $v(t,x):=u(it,x)$ solves the Schrodinger ...
- 1,904
0
votes
1
answer
235
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If we don't care about uniqueness, can we relax the coercivity condition in Lax-Milgram theorem?
Let $(H, \langle \cdot, \cdot \rangle)$ be a real Hilbert space and $\|\cdot \|$ its induced norm. Let $a: H \times H \to \mathbb R$ be a bilinear form. We say that
$a$ is coercive IFF there is $C>...
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7
votes
1
answer
525
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Groups acting on infinite dimensional CAT(0) cube complex
I have seen many examples where a finitely generated infinite group acts properly/freely by isometry on finite dimensional CAT(0) cube complexes. Examples of such groups are discussed in many articles....
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2
votes
0
answers
74
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Arbitrary base change of a parahoric subgroup in split case
Assume $R\subset R'$ are henselien discretly valued rings with fraction field $K$ and $K'$, $G$ is a semisimple split group over $K$. Consider the parahoric group scheme $\mathcal{P}_F$ over $R$ ...
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6
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0
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386
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An imaginary disaster scenario - second order arithmetic is inconsistent
I think my question is a natural follow up of What would be some major consequences of the inconsistency of ZFC?
Regarding the later question, I agree with the commentaries that probably an ...
- 3,318
0
votes
1
answer
106
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Weak convergence to product measure form conditional convergence of marginals
$\newcommand\Ac{\mathcal A}$
$\newcommand\BL{\operatorname{BL}}$
$\newcommand\reals{\mathbb R}$
$\newcommand\eps{\varepsilon}$
$\newcommand\pr{\mathbb P}$
$\newcommand\ex{\mathbb E}$
$\newcommand\...
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0
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42
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Eigenvalues of minors to Schrodinger matrices
Suppose that we have a graph $G$, define the hamiltonian $H$ on it as $$Hu(x) = \sum_{y\sim x}u(y).$$ Consider the operator $H+V$ where $V$ multiplies the value $u(x)$ in any vertex by the potential ...
-1
votes
2
answers
367
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An equivalent of the axiom of choice? [closed]
There is such a thing as a math course for relatively non-mathematically inclined people that is intended to challenge students' intelligence more than to teach them some mathematics. (It is true that ...
47
votes
10
answers
6k
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What is the most "concrete-feeling" equivalent formulation of the Continuum Hypothesis that you can think of?
There are many equivalent formulations of the Continuum Hypothesis, but I think the most standard one is that
there is no infinite cardinality lying strictly between the cardinality of the natural ...
2
votes
0
answers
92
views
Construct a Bregman divergence from Wasserstein distance
I was wondering whether one has studied the Bregman divergence arising from a squared Wasserstein distance.
More precisely, let $\Omega\subset \mathbb{R}^d$ be a compact set and $c\in \Omega\times \...
- 503
3
votes
0
answers
51
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Reference for Cauchy-type identity
For polynomials $f,g \in \mathbb{C}[x_1,\dots,x_n]$, the following inner product appears frequently in the literature of harmonic polynomials:
$$
\langle f,g \rangle = f(\partial/\partial x_1, \dots, \...
- 1,989
2
votes
1
answer
113
views
Natural density of thickly syndetic set
A syndetic set $S$
is a subset of the natural numbers $\mathbb{N}$ or integers $\mathbb{Z}$, having the property of "bounded gaps": that the sizes of the gaps in the sequence of natural ...
2
votes
1
answer
156
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Some identities from graph theory and probability
The other day I attended a seminar about probability. I took some notes and I am now revising it and trying to understand some steps that were omitted by the lecturer. To formulate my question, ...
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