All Questions
136,660
questions
6
votes
1
answer
725
views
Computer power in plane geometry
I often hear that modern computer programs "may prove any theorem in elementary Euclidean geometry". Of course, as stated it is false - say, they can not prove theorems about $n$-gons for ...
- 4,475
5
votes
4
answers
2k
views
Probability of random permutation having certain cycles
Are there any good references (either books or on-line) on the subject of the distribution of various cycle properties amongst permutations, particularly ones containing exact, closed-forms?
For ...
- 4,475
3
votes
2
answers
640
views
Paper about Sasaki-Einstein manifolds
can you give me a good paper (in the sense of a simple introduction) about Sasaki-Einstein manifolds?
Thank you and best regards
Florian M.
- 4,475
1
vote
1
answer
99
views
Extension of the projective norm to a cross norm
Let $\mathcal{H}$ be a finite-dimensional Hilbert space and $\mathcal{A}\subseteq\mathcal{B}(\mathcal{H})$ be an operator system. Is it possible to extend the projective norm (the greatest cross norm) ...
- 17.3k
1
vote
0
answers
18
views
On an integral equation of Volterra type
Consider the following integral equation
$$f(t) = A(t) + \int_0^t K(s,t)f'(s)ds,\quad \forall t\ge 0,\quad\quad\quad (\ast)$$
where $A: \mathbb R_+\to [0,1]$ and $K: \{(s,t): 0\le s\le t\}\to[0,1]$ ...
- 1,200
3
votes
0
answers
69
views
Periodic objects in Frobenius categories
Let $A$ be a finite dimensional Gorenstein algebra and $C$ the stable module category of $A$.
Question: Does there always exist an indecomposable periodic object $X$ in C, that is an object with $\...
- 22.2k
0
votes
0
answers
6
views
A non-Kolmogorov system with Lebesgue spectrum: New examples?
It is known that a Kolmogorov system has Lebesgue spectrum, while not every system with Lebesgue spectrum is Kolmogorov. Some of the examples of the latter case are mentioned in Example 9.5.12 of the ...
1
vote
0
answers
18
views
Does $\mathscr{H}^{d-1} (A)<+\infty$ for $A\subset \mathbb R^d$ imply $A$ is (Borel) measurable?
I'm reading section 1.3.1 The quadratic case in $\mathbb{R}^{d}$ at page 17 from Santambrogio's Optimal Transport for Applied Mathematicians. The PDF is freely available from here.
Let $\mu$ be a ...
- 145
0
votes
1
answer
167
views
Maximum number of colors for an optimal tiling which guarantees infinite paths
This question is a more applicable version of the question I've asked in mathexchange recently:
What is the maximum number of colors we can use to color $N^2$ square sub-tiles of $N×N$ square
block ...
- 3,012
5
votes
2
answers
99
views
+100
Hamiltonian, energy, and conservation laws of nonlinear PDEs
In many PDEs, I see the papers mention the energy of the PDE. And some papers and books mention Hamiltonians. I know that integrable systems have infinitely many conservation laws and these laws are ...
- 47.5k
4
votes
1
answer
60
views
On frequency decay of an integral transform of a function
Suppose $f \in C^{\infty}_c((-1,1))$ and assume that there exists constants $a,b>0$ such that
$$
\bigg|\int_{\mathbb R} f(t) \,e^{\tau t^2+i\tau t}\,dt\bigg| \leq a\,e^{-b|\tau|},$$
for all $\tau \...
2
votes
1
answer
64
views
Enlargement of filtration
Let $M_t$ be a continuous time real valued martingale, and $\mathcal F_t$ its natural filtration.
Suppose that $\mathcal F_t \setminus \mathcal F_s$ is nonempty for all $t > s$.
Let $\mathcal G$ be ...
0
votes
0
answers
41
views
Complete proof about Penrose tilings
It is well known that equivalence classes of Penrose tilings (say, by semikites and semidarts) are in bijection with binary sequences not containing 11 and modulo tail equivalence. However, I couldn't ...
2
votes
1
answer
139
views
Vanishing cycles exact sequence for degeneration of curves
Let $f : X\rightarrow D$ be a family of smooth projective curves over the complex unit disk $D$, degenerating to a nodal curve $X_0$ above $0\in D$.
Let $\eta\in D - \{0\}$ be a general point, and let ...
- 1,033
1
vote
1
answer
35
views
Alexander polynomial of the pretzel knot $P(2m+1,2n,2k+1)$
Is there a (closed) formula for the Alexander polynomial of the pretzel knot $P(2m+1,2n,2k+1)$, $m,k\ge 0 , n \ge 1$ ?
- 55
13
votes
3
answers
1k
views
Stable homology of arithmetic groups
$\DeclareMathOperator\SL{SL}\DeclareMathOperator\Sp{Sp}$Suppose that $F/Q$ is a number field.
Using automorphic forms, Borel computed the ($R$-) stable cohomology of $\SL_n(O_F)$, and as a result, ...
- 4,475
1
vote
1
answer
72
views
Infinitely many primes that split completely in an arithmetic progression
Let $d \geq 1$ be an integer. Dirichlet's theorem on arithmetic progression implies that the arithmetic progression $a, a+d, a+2d, \ldots$ contains infinitely many primes if and only if $\gcd(a,d)=1$.
...
- 41.8k
0
votes
0
answers
22
views
Problems about Chern-Yamabe flow
I'm reading a paper about Chern-Yamabe flow.
Chern-Yamabe flow
which is about the Chern-Yamabe problem in Chern-Yamabe problem
Here $\Delta$ is $\Delta=-\nabla^{\mu} \nabla_{\mu}$.
The flow is:
$$\...
- 329
15
votes
2
answers
3k
views
Question about functional derivatives
This page on Wikipedia defines the so-called functional derivative as follows: "Given a manifold $M$ representing (continuous/smooth) functions $\rho$ (with certain boundary conditions, etc.) and a ...
- 4,477
0
votes
1
answer
46
views
How to complete $i^*i_*\mathcal{F}\to \mathcal{F}$ into an exact triangle for a smooth divisor $i: X\hookrightarrow Y$?
Let $Y$ be a smooth algebraic variety and $i: X\hookrightarrow Y$ be a smooth divisor. We consider the derived functors $i^*: D^b_{coh}(Y)\to D^b_{coh}(X)$ and $i_*: D^b_{coh}(X)\to D^b_{coh}(Y)$. By ...
- 31.7k
1
vote
1
answer
44
views
Semi-orthogonal decomposition of Verra threefold
Let $X$ be a Verre-threefold, which is by definition a $(2,2)$ hypersurface in $\mathbb{P}^2\times\mathbb{P}^2$, it is a Fano threefold. What is the semi-orthogonal decomposition of $D^b(X)$? It ...
- 31.7k
5
votes
1
answer
344
views
Polynomial size embeddings of toric varieties from polytopes?
Background: Let $P$ be a integral polytope, and $X_P$ the toric variety associated to the normal fan.
$X_P$ is always projective, because the collection of characters corresponding to the points $\...
CommunityBot
- 1
0
votes
0
answers
33
views
Asking for reference about a relation related to Fourier transform
Sorry for the not-perfect question. I am asking for a reference for the following relation:
$$\int f . g. h ...= \int_{\xi_1 +\xi_2 +...=0} \hat{f}(\xi_1) \hat{g}(\xi_2)... d\xi_1 d\xi_2...$$
Could ...
- 229
31
votes
3
answers
6k
views
What are the applications of operator algebras to other areas?
Question: What are the applications of operator algebras to other areas?
More precisely, I would like to know the results in mathematical areas outside of operator algebras which were proved by ...
- 1,497
10
votes
1
answer
164
views
Fixed-point free diffeomorphisms of surfaces fixing no homology classes
One of my graduate students asked me the following question, and I can't seem to answer it. Let $\Sigma_g$ denote a compact oriented genus $g$ surface. For which $g$ does there exist an orientation-...
- 48k
0
votes
1
answer
52
views
Number of ways to write a finite set of cardinality n as the union of r distinct binary subsets
I want to know the number of ways to write a finite set of cardinality $n$ as the union of $r$ distinct two-element subsets. Is there a nice formula in binomial coefficients?
- 19k
0
votes
0
answers
46
views
A Newton identity and the primes--the Faber partition polynomials and modular arithmetic
Dress and Siebeneicher in their tale of the Burnside family express an opinion (1.2) that, if I read it correctly, leads me to believe that the classic Faber partition polynomials which satisfy
$\ln[A(...
- 8,262
1
vote
0
answers
61
views
Sudakov's lower bound type inequality for supremum of Chi-squared random variables
Let $\varepsilon$ be $n$-dimensional standard Gaussian veector, i.e., $\varepsilon \sim N_n(0, I_n)$. Let $\mathcal{P}$ be a subset of symmetric projection matrices in $\mathbb{R}^{n \times n}$ with $|...
- 197
2
votes
1
answer
88
views
Anti-Takagi: Given a Hermitian matrix $M$, is there a canonical form under $P \mapsto PMP^*$ where $PP^T = I$?
The Takagi decomposition provides a canonical form for a complex symmetric matrix $S$ under $U \mapsto USU^T$ where $UU^* = I$.
Question: Is there an anti-Takagi decomposition? I.e. Is there a ...
CommunityBot
- 1
7
votes
2
answers
492
views
Does the discriminant of an irreducible polynomial of a fixed degree determine the discriminant of the number field it generates?
In the quadratic case, it does. Given an irreducible quadratic polynomial $f(x)=ax^2+bx+c$, the discriminant of the quadratic number field $\frac{\mathbb{Q}[x]}{f(x)}$ is $\operatorname{sqf}(d)$ or $4\...
- 2,368
2
votes
0
answers
40
views
Orthonormal eigenspinors of the gauge-covariant dirac operator on $\mathbb{R}^4$, with extra conditions are possible?
Let $G$ be a simple Lie group, and $V$ a representation.
Consider $\mathbb{R}^4$ with its flat Euclidean metric. Let $P$ be the trivial $G$-bundle on $\mathbb R^4$ equipped with some (non-trivial) ...
- 40.3k
0
votes
0
answers
22
views
Question regarding properties of map which produces measures that are invariant to orthogonal rotation
Let $\mathcal{M}_1$ denote the set of probability measures on the unit ball in $\mathbb{R}^d$ (which comes with its Borel $\sigma$-field). Denote by $\sigma$ the uniform measure on the orthogonal ...
- 245
1
vote
0
answers
46
views
Some $p$-adic congruences involving permutations
Motivated by my study of determinants and permanents, here I present several conjectures on $p$-adic congruences involving permutations.
As usual, we let $S_n$ be the symmetric group consisting of all ...
- 13k
2
votes
1
answer
171
views
Game on groups (generalization of spinning switches puzzle)
Alice and Bob are playing a game as follows:
Initially
There're two subgroups $A,B$ of Sym(n) known to both Alice and Bob
There're $n$ slots $S_1, \cdots, S_n$ and $n$ boxes $B_1, \cdots, B_n$. ...
CommunityBot
- 1
7
votes
3
answers
711
views
Is there a "geometric" language that describes the equivalence groupoid of a foliated manifold?
Sitting on the couch in my office is a certain groupoid. It's waiting for me to say something to it. My problem is that I don't know its language. My question here is for some suggestions.
Here, ...
- 31.2k
6
votes
0
answers
84
views
The distribution of certain Galois groups
Let $f(x)$ be a polynomial of degree $d$ with integer coefficients. Let $G_p^+$ be the Galois group of the polynomial $f(x)-y$ over $\overline{\mathbb{F}}_p(y)$ and $G_p$ be the Galois group of the ...
- 5,269
2
votes
1
answer
71
views
Does pointwise convergence yield the convergence under Skorokhod topology?
Let $D_+$ be the set of non-increasing functions $f: [0,T]\to [0,1]$ that are right-continuous. Let $(f_n)_{n\ge 1}\subset D_+$ be a sequence of continuous functions s.t. $\lim_{n\to\infty }f_n(t)$ ...
- 78.4k
31
votes
1
answer
2k
views
Connes's absolute geometry and Lurie's spectral algebraic geometry
Alain Connes and Caterina Consani seem to be currently working on "absolute algebraic geometry", which is a kind of "algebraic geometry over the sphere spectrum" (https://arxiv.org/...
- 9,277
0
votes
0
answers
49
views
Automorphism group and independent sets in vertex-transitive graph
Suppose a graph $G$ is vertex-transitive. Then, is there any relation, or better how can the automorphism group $\operatorname{Aut}(G)$ aid in the computation of independent sets of $G$.
I hope the ...
- 51.7k
2
votes
1
answer
59
views
A compact family of holomorphic functions and their corresponding compact ranges?
Let $C = [0,1]^n \subset \mathbb{R}^n$ be the closed unit cube. For some open set $V \subset \mathbb{C}^n$ such that $C \subset V$, denote by $\mathcal{F}$ some compact family of holomorphic functions ...
- 481
3
votes
0
answers
88
views
Faithfulness of parabolic induction
I've only recently begun to study the representation theory of $p$-adic groups, so the following question might be quite silly.
Let $F$ be a non-archimedean local field of residue characteristic $p$, $...
- 131
12
votes
5
answers
5k
views
Number of spanning forests in a graph
Hello,
I have two questions that have been bugging me recently. The first is about the number of spanning forests in a graph and the second is about enumerating these with edge labels.
Q1: I am ...
- 2,015
1
vote
0
answers
28
views
How can I create a cover for H's weight space?
$$
\mathcal{B}:=\left\{\mathrm{B}_{\boldsymbol{w}} \mid \mathrm{B}: \mathbb{R}^{d_{1}} \rightarrow \mathbb{R}^{q}, \operatorname{Lip}\left(\mathrm{B}_{\boldsymbol{w}}\right) \leq L_{B} \&\|\...
- 111
8
votes
1
answer
456
views
Existence of a vector field with a finite number of limit cycles.
The following question is related to the Seifert conjecture.
Let $M$ be a closed manifold with zero Euler characteristic. Is it true that each homotopy class of nowhere-zero vector fields on $M$ ...
- 481
10
votes
0
answers
182
views
Surprisingly only real points on intersection of certains quadrics
Let $G$ be a finite group and let $X_g$ be variables indexed by $G$. Consider the complex algebraic set defined by
\begin{align}
X_e &= 0\\
X_g &= X_{g^{-1}}\;\;\text{ for all }g\in G,\\
X_g &...
- 16.1k
4
votes
1
answer
231
views
Quadratic extensions of cyclotomic numbers by absolute values of elements
Summary
I was wondering whether there is an explicit description of the extension $\mathbb{Q}(\zeta_n, |z|)/\mathbb{Q}$ obtained from a cyclotomic field $\mathbb{Q}(\zeta_n)$ by adjoining any finite ...
CommunityBot
- 1
3
votes
0
answers
137
views
Bailey's lemma in number theory
A pair of sequences $(α_n,β_n)$ is called a Bailey pair if they are related by
$$\beta_n=\sum_{r=0}^n\frac{\alpha_r}{(q;q)_{n-r}(aq;q)_{n+r}}$$
or equivalently
$$\alpha_n = (1-aq^{2n})\sum_{j=0}^n\...
- 31
2
votes
0
answers
93
views
Compactness of a nonlinear operator
Let $H^{1}_{0}(0;\pi)=\{f\in L^{2}(0; \pi): f^{\prime}\in L^{2}(0; \pi)\ \text{and}\ f(0)=f(\pi)=0 \} .$ equipped with the following norm $$\|f\|=\Big(\int_{0}^{\pi}|f'(x)|^2dx \Big)^{\frac{1}{2}}$$
...
- 27
1
vote
0
answers
47
views
Ramsey-theoretic properties of Erdős cardinals
The $\beta$-Erdős cardinal is defined as the least ordinal $\eta$ such that $\eta \to (\beta)^{\lt \omega}_2$, that is for every function $f: [\eta]^{\lt \omega} \to 2$, there is an $f$-homogeneous ...
- 581
5
votes
1
answer
174
views
Points of differentiability of squared distance from a point in metric spaces
I posted this same question on MSE with no answer.
Let $I:=(0, + \infty)$ and let $(X,d)$ be a complete and separable metric space. In this setting we say that $u : I \to X$ is absolutely continuous ...
- 16.7k