All Questions

Filter by
Sorted by
Tagged with
1 vote
0 answers
16 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 $K: \{(s,t): 0\le s\le t\}\to[0,1]$ ...
user avatar
  • 1,200
0 votes
0 answers
5 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 ...
user avatar
1 vote
0 answers
17 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 ...
user avatar
  • 145
0 votes
0 answers
39 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 ...
user avatar
0 votes
1 answer
34 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 $P(2m+1,2n,2k+1)$, $m,k\ge 0 , n \ge 1$ ?
user avatar
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: $$\...
user avatar
0 votes
1 answer
45 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 ...
user avatar
  • 8,107
1 vote
1 answer
70 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 $\gcd(a,d)=1$. ...
user avatar
0 votes
0 answers
31 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 ...
user avatar
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 ...
user avatar
  • 1,526
0 votes
1 answer
50 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?
user avatar
  • 11
0 votes
0 answers
21 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 ...
user avatar
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 ...
user avatar
10 votes
1 answer
156 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-...
user avatar
  • 101
6 votes
0 answers
83 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 ...
user avatar
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(...
user avatar
  • 8,262
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) ...
user avatar
  • 717
0 votes
0 answers
48 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 ...
user avatar
  • 1,741
1 vote
0 answers
46 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 ...
user avatar
2 votes
0 answers
24 views

Ratio between relative totient function and totient function

Define $\varphi(n,x)$ as the number of elements in the interval $[1,x]$ that is relatively prime to $n$. Is it true that there exists some constant $c_1,c_2>0$, such that if $x\geq c_1\log n$, then ...
user avatar
  • 573
0 votes
1 answer
29 views

Existence of a particular positive definite and radially unbounded function

Let $A \subset \mathbb{R}^{n}$ be a closed set. Does there exist an open set $O$ containing $A$, and a smooth function $f : O \to \mathbb{R} $ such that $f(x) = 0$ for all $x \in A$, $f(x) > 0$ and ...
user avatar
  • 169
3 votes
0 answers
77 views

Eriksson's thesis "Strongly convergent games and Coxeter groups"

The diamond lemma has recently come up in my teaching, and as always I've been looking for nice and simple applications. This has reminded me of the thesis Kimmo Eriksson, Strongly convergent games ...
user avatar
5 votes
0 answers
89 views

Square root in number field

I'm trying implement an algorithm that, for an element $b$ of a number field $\mathbb{Q}(\alpha)$, if it is a square in $\mathbb{Q}(\alpha)$ (i.e., $\exists x\in\mathbb{Q}(\alpha):x^2=b$), computes ...
user avatar
3 votes
0 answers
25 views

Isometric embedding of a 2-dimensional orbifold with constant curvature and three cone points

There are classical surfaces of revolution, shaped like footballs, that have constant positive curvature, except for their two cone points. How about such a surface with three cone points? To give ...
user avatar
  • 4,361
2 votes
0 answers
64 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 $\...
user avatar
  • 22.2k
4 votes
1 answer
59 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 \...
user avatar
  • 2,905
0 votes
0 answers
12 views

A two-dimensional variant of Bessel stochastic differential equation

Let $Z$ be a complex Brownian motion starting at $0$. The stochastic integral $$W = \int_0^t \frac{Z_s}{|Z_s|} \mathrm{d}Z_s.$$ yields a complex Brownian motion (starting at $0$). The natural ...
user avatar
1 vote
0 answers
27 views

$H^2(\partial \Omega)=L^2(\partial\Omega)$ closure of $A^{\infty}(\Omega)$ for $C^{\infty}$ bounded domain $\Omega$

Let $\Omega\subset\mathbb{C}^n$ be a $C^{\infty}$ bounded domain and $A^{\infty}(\Omega)=\mathcal{O}(\Omega)\cap C^{\infty}(\overline{\Omega}).$ I want to show that the Hardy space $H^2(\partial\...
user avatar
  • 73
1 vote
1 answer
43 views

Counting spanning trees of $K_{b+1,w+1}$ with certain properties or calculating a combinatorial sum

For $b,w \geq 0$ let $K_{b+1,w+1}$ be the complete bipartite graph with vertices $a_1,...,a_{b+1}$ on the left hand side and $c_1,...,c_{w+1}$ on the right hand side. For given $1 \leq d \leq w$ and $...
user avatar
  • 353
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} \&\|\...
user avatar
  • 111
6 votes
0 answers
79 views

Equivalent forms of Fourier restriction conjecture

this question is posted in mathstackexchange, but it seems that no one answers it. Sorry to the administrator if this question is not appropriate on Mathoverflow. I'm reading Pertti Maattila's book ...
user avatar
2 votes
1 answer
62 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 ...
user avatar
  • 1,551
1 vote
0 answers
24 views

3-Sasakian manifolds and contact Fano Kähler-Einstein manifolds

Let $(M,g)$ be a Riemannian manifold. The Riemannian cone of $M$ is $C(M) = M \times {\Bbb R}^{>0}$ with the metric $t^2 g + dt\otimes dt$. A manifold is called Sasakian if its cone is Kähler, ...
user avatar
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. $\lim_{n\to\infty }f_n(t)$ ...
user avatar
  • 185
7 votes
2 answers
268 views

How restrictive is having zero Chern numbers for a compact complex manifold ? Same for negative Chern number?

In complex dimension $2$, if a surface $S$ is a blowup of a surface $S'$, one has the following relation between their Chern numbers : $c_1^2(S) + 1 = c_1^2(S')$ $c_2(S) - 1 = c_2(S')$ By using this ...
user avatar
  • 71
3 votes
0 answers
66 views

Flat essential ring extensions

We call a ring extension (where $R$ and $S$ are commutative) $R \subset S$ essential if for every ideal $I$ of $S$ we have that $I \cap S \neq 0 \implies I \cap R \neq 0$. Suppose now that $R \subset ...
user avatar
  • 255
1 vote
1 answer
94 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) ...
user avatar
  • 161
1 vote
0 answers
41 views

Canonical basis of the invariant part of $O_q(\mathfrak g)^{\otimes N}$

Let $\mathfrak g$ be a semi-simple Lie algebra (We can assume $\mathfrak g=sl(n)$ for simplicity) and let $O_q(\mathfrak g)$ be the corresponding quantum algebra of functions. Then $O_q(\mathfrak g)^{\...
user avatar
  • 2,070
2 votes
0 answers
77 views

Vertex operator algebras and modular fusion categories

Let $\mathcal{V}$ be a vertex operator algebra (VOA), and let $\mathcal{C}=Rep(\mathcal{V})$ be the tensor category of (ususal) $\mathcal{V}$-modules. It is a well-known open-problem whether every ...
user avatar
-3 votes
0 answers
39 views

Division rings containing $\mathbb{C}$ of given degree

Let $N$ be any positive integer. Can the division rings (= skew fields) $D$ containing $\mathbb{C}$ for which $[ D : \mathbb{C} ] = N$ be classified ? For, example, what about $N = 3$ ? More generally,...
user avatar
  • 3,265
3 votes
0 answers
60 views

Confused about the proof of a lemma about deleted products

I am confused about the proof of Lemma 2.1 in the paper Obstructions to the imbedding of a complex in a euclidean space. I: The first obstruction, by A. Shapiro, Ann. Math. 66 (1957). Let $K$ be a ...
user avatar
0 votes
0 answers
44 views

Is definability in $V$ in $\sf Ack+MK$ expressible in its language?

Recall Ackermann set theory. If we extend Ackermann's set theory by adding all axioms of $\sf MK$ to it. We shall denote the universe of all elements by $W$, while $V$ is the primitive constant symbol ...
user avatar
3 votes
0 answers
136 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\...
user avatar
1 vote
0 answers
46 views

Tiling a rectangle with squares

Recently, the German science journal Spektrum put online a riddle about squares being tiled to a rectangle: The task was to determine the area of the rectangle tiled with $8$ squares, of which the ...
user avatar
2 votes
1 answer
76 views

Definite integral of power of sine ratio

I stumbled on the following rather appealing trigonometric definite integral, \begin{equation} \int_0^y \left(\frac{\sin x}{\sin (y-x)}\right)^a \mathrm{d}x = \pi \frac{\sin(ya)}{\sin(\pi a)} \end{...
user avatar
  • 3,120
2 votes
0 answers
46 views

Index-coclosure for monoidal categories, generalizing products and lextensive coproducts

I found a kind of monoidal structure that generalizes cartesian product and lextensive coproduct, and I'm wondering if anyone has seen it before and/or can tell me about it. I'm calling this structure ...
user avatar
  • 7,877
2 votes
0 answers
21 views

Transition maps between coordinate charts on the Grassmann manifold

Let $\mathbf{Gr}_{n,k}$ be the manifold of $k$-dimensional subspaces of $\mathbb{R}^n$, and let $\mathbf{col}$ be the map that takes a matrix in $\mathbb{R}^{n\times k}$ to its columnspace. The map \...
user avatar
4 votes
1 answer
271 views

Curves and semi-abelian varieties

Fix $k$ a number field, and let $C$ be a smooth geometrically integral affine curve over $k$. We can "associate" to $C$ a semi-abelian variety in the following way: One knows that $C$ is a ...
user avatar
9 votes
1 answer
307 views

Characteristic classes of non-linear sphere bundles

It is well known that the diffeomorphism group of there sphere $\operatorname{Diff}(S^n)$ has the homotopy type of a product $X:=O(n+1)\times \operatorname{Diff}_{\partial D^n}(D^n)$ of the orthogonal ...
user avatar
  • 5,706
2 votes
0 answers
80 views

Configurations of points in a spectrum

I am wondering if the following construction has appeared in the literature: Let $X$ be a pointed space. Define the singular configuration space of $X$, $F^*(X,k)$ ,to be $\{(x_1,\dots,x_k)| x_i=x_j \...
user avatar
  • 3,254

15 30 50 per page
1
2 3 4 5
2734