# All Questions

136,660 questions
Filter by
Sorted by
Tagged with
1 vote
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 $K: \{(s,t): 0\le s\le t\}\to[0,1]$ ...
• 1,200
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 ...
1 vote
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 ...
• 145
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 ...
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 $P(2m+1,2n,2k+1)$, $m,k\ge 0 , n \ge 1$ ?
22 views

• 111
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 ...
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 ...
• 1,551
1 vote
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, ...
• 7,810
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. $\lim_{n\to\infty }f_n(t)$ ...
• 185
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 ...
• 71
66 views

• 2,070
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 ...
• 23.2k
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,...
• 3,265
60 views

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 ...
• 9,844
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 ...
• 5,318
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\...
• 31
1 vote
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 ...
76 views

### Definite integral of power of sine ratio

I stumbled on the following rather appealing trigonometric definite integral, \int_0^y \left(\frac{\sin x}{\sin (y-x)}\right)^a \mathrm{d}x = \pi \frac{\sin(ya)}{\sin(\pi a)} \end{...
• 3,120
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 ...
• 7,877
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 \...
• 21
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 ...
• 491
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 ...
• 5,706
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 \...