# All Questions

**7**

votes

**0**answers

35 views

### Why should intersection cohomology and quantum cohomology be related for a symplectic resolution?

In http://arxiv.org/pdf/1410.6240.pdf M. McBreen and N. Proudfoot conjectured a precise relationship between the quantum cohomology of a symplectic resolution and the intersection cohomology of the ...

**1**

vote

**0**answers

11 views

### Multivariate analogue of Jackson's inequality's modulus of continuity form

According to Jackson's inequality, there is $c > 0$ s.t. for any continuous function $f: S^1 \rightarrow \mathbb{R}$ (where $S^1 := \mathbb{R} / \mathbb{Z}$ is the circle) and integer $n$ there is ...

**1**

vote

**1**answer

37 views

### Minimal Support Solutions of a Linear System (Dissertation)

For a given $n \times m$ matrix A with $m>>n$ and a given vector $\vec b \in \mathbb{F}^{n \times 1}$, and given that $A\vec{x}=\vec{b}$ for at least one $\vec{x} \in \mathbb{F}^{m \times ...

**-1**

votes

**0**answers

36 views

### Twisted Hodge numbers in a family

It is well known (e.g. Voisin's book) that for a smooth family $\pi: \mathcal{X} \to B$ of smooth projective varieties (and projectively normal) over $\mathbb{C}$, the Hodge numbers $h^{p,q}(X_b)$ are ...

**0**

votes

**0**answers

29 views

### Additional condition to the Bollobas theorem in extremal set theory

The Bollobas'1965 theorem is the following:
If $A_1,...,A_n$ and $B_1,...,B_n$ are two sequences of subsets of $X=\{1,...,r\}$ such that $A_i\cap B_j = \emptyset$ if and only if $i=j$, then
...

**1**

vote

**0**answers

50 views

### Calabi-Yau theorem on Arithmetic Variety

Let $\mathcal X\to Spec(\mathcal O_K)=C$ be an arithmetric projective variety over $C$ , where $\mathcal O_K$, ring of number filed $K$. Let $\omega$ be a Kaehler current of $\mathcal X(\mathbb C)$. ...

**1**

vote

**0**answers

11 views

### What is the isomorphism type of the ring of invariants of the matrix algebra under the Klein four group?

If the group $G=\mathbb{Z}/2\mathbb{Z}$ acts on $\mathbb{C}[x]$ via $x\mapsto -x$, then we have $\mathbb{C}[x]^G=\mathbb{C}[x^2]$. If $G$ acts on $A=M_2(\mathbb{C}[x])$ via $\begin{bmatrix}a(x) & ...

**1**

vote

**0**answers

12 views

### Uniform convergence of long geodesic to Liouville measure

Here is the set up : let $(S,g)$ an hyperbolic surface and $L_g$ the associated volume measure. By the shadowing lemma there exist sequences of long closed geodesics, $\gamma_n$ which approximate the ...

**1**

vote

**0**answers

47 views

### A generalized ellipse

We know that an ellipse is the locus of all point $z$ in the plane with $$|z-a|+|z-b|=\lambda$$
where $a,b$ are two given points in the plane and $\lambda$ is a constant.
Now we consider the ...

**4**

votes

**1**answer

64 views

### Schur multiplier of $Sp(2g, \mathbb{Z}_2)$ for $g \geq 3$

This question is about the computation of $H_2(Sp(2g, \mathbb{Z}_2), \mathbb{Z})$, where $Sp(2g, \mathbb{Z}_2)$ is the group of symplectic $2g \times 2g$ matrices over $\mathbb{Z}_2$.
With respect to ...

**2**

votes

**1**answer

66 views

### Topology in space of test functions $\mathcal{D}(\Omega)$ and space of distributions $\mathcal{D}'(\Omega)$

We can concluded that $\mathcal{D}(\Omega):=\bigcup_{K \in \mathcal{K}(\Omega)} \mathcal{D}_K(\Omega)$ (where $\mathcal{K}(\Omega)$ denotes the union of all compacts set content in a open subset ...

**0**

votes

**0**answers

39 views

### Continuation of an almost periodic sequence on the Bohr group

A sequence $x={(x_k)}_{k \in \mathbb{Z}}$ of real numbers is a uniformly continuous function on the discrete topological group $\mathbb{Z}$. Therefore it admits a unique continuous extension $\bar x$ ...

**5**

votes

**0**answers

32 views

### General properties of the Ruelle operator

Recently I have read Parry and Pollicott's book, Zeta functions and the periodic orbit structure of hyperbolic dynamics.
I have been interested in some technical properties of the ...

**0**

votes

**0**answers

4 views

### Understanding the steps taken in a calculation of the maximum profile likelihood of a simple ODE, given some data

I'm trying to understand a calculation made in a paper (section 2 from the supplementary contents of ...

**11**

votes

**1**answer

517 views

### Is deciding if one planar graph is dual to another really NP-hard (Wikipedia claim)?

Wikipedia claims (permanent link) without reference:
Testing whether one planar graph is dual to another is NP-complete.
Another claim with reference:
For any plane graph G, the medial graph ...

**2**

votes

**0**answers

26 views

### Local product structure of determinantal variety

The variety $X_n$ of singular $n\times n$ real matrices is stratified by smooth strata $X_{n,k}$ where $k$ is the rank. Choose a rank $k$ matrix $A\in X_{n,k}$. Is there a local diffeomorphism sending ...

**1**

vote

**0**answers

12 views

### What is the topology of teichmuller space of a given hyperhahler manifold

From verbitsky's paper, I read that the teichmuller space of a given hyperkahler manifold is not hausdorff but it's smooth. I am a bit confused about how to put a natural topology on the teichmuller ...

**1**

vote

**1**answer

80 views

### Duality argument to get $L^\infty-L^2$ inequality

In page 79 of Davies's book on Heat Kernels and spectral theory, the author proves that
$$\lVert e^{-Ht}f \rVert_2 \leq c_1t^{-\mu/ 4}\lVert f \rVert_1$$
where the norms are $L^p$ norms. He states
...

**5**

votes

**1**answer

139 views

### $K$ theory and singular cohomology

For cell complexes${}^1$ $X$ we have an isomorphism
$$
K^*(X)\otimes \mathbb{Q}\cong H^{*}(X;\mathbb{Q}),
$$
which is induced by the Chern character.
What is the analogous statement for $KO(X)$?
...

**1**

vote

**2**answers

37 views

### Sum of subsequence, over index set of non-zero density, of monotone divergent sum also divergent?

For a subset $S$ of the natural numbers $N$ and $n\in N$ let $|S\cap n|$ be the number of members of $S$ that are less than $n$. Suppose $S$ does not have upper asymptotic density $0$. That is, ...

**7**

votes

**0**answers

31 views

### Holomorphic vector fields on compact complex manifolds with trivial canonical bundle

Let $M$ be a compact complex manifold whose canonical bundle $K_M$ is holomorphically trivial. Is it possible for $M$ to admit a non-zero holomorphic vector field with zeroes? Equivalently, using a ...

**2**

votes

**0**answers

59 views

### Lefschetz on étale fundamental group for quasi-projective varieties

If $X$ is a smooth projective variety of dimension at least $3$ over $\mathbb{C}$, Lefschetz's Hyperplane theorem says that for every hyperplane section $H$
$$\pi^1(H)\to\pi^1(X)$$
is an isomorphism, ...

**1**

vote

**0**answers

16 views

### Boundary of pseudospectra

Suppose:
$B_i \in \mathbb{C}^{n \times n}$, $0<w_i\in \mathbb{R}$ $(i = 0,1,2,\ldots,m)$
${\rm P}(x) ={\rm{B}_m} x ^m + \cdots + B_1 x + B_0$ is a matrix polynomial, and $x $ is a complex ...

**0**

votes

**0**answers

24 views

### to find topological properties under a metric on a set [on hold]

we define a metric d on a set of composition operators on L2. I would like to find connected component and path connected component and other topological properties by d . Is there any book or paper ...

**0**

votes

**0**answers

42 views

### Exercise 5, chapter 4 in Do Carmo's “Riemannian Geometry” [on hold]

This is probably a silly question and maybe in the end the answer is trivial but I can't see it.
The problem is the following.
Let $M$ be a Riemannian manifold, $\gamma \colon[0,l]\to M$ be a ...

**2**

votes

**0**answers

77 views

### The homology of $\varinjlim SO(p,q)$

Is there a way to explicitly compute the homology of the space
$$
\varinjlim_{(p,q)} SO(p,q)^+,
$$
where each $SO(p,q)$ is the indefinite special orthogonal group, and $SO(p,q)^+$ its identity ...

**0**

votes

**1**answer

47 views

### About $c(A)$ in $c(A)|A|\leq |A^{-1}A|$

Let $G$ be a finite group, $\emptyset\neq A\subseteq G$, $A^{-1}:=\{ a^{-1}:a\in A\}$, and put $$c(A):=\max\{t\in \mathbb{Z}: t|A|\leq |A^{-1}A|\}$$
It is clear that $1\leq c(A)\leq ...

**-3**

votes

**0**answers

39 views

### Negative Arithmetic Line bundle [on hold]

Let $X$ be an arithmetic variety and $L$ be an Arithmetic Line bundle, then how can we define a negative Arithmetic line bundle ?

**1**

vote

**0**answers

32 views

### How to solve the following bivariate recurrence?

$$F(n,r) = (1-w(r))F(n-1, r) + w(r-1)F(n-1, r-1)$$
where $w(r)$ is monotonically non-increasing in $r$ and $0 \leq w(r) \leq 1$ with $0 \leq r$
Initial condition:
\begin{eqnarray}
F(0, r) & = ...

**4**

votes

**1**answer

115 views

### Uniformizer for splitting field of p^{1/p^n} over p-adics

Does anyone know an explicit uniformizer for $\mathbf{Q}_p(\zeta_{p^n}, p^{\frac{1}{p^n}}) / \mathbf{Q}_p$? I was reading the question "adding an n-th root to Q_p" where dke mentions this question but ...

**7**

votes

**1**answer

75 views

### Admissibility in Heegaard Floer, especially with torsion Spin^c structures

I'm confused about the relationship between strong admissibility and weak admissibility for pointed diagrams in Heegaard Floer theory. For reference, here are Ozsváth-Szàbo's original definitions:
...

**3**

votes

**0**answers

51 views

### Numerical and topological density

Let $\mathbb{N}$ denote the set of positive integers, and let's say that $A\subseteq \mathbb{N}$ is numerically dense if $$\text{lim inf}_{n\to\infty}\frac{|A\cap\{1,\ldots,n\}|}{n} = 1.$$
Is there a ...

**3**

votes

**1**answer

122 views

### Convergence of a triple sum involving the imaginary part of the Riemann zeta function's non trivial zeros

Let $N>0$ an integer, $k>0$ a real parameter and let $\rho = \beta +i \gamma$ a non trivial zero of the Riemann zeta function. For a work I need to find the best possible $k$ such that ...

**0**

votes

**0**answers

71 views

### What is the upper-bound for this?

I am looking at a paper and am trying to understand how this bound was driven. The first part is clear, but not sure how you can extend it to the second part. So here is the first part:
Assume
...

**4**

votes

**1**answer

126 views

### A solution for this equation with a certain condition

Let $(S^n(1),g)$ be the round sphere and $J_{\delta , \beta}$ be an almost complex structure on $TS^n(1)$ with the definition
\begin{equation}
J_{\delta , \beta}(X^h)=\beta X^h + \alpha X^v,\\
...

**-1**

votes

**0**answers

13 views

### weakly p- summable sequence

Let $ (x_{n}) $ a weakly $ p- $ summable sequence in $ X $ and $ ( x^{\ast}_{n})$ a weakly null sequence in $ X^{\ast} $ and $ i_{n} : Y_{n}\rightarrow X$ be the natural injection and $ p_{n} : ...

**2**

votes

**0**answers

44 views

### Darboux-like coordinates on a Kähler manifold

If $(M, g, J, \Omega)$ is a Kähler manifold, do there exist local coordinates in which 2 out of the 3 geometrical structures look nice? I have Darboux coordinates in which $\Omega$ looks nice, but ...

**5**

votes

**1**answer

101 views

### The unique positive real root of summation function

update: add one condition according to answer below.
I post this question in MSE a week ago. I thought this should be an easy freshman exercise, but it turns out not easy...
The original question ...

**9**

votes

**3**answers

457 views

### Collection of dense subsets as a “fingerprint” for Hausdorff topologies?

Let $(X,\tau)$ be a Hausdorff space and let ${\cal D}$ denote the collection of dense subsets of $(X,\tau)$. Is it possible that there is another Hausdorff topology $\tau_1 \neq \tau$ on $X$ such that ...

**2**

votes

**0**answers

79 views

### Degree formalism for line bundles on Deligne-Mumford stacks

Let $k$ be an algebraically closed field and let $\mathcal{C}$ be proper, Cohen-Macaulay, purely $1$-dimensional Deligne-Mumford stack over $k$. From looking at section 4.3 on page 135 of the paper ...

**2**

votes

**0**answers

136 views

### Groups with isomorphic quotients [on hold]

Assume we have a finitely presented group $G$ and a non-trivial normal subgroup N. How can one decide that $G/N$ is isomorphic to $G$ or not? $G$ is given as a presentation and $N$ as a set of words.

**20**

votes

**0**answers

183 views

### What is the “real” meaning of the $\hat A$ class (or the Todd class)?

In the Atiyah-Singer index theorem as well as in the Grothendieck-Riemann-Roch theorem, one encounters either the $\hat A$-class or the Todd class, depending on the context. I want to focus on the ...

**1**

vote

**0**answers

51 views

### Smooth points of the secant variety with a given tangent space [on hold]

Let $X\subseteq\mathbb{P}^{N}$ be an algebraic variety of dimension $n$. Let $(x,y)\in X\times X-\Delta_{X}$ and $z\in\langle x,y\rangle\subseteq SX$, where $SX$ is the secant variety of $X$. I want ...

**2**

votes

**0**answers

108 views

### About the limitation by size

This could be a big post, so I'll try to summarize my thoughts and divide them into several questions.
When working in category theory, I used to choose the following definition. A category $C$ is ...

**5**

votes

**1**answer

105 views

### Simply connected noncompact surfaces

Is there a theorem saying that any noncompact, simply connected topological surface is homeomorphic to the plane ? There seems to be many well-known results about the classification of compact ...

**1**

vote

**1**answer

30 views

### Relative local compactness for locales?

I am looking for informations on the relative version of local compactness for locales:
If $f:X \rightarrow Y$ is a morphism of locales I want to say that $f$ is relatively locally compact if ...

**6**

votes

**4**answers

396 views

### Direct axiomatization of ordinal and cardinal numbers

Again, this question is related (**) to a previous one:
in standard books on basic set theory, after stating the axioms of ZFC, ordinal numbers are introduced early on. Afterwards cardinals appear: ...

**0**

votes

**0**answers

14 views

### Properties of a map regarding the space of invariant probability measures for controlled Markov process

Let us consider a controlled Markov process with the transition kernel $p(dy|x,\theta)$ ($\theta$ being the control parameter. Now, consider the map
$\theta \to I(\theta)$ where $I(\theta)$ is the ...

**2**

votes

**0**answers

90 views

### Deligne-Simpson problem for classical groups

Additive Deligne-Simpson problem was partially prooved by Kostov. Also there is Crawley-Boevey's approach to the question. The problem is about existence of a solution of the equation
$$
A_1 +...+A_n ...

**4**

votes

**2**answers

76 views

### Does the truncated Hausdorff moment problem admit absolutely continuous solutions?

Let $\mu$ be a (Borel) probability measure on $[0,1]$ and define $m_j(\mu) = \int x^j\,\mu(dx)$. Let $k$ be a positive integer and consider the set $\mathcal C_{\mu,k}$ of probability measures $\nu$ ...