**3**

votes

**2**answers

133 views

### Triangles in rigid Riemann surfaces

Edit: We thank Vladimir Matveev for his comment on this post which leeds us to revise the question as follows:
Assume that $M_{g}$ is a compact Riemann surface with constant negative cuvature (That ...

**0**

votes

**1**answer

87 views

### section of reduced structure map

Let $R$ be a commutative ring whose characteristic is either prime or $0$, such that $R/N$ is an integral domain, where $N$ is the nilradical, and $p: R \rightarrow R/N$ the canonical map. Is there a ...

**0**

votes

**0**answers

44 views

### Stabilizer subgroup in adjoint action [migrated]

Given $b \in \mathfrak{su}(n)$, how can I find the stabilizer $\text{stab}(b)$ for the adjoint action of $SU(n)$ on $\mathfrak{su}(n)$ given by $Ad_U(b) = UbU^{\dagger}$ without using coordinates? The ...

**5**

votes

**1**answer

329 views

### Loop space generalization

Let $X$ be a based connected space. The space of based continuous morphisms $Top_{\ast}(S^1,X)$
is the space of loops $\Omega X$. Since $S^1$ is homotopy equivalent to the Eilenberg-Mac Lane Space ...

**3**

votes

**0**answers

49 views

### Dimension reduction for low-order moments of Rademacher-weighted sums of vectors

Let $x_1,\dots,x_n$ be vectors in a Euclidean space $H$. Let $\varepsilon_1,\dots,\varepsilon_n$ be independent Rademacher random variables (r.v.'s), so that $P(\varepsilon_i=\pm1)=1/2$ for all $i$.
...

**-3**

votes

**0**answers

55 views

### The Zariski closure of subset connected [on hold]

Let $R$ be a commutative ring and $\{P_i\}_{i\in I}$ be an arbitrary subset of $Spec(R)$ (where $Spec(R)$ is the set of all prime ideals of $R$) such that $\dfrac{R}{\bigcap_\limits{i\in I}{P_i}}$ is ...

**1**

vote

**1**answer

38 views

### Nuclear norm (convex) minimization with complex-valued matrices?

Rank minimization subject to some constraints can be accomplished in many cases through the nuclear norm.
\begin{align}
\min_{X}.\,\,& \left\|X\right\|_* \\
\text{s.t. }& X\in\mathcal{C}
...

**1**

vote

**0**answers

56 views

### What can we say about the boundary of the level set of a Sobolev function?

I'm a beginner of the area of free boundary problem. Let me first give some background:
$\Omega \subset \mathbb{R}^n$ is an open connected set, and locally $\partial \Omega$ is a Lipschitz graph.
...

**0**

votes

**0**answers

16 views

### Eigenvalue Problem — prove eigenvalue for A^2 + I [migrated]

This is a proof I've been trying to figure out since the problem was presented to me.
We are given that $\lambda$ is an eigenvalue for a matrix $A$ and the vector $u$ is the eigenvector corresponding ...

**-2**

votes

**1**answer

34 views

### expected value of cosine wirh Gaussian phase

Is there a solution to the expected value/variance for a Gaussian with random phase:
$$\cos(\omega_0 t + \phi), \qquad \phi \sim \cal{N}(0,\sigma^2) $$
?
For $t=0$, the solution is for example ...

**0**

votes

**0**answers

17 views

### Concave functions on a cone over Alexandrov space

Let $\Sigma$ be an Alexandrov space of curvature $\geq 1$ without boundary. Let $X$ be the cone over $\Sigma$. $X$ is well known to be non-negatively curved.
Let $f\colon X\to \mathbb{R}$ be a ...

**2**

votes

**1**answer

68 views

### Existence of a countable linear combination with positive coefficients

Consider a (Hausdorff and complete) locally convex topological vector space $V$ and a countable subset $(v_k)_{k=1}^\infty \subset V$ of non-zero vectors.
$(*)$ Under what conditions on this ...

**1**

vote

**0**answers

33 views

### The best constant in Poincare-liked inequality in $BV$ and $BD$ space

This question has been posted on Math Stack exchange for a while and received no response. So I decide to move it here to get more attention.
Let $\Omega\subset \mathbb R^N$ be open, bounded and with ...

**2**

votes

**0**answers

30 views

### Existence of martingales given some constraint on laws

Let $X=(X)_{0\le t\le 1}$ be a continuous martingale starting at $0$, then denote by $\mu$ and $\nu$ the probability laws of $\int_0^1X_t \mathrm{d}t$ and $X_1$. Then it is easy to see that the couple ...

**2**

votes

**2**answers

51 views

### Minimal edge color on constraints

Is minimum colors needed to assign colors to edges of complete graph $K_n$ so that every even simple cycle contains an odd number ($>1$) of colors much larger than $(\log n)^\beta$ or ...

**1**

vote

**0**answers

124 views

### divisible 2nd cohomolgy group $H^2(G,\mathbb{Z}G)$

Recall that a (nontrivial) abelian group $A$ is called divisible if the multiplication by $n$ is surjection $A\to A$ for all $n\in\mathbb{Z}_{>1}$.
My question is the following.
Is there a ...

**-2**

votes

**0**answers

86 views

### Fiber bundle with no connection for the fibers [on hold]

Suppose I have a fiber bundle $\pi: E \mapsto M$ where the base manifold $M$ is smooth, but I cannot define a connection on the fiber $F$ because in a small neighborhood of a fiber at an $x \in M$ ...

**1**

vote

**0**answers

33 views

### dense subgroup in pro v topology

Let $V$ be a extension closed pseudovariety. The pro-$V$ topology on a group $G$ is the unique group topology such that the set of normal subgroups $N$ with $G/N$ in $V$ is a fundamental system of ...

**0**

votes

**0**answers

24 views

### How to construct examples of functions in the Spaces of type $\mathcal{S}$ [on hold]

We know that there are $3$ types of $\mathcal{S}$-type Spaces, namely $\mathcal{S}_\alpha\:,\: \mathcal{S}^\beta\:,\:\mathcal{S}_\alpha^\beta$.
$\mathcal{S}_\alpha: |x^k\varphi^{(q)}(x)|\le ...

**0**

votes

**0**answers

69 views

### Find motivation for calculating $\int_{2}^{X} A^2(t) A(\alpha t)dt$

I read a thesis of Kong Kar Lun (student of Tsang K.M) about the some mean value theorems for certain errors terms in analytic number theory and in which he gave the asymptotic formulas of the ...

**1**

vote

**1**answer

45 views

### Two geodesics with angle $\pi$ in Alexandrov space

Let two geodesic segments in an Alexandrov space with curvature bounded from below start at the same point and the angle between them equals $\pi$. It is possible that these segments are not the two ...

**1**

vote

**0**answers

41 views

### Interest to know explicit values of certain coefficients

Sorry if my question is stupid but it comes to my mind whenever I read about the theory of symmetric power $L$ functions. Out of curiosity, I did a web search and found only the explicit expression of ...

**-1**

votes

**0**answers

158 views

### Straight line complexity of $k!a$ where $(a,p)=1$

In Qi Cheng's paper, an algorithm is provided to calculate $k!a$ at some random $a∈ℕ$ ($a$ is not input to algorithm, only $k$ is), what is the probability that given a prime $p$ such that ...

**11**

votes

**1**answer

273 views

### Automorphisms of finite order in $Out(\widehat{F_2})$

Let $\widehat{F_2}$ be the pro-$\ell$ completion of the free group of rank 2, where $\ell$ is some prime.
Every outer automorphism of $F_2$ induces an outer automorphism of $\widehat{F_2}$, hence an ...

**1**

vote

**0**answers

28 views

### 9-point stencil “equivalent” for advection equation [on hold]

So I inherited from some people a code that solves the advection-diffusion-reaction equation for a particular system. The original code was first implemented in 1D which worked fine in cartesian ...

**1**

vote

**0**answers

50 views

### How to calculate $det(X^TX)$ efficiently, update one column of X each time [on hold]

$X_{1} = (A, b)$, where $X_{1}$ is a $n\times p$ matrix, $A$ is a $n\times (p-1)$ and $b$ is $n\times1$.
First calculate $\det(X_{1}^T X_{1})$, then update $b$ with $c$, st. $X_{2} = (A, c)$ and ...

**5**

votes

**1**answer

129 views

### Do Iwahori-Hecke algebras come from cohomology classes?

Let $W$ be a Coxeter group. The Iwahori-Hecke algebra $H_q(W)$ is a deformation of $k W$.
Question: is there some way to interpret the deformation $H_q(W)$ as a cohomology class? It doesn't ...

**5**

votes

**2**answers

163 views

### Invariants of the maximal unipotent subgroup of GL(n) acting on the space of n by n matrices

Let $G=GL(n,\mathbb{C})$ and let $U\subset G$ be a maximal unipotent subgroup. (For example,assume that U is the set of upper triangular matrices with ones in the diagonal.) Now let ...

**5**

votes

**0**answers

154 views

### Examples of Rankin-Selberg L-functions from Eisenstein series

I've been digging for awhile to not much success, so I figure I would try here:
I am looking for some references which compute explicitly examples of Rankin-Selberg L-functions from the constant ...

**4**

votes

**1**answer

166 views

### highest weight the half-sum of positive roots

Sorry if this one is already asked - couldnt find anything about it.
If I take the irreducible representation of $GL_n$ whose highest weight is the half-sum $\rho$ of positive roots, it has ...

**-2**

votes

**0**answers

104 views

### question on collatz trajectories/pattern in form $2^n - 1$ [on hold]

I recently noticed this remarkable general "wedge" pattern in base 2 for Collatz iterates starting with values $2^n - 1$, displayed here for $n=20$.
Has this been noticed or analyzed before?
...

**8**

votes

**1**answer

355 views

### Analytical formula for topological degree

At the first page of the following article http://arxiv.org/pdf/1004.1018v1.pdf [edit: the formula on the arXiv differs from the formula in the published paper, and the formula displayed below is the ...

**4**

votes

**1**answer

157 views

### Bézout's theorem for arcs in the plane

Consider two polynomials $p,q \in {\mathbb R}[x,y]$, both of degree $d$. Let $\gamma_p$ and $\gamma_q$ be the two curves in ${\mathbb R}^2$ that are defined by these polynomials, and assume that these ...

**1**

vote

**0**answers

26 views

### Perimeters of the cells of a convex tessellation

Let $C$ be a compact, convex region in $\mathbb{R}^2$, and say we have scalars $a_i, b_i, c_i$ for $i\in\{1,\dots,n\}$. Consider the tessellation $R_1,\dots,R_n$ of $C$ defined by letting ...

**1**

vote

**0**answers

43 views

### Complexity of sparse matrix-vector multiplication?

I have a vector $\mathbf{x}$ of size $m\cdot n$ of zeros and ones, i.e., $\mathbf{x}\in\{0,1\}^{m\cdot n}$ and a matrix $\mathbf{A}$ of size $\left(m\cdot n+m+n+1\right)\times\left(m\cdot n\right)$ of ...

**4**

votes

**1**answer

137 views

### An example for a construction on monads/operads?

Suppose that $C$ is a bicategory. (I only need a monoidal category, i.e. one object bicategory, but I will stick with bicategories, since theory of monads is more commonly stated in that setting). A ...

**0**

votes

**0**answers

54 views

### Time varying markovian birth death process

In a markovian birth death process, birth rate and death rate are $\lambda(t)$ and $\mu(t)$ respectively with Poisson arrival and departure. The population at $t$ is defined as $N(t)$.
The problem ...

**4**

votes

**2**answers

80 views

### Do subgaussian variables obey the slightly-stronger-than-Chernoff tail bound?

If $X \sim Normal(0,1)$, then we have the tail bound:
$$ (*) \qquad\Pr[X > t] \leq \mathcal{O}\left(\frac{e^{-t^2/2}}{t}\right) .$$
Now for general variables $X$, a nice condition is that $X$ be ...

**-1**

votes

**3**answers

204 views

### A question about sentences undecidable in Peano's Arithmetic

Many examples are now known of sentences undecidable in Peano's Arithmetic (PA) assuming that PA is consistent. Are some or all of these sentences also undecidable in Second Order Arithmetic (SOA) if ...

**9**

votes

**1**answer

494 views

### Are there discontinuities in the large cardinal hierarchy?

Suppose that $\Phi(x,y)$ is a formula in the language of set theory so that for each natural number $n$, the axiom $\exists x\Phi(n,x)$ is a large cardinal axiom (for example consider $n$-huge ...

**12**

votes

**3**answers

1k views

### Intuition behind the Kodaira Vanishing Theorem?

As the question suggests, what is the intuition behind the Kodaira Vanishing Theorem? The Kodaira Vanishing Theorem says that the cohomology groups $H^q(M, L \otimes K_M)$ vanish for $q \ge 1$ when ...

**2**

votes

**1**answer

99 views

### Where can I find a copy of Moussatat's 1976 thesis “On the Asymptotic Theory of Statistical Experiments and Some of Its Applications”?

It was apparently written at Berkeley under the direction of Le Cam, and it is cited in a number of contributions to mathematical statistics, for example in Strasser's (1985) book "Mathematical Theory ...

**0**

votes

**1**answer

69 views

### Can I use Birkhoff's Ergodic Theorem for Vector Valued Process?

I have a stationary process $\{u_n\}$ and I have a function $f:\mathbb{R}^L\to \mathbb{R}^+$. I want to evaluate the following limit $$\lim_{n\to \infty}\frac{1}{n}\sum_{k=1}^n g(f(\mathbf{u}_{k}))$$ ...

**1**

vote

**0**answers

22 views

### PDF of points at the intersection of a sphere and hyperboloid in n dimensions

I'm studying a statistical mechanics problem and I have two conserved quantities:
$$ E = \sum_{k=0}^{M} \left[ a_1^2(k) + a_2^2(k) + b_1^2(k) + b_2^2(k)\right] $$
$$ H = \sum_{k=0}^{M} 2 k \left[ ...

**5**

votes

**1**answer

61 views

### Continuous section inside a family of rank-varying operators

Good morning everybody,
my question is as follows: let $K$ be a compact set and assume $F:K\to L(\mathcal H,\mathbb R^{m+1})$ be a continuous map from the compact set $K$ to the space of linear ...

**4**

votes

**0**answers

33 views

### $w(\mathbb{R}P^q) = 1$ if and only if $q = 2^k - 1$ for some $k$ [migrated]

How do I show that $w(\mathbb{R}P^q) = 1$ if and only if $q = 2^k - 1$ for some $k$?

**-3**

votes

**0**answers

42 views

### non local indecomposable commutative ring

Let $R$ be a commutative ring and $\{P_i\}_{i\in I}$ be an arbitrary subset of $spec(R)$ such that $\frac{R}{\bigcap_{i\in I}{P_i}}$ is an indecomposable ring, is any relation among $P_i$'s? Thanks a ...

**3**

votes

**0**answers

102 views

### Does there exist a smooth compact manifold whose boundary is $\mathbb{R}P^3$? [on hold]

As the question suggests, does there exist a smooth compact manifold whose boundary is $\mathbb{R}P^3$?

**2**

votes

**0**answers

63 views

### $Pin^{+}(4k)$ and $Pin^{-}(4k)$ are isomorphic [Reference Request]

This is some sort of "follow-up" to the (unanswered) question posted here.
Let's denote $$\varphi :O(2n)\rightarrow O(2n);A\mapsto det(A)\cdot A.$$
Then $\varphi $ is an automorphism of $O(2n)$, and ...

**0**

votes

**1**answer

112 views

### Continuous maps on compact topological spaces which induce compact (Fredholm) operators

Let $X$ be a compact Hausdorff space. A continuous map $f:X \to X$ defines a bounded linear operator $T_{f}$ on the Banach space $C(X)=\{\phi:X\to \mathbb{C} \mid \phi\; \text{is continuous}\}$ with ...