# All Questions

**1**

vote

**0**answers

21 views

### Divergence of a second order tensor [on hold]

I think that I have found 2 seemingly conflicting sources relating to the divergence of a second order tensor. I am not sure which is correct.
Suppose you would like to compute the components of a ...

**6**

votes

**0**answers

109 views

### Bott-Samelson theorem for simplicial sets

Let $X\in \mathrm{sSet}$ and $FX$ be the Milnor's construction (model for $\Omega\Sigma |X|$) - in each dimension $n$ this is the free group on $X_n$ with one relation $*=1$. I'm interested in ...

**4**

votes

**1**answer

141 views

### Looking for Severi varieties

Let $K$ be an algebraically closed field of characteristic $0$, and let $\mathbb{O}$ be the Cayley algebra over $K$. Let
$$
\mathfrak{J}_{3}=\{A\in\mathcal{M}_{3}(\mathbb{O}):A\text{ is Hermitian}\},
...

**0**

votes

**0**answers

18 views

### Find optimal value for a regularization parameter in generalized eigenvalue problem

Consider the generalized eigenvalue problem :
$ \Sigma_{XY} \Sigma_{YX} {W} = \lambda \Sigma_{XX} {W} $
where $\Sigma_{XX} $ and $\Sigma_{XY}$ are sample covariance matrices are of the matrices ...

**0**

votes

**0**answers

24 views

### Gradient Descent with L2 Norm Regularization [on hold]

So I've worked out Stochastic Gradient Descent to be the following formula approximately for Logistic Regression to be:
$
w_{t+1} = w_t - \eta((\sigma({w_t}^Tx_i) - y_t)x_t)
$
$p(\mathbf{y} = 1 | ...

**-5**

votes

**0**answers

44 views

### Should I learn information technology? [on hold]

I am 18 y.o., and I want to go to university. I will study
Software Engineering (Computer science), but I do not know yet what I want to be exactly.
I love math, phisycs, psychology, philosophy, ...

**1**

vote

**1**answer

68 views

### Isotopy class of closed 2-ball embedded in R^3

My intuition tells me that any two topological embeddings of the closed 2-ball (aka unit disk) into $R^3$ are isotopic in $R^3$. Is this correct? Maybe easy to prove?
It seems like it should be easy ...

**8**

votes

**2**answers

202 views

### Topological Derivation of Leray Spectral Sequence

I'm interested in computing - to the extent possible - the Leray spectral sequence for a particular map which is almost, but not quite, a fiber bundle (e.g. a Seifert fiber space). The hardest step ...

**6**

votes

**1**answer

150 views

### Chern-Simons forms, characteristic numbers, and boundary terms?

For any principal $G$-bundle $P \to M$ with principal connection $\omega$, given a $G$-invariant polynomial $p: \mathfrak{g} \to \mathbb{R}$ we can construct a form $p(F_\omega)$ on $P$ which descends ...

**2**

votes

**1**answer

286 views

### One-dimension Algebraic groups

I am searching for a possible analogue of a result in algebraic groups in a non-commutative setting, so I am looking for different proofs of the following :
Let $K$ be an algebraically closed field. ...

**-2**

votes

**0**answers

45 views

### Finding topological properties under a metric on set of composition operators of L2 [on hold]

We define a new metric on all composition operators in $L^2$:
$$ d_R (A,B)= \sqrt{\Vert P_{R(A)}- P_{R(B)} \Vert^2+\Vert A-B\Vert^2 }. $$
Now we would like to find ...

**1**

vote

**0**answers

46 views

### Difference in the Four Color Theorem [on hold]

How is proving that any planar graph with maximum degree of four has a four coloring, different from proving the four color theorem? If they are different, then how would one prove it?

**28**

votes

**3**answers

2k views

### What is a foliation and why should I care?

The title says everything but while it is a little bit provocative let me elaborate a bit about my question. First time when I met the foliation it was just an isolated example in the differential ...

**0**

votes

**0**answers

44 views

### CLT for sums of an infinite sequence of rv with an asymptotic distribution

Excuse me if the question is ill-posed. I'll do my best to explain the problem.I have a vector $(x^{(n)}_1, x^{(n)}_2, \ldots x^{(n)}_n),$ whose individual components can be shown to be asymptotically ...

**9**

votes

**1**answer

207 views

### Counting isomorphism classes in open subsets of Bun_G

Let $G$ be a split semisimple algebraic group and let $C$ be a curve of genus $g$ over $\mathbb F_q$. Assume $g \geq 2$.
The number of $\mathbb F_q$-points of $\# \operatorname{Bun}_G(C)$, where each ...

**11**

votes

**0**answers

230 views

### Have Grothendieck's notes in Montpellier already been investigated?

Grothendieck, who passed away on November 13, 2014, left a huge amount (around 20.000 sheets) of personal notes in the University of Montpellier that he thought he was the only one to be able to ...

**1**

vote

**0**answers

238 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 $\mathcal X(\mathbb C)$. ...

**8**

votes

**0**answers

78 views

### Existence/characterization/properties of $C^*$-algebras which “are” quantization of compact symplectic manifolds?

Understanding of "quantization" achieved much progress recent years, especially after Kontsevich breakthrough on deformation quantization, where he proved one-to-one correspondence between Poisson ...

**2**

votes

**0**answers

50 views

### End points of continua

Whyburn (1942) defined an end point x of a continuum X to be any point having arbitrarily small neighborhoods each of whose boundaries contains a single point. Thus, he defines end point locally.
...

**10**

votes

**0**answers

88 views

### Is the restriction map for embeddings of manifolds with boundary a fibration?

Let $M$ and $W$ be smooth manifolds (possibly with boundary) and $V\subseteq W$ a submanifold. We have a map between embedding spaces
$$Emb(W,M)\rightarrow Emb(V,M)$$ given by restriction.
Richard ...

**1**

vote

**0**answers

51 views

### Stochastic calculus in $L^1$

Does there exist a more general (than Malliavin or Itô) "Stochastic calculus" defined on $L^1$ space, or some Orlicz space between $L^2$ and $L^1$?
For examples: are there:
Ito ...

**5**

votes

**0**answers

107 views

+50

### When is a map from a logarithmic tangent bundle to a normal bundle surjective?

Suppose that $X$ is a smooth algebraic variety with free divisor $D$ so that the logarithmic tangent bundle $\mathcal{T}_X(-\log D)$ is locally free. Suppose moreover that $\iota\colon Y\to X$ is a ...

**6**

votes

**3**answers

193 views

### Uniform bounds on the number of integer points on a family of elliptic curves

Let $P(x,y)$ be a binary cubic polynomial with integer coefficient. Let $n$ be an integer. Suppose the (complex) curve $P(x,y)=n$ is nonsingular, so is an elliptic curve. Is there any bound on the ...

**10**

votes

**3**answers

232 views

### Random links and $3$-manifolds

In Jeffrey Weeks book "The Shape of Space" he explaines at the end of Chapter 18 (on page 255) the following about the geometrization conjecture:
A non-trivial connected sum $M_1\# M_2$ admits a ...

**2**

votes

**1**answer

61 views

### Weighted global Holder property for Brownian motion paths

It is well-known that the Brownian motion (Wiener process) is almost sure locally $\alpha$-Holder for any $\alpha<1/2$. That is, with probability 1
$$
...

**6**

votes

**1**answer

270 views

### Solutions of equations characterizing a complex structure

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,\\
...

**10**

votes

**2**answers

251 views

### Representation theorem for modular lattices?

Birkhoff's representation theorem implies that every distributive lattice embeds into the lattice of subsets of a set. Is there also some representation theorem for modular lattices?
For example, I ...

**34**

votes

**5**answers

2k views

### The sequence $a_{n+1}=\left\lceil \frac{-1+\sqrt{5}}{2}a_{n}-a_{n-1} \right\rceil$ is periodic

Let $(a_{n})_{n \ge 1}$ be a sequence of integers such that for all $n \ge 2$:
$0\le a_{n-1}+\frac{1-\sqrt{5}}{2}a_{n}+a_{n+1} <1$.
Prove that the sequence $(a_{n})$ is periodic.
...

**5**

votes

**1**answer

548 views

### Algebraic Geometry needed for Kähler-Einstein metric

I am a Master's student interested in Differential Geometry / Geometric Analysis. Currently active research is going on in Kähler-Einstein / Extremal Kähler metric. I was wondering how much Algebraic ...

**0**

votes

**1**answer

76 views

### Generalized Lax-Milgram for Weak Formulation of 1D Linear Schrodinger

I am interested in the variational formulation of the 1D Schrodinger equation:
$i u_t- \beta u_{xx} = 0 $ and $u(x,0)=u_0(x)$ which upon integration by parts yields:
$i(u_t,v) + \beta (u_x,v_x) = 0$ ...

**4**

votes

**2**answers

139 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 $\nu$ ...

**7**

votes

**1**answer

319 views

### Does $|A+A|$ concentrate near its mean?

Fix $N$ to be a large prime. Let $A \subset \mathbb{Z}/N\mathbb{Z}$ be a random subset defined by $\mathbb{P}(a \in A) = p$, where $p = N^{-2/3 + \epsilon}$ for some fixed $\epsilon > 0$. My ...

**24**

votes

**0**answers

255 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 ...

**2**

votes

**1**answer

105 views

### Brownian motion - probability of striking a sphere in $\mathbb{R}^n$ (a clarification)

This is primarily in reference to this question on MO. Serguei Popov's answer gives an explicit formula for the probability of a Brownian particle starting at the origin in $\mathbb{R}^n$ hitting the ...

**9**

votes

**3**answers

495 views

### Applications of Jordan algebras

Jordan algebras are non-associative algebras satisfying a somewhat strange (to me) list of axioms, see wikipedia. Basic examples are real symmetric and complex hermitian matrices with the product ...

**0**

votes

**1**answer

32 views

### Uniqueness and invariance of the LDLT decomposition

A real symmetric positive semi-definite matrix $A$ can be decomposed in the form
$A = P^TLDL^TP$,
where $P$ is a permutation matrix, $L$ is a lower unit triangular matrix and $D$ is a diagonal ...

**21**

votes

**0**answers

375 views

+50

### Is the Flajolet-Martin constant irrational? Is it transcendental?

Facebook has a new tool to estimate the average path length between you and any other person on Facebook. An interesting aspect of their method is the use of the Flajolet-Martin algorithm.
In the ...

**0**

votes

**0**answers

35 views

### Applications of systems with multiple time

A dynamical system with multiple time is an action of a group $\mathbb{Z}^d$ or $\mathbb{R}^d$ on a metric space.
I am interested in informative examples and applications of such systems. I know ...

**0**

votes

**0**answers

20 views

### Hypergeometric distribution with a priori probabilities of the balls

If we have an urn with $N$ balls of two colours ($D$ red and $N-D$ black balls respectively), then the probability of having $k$ red out of $n$ balls drawn at once without replacement follows the ...

**11**

votes

**1**answer

211 views

### The Weyl group of E8 versus $O_8^+(2)$

Right now Wikipedia says:
The Weyl group of $\mathrm{E}_8$ is of order 696729600, and can be described as $\mathrm{O}^+_8(2)$.
The second part feels wrong to me. $\mathrm{O}^+_8(2)$ is the ...

**0**

votes

**0**answers

80 views

### Strong Lebesgue regularity lemma

I have troubles in understanding how the Strong Lebesgue regularity lemma implies the Rademacher differentiation theorem. An informal proof was given by Tao on his blog: ...

**1**

vote

**0**answers

15 views

### The complexity of Max-K interval selection

I came up with the following problem, but do not know how to analyze it.
Let $S$ be an ordered set of integers with size $n$ (i.e., $S=\{1,2,...,n\}$). An interval $INV(a,b)$ covers the elements in ...

**9**

votes

**0**answers

256 views

+200

### Dimensions of dual vector spaces

Let $V_F$ be an infinite dimensional right $F$-vector space (over a field $F$, or even over a division ring). The dual space $V^{\ast}={\rm Hom}(V,F)$ is naturally a left $F$-vector space (coming ...

**0**

votes

**0**answers

14 views

### Potential theory solution for Variable coefficient Poisson with Dirichlet Boundary conditions

I am looking for a potential theory representation for the following equation in $2$D:
$$\vec{\nabla} \cdot \left(a(x) \vec{\nabla}u\right) = 0 \,\, \forall x \in \Omega \,\, (\spadesuit)$$
$$u = g ...

**-1**

votes

**0**answers

181 views

### Is there any significance in Heegner numbers (or class number 1) representation symmetry?

$\mathrm{A003173}(n) = 1+((1 + \sqrt{3})^{n-1} - (1 - \sqrt{3})^{n-1})/(2\sqrt{3})$
for n = 1,2,3,4.
$\mathrm{A003173}(n) = 19+24((1 + \sqrt{3})^{n-6} - (1 - \sqrt{3})^{n-6)})/(2\sqrt{3})$
for n = ...

**0**

votes

**0**answers

43 views

### Necessary condition for decouplings for surfaces in $\mathbb{R}^4$

I'm currently studying the paper Decouplings for surfaces in $\mathbb{R}^4$ written by Bourgain and Demeter. This paper is available in here.
As an example of nondegenerate $2$-dimensional surfaces ...

**-2**

votes

**0**answers

30 views

### Correct definition of submodularity

I am currently looking at a paper whose submodularity definition is different from whatever I thought I knew. In this paper, the authors consider a function $\Pi_2(q;a^r)$, where $q$ is composed of ...

**-1**

votes

**0**answers

29 views

### Norm of a linear operator in a tight frame

My question certainly has a simple answer, but I am not sure about how to formalize my thoughts; to put it simply, I am looking for the norm of a linear operator that is a composition of 2 linear ...

**4**

votes

**2**answers

555 views

### What is the mathematical significance of the IHES logo?

The logo of the IHES
http://www.ihes.fr/jsp/site/Portal.jsp
(upper left) is lovely, but what exactly does represent mathematically?
(There's a slightly larger version at
...

**9**

votes

**3**answers

237 views

### Construct discrete series of SL(2,R) as kernel of twisted Dirac operators

I’m studying the paper of (Baum-Connes-Higson, ex 4.25), and I would like to give an explicit computation for the Connes-Kasparov conjecture for SL(2,R).
The idea is that each non-trivial ...