# All Questions

**-1**

votes

**0**answers

15 views

### Mean squared error of a noisy random variable

Assume we have a distribution D, and a random variable X from this distribution. We want to estimate E(D) through X. Obv E(X) is an estimator for E(D). The question is that does the MSE (=mean ...

**0**

votes

**0**answers

18 views

### Explicit solution for a first order non-linear ODE

Is there any explicit solution to the following ODE?
$G'(z) =aG(z)+bG(z)^α-c$
$G(0) = d_0 $
my range of $\alpha$ is something like $(0.2,9)$

**3**

votes

**0**answers

71 views

### Does the Pfaffian have a geometric meaning?

While reviewing the proof of Gauss-Bonnet in John Lee's book, I noticed the following paragraph:
"
...In a certain sense, this might be considered a very satisfactory generalization of Guass-Bonnet. ...

**3**

votes

**0**answers

33 views

### Diophantine approximation in $\mathbb{G}_m^r$ with approximants restricted to a finiteley generated subgroup

Faltings, in the same paper (Diophantine approximation on abelian varieties, Ann. Math. 1991) in which he proved his famous ``big theorem,'' proved also that at any place $v$ of a number field $K$ and ...

**3**

votes

**1**answer

39 views

### nonnegativity conditions for a polynomial in two variables

Let
$$P(X,Y)= c_{22}X^2Y^2 +c_{21}X^2Y +c_{12}XY^2 +c_{20}X^2 +c_{11}XY+c_{02}Y^2+c_{10}X+c_{01}Y+c_{00}$$
be a polynomial of two variables $X$ and $Y$ with real coefficients $c_{ij}$.
What are the ...

**0**

votes

**0**answers

33 views

### Computing a projection of a $p$-adic plane curve

Given a prime $p$ and a polynomial equation $f(x,y)=0$ with rational coefficients, I would like to obtain a precise description of the set of all numbers $y\in\mathbb Q_p$ such that the equation has a ...

**0**

votes

**0**answers

25 views

### Binary motives in the decomposition of a minimal Pfister neighbor

Let $\alpha \in H^n(k,\mu_2)$ and $X_\alpha$ be the respective Pfister quadric. Its well known due to Rost that the Motive $M(X_\alpha)$ decomposes as a sum of twisted Rost motives $R_\alpha$ such ...

**1**

vote

**3**answers

96 views

### What do you call a fixed point theorem for a mapping from a subset of a space to the whole space?

There are a number of fixed point theorems in which we have a map from some subset of a (metric, topological, ...) space to the whole space. (Usually, there is some condition regarding the behavior ...

**1**

vote

**0**answers

43 views

### discriminant of smooth quartic del Pezzo surface in $\mathbb{P}^4$

I can't understand the proof of Lemma3.3 in Stability of genus 5 canonical curves.
Let $C$ be a complete intersection of three quadrics in $\mathbb{P}^4$ and let $\Lambda$ be the net of quadrics ...

**1**

vote

**0**answers

35 views

### question about the tightness of probability measures for a general topological space

Let $(E,\mathcal{X})$ be a topological space and denote by $\mathcal{F}$ its collection of Borel subsets referred to $\mathcal{X}$. Now let $\mathcal{P}$ be the set of all probabilities on ...

**0**

votes

**0**answers

25 views

### Bases for free pro-p groups

Let $p$ be a prime number, $F$ a free nonabelian finitely generated pro-$p$ group, $L \lhd_o F$ and $Y$ a basis for $L$ with $y \in Y$. Is there a basis $X$ for $F$ such that $y$ is in the abstract ...

**7**

votes

**1**answer

94 views

### Variety of nilpotent Lie algebras or $p$-groups

Here's a couple of analogous questions, one in terms of finite-dimensional complex Lie algebras and one in terms of finite $p$-groups; I'd be interested in an answer to either:
1) Let $\mathcal{L}$ ...

**1**

vote

**1**answer

63 views

### Differentiability of Nemytskii operator on Sobolev space

I am trying to consider hypothesis on $g$ such that the operator
$$ H_0^1 (\Omega) \to L^2(\Omega), \qquad v \mapsto g(v) $$
is $\mathcal C^1$. As additional hypothesis $\Omega$ is bounded and $g(0) = ...

**17**

votes

**1**answer

257 views

### Random points on the unit sphere

Suppose you have $n$ points picked uniformly at random on the surface of $\mathbb{S}^d,$ and let the volume of the convex hull of these points be $V_{n, d}.$ Clearly, $V_{n, d}$ converges to the ...

**-2**

votes

**0**answers

35 views

### The relationship between arc lengths and area of sectors [on hold]

How are arc length and area of a sector related to proportionality?

**-3**

votes

**0**answers

18 views

### Feasibility of a linear program with linear single constraint [on hold]

A linear program with a single constraint minimize
$z = c_{1}x_{1} + c_{2}x_{2} +· · ·+c_{n}x_{n}$
subject to $a_{1}x_{1} + a_{2}x_{2} +· · ·+a_{n}x_{n} ≤ b$, $x_{1}, x_{2}, . . . , x_{n} ≥ 0.$
(a) ...

**-3**

votes

**0**answers

27 views

### Relationships between different properties and parts of a circle [on hold]

What are the relationships among radii, chords, tangents, and inscribed and circumscribed angles of a circle?

**1**

vote

**0**answers

20 views

### numerical method (implicit , backward difference or forward difference) for nonlinear pde

$\newcommand{\lbar}{\underline{\lambda}}$
In this linear PDE:
\begin{cases}
B_t+b^Q(r,t)B_r+\frac{1}{2}d^2(r,t)B_{rr}+(\mu(\lambda,t)+\alpha \sigma (t))(\lambda -\lbar)B_{\lambda} \\ ...

**3**

votes

**0**answers

90 views

### Non-trivial bounds for polynomials at a fixed point

Let $f$ be a polynomial of degree $d$. Of course $|f(z)|\sim C|z|^d$ as $|z|\rightarrow\infty$ but also, since any polynomial is completely determined by its values at any $d+1$ points, we may ask how ...

**-2**

votes

**0**answers

30 views

### What does mean the exact value of derivative [migrated]

i'm starting my calculus's journey and i have a question.
What does mean the exact value of a derivative
Take an easy example we have a derivative of $f(x)=x^2$, that is $f'(x)=2x$.
Someone would ...

**-1**

votes

**0**answers

43 views

### Estimate of a Sobolev norm of p-form [on hold]

$\underline{\mathrm{NOTATIONS}}$
Let $(M,g)$ be a compact connected Riemannian malifold of $d$ dimensional.
$A^p(M)$ denotes the set of $p$-forms on $M$.
$g_{\wedge^p}$ denotes the fiber metric on ...

**-4**

votes

**0**answers

50 views

### Proving, that closure off set is equal this set iff set is closed [on hold]

I've started intorduction to topology course and I need help with one of the problems:
Let $A \subset(X,T). $ Prove that $cl(A) = A\iff A$ is closed.
It may looks trivial, but I had a little ...

**4**

votes

**1**answer

134 views

### If every nonseparable metric space contains a sequence of subsets with no convergent subsequence, does the Continuum Hypothesis hold?

If every nonseparable metric space contains a sequence of subsets with no convergent subsequence, does the Continuum Hypothesis hold?
This question is submerged in the discussion of Fedor Petrov's ...

**5**

votes

**1**answer

137 views

### On a weak tree property for inaccessible cardinals

Suppose that $\kappa$ is inaccessible and consider a tree of height $\kappa$ whose levels have size strictly below some cardinal $\gamma < \kappa$. Does this type of tree always have a ...

**1**

vote

**0**answers

154 views

### Two questions on canonical line bundle over $\mathbb{C}P^{n}$

The canonical line bundle over $\mathbb{C}P^{n}$ is denoted by $\ell_{n}$. It is well known that: $$(\ell_{1}\otimes \ell_{1})\oplus1 \simeq\ell_{1}\oplus\ell_{1}$$ Both sides are isomorphic ...

**0**

votes

**0**answers

63 views

### Dense free subgroups

Let $F$ be a free pro-$p$ group (for a prime number $p$) on a finite set $X$, $\Phi$ the abstract subgroup generated by $X$, $\{1\} \neq N \lhd_c F$. Is it possible that $\Phi \cap N = \{1\}$?

**-1**

votes

**0**answers

22 views

### Try to prove that a discrete distribution function is a singular distribution function [on hold]

actually it is the 6th question in the section 3 of chapter 1 in the book named by The course in probability theory,
author:Chun Kai Lai.
someone asserts that the derivate of the discrete function on ...

**-1**

votes

**0**answers

70 views

### Compact elements of $\mathrm{GL}_n(F)$, where $F$ is a nonarchimedean local field

Let $F$ be a nonarchimedean local field and let $\mathcal{O}$ be its ring of integers. An element $g$ of $\mathrm{GL}_n(F)$ is called compact if the cyclic subgroup that it generates has compact ...

**4**

votes

**3**answers

162 views

### Contractibility of space of embeddings of a disc

I'm pretty sure that both of the following spaces are contractible. However, I can't seem to find a proof or a reference. Can anyone provide one? Let $D^2$ be the unit disc in $\mathbb{R}^2$.
The ...

**-2**

votes

**0**answers

128 views

### Connes on Integers / Primes and Quantum Field Theory / Elementary Particles [on hold]

I was browsing Matthew Watkins' Number Theory and Physics archive, and came across the following quote by Alain Connes (without context, aside from the fact that it was in a popular book called Dr. ...

**22**

votes

**1**answer

1k views

### Did Leibniz really get the Leibniz rule wrong?

A couple of posts ([1], [2]) on matheducators.SE seem to suggest that Leibniz originally got the wrong form for the product rule, perhaps thinking that $(fg)'=f'g'$. Is there any actual historical ...

**6**

votes

**2**answers

149 views

### curvature flow for loops in S^2

Consider the unit 2-sphere and a smooth simple closed curve c embedded in it. I would guess that under the well studied parabolic equation which evolves the curve according to its curvature vector, ...

**7**

votes

**1**answer

56 views

### co-dimension one minimizing verifolds

It is known that a minimizing co-dimension verifolds within a manifold may need to be singular. I think a famous example first partially analyzed by Jim Simons is the cone on in the 8-ball of the ...

**-1**

votes

**0**answers

16 views

### Online algorithm for nested optimization problem(with locally optimization) [on hold]

How to construct a sequence ${x_t;\theta_t}$, which is online algorithm for following optimization problem:
$\arg\min_\theta \sum_t\min_{x_t}\ell_t(x_t;\theta)$
For simply, we can assume ...

**5**

votes

**1**answer

60 views

### Tori in Compact Riemannian Symmetric Spaces

The closure of a 1-parameter subgroup in a compact Lie group is a torus. To what extent does this result generalize to compact Riemannian symmetric spaces? In other words, is the closure of a geodesic ...

**11**

votes

**0**answers

231 views

### same paper was published in the same journal twice

I just realized that the paper "Hitchin's connection and differential operators with values in the determinant bundle" by Xiaotao Suna and I-Hsun Tsai and was published twice in the Journal of ...

**1**

vote

**1**answer

69 views

### Question about B. Host paper 'Nombres, normaux entropie, translations'

I put this question on mathstack but it seems more suitable to put it here:
I was reading this paper and I got stuck in a detail left for the reader that I couldn't figure out:
Let $X = ...

**0**

votes

**0**answers

49 views

### Graham's Number and Ramsey Theory [on hold]

I had a few questions regarding Graham's number and Ramsey theory.
I understand what Graham's number is and what it is attempting to solve. My question is, is a hypercube with dimension equal to ...

**-4**

votes

**0**answers

32 views

### What is the difference between Representation and Fibre Bundle? [on hold]

When a Group G have a homomorphism to General Liner Group GL(n, K), we call GL(n, K) Liner Representation.
When a Space X have a map to another Space Y, We call the inverse image of y, or f~-1(y), ...

**7**

votes

**1**answer

569 views

### Did differential geometry undergo a notation change?

As a graduate student, I found the old books of differential geometry used a different set of notation from modern textbooks. For example, Chern and Milnor defined the curvature 2-form by ...

**0**

votes

**0**answers

30 views

### Sumset of parallel arithmetic progressions in Euclidean space

Let $A$ be a finite subset of $\mathbb R^n$ such that dim $A=n$. It is a known result from Freiman that the size of the sumset $A+A$ is lower bounded as
$|A+A|\geq (n+1)|A|-\frac{n(n+1)}{2}$
where ...

**2**

votes

**1**answer

201 views

### Stalks of étale sheaves

I want to prove that $0 \to F\to G\to H \to 0$ is an exact sequence of étale sheaves. I understand that it is enough to show that $0\to F_{\bar{x}}\to G_{\bar{x}}\to H_{\bar{x}}\to 0$ is exact at ...

**1**

vote

**0**answers

30 views

### Asymptotic decay for the inhomogeneous Schrödinger equation

Let $H$ be an Schrödinger operator $(-\Delta+V(x))$ with a radially symmetric and smooth potential and $H$ is e.s.a. on $C^\infty_0(\mathbb{R}^n)$, furthermore let $\lambda$ be an element of the ...

**0**

votes

**0**answers

60 views

### Generalization of the alternating sign test for convergence of a series? [on hold]

I'm struggling with a series of the form $$\sum_n |a_n|\, s_n $$ where
$s_n$ is the sign of a simple function of $n$. The
$|a_n|$ monotonically decrease and are relatively simple functions of ...

**1**

vote

**0**answers

100 views

### Hilbert triples

I apologize if this is a trivial matter for the analysts out there, but I looked this up in a number of books (e.g., H. Brezis, J. Wloka) and could not find a satisfying discussion, one that would ...

**1**

vote

**0**answers

148 views

### Is there a relationship between the standard conjectures and Langlands program? [on hold]

I would like to know are there connections between Standard conjectures on algebraic cycles and Langlands program (in the light of Motives, I assume)?
What implications would a development of the ...

**5**

votes

**1**answer

114 views

### A perturbation question for the intersection of C*-subalgebras

This feels like something I may have asked before (in which case, apologies) and it also might be some kind of "standard counterexample in a book on C* algebras".
Let M be a unital C*-algebra and let ...

**1**

vote

**0**answers

134 views

### Old Math books, will research and sell most [on hold]

I saw an earlier thread on selling old math books. My dad was a professor at Manhattan College for many years, he passed away a couple of years ago and has tons of old math books. I promised him I ...

**2**

votes

**0**answers

54 views

### $L^p$ estimates for Ornstein-Uhlenbeck: what is known beyond hypercontractivity?

Consider an infinite-dimensional Gaussian random vector $X$, and a positive random variable $f(X) \in L^p, p > 1$. Let $f(X) \sim \sum_n f_n(X)$ be its (formal) chaos expansion. Let $(U_\rho, \rho ...

**1**

vote

**1**answer

106 views

### L-function of twist

I'd like to ask the following easy question, since I can't find a reference.
Let $X/\mathbb{Q}$ a smooth projective variety. How does one express the $L$-function of the twist, $L(H^i(X)(r), s)$ in ...