# All Questions

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 ...
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 $(0.2,9)$
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. ...
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 ...
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 ...
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 ...
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 ...
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 ...
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 ...
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 ...
25 views

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 ...
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 $\mathcal{L}$ ...
63 views

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 ...
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), ...
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 ...
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 ...
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 ...
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 ...
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 ...
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 ...
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 ...
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 ...