0
votes
0answers
2 views

Examples of non isometric surfaces having the same curvature function

I think it is really natural to believe, after doing Riemannian geometry for a little time, that sectional curvature encodes the all local geometry of a Riemannian manifold. One of the first thing one ...
-2
votes
0answers
5 views

dimension of intersection of two subspaces

w1={(0,x2,x3,x4,x5) where all x's belongs to R} & w2={(x1,0,x3,x4,x5),all x's belongs to R} are subspaces of R5 hen dim(w1 intersection w2)=
0
votes
0answers
3 views

relationship of SDE in Langevin equation form and Ito form

A formal SDE can be written in a way as (ito form): $dx(t)=ax(t)dt+dw(t)$ where $w(t)$ is brownian motion. Another way is to write the SDE (Langevin equation form) is $\frac{dx(t)}{dt}=ax(t)+w(t)$ ...
0
votes
0answers
6 views

How prove this numerical solution we parameterize the integral equations

Let $\Omega\subset R^2$ be a simply connected bounded domain with infinitely differentiable boundary $\partial\Omega$and unit normal vector $v$ directed into the exterior of $\Omega$ ...
0
votes
0answers
7 views

query about quasi-simple algebraic groups over local fields

Suppose that $G$ is an absolutely quasi-simple algebraic group over a non-archimedean local field $k$ (of either zero or positive characteristic). Is it known whether or not it is necessarily the case ...
0
votes
0answers
20 views

a question about minimal nonnilpotent groups1

Let G be a minimal nonnilpotent group with cyclic Sylow p-subgroup P and normal Sylow q-subgroup Q[see Huppert,Endlich gruppen I].If Q is Abelian and q>2,then can we get that G is a minimal nonabelian ...
1
vote
0answers
43 views

The angular distribution of the $(a,b)$ in $p = a^2+b^2$, and the distribution of the lattices corresponding to prime ideals

Here is a really basic question which I wished I understood better about the primes of the Gaussian field $\mathbb{Z}[i]$. But I was curious about the possibility of generalizing it to other (real ...
0
votes
0answers
19 views

Complexity of Untwisting Polygons

What is the complexity of the following task: given a sequence $p_1, ..., p_n, p_1$ that defines a closed polyline in the euclidean plane, what is the complexity of finding a reordering of the points, ...
-2
votes
0answers
23 views

Mean squared error of a noisy random variable [on hold]

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
0answers
34 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 $(0.2,9)$
6
votes
0answers
108 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
0answers
38 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
1answer
94 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
0answers
47 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
0answers
28 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
3answers
110 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 ...
2
votes
1answer
72 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
0answers
45 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
0answers
26 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
1answer
112 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
1answer
73 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) = ...
19
votes
1answer
297 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
0answers
37 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
0answers
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
0answers
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
0answers
21 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} \\ ...
4
votes
0answers
102 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
0answers
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
0answers
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 ...
-5
votes
0answers
55 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
1answer
148 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
1answer
144 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
0answers
168 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
0answers
64 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
0answers
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
0answers
71 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
3answers
163 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
0answers
133 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. ...
26
votes
1answer
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
2answers
155 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
1answer
65 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
0answers
17 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
1answer
62 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
0answers
241 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
1answer
72 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
0answers
50 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
0answers
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
1answer
575 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
0answers
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
1answer
208 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 ...

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