# All Questions

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1 views

### Stable analytic manifold under simple action

For an integer m > 1, let us define the action
f: X_i ---> (1+X_i)^m - 1
on variables X_1,...,X_N. Consider the analytic manifild V(I) defined by the ideal I in
C[[X_1,...,X_N]], where C is the ...

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2 views

### Homotopy type of a locally contractible compact

Does a locally contractible compact space have the homotopy type of a finite CW complex? (I think it probably does, but I need a reference anyway.)

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2 views

### Do compact groups acting irreducibly have finite subgroups which do the same?

Let $G$ be a closed subgroup of $U(n,{\bf C})$, not necessarily connected. Regard ${\bf C}^n$ as a complex $G$-module $M$.
Q. Suppose $M$ is irreducible as a $G$-module (equivalent, I think, to ...

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6 views

### A question on periodic points and recurrent points

I have been reading the book "dynamical systems and semisimple groups an introduction". In this book, a point of a topological $G$-space $X$ is a periodic point if $G/G_x$ is compact, where $G$ is a ...

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22 views

### Approximating rational values in ]0,1[ by a sum or difference of unit fractions

Let $U=\{\frac{1}{n}: n\in\mathbb{N}\} \cup \{-\frac{1}{n}: n\in\mathbb{N}\}$ be the set of positive and negative unit fractions.
Are there positive integers $m<n \in \mathbb{N}$, such that for ...

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38 views

### DEF vs DIFF for projective bundles over $\mathbb{P}^3$

In this MO-question I asked about deformations of vector bundles, and from the answer given by Mohan it appears that there are several deformation classes of rank two bundles with trivial Chern ...

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25 views

### How to Express Undirected Integration

Is there an agreed way of expressing undirected integration in formulas?
my idea of doing so would be to use the absolute value of the differential
$$\int_a^b f(x)|dx| = \int_b^a f(x)|dx|$$
but I ...

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23 views

### Semantic web and math learning / research communities

I have been looking at certain Math Overflow questions about amateur mathematicians doing research. Quite truthfully, over the past few years, I have been scanning the Internet for a solution to this ...

**3**

votes

**1**answer

59 views

### Can the boundedness of $A^2$ imply the boundedness of $A$?

Given an operator $A \in \mathcal L(B)$, $B$ being a Banach space, I came across the following question: assume $dom(A)=range(A)$, $dom(A)$ dense in $B$.
Under which conditions is it possible to ...

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**1**answer

16 views

### Coloring graph such that the coloring classes are not maximal independent sets

Let $G$ be a (finite or infinite) simple graph. We let $\mathrm{Ind}(G)$ be the collection of independent sets. For any cardinal $\kappa$ and coloring map $\chi: G\to \kappa$ we have ...

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29 views

### convergence of $e^{it\Delta}f$

I heard of a conjecture that
$e^{it\Delta}f\rightarrow f$ a.e. as $t\rightarrow 0$ for $f\in H^{1/4+\epsilon}$
but couldn't find a proper reference.

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29 views

### Function Related to Jordan Curves

I am looking for a solution to the following problem:
given
a Jordan curve $c(s) = (x(s),y(s))$ with $\dot x(s)^2+\dot y(s)^2 = 1$ and $c(s+L)=c(s)\,$
an integrable function $g(s): c(s)\mapsto ...

**0**

votes

**1**answer

55 views

### Proving that the eigenvalues of a certain matrix product are positive

Let $A$ be an $m \times n$ matrix, and define:
\begin{align*}
U &= {\rm diag} \{ \frac{1}{\beta_j} \}, \beta_j = \sum_{k=1}^m |a_{kj}|, j = 1 \dots n \\
V &= {\rm diag} \{ \frac{1}{\alpha_i} ...

**2**

votes

**1**answer

33 views

### When will the mirror of a K3 surface be an elliptic K3?

Let $f:Y\rightarrow\mathbb{P}^1$ be an elliptic $K3$ surface, then the holomorphic 2-form $\Omega_Y$ vanishes when restricted to an elliptic fiber $f^{-1}(b)$ with $b\in\mathbb{P}^1$. After a ...

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**0**answers

47 views

### Interpolation between L^1 and Sobolev Space

Suppose $D^\alpha$ is fractional differentiation of order $\alpha$ on the real line. Is it true that
$||D^\alpha f||_{L^\frac{2 \beta}{2 \beta - \alpha}({\mathbb R})} \leq C_{\alpha,\beta} ...

**3**

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70 views

### Generalization of a theorem of Øystein Ore in group theory?

Theorem (Øystein Ore, 1938): A finite group $G$ is cyclic iff its lattice of subgroups $\mathcal{L}(G)$ is distributive.
Proof: see below.
Let $(H \subset G)$ be an inclusion of finite groups and ...

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**0**answers

46 views

### Paritial order help? [on hold]

Let (A,≼A) and (B,≼B) be partially ordered sets. Define C = A×B and define the relation ≼' on C by (a,b)≼'(a′,b′) ⇐⇒ (a≼A a′)∧(b≼B b′).
(a) Prove that ≼' is a partial order on C.
$[(a,b) ...

**1**

vote

**2**answers

196 views

### Are linear algebraic groups rigid?

The underlying variety of a linear elgebraic group (say, over an algebraically closed field) is affine, so doesn't have nontrivial (infinitesimal) deformations. I'm curious to know whether it's ...

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87 views

### irreducible etale cover of a blowup

Let $X\rightarrow Y$ be a smooth morphism of schemes and consider the blowup of the fiber product along the diagonal, $W:=Bl_{\Delta}X\times_Y X$.
Does there exist a collection of smooth morphisms of ...

**2**

votes

**0**answers

38 views

### second smallest eigenvalue Laplacian - submodular set function

Let $G$ be a connected unweighted undirected graph. In addition, let $\lambda_2(L(G))$ be the second smallest eigenvalue of the Laplacian matrix of graph $G$. Is $\lambda_2(L(G))$ a submodular set ...

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101 views

### Symmetry on a sphere

Let $u$ be a smooth function on the sphere $S^2$. Suppose there exists $C>0$ such that for all $R \in SO(3)$, the area of every nonempty connected component of $\{x\in S^2: u(x)> u(Rx)\}$ is at ...

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37 views

### Is a inverse limit of indecomposable again indecomposable?

In truth, I do not need in the general case.
Let $R$ be a ring (associative, with unit element) and let $\mathrm{Comp}(R)$ the category of complexes over $\mathrm{Mod}\ R$.
If $\mathbb{N}$ is the ...

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**0**answers

55 views

### Euclidean geometry problems for thirs year students [on hold]

Can you suggest some books with exercises related to Euclid's Elements, or to Euclidean Geometry, as an aid to an undergraduate course on Euclidean Geometry and its history? I need exercises that ...

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23 views

### Simple monotone differential operators

Where one may find any reference to lemmas the following kind:
If x(t) is C1 in [0,T], x(0)>0, dx/dt + c(t)x(t) >0 in [0,T] then x(t) > 0 in [0,T].
There is a version with weak inequalities.
This ...

**2**

votes

**2**answers

168 views

### Reference for Skinner-Urban on the Iwasawa main conjecture for $GL_2$

Does anyone know the existence of an expository paper or a report discussing the work of Skinner-Urban
"The Iwasawa main conjecture for $GL_2$"?
I am interested in partucular in the case of elliptic ...

**-4**

votes

**0**answers

47 views

### How to compute difference between two mathematical forms? [on hold]

My apology for the unclarity of the previous version.
This question focuses on the form————
Is there any way to compute the difference between two forms (or to say, format? I'm Really sorry for my ...

**2**

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**0**answers

129 views

### Prime order elements in $GL(n,\mathbb{Z})$

What is known about the elements of prime order $p$ in $GL(n,\mathbb{Z})$? All I know at this point is that $GL(n,\mathbb{Z})$ has an element of prime order $p$ iff $p-1\leq n$ and that there are only ...

**-1**

votes

**2**answers

100 views

### Is real analytic function good enough (see problem)? [on hold]

Let $f \colon \mathbb{R}\to \mathbb{R}$ be real analytic and let $A\subseteq \mathbb{R}$ be such that the set $A'$ of all accomulation points od $A$ is not empty. If $f(a)=0$ for all $a \in A$ is then ...

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**0**answers

72 views

### Theorem related to Goldbach's Conjecture [on hold]

Theorem. Let $x > 4$ be an even number. Let $P_x$ be the set of primes less than $\sqrt{x}$. Let $p$ be a prime less than $x$. Therefore, if $p \not\equiv x \pmod{q}$ for every $q \in P_x$, then ...

**-1**

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**0**answers

88 views

### Translation of the paper titled “ Quelques remarques sur les groupes de Lie algebriques reels” [on hold]

Is there any English translation of the following paper which is written in French.
http://projecteuclid.org/euclid.jmsj/1260975679

**1**

vote

**1**answer

131 views

### Quadratic Gauss sums: Explicit determinations?

Can anyone please tell me (give me a reference, preferably) if there is any explicit determination of sums of the form $g(n,\chi):=\sum_{r=1}^{q}\chi(r)e(\frac{rn+r^2}{q})$ where $\chi$ is a Dirichlet ...

**0**

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**0**answers

90 views

### Applications of infinite permutations [on hold]

I was looking at approximation in the forlmula of Products of necklaces:
$n \to \infty$ we have $\prod_{p=1}^n N(p,a) \approx \frac {a^n} {n!}$ $\prod_{p=1}^n \frac {1-a^p} {1-a}$. (The number of ...

**1**

vote

**2**answers

126 views

### Cubic Cayley (undirected) graphs

The generators of a cubic Cayley graph always include at least one involution, since 3 is odd. There are then two possibilties for the other two generators: either (A) they are also (distinct) ...

**3**

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**0**answers

67 views

### Invariant definition of the space of symbols on a vector bundle

Normally a symbol on a vector bundle $E$ is defined as a smooth function on $E$ which in each trivializing chart fulfills the usual symbol estimates
\begin{equation}
\sup_{x \in U, \xi \in ...

**2**

votes

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42 views

### On the uncountability of a subset of U-numbers of type $\leq m$

We say that $\xi\in \mathbb{R}$ is an $m$-ultra number if there exists a sequence $(\alpha_n)_n$ of $m$-degree real algebraic numbers, such that
$$
|\xi-\alpha_n|<(\exp^{[3]}(H(\alpha_n)))^{-n},\ ...

**2**

votes

**1**answer

127 views

### Existence of real modular function with specific behavior as $q\to 0$

I am looking for a real modular function $F(q,\bar{q})$ such that in the limit of small $q,\bar{q}$ it behaves as:
$F(q,\bar{q})=(a_0 + a_1 (q + \bar q)+...)\log q \bar q+ (b_0 + b_1 (q + \bar ...

**4**

votes

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144 views

### Counterexamples to Elkik's theorem in the non-Noetherian case

Elkik in Solutions d'equations a coefficients dans un anneu Henselian, Theorem 7 proves that:
Let $A$ be a Noetherian ring that is Henselian with respect to a principal ideal $(a)$.
That is, if ...

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vote

**0**answers

52 views

### “Direct” proof (without hypercontractivity) of equivalence of moments?

Let $(x_i)_{i \in \mathbb{N}}$ be a family of independent $\pm 1$ centered Bernoulli random variables, and let $p, q > 1$. There exists a constant C such that for every (finite) linear combination ...

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votes

**2**answers

82 views

### Blow up solution for a Riccati's equation

I apologize for the problem too simple, but I'm not able at solving it.
Consider the Cauchy problem
$$
\left\{
\begin{array}{l}
\dot x=x(t)^2+t\\
x(0)=0
\end{array}
\right.
$$
Show that its solution ...

**1**

vote

**1**answer

187 views

### Is it possible to sum the divergent series with prime coefficients?

It is known that the series $$ P := \sum_{n=1}^{\infty} p_{n} \qquad \text{where } p_{n} \text{ is the n'th prime} $$ cannot be summed by means of (prime) zeta function regularization. (The result was ...

**3**

votes

**2**answers

241 views

### Is a Morse function always the height function of some embedding? [on hold]

Pictures in introductory texts to Morse theory are often drawn as to interpret a Morse function as a height function. Typically, an embedding of a torus into $\mathbb{R}^3$ is drawn, and the Morse ...

**0**

votes

**1**answer

191 views

### A Problem Concerning Odd Perfect Number

Briefly, prove that every odd number having only three distinct prime factors cannot be a perfect number.
I know there are results much stronger than the one above, but I am looking for an answer ...

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**0**answers

64 views

### A Iterated Function Problem, f(f(x))=x^2+x [duplicate]

Suppose f(x) is a function defined on complex plane, satisfying f(f(x))=x^2+x, does it admit a solution?
And what if f(x) is defined on R?

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25 views

### Is polynomial chaos expansion interesting to surrogate surface?

I'm currently studying polynomial chaos. I want to use it for approximate surfaces but i'm not sure it's possible ? My surface is recursively defined like this : $$ F(x,t) = \underset y \sum ...

**8**

votes

**1**answer

96 views

### What is the maximal number of distinct values of the product of n permuted ordinals

Because addition and multiplication of two order types are non-commutative operations, we have that for every integer n, given n ordinals, there are at most n! distinct possible values for the sum (or ...

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52 views

### A question on $p$-approximation property

We say that a subset $K$ of a Banach space $X$ is relatively $p$-compact($1\leq p<\infty$)if there exists a $p$-summable sequence $(x_{n})_{n=1}^{\infty}$ in $X$ such that $K$ is contained in ...

**8**

votes

**4**answers

424 views

### A metric space of geometric shapes

My research involves geometric shapes in $R^2$, and I need a metric with several properties such as:
Families of similar shapes, such as squares, are closed in this metric. Also more general ...

**1**

vote

**2**answers

62 views

### Complementary integrable vector fields

Let $(M,g)$ be a Riemannian manifold. Assume that $X$ is a non vanishing vector field tangent to $M$.(Or assume that we have a one dimensional foliation of $M$). Under what geometric ...

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25 views

### signed area between a curve and a straight line $x_1$=$x_2$ [on hold]

Prove $\int_0^1$($x_1$ d$x_2$-$x_2$d$x_1$)=$\int_0^1\int_0^1$ d$x_1$ d$x_2$.
Using the rules of differential form we can get
d($x_1$ d$x_2$-$x_2$_d$x_1$)=2 d$x_1$ d$x_2$ and thus the Question. How ...

**2**

votes

**0**answers

71 views

### What is the relation between the length of period of simple continued fraction expansion of quadratic algebraic numbers $\sqrt{A}$ and the integer $A$

What is the relation between the length of period of simple continued fraction expansion of quadratic algebraic numbers and the integer
As we know,$\sqrt{2} = [1;2,2,2,2,…]$; while $\sqrt{14}= ...