# All Questions

**1**

vote

**0**answers

112 views

### Is liminf|(n*sinn)|=0 as n tends to infinity? [duplicate]

One of my friends asked me that is $\varliminf |nsinn|=0$?
I think maybe it has some relations with Number theory, But I don't know how to solve it. If you know the answer, please tell me since it ...

**1**

vote

**1**answer

54 views

### Dropping rank of IA automorphisms

Is there a natural way to map a given IA automorphism $\alpha\in Aut(F(X_n))$
to $Aut(F(X_{n-1}))$?
Think about braids. A pure braid on $n$ strands can be naturally mapped to a braid
on $n-1$ strands ...

**2**

votes

**1**answer

150 views

### Derived category of a hypersurface

Let $X$ be a smooth projective variety over $\mathbb{C}$, and $H \subset X$ be a smooth hypersurface.
Many properties of an ambient variety $X$ could somehow inherit to the hypersurface $H$, I was ...

**5**

votes

**1**answer

119 views

### Examples of Maass forms with eigenvalue 1/4

For what I have heard, Maass forms of (Laplacian) eigenvalue $1/4$ on modular surfaces are somewhat special. But I don't know where to look for explicit examples. (In fact, one form came here on MO ...

**7**

votes

**1**answer

163 views

### Why can't one modify Kaplansky's proof to conclude that every projective module is a direct sum of its finitely generated projetive submodules?

Due to the examples given in the answer to this question, I know that the conclusion is of course incorrect. But by reading Kaplansky's proof of theorem 1 in this paper and replacing every occurrence ...

**2**

votes

**0**answers

35 views

### Stabilization of the pencil of skew symmetric matrices by the orthogonal group

Good morning everybody.
During my researches I've come across the following question.
Let $A$ and $B$ be a couple of square $k\times k$ skew symmetric matrices on $\mathbb R$. Let us consider the ...

**0**

votes

**1**answer

89 views

### ($^{\omega}2$,<) is not well-order. [on hold]

Let < be a lexicographic order on $^{\omega}2$ or in other words given distinct functions $f,g$ from $\omega$ to 2, let $f<g$ if and only if $f(n)=0$ and $g(n)=1$, where $n$ is the lease ...

**2**

votes

**1**answer

64 views

### Test functions with “wrong” topology not locally convex?

I didn't find it in any book, although it seems that this should be standard: Endow the space $C^\infty_c(M)$ of compactly supported functions with the inductive topology coming from the embeddings
$$ ...

**0**

votes

**0**answers

61 views

### proof non diagonalizable matrix is not an inner product [on hold]

Given $ A \in M_n(\Bbb C) $ and $ <x,y>_A = x^TA\overline y $
I need to proof that if A is non diagonalizable then $<.,.>_A$ is not an inner product.
I thought about:
Let A be non ...

**0**

votes

**0**answers

50 views

### reference needed for some well know results on cohomology of the orbit spaces

The following results are well known
If the group $\mathbb Z_2$ acts freely on a mod $2$ cohomology $n$-sphere $X$, then the orbit space
$X/\mathbb Z_2$ is a cohomology real projective $n$-space.
...

**2**

votes

**1**answer

65 views

### How to get transition matrix of markov process?

I am monitoring a Markov process with ~21 states. I know all the states, initial state and what states transitions can/cannot be, so that zero elements of transition matrix are known.
I know the ...

**2**

votes

**0**answers

43 views

### Maximal $k$-chordal subgraph

Recall that a graph is called $k$-chordal if any cycle $C$ of length $> k$ contains a chord, i.e. an edge joining to non-consecutive vertices in $C$. Let $f(n, k)$ be the minimal number of edges ...

**6**

votes

**2**answers

288 views

### Vector bundle for prescribed Stiefel-Whitney classes

I hope this is not trivial.
Let $B$ be a nice topological space (paracompact, CW-complex or whatever you think is nice)
For $i=1,\ldots,n$ let $x_i \in H^i(B,\mathbb{Z}_2)$ be certain cohomology ...

**2**

votes

**1**answer

65 views

### Looking for a canonical (matroid polytope) subdivision of the hypersimplex

A matroid polytope is the convex hull of the indicator vectors of the bases of a matroid, and a matroid polytope subdivision (MPS) is a polyhedral subdivision of a matroid polytope whose cells are ...

**4**

votes

**1**answer

162 views

### Link for “A spine for Teichmüller space”, preprint by Thurston

Can someone please give any link or mention any source where I can find the following preprint.
W.Thurston, A spine for Teichmüller space, preprint, three pages, 1986.

**0**

votes

**0**answers

38 views

### partition of an integer n into atmost k =O(log n) parts

Suppose you have a partition p of n into atmost k parts, say $$\{i_1, i_2, ..., i_j, ..., i_{k-1}\}$$
For example $\{1, 4\}$ is a partition of 10 into 3 parts (in this notation i am specifying the ...

**5**

votes

**0**answers

70 views

### Concentration of sum of powers of normals

Let $Z_1,Z_2,\ldots,Z_n$ be i.i.d. copies of a random variable $Z$ distributed as $\frac{1}{\sqrt{2}}X+i\frac{1}{\sqrt{2}}Y$ with $X$ and $Y$ independent standard Normal random variables ...

**0**

votes

**0**answers

72 views

### Manifolds supporting finite order diffeomorphisms (a local construction?)

The following question is mainly inspired by this previous one Which manifolds admit a diffeomorphism of order $n$? and some answers given there.
For $d\geq 2$, let $\mathbb{B}^d$ denote the closed ...

**1**

vote

**1**answer

84 views

### Classes of dynamical systems

A consequence of birkhoff ergodic theorem tells us that ergodicity is equivalent to:
$\forall A,B \in \mathcal{B} \ \frac{1}{N}\sum_{n=0}^{N-1}\mu(A\cap T^{-n}(B))\stackrel{N\to ...

**7**

votes

**0**answers

192 views

### I think I have a category enriched in $(\infty,n-1)$-categories. Is it an $(\infty,n)$-category?

I have recently been thinking about some mathematical gadgetry that should together combine into an $(\infty,n)$-category (actually, an $(n,n)$-category) for $n = 4$. I don't know what axioms I need ...

**2**

votes

**1**answer

118 views

### Meaning of “ Open Book cannot be Stabilized Further”?

I'm going over some old notes on Giroux's theorem on the equivalence ( bijection, actually) between open books ( up to positive stabilization) for 3-manifolds and contact structures ( up to isotopy.) ...

**1**

vote

**0**answers

19 views

### Maximal Cohen-Macaulay modules of type one

Does any body know an example of a Noetherian local ring (R,m) which admits a maximal cohen-macaulay module of type one, but the ring R itself is not CM.
If C is the maximal CM module then the type ...

**1**

vote

**1**answer

65 views

### How to calculate the maximum number of rainbows for arbitrary graphs?

This question is inspired by problem 1 of the combinatorics test of the 2012 third round iranian olympiad which is as follows:
We've colored edges of $K_n$ with $n-1$ colors. We call a vertex rainbow ...

**3**

votes

**1**answer

72 views

### Numerical Evaluation of Some Triple Integral involving Negative Powers

Let $\beta_i\in (-1/2,0)$, $i=1,2,3,4$. I'm interested in obtaining numerical value of the following integrals:
$$
\int_{0<u_1<u_2<u_3<1} (1-u_1)^{\beta_1}(1-u_2)^{\beta_2} ...

**4**

votes

**1**answer

138 views

### Property of IA automorphisms of free groups

For $n \in\mathbb{N}$ define:
$X_n=\{x_1,\ldots,x_n\}$,
$F(X_n)$ the free group on $X_n$,
$\varphi:F(X_n)\to F(X_{n-1})$ an epimorphism defined by $x_i\stackrel{\varphi}{\mapsto} x_i$ for $1\le ...

**-2**

votes

**0**answers

106 views

### Show that $\int_0^\infty \sin\left(x^2\right)dx$ converges, but that $\int_0^\infty \sqrt{\sin^2\left(x^2\right)}dx$ does not. [on hold]

Show that $\int_0^\infty \sin\left(x^2\right)dx$ converges, but that $\int_0^\infty \sqrt{\sin^2\left(x^2\right)}dx$ does not.
The first part I think I proved using triangles, but I could not prove ...

**3**

votes

**0**answers

120 views

### smoothing a current

Let $M$ be a smooth oriented manifold of dimension $n$ and $T$ a current of dimension $k$ on $M$. Let $\phi:P\times M \to M$ be a proper smooth family of diffeomorphisms of $M$ (i.e. $P$ is a smooth ...

**3**

votes

**0**answers

102 views

### Stable homotopy of spheres non-locally

Are there any results/conjectures about the stable homotopy groups of spheres that relate the picture at different primes? Something like Gauss's reciprocity law in number theory?
I know about the ...

**-3**

votes

**0**answers

46 views

### Why is a principal prime ideal of $\mathrm{PID}[x]$ not maximal? [migrated]

Let $R$ be a PID and let $f(x)\in R[x]$ be an irreducible primitive polynomial. I want to show that the prime ideal $(f)<R[x]$ is not maximal. It would be enough to find a prime $p\in R$ such that ...

**2**

votes

**1**answer

72 views

### Distance between two networks

Suppose you have networks A and B, each with a set of nodes and edges. You want to measure how similar the networks are to each-other. None of the nodes or edges are labelled. What are the metric(s) ...

**7**

votes

**5**answers

700 views

### Advice on choosing an area of specialization

I'm not sure if this is an appropriate question for MO, but I figured it couldn't hurt to ask. I'm a second year graduate student, currently gearing up to construct a committee and syllabus for my ...

**0**

votes

**0**answers

45 views

### Product Approximations [on hold]

Can anybody let me know about the approximation of the following product expression:
$$
\prod_{i=1}^{n} (1+ \frac {x} {i})
$$
Regards

**6**

votes

**1**answer

173 views

### Sets of cardinalities of bases without choice

For a vector space $V$, let $BS(V)$ be the set of cardinalities (not necessarily $\aleph$s) of bases of $V$. Of course, in ZFC each $BS(V)$ is a singleton, but supposing the axiom of choice fails, it ...

**0**

votes

**0**answers

49 views

### Inverse problem with a rank-1 update

I hope you can help me out with this. I have to find the solution x to an inverse system
$$
x=A^{-1}b
$$
This inverse problem is basically a least square problem with a rank-1 update.
$$
...

**1**

vote

**1**answer

31 views

### How to compute the limit of skewness function?

The skewness function of a list of values is:
where
$m_k=\sum_{i=1}^N (x_i-u)^k$
$u=E[x]$
The image shows the meaning of this function related to the shape of the distribution of its x values ...

**0**

votes

**2**answers

101 views

### lift of Riemannian metric to branched double cover

Let $\hat{M}$ be a branched double cover of $M$. Is there a way to lift a Riemannian metric $g$ on $M$ to get a smooth Riemannian metric $\hat{g}$ on $\hat{M}$. Moreover, if $g$ has nonnegative ...

**-1**

votes

**0**answers

81 views

### some question about Geometric invariant theory [on hold]

I am studying "Geometric invariant theory(GIT)". But I do not have a strong background and I want to get some motivation. I just know GIT is one method for studying moduli space.
Which results are ...

**0**

votes

**1**answer

78 views

### How to compute the expectation of $\frac{Y^L}{Y^L + (N-Y)^L}$ where Y is Binomial(n,p)

How to compute the expectation of $\frac{Y^L}{Y^L + (N-Y)^L}$ where $Y$ is Binomial(n,p)? If it is not exactly computable, then are their ways to approximate this qty?

**-3**

votes

**0**answers

46 views

### Math test and help [on hold]

Which statements are true for both translations and rotations?
A. Transformed figures are congruent.
B. Resulting line segments are parallel.
C. Angle measures are preserved.
D. Figure ...

**0**

votes

**0**answers

54 views

### when does a “triangulated” functor factor over the homotopy category?

The setup is as follows:
We have the category $C$ of chain complexes over some additive/abelian category and want to pass to the category $K$ of chain complexes modulo homotopy.
So we have an ...

**1**

vote

**0**answers

26 views

### Intersection subgroup in cyclic group [migrated]

Let $G$ be a finite group and $x$ a non-trivial element of $G$. If, for every non-trivial element $y\in G$, $\langle x \rangle \cap \langle y \rangle \neq \{1\}$, then is $G$ cyclic or generalized ...

**-3**

votes

**0**answers

32 views

### A non-liner second order differentice equation with two parameters [on hold]

Can we analysis the effect of the two parameters on the equation? And how can we determine the range of parameters to obtain the solutions for the equation? And
it is much gratitude for you help even ...

**0**

votes

**1**answer

103 views

### Fixed point problem with a monotone vector as a fixed point?

$F : [0,1]^n \to [0,1]^n$ is a continuous and monotonic function. Therefore it has a unique fixed point $x^* \in [0,1]^n$.
I need to show that its elements are ordered, i.e. $x_1^* \leq \dots x_n^*$. ...

**2**

votes

**0**answers

45 views

### Polytopes generated as the convex hull of the orbit of a finite reflection group acting on a given vector

Consider a finite reflection group acting on $\mathbb{R}^N$. Pick a vector $x \in \mathbb{R}^N$ and look at the convex hull of its orbit. This is a polytope.
There are some interesting particular ...

**1**

vote

**1**answer

97 views

### Why can't there be a problem both in P and NPC [on hold]

In this illustration, P and NPC are two disjoint set.
We know that NPC is non-empty. If P $\cap$ NPC $=\varnothing$, then there are elements in NP which are not in P. Doesn't this imply that P ...

**0**

votes

**0**answers

72 views

### Integer sequences such that each term forms k-consecutive composite integers

For $\mathbb{N} \ni k>3$, let $\{a_n\}_{n=1}^{\infty}$ be an increasing positive integer sequence such that for each $n$, $(a_n, a_n+1,\ldots,a_n+(k-1))$ is a $k$-tuple of composite positive ...

**2**

votes

**1**answer

106 views

### Criterion for R-equivalence of two points on cubic surfaces over $\mathbb{Q}$

The definition of R-equivalence is given in the paper as Definition 4.1. Coarsely speaking, given a field $K$ and a cubic surface over $K$, two points $x,y$ are R-equivalent over $K$ if they can be ...

**4**

votes

**2**answers

315 views

### Properties of vector spaces without AC

With AC, it is easy to see that any vector space is injective, and free, therefore alse flat and projective.
Without AC, vector spaces can be not free. Are they must be projective modules? Flat ...

**7**

votes

**3**answers

787 views

### Is there a homology theory that gives a *necessary and sufficient* condition for homotopy equivalence?

Is there a (non-Abelian) homology theory that realizes the following:
Let $X,Y$ be manifolds with complexes $C(X),C(Y)$.
Then $X$ and $Y$ are homotopy-equivalent if and only if $C(X)$ and $C(Y)$ ...

**-4**

votes

**0**answers

72 views

### how to make Contravariant and Covariant tensors applicable to problems of curvatures in halfspace problems? [on hold]

Consider a material halfspace and assume it to be made of infinite number of layers of same material, such that when the material is loaded at the top surface, how to quantify the variation of ...