1
vote
0answers
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
1answer
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
1answer
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
1answer
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
1answer
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
0answers
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
1answer
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
1answer
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
0answers
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
0answers
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
1answer
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
0answers
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
2answers
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
1answer
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
1answer
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
0answers
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
0answers
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
0answers
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
1answer
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
0answers
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
1answer
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
0answers
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
1answer
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
1answer
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
1answer
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
0answers
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
0answers
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
0answers
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
0answers
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
1answer
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
5answers
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
0answers
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
1answer
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
0answers
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
1answer
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
2answers
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
0answers
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
1answer
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
0answers
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
0answers
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
0answers
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
0answers
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
1answer
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
0answers
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
1answer
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
0answers
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
1answer
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
2answers
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
3answers
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
0answers
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 ...

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