# All Questions

**5**

votes

**0**answers

47 views

### Chern-Simons form and Rarita-Schwinger operator

The Rarita-Schwinger (RS) operator naturally generalizes the Dirac operator and in Physics it describes particles with spin-3/2.
I was wondering if there exists any reference concerning the ...

**0**

votes

**0**answers

10 views

### Ergodic, non-atomic measure on the circle which are $\times 2$ and $\times \frac12$ invariant

There any many ergodic, $T$-invariant, non-atomic measures on the space $X = [0,1)$, where $Tx = 2x \pmod 1$ is the doubling map.
My question is: are any such measures also $T^{-1}$-invariant? BYO ...

**0**

votes

**0**answers

12 views

### $\int_{\mathbb{R}\setminus\left[-1,1\right]}\hat{f}^{2}\left(\xi\right)\left(\xi^{2}-1\right)d\xi $

Assume that $\hat{f}\in L^2(\mathbb{R})$, how can we compute this integral $$\int_{\mathbb{R}\setminus\left[-1,1\right]}\hat{f}^{2}\left(\xi\right)\left(\xi^{2}-1\right)d\xi
$$
where $\hat{f}$ is ...

**0**

votes

**1**answer

55 views

### How to write $\mathbb{C}[G/U_-]=\oplus_{\lambda} V_{\lambda}$ explicitly?

Let $G=GL_n$ and $U_-$ the set of all lower unipotent triangular matrices. Then by Gauss Decomposition, we have $G = U_-B$, where $B$ is the set of all upper triangular matrices. The group $U_-$ acts ...

**0**

votes

**0**answers

31 views

### Free cocompact action of discrete group gives a covering map [migrated]

I'd like to find a short proof of the following seemingly basic fact encountered on the second page of Atiyah's paper "Elliptic operators, discrete groups, and von Neumann algebras." ...

**6**

votes

**3**answers

156 views

### Logarithms of matrices in the disk-algebra

It is easy to see that within the disk algebra $A(D)$
$$\Delta(z):= \begin{pmatrix} 1&0\\z&1 \end{pmatrix}\; \begin{pmatrix} 1&1\\0&1 \end{pmatrix}=
\begin{pmatrix} ...

**7**

votes

**2**answers

256 views

### Link between the hairy ball theorem and the fundamental theorem of algebra

I read in the book "Concepts of modern mathematics" by Ian Stewart that it was possible to proof the fundamental theorem of algebra using the hairy ball theorem (complete reference to the page is in ...

**3**

votes

**1**answer

73 views

### Minimal zero-dimensional Hausdorff spaces

A topological space $(X,\tau)$ is said to be zero-dimensional Hausdorff (zdH) if for $x\neq y\in X$ there is $C\subseteq X$ clopen (closed and open) such that $x\in C$, but $y\notin C$.
We say a zdH ...

**-1**

votes

**0**answers

10 views

### Sequences of random variables converging in probability to the same limit a.s [migrated]

Let $(X_n)_{n \geq 1}$ and $(Y_n)_{n \geq 1}$ be two sequences of random variables s.t. $X_n$ converges to X and $Y_n$ to $Y$ both in probability. Furthemore, $X$ = $Y$ a.s. How can I prove that, for ...

**21**

votes

**1**answer

404 views

### Complex manifold with subvarieties but no submanifolds

I previously asked this question on MSE and offered a bounty but received no responses.
There are examples of compact complex manifolds with no positive-dimensional compact complex submanifolds. ...

**-3**

votes

**0**answers

133 views

### Can the work of Hardy & Ramanujan about partitions shed light on Hardy-Littlewood's k-tuple conjecture? [on hold]

If I'm not mistaken, Hardy and Ramanujan produced an asymptotic formula for the number of partitions of an integer that was later shown to be an exact formula by Selberg.
But Hardy also formulated ...

**0**

votes

**1**answer

61 views

### Is there an entire solution for the Van der pol equation?

Is there a non constant entire function $\gamma(t)=(x(t),y(t)): \mathbb{C} \to \mathbb{C}^{2}$ which satisfy the following Vander pol dififferential equation?
$$\begin{cases}\dot{x}=y-x^{3}\\\dot ...

**5**

votes

**1**answer

212 views

### Rotation invariance of an integral

Consider the integral depending on 2 parameters
$$f(\tau,x):=\int_{-\infty}^{+\infty}\frac{dp}{\sqrt{p^2+1}}e^{-\sqrt{p^2+1}\tau+ipx},$$
where $\tau >0,x\in \mathbb{R}$. This integral absolutely ...

**6**

votes

**0**answers

102 views

### When does a CAT(0) group contain a rank one isometry

Let $G$ be a CAT(0) group which acts on the CAT(0) space $X$ properly and cocompactly via isometry. Let $g \in G$ be a hyperbolic isometry of $X$. Then $g$ is called $\textbf{rank one}$ if no axis of ...

**0**

votes

**0**answers

38 views

### Upper bound on the norm of the inverse of matrices with zero limit

Posted here too, with no answer yet:
http://math.stackexchange.com/questions/1766281/upper-bound-on-the-norm-of-the-inverse-of-matrices-with-zero-limit
Let $\{L(\sigma)\}_{\sigma}$ be a family of ...

**8**

votes

**0**answers

126 views

### Rationality of a certain real algebraic variety

Let $A_n$ denote the vector space of $n\times n$ antisymmetric matrices over ${\mathbb{Q}}$, where $n$ is even.
Let $A_n^*\subset A_n$ denote the affine ${\mathbb{Q}}$-subvariety of invertible ...

**0**

votes

**0**answers

12 views

### Example of a Schur-nontrivial group with no abelian subgroup of the form $H\times H$?

A group $G$ is Schur-nontrivial if the Schur multipler $H^2(G,U(1))$ is not the trivial group.
I am trying to find an example of a Schur-nontrivial group which does not contain a subgroup of the form ...

**1**

vote

**0**answers

112 views

### What is the ring of integers in $\mathbb Q^c\otimes_K K_\mathfrak p$?

Let $K$ be a number field with ring of integers $\mathcal O_K$ and $\mathfrak p$ a prime of $K$. Let $\mathbb Q^c$ be the algebraic closure of $\mathbb Q$ in $\mathbb C$.
If $L$ is a number field ...

**-4**

votes

**0**answers

16 views

### Defining a function on a subset but it's domain is the whole set [on hold]

Under what conditions would it be acceptable to define a function only on a subset of it's domain?
Let $\{\mathbf{u}_1,\mathbf{u}_2\}$ be a basis for $\mathbb{R}^2=\mathbb{R}_1\times\mathbb{R}_2$ (in ...

**3**

votes

**3**answers

218 views

### What is the group of automorphisms of $l^{\infty}$?

What is the group of automorphisms of $l^{\infty}$?
I think it would be the permutations of the integers. Is this right?

**3**

votes

**0**answers

81 views

### Auslander-Reiten-Quivers of representation-finite algebras having different 3-dimensional forms

I am looking for references, where I can find (pictures of) connected Auslander-Reiten-Quivers of representation-finite $k$-algebras ($k$ is a (preferably, but not necessarily finite) field) with one ...

**5**

votes

**0**answers

180 views

### Average size of $p$-part of the Tate-Shafarevich group for elliptic curves

Let $E/\mathbb{Q}$ be an elliptic curve defined over $\mathbb{Q}$. For a given prime $p$, the $p$-Selmer group $\operatorname{Sel}_p(E)$ of $E$ and the $p$-part of the Tate-Shafarevich $Ш_E[p]$ group ...

**6**

votes

**1**answer

122 views

### Time-Energy Uncertainty Relation in relativistic Quantum Mechanics

There is an old intriguing result in non-relativistic QM, stating (roughly) that there is an Heisenberg Time-Energy Uncertainty Relation.
Unfortunately, in QM time is not an operator like space, ...

**4**

votes

**1**answer

102 views

### Upper Bound for the Difference of Even Probability and Odd Probability in Hypergeometric Distribution

Let $X$ be a random variable following the hypergeometric distribution with parameters $N,K,n$, where
\begin{equation}
Pr(X=k) = \frac{\binom{K}{k}\binom{N-K}{n-k}}{\binom{N}{n}}.
\end{equation}
To ...

**2**

votes

**0**answers

33 views

### What is the densest bipartite graph with unique Hamiltonian cycle?

In a prior post regarding perfect matching, it was stated that the densest graph with a perfect matching cannot have more than $n^2$ edges, if graph has $2n$ vertices.
Analogously, what is the ...

**4**

votes

**1**answer

118 views

### What explains the asymptotic and the pattern in this sequence related to Riemann zeta zeros?

As the starting point for my experiment I assumed that the imaginary parts of the Riemann zeta zeros are of the form:
$$\Im \{ \rho_n \} = \frac{2\pi}{\log x_n}$$
where $x_n$ is unknown. Therefore I ...

**22**

votes

**3**answers

2k views

### Adapting arguments and plagiarism

I'm currently working on my PhD thesis. I have several suggested problems to work on, some of them are very similar to some problems that my advisor have worked before and published already, either in ...

**3**

votes

**0**answers

70 views

### Intuition for density comonad in relation to lifting problems

In Emily Riehl's Categorical Homotopy Theory, there is a section on Garner's Small Object Argument which I'm trying and failing to understand. Originally I followed most of Garner's paper, using the ...

**2**

votes

**0**answers

48 views

### Functions of form $f(z)/f(z^*)$

I am doing my research in mathematical physics, and in the process I am getting functions of complex variable $z$ of form
$F(z) = \frac{f(z)}{f(z^*)}$
In my case $f$ doesn't have any interesting ...

**5**

votes

**0**answers

76 views

### Complex Stone-Weierstrass Type problem

I have come across this problem which resembles complex Stone-Weierstrass theorem except for a problem that the conjugate of the functions are not necessarily in the sub algebra.
Suppose $\Omega$ is ...

**9**

votes

**0**answers

112 views

### Definition of discrete spectrum and continuous and basic properties

I apologize if this is too basic for MO.
I have an embarrassing admission to make: I don't know the actual definition of the discrete/continuous spectrum of a reductive group $G/\mathbb{Q}$ (in the ...

**7**

votes

**1**answer

104 views

### Two notions of bundle of C* algebras

One can find in the literature two notions of $C^*$-algebra over a topological space $X$.
The first is as the data of an open surjection $ \pi: B \rightarrow X$ together with the structure of a ...

**2**

votes

**0**answers

82 views

### First Chern class and second Chern class in Quantizable Kaehler manifolds

Assume that $(X,\omega)$ is a K\"ahler manifold and $L\to X$ be a pre-quantum line bundle, then is there any relation between first Chern class and second chern class?

**0**

votes

**0**answers

131 views

### Weak formulation of a nonlinear problem with test functions in a dense subspace of $H_0^1$

Let $\Omega\subset \mathbb R^{d=3}$ is a bounded and Lipschitz domain. Let $u\in H_g^1(\Omega)$ satisfy the weak formulation
$$a(u,v)+\int\limits_{\Omega}{f(u)vdx}=0,\,\forall v\in H_0^1(\Omega)\cap ...

**9**

votes

**3**answers

389 views

### “Spatial (geometrical)” realization of Elementary topos?

It is well known that (Grothendieck) Topos (in fact, Model topos too) has many good geometrical properties. In many senses reflects general forms of generic geometry.
Note: Grothendieck view of Topos ...

**7**

votes

**1**answer

376 views

### what is the universal cover of GL(2,R)?

In the theory of Bridgeland stability conditions one has an action of the universal cover $G'$ of $G = GL^+(2,\mathbb R)$.
What is G'?
I know there is concrete description in terms of pairs ...

**6**

votes

**5**answers

245 views

### Probability theory without deductive closure

Human knowledge is not deductively closed. Uncertainty can arise from that just as much as from lack of brute facts. (When a Harvard graduate was reported to have thought that the earth is farther ...

**3**

votes

**1**answer

104 views

### Two point function of a free scalar field in Euclidean space-time

This question was previously asked here
http://physics.stackexchange.com/questions/251927/two-point-function-of-a-free-massless-scalar-field-in-euclidean-space-time
though I did not get there an ...

**6**

votes

**1**answer

949 views

### How much of modern algebraic geometry is there in modern complex(algebraic, analytic, differential) geometry?

Good day to you, people of mathoverflow. I'll get to the point. I wonder how much of modern(abstract) algebraic geometry is there in modern complex geometry?
What do I mean by complex geometry? ...

**5**

votes

**0**answers

75 views

+50

### K theory for pre $C^*$-algebras

In noncommutative geometry when one want to go to the differentiable level, one is forced to work with algebras which are no longer $C^*$. It is nice if we don't loose much information by the ...

**1**

vote

**1**answer

104 views

### Does a $W^*$ envelope exist?

I know that an operator algebra $A$ has a "minimal" $C^*$-algebra $C$ containing $A$, which is known as the $C^*$ envelope of $A$. The existence of such a minimal $C^*$-algebra generated by $A$ (a ...

**1**

vote

**1**answer

67 views

### $p$-adic Dirac measure as a weak limit

The standard Dirac delta is a generalised function (or measure, or distribution...) $\delta(x)$ which can be seen as a weak limit functions $\delta_n(x)$ spiked at the the origin, in the sense that:
...

**4**

votes

**2**answers

152 views

### Can one satisfaction class code another?

Let $M$ be a model of ${\sf ZFC}$. A satisfaction class $S$ for $M$ is subset of $M$'s ordered pairs which satisfies in $M$ the standard Tarskian compositional axioms. E.g.:
$M\vDash \forall \phi, ...

**5**

votes

**0**answers

142 views

### K theory as the fundamental group

There are several ways in which one can define $K$-theory for $C^*$-algebras: for $K_0(A)$ group two aproaches: algebraic (using idempotents) and topological (using projections, i.e. self-adjoint ...

**17**

votes

**0**answers

665 views

+150

### A curious eigenvalue inequality

Suppose $A, B$ are positive definite Hermitian matrices, $U$ is a unitary matrix such that $AUB$ is Hermitian. The spectral radius of a square matrix $X$ is denoted by $\rho(X)$. In my study, I ...

**8**

votes

**1**answer

263 views

### Interpreting Robinson arithmetic in a very weak set theory

It is known that adjunctive set theory interprets Robinson arithmetic, and that extensionality is not needed for that. (Montagna and Mancini, "A minimal predicative set theory", Notre Dame Journal of ...

**4**

votes

**1**answer

198 views

### Application of Factorization Theory to Oscillatory Integral Estimates

In the article "Some New Estimates on Oscillatory Integrals" by Bourgain in the book Essays in Honor of Elias M. Stein, Bourgain considers operators of the form
...

**11**

votes

**2**answers

350 views

+100

### A matrix norm inequality

Suppose that $A, B$ are Hermitian positive definite matrices of the same order and $0\le p\le 1$. Using a standard approach in matrix analysis, one can show that
$\|A^{1-p}B^p\|\ge \|A\sharp_p B\|$, ...

**10**

votes

**1**answer

379 views

### Differential geometric interpretation of cohomology

I'm not sure whether this is an appropriate question for this forum. I'm afraid that this is not a research level question however:
1. It's about reference request therefore the answer does not ...

**11**

votes

**1**answer

278 views

### Singular in $V$ regular in $HOD$

Prikry forcing can be used to produce a model $V$ of $ZFC$ such that fo rsome cardinal $\kappa$ we have:
(1) $\kappa$ is singular in $V$ of cofinality $\omega,$
(2) $\kappa$ is regular (and in fact ...