# All Questions

**2**

votes

**0**answers

18 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 ...

**1**

vote

**0**answers

37 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 ...

**0**

votes

**0**answers

4 views

### When is the sum of complemented subspaces complemented?

Let $X$ be a Banach space.
Question. Suppose $X_1,...,X_n$ are complemented subspaces of $X$. When is the sum $X_1+...+X_n$ complemented? Further, suppose we know some projections $P_1,...,P_n$ onto ...

**0**

votes

**0**answers

25 views

### What is the complexity of Hodge conjecture

This question is motivated by the following quote from serre in his interview given
here:
... The Hodge conjecture, too,
but for a different reason: it is not clear at all whether
the answer ...

**0**

votes

**0**answers

15 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 ...

**2**

votes

**1**answer

13 views

### Logarithms of matrices in the disk-algebra

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

**0**

votes

**0**answers

44 views

### Parallel unit vector fields on spheres

Let $(S^n,g)$ be the standard unit sphere. I would like to write a corollary of my calculations on harmonic vector fields with respect to a class of Riemannian metrics on $TS^n$ induced by isotropic ...

**2**

votes

**2**answers

103 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

29 views

### GEOMETRY PROVE CHORDS CONGRUENT [on hold]

Attached is the picture of the proof. My teacher gave us some hints which were: draw radii, use triangles, and use addition property.

**1**

vote

**0**answers

18 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 ...

**0**

votes

**0**answers

41 views

### Inverse trace theorem for partial trace

A general result is that for a lipschitz bounded domain $\Omega$ in $R^n$, for $u^*\in W^{1-\frac{1}{p},p}(\partial\Omega)$, $1<p<\infty$, there exists $u\in W^{1,p}(\Omega)$ such that ...

**3**

votes

**0**answers

33 views

### Are rays in Carnot groups straight?

A famous open problem in Geometric Control Theory and in the study of sub-Riemannian manifolds is whether constant-speed length minimizers in a sub-Riemannian manifold are always smooth (see also this ...

**127**

votes

**33**answers

73k views

### Best Algebraic Geometry text book? (other than Hartshorne)

I think (almost) everyone agrees that Hartshorne's Algebraic Geometry is still the best.
Then what might be the 2nd best?
It can be a book, preprint, online lecture note, webpage, etc.
One suggestion ...

**3**

votes

**4**answers

317 views

### unitization-process of unital- and non-unital $C^*$-algebras

I have a small question about unitization of (unital) $C^*$-algebras. I first asked on math.stackexchange because it is basic theory, but I still have no suitable answer, the link ...

**25**

votes

**3**answers

2k views

### Interactions between (set theory, model theory) and (algebraic geometry, algebraic number theory ,…)

Set theory and model theory have many applications outside of logic, in particular in algebra, topology, analysis, ...
On the other hand model theory, in particular after Hrushovski, found many ...

**16**

votes

**1**answer

362 views

### Axiomatizations of the real exponential field

According to Marker's "Model Theory: An Introduction", the real exponential field has a $\forall\exists$ axiomatization (because it is model complete) but no-one has any idea what such an ...

**61**

votes

**5**answers

8k views

### What is sheaf cohomology intuitively?

What is sheaf cohomology intuitively?
For local systems it is ordinary cohomology with twisted coefficients. But what
if the sheaf in question is far from being constant?
Can one still understand ...

**11**

votes

**6**answers

3k views

### Is there a relationship between model theory and category theory?

According to Chang and Keisler's "Model Theory", Model Theory = Universal Algebra + Logic. Model theory generalized Universal Algebra in the sense that we allow relation while in Universal Algebra we ...

**9**

votes

**2**answers

1k views

### The derived category of the heart of a t-structure

Suppose $\mathcal{D}$ is a triangulated category and that we are given a $t$-structure $(\mathcal{D}^{\leq 0},\mathcal{D}^{\geq 0})$ on $\mathcal{D}$. The heart of the $t$-structure, ...

**9**

votes

**2**answers

326 views

### A question about the dispersion points of connected metric spaces

Let $C$ be an infinite, separable and connected metric space. If $C$ becomes totally disconnected when one of its points $p\in C$ is removed, does every closed ball of $C$ with
positive radius and ...

**0**

votes

**0**answers

29 views

### $z^n-t^m=x^3+y^3$ and Vojta's more general abc conjecture

In A more general abc conjecture, p. 7 Paul Vojta conjectures:
If $x_0,\ldots x_{n-1}$ are nonzero coprime integers satisfing $x_0 + \cdots x_{n-1}=0$
$$ \max\{|x_0|,\ldots |x_{n-1}|\} \le C ...

**0**

votes

**0**answers

14 views

### Milnor numbers and mixed multiplicities

section 6 of the link
Teissier showed that Milnor numbers of a hypersurface $(X,0)$ with isolated singulraity at 0 is same as mixed multiplicities of the Hilbert polynomial of the filtration ...

**7**

votes

**0**answers

81 views

### Holomorphic contractibility of ${\rm GL}(H)$?

Kuiper's theorem is well-known to give the triviality of the homotopy groups of ${\rm GL}(\mathcal{H})$ for $\mathcal{H}$ a (separable) infinite-dimensional complex Hilbert space. Work of Palais later ...

**0**

votes

**0**answers

14 views

### Quadratic characteristic and constancy

Consider a change of measure on $\mathcal{F}_{t}$ defined by the restriction of two probability measures of the form
\begin{align}
\frac{dQ_{t}(\theta)}{dP_{t}}=\exp^{ \theta A_{t}-\kappa(\theta) ...

**3**

votes

**1**answer

84 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

11 views

### Integral of error function (erf) and gaussian [on hold]

answer given in mathoverflow for integral of erf and gaussian, what will be the right hand side limits if the lefthand side limits are not from -inf to +inf but from a to b.

**3**

votes

**1**answer

81 views

### Transfer map in group cohomology

Let $H$ be a subgroup of a finite group $G$, and let $M$ be a $G$-module. Are there any simple conditions on $H,G$ and $M$ which would ensure that the transfer map $H^p(H,M)\to H^p(G,M)$ is the zero ...

**7**

votes

**2**answers

86 views

### Is there a subset of $\Sigma_n$ s.t. each pair of elements is once in each pair of positions?

Is there a subset $A \subset \Sigma_n$ such that for each pair $(x, y)$ and each pair $(i, j)$, there is exactly one permutation $\sigma \in A$ such that $\sigma(i) = x$ and $\sigma(j) = y$? Remark ...

**2**

votes

**1**answer

72 views

### Henkin semantics for Second-order Logic

I know that the natural numbers can be categorically characterized in second-order logic with the standard semantics. However, I could not find an example of a non-standard Henkin structure (one that ...

**-3**

votes

**1**answer

66 views

### Eigenvalues of cyclic tridiagonal matrix [on hold]

Following matrix is result of special kind of balanced signed graph of order $n$. In the Matrix $n_1,n_2,..,n_k$ are positive integers, which satisfy $\sum
n_i=n.$ Prove that matrix has two zero ...

**-4**

votes

**0**answers

56 views

### Eigenvalues of tridiagonal matrix [on hold]

The following matrix $T$ is result of my research on special type of balanced signed graphs of order(No. of nodes) $n$. In the matrix $T$ $n_1,n_2,...,n_k$ are positive integers such that $\sum ...

**21**

votes

**2**answers

1k views

### Can we trap light in a polygonal room?

Suppose we have a polygonal path $P$ on the plane resulting from removal of an one of a convex polygon's edges and a ray of light "coming from infinity" (that is, if we were to trace the path ...

**2**

votes

**0**answers

53 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 ...

**9**

votes

**1**answer

77 views

### Which simplicial objects are Čech nerves?

In 1-categories, a regular epimorphism is a coequalizer of some parallel pair. An effective epimorphism is one which coequalizes its kernel pair. In the presence of kernel pairs, regular and ...

**3**

votes

**1**answer

88 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 ...

**0**

votes

**1**answer

48 views

### Polynomials pulled back by momentum maps

Let $G$ be a Lie group acting Hamiltonian on some real analytic symplectic manifold $(M, \omega)$, with an $G$-equivariant momentum map $\Phi \colon M \to \mathfrak{g}^*$.
Assuming I can find ...

**3**

votes

**1**answer

221 views

### Compact non-nuclear operators

I am not sure if this question makes sense, or if it is trivial, but does there exists an infinite dimensional Banach space (necessarily without the approximation property) such that no compact, ...

**4**

votes

**0**answers

56 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, ...

**0**

votes

**0**answers

24 views

### Lowest upper bound of resultant [on hold]

Given two polynomials $f(x)=X^N+1$ and $g(x)= \sum_{i=0}^{N-1} s_i x^i$, where $s_i$'s are $\eta$-bit integers, let $res=resultant(f(x),g(x))$.
What is the upper bound of $\log_2 res$?.
Below my ...

**1**

vote

**0**answers

36 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 ...

**1**

vote

**1**answer

100 views

### What is difference between length of proof and length of its presentation in Peano Arithmetic?

In this paper http://www.sciencedirect.com/science/article/pii/0304397584901117 page $19$ or $29$ it seems to imply there is a difference between length of proof and length of its presentation in ...

**0**

votes

**0**answers

24 views

### How to numerically evaluate a integral whose limits are functions of x (using Gauss quadrature rule)? [on hold]

I am trying to numerically evaluate an integral $\int_q^1 \ln (\sum_i \alpha_ix_i) dq$, in which $\ln (\sum_i \alpha_i x_i)$ is related to $q$ via the following:
$z_i=(1-q)\frac{\alpha_ix_i}{\ln ...

**4**

votes

**0**answers

67 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 ...

**3**

votes

**0**answers

35 views

### Vanishing of power of nilpotent operator $\mathrm{ad} \, \;e$ in different characteritics

This question needs some background:
(1) In his influential 1959 paper here, Kostant studied the adjoint representation of a simple Lie algebra $\mathfrak{g}$ over $\mathbb{C}$ (which can be ...

**10**

votes

**2**answers

355 views

### From Weyl groups to Weyl groupoids?

I'm trying to find a framework where the choices in the classical construction of a root system of a semi-simple lie algebra are not needed.
Let $\mathfrak{g}$ be a semisimple lie algebra.
...

**8**

votes

**0**answers

91 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 ...

**2**

votes

**1**answer

123 views

### The spectrum and the tangent space of the algebra of holomorphic functions on a Stein manifold

Let $A$ be a Fréchet algebra over ${\mathbb C}$, and let us call the spectrum ${\tt Spec}[A]$ of $A$ the set of all characters, i.e. continuous multiplicative linear functionals $s:A\to{\mathbb C}$, ...

**3**

votes

**2**answers

86 views

### Unit-Distance Polyhedra

What polyhedra are known to have two vertices adjacent if and only if they are of distance $d$ apart, for fixed $d$? For example, regular Platonic solids satisfy this condition, so I am looking for ...

**3**

votes

**2**answers

202 views

### Nice way to express $H^{-1}(\mathbb{S}^1)$

I am looking for a good way to write down what $H^{-1}(\mathbb{S}^1)$ is. What do I mean by this: Well, certainly one can define this space via charts that map from the sphere into $\mathbb{R}^2$ and ...

**-3**

votes

**0**answers

27 views

### Number of trials until completion? [on hold]

You've got a discrete uniform distribution - what is the expected number of trials until each point is hit at least once?
I started my thinking with maybe a Geometric distribution representing each ...