# All Questions

**1**

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11 views

### definable curves in definable sets

Suppose I have an UNBOUNDED subset X of $R^n$ ($R$=real numbers), definable in the o-minimal structure $R_{an, exp}$. It is possible to find an UNBOUNDED, ANALYTIC and DEFINABLE curve (i.e. ...

**0**

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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**

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**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**

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**0**answers

23 views

### Generating-bijective groups

We may say that two finitely generated groups $G$ and $H$ are generating-bijective when there exist homomorphisms $\phi:G\rightarrow H$ and $\psi:H\rightarrow G$ such that, for each ordered generating ...

**0**

votes

**1**answer

60 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**

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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." ...

**5**

votes

**0**answers

51 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**

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**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**

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 ...

**0**

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**0**answers

26 views

### Differential categories vs McBride's notion of derivative

Has anyone done an analysis to see if Blute, Cockett, and Seely's differential categories suffice to provide a notion of 1-hole context in the symmetric monoidal setting?

**0**

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**0**answers

27 views

### Expected value and variance of a stochastic process

I would like to ask if there is a way to find the expected value and the variance of the following process
$$
dv_t=(a-be^{\alpha v_t})dt+\sigma dW_t, \quad v_t=v_0
$$
where $a\in (-\infty,+\infty), ...

**0**

votes

**0**answers

21 views

### Morse function on slicing disk complement determines ribbon?

It is well-known that given a ribbon knot and the corresponding slicing disk in the 4-ball, the distance function (maybe squared) to the origin defines a Morse function in the complement of the ...

**-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 ...

**-5**

votes

**0**answers

44 views

### I've read on the internet that 43 is congruent to 8 modulo 5+3w, can you explain? [on hold]

Initially, I wanted to determine whether 19 is a cube modulo 43.
Reading online, I've come across this computation that includes
a step that I don't understand. The step in question states that
(43 ...

**1**

vote

**0**answers

115 views

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

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 ...

**0**

votes

**0**answers

24 views

### Eigenvalues of signed networks [on hold]

A signed graph is a graph where edges can be positive or negative. A cycle in a signed network is called balanced cycle when it has an even number of negative edges otherwise it is called unbalanced ...

**-3**

votes

**0**answers

26 views

### Discrete circular distribution [on hold]

Having a distribution of a discrete number N of angular values [0:360], not necessarily all adjacent, ordered in time. How to determine the maximum and the minimum of this distribution?
Please look ...

**2**

votes

**0**answers

37 views

### Non-Haken hyperbolic 3-manifolds without nonorientable surfaces

It is well known that there exist infinitely many (non homeomorphic) non-Haken closed hyperbolic 3-manifolds. These can be obtained for example doing Dehn surgery on the figure eight knot complement. ...

**-1**

votes

**0**answers

15 views

### Necessity of expansiveness for existence absolutely continuous invariant measures for piecewise smooth maps of an interval

A map $\tau:[0,1]\to[0,1]$ is piecewise smooth (or $C^r$) if there is a partition of $[0,1]$ into intervals, $[0,1]=\cup I_n$, (which can be either finite or countable) such that the restriction of ...

**4**

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**0**answers

59 views

### Are the canonical maps from $\Omega^1_k(C^\infty(M))$ into $\Omega^1(M)$ and into $\Omega^1_k(C^\infty(M))^{**}$ compatible?

Let $M$ be a smooth manifold and let $A=C^\infty(M)$. In this question, it is observed that the map $\Omega^1_k(A)\to \Omega^1(M)$ from the Kähler differentials of $A$ to the 1-forms of $M$ is not an ...

**0**

votes

**0**answers

15 views

### Inverse of a correlation matrix that has arcsin elements

Let $C$ be a correlation matrix whose off-diagonal elements are defined as follows: $C(i,j)=arcsin(\sqrt{k^{|i-j|}}$ where $k<1$. Can the inverse of the correlation matrix be derived analytically?
...

**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 ...

**21**

votes

**1**answer

413 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. ...

**2**

votes

**0**answers

138 views

### Is this Grothendieck trace map an isomorphism?

Let $A$ be a commutative ring and let $S := \operatorname{Spec}(A)$. Let
$$ g : Y \to X $$
be a proper, birational morphism of separated schemes of finite type over $S$, where $X$ is affine and ...

**0**

votes

**0**answers

28 views

### Fourier tranform of the Euclidean norm [migrated]

where can I find the Fourier transform of the power of the Euclidean norm?, that is:
$$\mathcal{F}[\|x\|^{p}](\omega) = \int_{\mathbb{R}^{d}}\exp(-2\pi i \langle\omega, x\rangle) \|x\|^{p} dx$$
...

**3**

votes

**1**answer

75 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 ...

**-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 ...

**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 ...

**5**

votes

**1**answer

213 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 ...

**1**

vote

**0**answers

88 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 ...

**8**

votes

**0**answers

127 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

49 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 ...

**1**

vote

**1**answer

72 views

### Does similarity imply symmetric similarity?

Let $F$ be an infinite field. If two $n×n$ matrices $A,B$ are similar in $M_n(F)$, then are they also similar via a symmetric matrix (that is, is there a symmetric matrix $Q∈GL_n(F)$ such that ...

**7**

votes

**2**answers

257 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**

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**0**answers

78 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 ...

**6**

votes

**3**answers

157 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} ...

**2**

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**0**answers

34 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 ...

**-3**

votes

**0**answers

17 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.

**0**

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**0**answers

23 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) ...

**-4**

votes

**0**answers

36 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.

**7**

votes

**0**answers

135 views

### Holomorphic contractibility of 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**

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**0**answers

41 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 ...

**3**

votes

**1**answer

111 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 ...

**4**

votes

**1**answer

103 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 ...

**3**

votes

**3**answers

219 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?

**0**

votes

**0**answers

26 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 ...

**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 ...

**2**

votes

**2**answers

138 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 ...

**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 ...

**0**

votes

**0**answers

29 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 ...