**2**

votes

**1**answer

216 views

### Embedding of $\ell_p$ into infinite direct sums

Let $p\in (1,\infty)$ and let $q$ be conjugate to $p$. Is there a subspace of $\ell_1(\ell_p)$ isomorphic to $\ell_q$? Of course, I am uninterested in the case $p=2$.

**8**

votes

**0**answers

443 views

### Dual versions of “folding” symmetric ADE Dynkin diagrams?

Start with the Dynkin diagram of an irreducible root system, typically associated with a simple
Lie algebra over $\mathbb{C}$ or a simple algebraic group. Most of the simply-laced ADE
diagrams ...

**5**

votes

**1**answer

336 views

### Projections which are not completely bounded

There are 'canonical' examples of maps on operator spaces which are not completely bounded. Nevertheless, I couldn't produce any examples of bounded projections on relatively easy to understand ...

**5**

votes

**1**answer

1k views

### The Guinand-Weil explicit formula without entire function theory

I'll admit from the outset that this question is slightly vague. The actual question appears at the end of the post.
The explicit formula of Guinand and Weil can be written in the following way:
For ...

**3**

votes

**0**answers

272 views

### quasi periodic continued fractions and powers of e, tanh, tan

It is well known that some transcendental numbers like e.g. rational multiples of $e^{2/n}$ with $n\in\mathbb N $, when written as regular continued fractions (R.C.F.), yield what can be called a ...

**3**

votes

**2**answers

241 views

### Does every embedding of one unipotent group (over R) in another extend to an embedding of the respective upper triangular matrix groups?

Let $T^*$ denote upper triangular matrices (of the appropriate size) with positive diagonal entries and $\mathrm{UT}$ upper triangular matrices with all diagonal entries equal to 1.
Does every ...

**1**

vote

**0**answers

103 views

### Reference needed for: Automatic generation of relevant mathematical exercises (similar to ones written by human) with the help of machine learning

I am doing research on automatic generation of relevant mathematical exercises (similar to ones written by human) with the help of machine learning.
I have found several research papers on ...

**6**

votes

**1**answer

318 views

### Generalizations/applications of a formula for the Dedekind zeta function?

I saw the following nice formula in an unpublished paper of H. Cohen. Let $L$ be a quartic field, whose Galois closure $\widetilde{L}$ has Galois group $S_4$. Denote the cubic resolvent field of $L$ ...

**6**

votes

**0**answers

313 views

### elliptic curves over function fields

Let $K$ be a number field, $E$ an elliptic curve over $K$, and $p$ an odd prime. If $v$ is a place of $K$, we know by the Kummer injection that $E(K_v)/pE(K_v) \hookrightarrow H^1 (K_v, E[p])$ for ...

**2**

votes

**4**answers

1k views

### Fractals as solution to optimization problem?

What's the scientific reason for fractals being present in nature at such a large scale? Is it perhaps the solution of an optimization problem? For example, would the fractal based shape of certain ...

**5**

votes

**1**answer

563 views

### Rational approximation to a set of reals

Are there any well known algorithms for finding good rational approximations to sets of real numbers?
Given just two real numbers, I can use continued fractions to find a rational approximation to ...

**4**

votes

**3**answers

367 views

### Functions on hyperbolic space and modular curves

The decomposition of $L^{2}\left(S^{2}\right)$ under $SO\left(3,\mathbb{R}\right)$ is well-known.
Focus now on the hyperbolic plane $H$ presented as the quotient ...

**3**

votes

**1**answer

186 views

### What are the fixed points of the jacobian acting on the compactified jacobian ?

Let C be an integral projective curve over $\mathbb{C}$ and Jac(C) be its jacobian.
Let $\overline{Jac(C)}$ be the compactified jacobian of C (the moduli space of rank 1 torsion free
sheaves of degree ...

**6**

votes

**1**answer

318 views

### Topological dynamics and Turing complete automata

One can look at, say, Conway's Game of Life in at least two ways:
1) as a cellular automaton; and
2) as a discrete topological dynamical system (on an underlying Cantor set).
Famously, Conway ...

**3**

votes

**2**answers

27k views

### Correlation between 3 variables

For correlation measurement betweeen 2 variables, I use Pearson formula.
What formula can use to find degree of correlation between 3 variables ? My variabes are not symmetric: The correlation in ...

**2**

votes

**2**answers

365 views

### Looking for criterion for $\mathbb{Z}G$-modules to be projective

Given a finite group $G$ and a (finitely generated) $\mathbb{Z}G$-module $M$, assume that for each prime $p$ dividing the order $|G|$ of $G$ the $\mathbb{Z}_pG$-module $M^{\mathbb{Z}_p} = ...

**9**

votes

**2**answers

752 views

### Properties of Some Random Graphs

Working in a problem the following family of graphs appears naturally. Consider the set $A_{n}=\{1,2,3,\ldots,n\}$ and let $\mathcal{C_{n}}$ be the set of all permutations of $A_{n}$ of order $n$ ...

**2**

votes

**1**answer

250 views

### Discreteness of a group of hyperbolic isometries

Referring to A question about hyperbolic double torus, there is non-discrete $\Gamma= \left< a,b,c,d~~|~~[a,b][c,d] \right> \subset PSL_{2}(\mathbb{R})$ where $a,b,c,d$ are hyperbolic elements. ...

**16**

votes

**3**answers

1k views

### Cutting convex sets

Any bounded convex set of the Euclidean plane can be cut into two convex pieces of equal area and circumference.
Can one cut every bounded convex set of the Euclidean plane into an arbitrary number ...

**6**

votes

**1**answer

456 views

### Wrapping a convex polyhedron with string

This is a meta-question, rather than a specific mathematical question.
I am seeking a mathematical definition that captures the following physical idea.
Suppose you have a convex polyhedron $P ...

**1**

vote

**2**answers

291 views

### Berry Esseen inequality for multidimensional distributions

The classical Berry-Esseen theorem asserts that if $f$ and $g$ are the characteristic functions of two distribution functions $F(t)$ and $G(t)$ respectively then with $T$ arbitrary
$$
\sup_{t \in ...

**4**

votes

**2**answers

1k views

### Dualizing sheaf on varieties

Hi,
there is Corollary III,7.12 in Hartshorne which says that:
If $X$ is a projective nonsingular variety over an algebraically closed field $k$, then the dualizing sheaf is isomorphic to the ...

**2**

votes

**3**answers

152 views

### Simultaneous “Monomialization” of a set of operators.

We all know that a set of commuting diagonalizable matrices can be simultaneously put in diagonal form. My general question is:
Under what conditions can a set of (diagonalizable) matrices be ...

**4**

votes

**1**answer

411 views

### Representations of infinite dimensional Lie algebras as vector fields on manifolds

Suppose I have e.g. the Witt algebra,
$\left[l_n,l_m \right] = -(n-m)l_{n+m}$.
I want to realize the $l_n$ as vector fields on some manifold. The classical example is when the $l_n$ span the Lie ...

**1**

vote

**3**answers

1k views

### Generalization of eigenvalues/vectors to modules?

What is the generalization of eigenvalues/vectors to modules?
To be specific, given a "vector" v in a module over some ring, and a linear "operator" O from the module to itself (please feel free to ...

**4**

votes

**4**answers

443 views

### A question about the additive group of a finitely generated integral domain

Let $R$ be an integral domain of characteristic 0 finitely generated as a ring over $\mathbb{Z}$. Can the quotient group $(R,+)/(\mathbb{Z},+)$ contain a divisible element? By a "divisible element" I ...

**10**

votes

**3**answers

2k views

### Has Vaught's Conjecture Been Solved?

Today I found myself at the Wikipedia page on Vaught's Conjecture,
http://en.wikipedia.org/wiki/Vaught_conjecture
and it says that Prof. Knight, of Oxford, "has announced a counterexample" to the ...

**4**

votes

**1**answer

672 views

### Symplectic groups Sp_{2m}(2) as 2-transitive permutation (i.e. Galois) groups

Hello,
I am looking for information about the symplectic groups $Sp_{2m}(2)$ as permutation group acting on quadratic forms.
Consider the block matrices
...

**0**

votes

**1**answer

161 views

### How to compute SE(2) group exponential and logarithm?

I want the rodrigues like formula using sin and cos , not a matrix series expansion.
I've found some references for se(n) , n > 3 in :
ftp://ftp.cis.upenn.edu/pub/papers/gallier/rodrig.pdf

**4**

votes

**1**answer

600 views

### Polyline Averaging

I'm trying to find a method that can take a collection of polylines, each described by a list of connected points on a plane, and find an "average" path through them. The input lines do not loop.
...

**10**

votes

**0**answers

420 views

### Zero-free theta functions in the upper half plane

Problem $1$. Which full rank lattices $\Lambda \subset \mathbb R^d$ have their corresponding theta function $\theta_{\Lambda}(\tau):= \sum_{\bf n \in \Lambda } e^{\pi i \tau ||n||^2} $ zero-free in ...

**1**

vote

**1**answer

262 views

### System of polynomial equations: P(x)=P(y) rather than P(x)=0

$P$ is a system of polynomials in $n$ variables over $\mathbb{Q}$. $Q$ is a singe such polynomial. Let $V$ be the zeros of $Q$. I know from some symmetry argument that for every $y \in [0,1]^n ...

**3**

votes

**1**answer

490 views

### Does there exist a surreal long line?

Does there exist a totally-ordered-without-endpoints proper class $L$ such that every closed interval in $L$ does have the order type of a closed interval in the Conway's surreal numbers, but $L$ as a ...

**5**

votes

**4**answers

2k views

### How to solve a generalization of the Coupon Collector's problem

The coupon collector's problem is a problem in probability theory that states the following (from wikipedia):
Suppose that there are $n$ coupons, from which coupons are being collected with ...

**1**

vote

**0**answers

83 views

### Is scalarwise measurability determined by the strong dual?

Since this question has not received an answer so far, I try to reformulate the question in a simpler manner as follows: Do there exist $E,F,\ell,f$ such that
$E$ and $F$ are separable (real) ...

**0**

votes

**1**answer

114 views

### Is the first eigenvalue of a parabolic ends of a Riemanian manifold 0?

Is the first eigenvalue of a parabolic ends of a Riemanian manifold 0?

**2**

votes

**2**answers

637 views

### Classification of certain algebraic curves

Let $C$ be a complex algebraic curve. It is well known that if $L$ is a special divisor on $C$, i.e., $h^0(L) > 0$ and $h^1(L) > 0$, then
$$
h^0 (L) \le \frac{1}{2} \deg L + 1.
$$
Assume that ...

**3**

votes

**2**answers

569 views

### On the generalisation of Bernstein's theorem on monotone functions

Bernstein's theorem states that for any completely monotone function $f$: $f \in C^{\infty}[0,+\infty)$, $(-1)^n f^{(n)}(t) \geqslant 0$ there is a finite Borel measure $\mu$ such that
$$ f(t) = ...

**1**

vote

**1**answer

676 views

### Estimating parameters of a mixture of normal distributions.

I want to estimate the parameters $\mu_i$ and $\sigma^2_i $ of a countable mixture of Gaussians with assumed equal weights, variance and identically spaced means. I intially thought that the Fourier ...

**1**

vote

**1**answer

578 views

### Matrix Inversion Lemma for Infinite Matrices

Assume all matrices are real. Suppose $A$ is a positive definite matrix of size $n \times n$, while $H$ is a $\infty \times n$ matrix and $D$ is an infinite matrix with a diagonal structure, that is ...

**2**

votes

**2**answers

434 views

### The exceptional locus of a minimal resolution of singularities

Let X be a surface. (A surface is an excellent integral normal separated 2-dimensional scheme.)
Let $\psi:Y\longrightarrow X$ be a minimal resolution of singularities and let $E$ be an irreducible ...

**1**

vote

**1**answer

127 views

### For what values of the parameter does this function have an elementary anti-derivative?

I am a grad student working on some independent research trying to derive some exact formulas for a particular class of power series. During my study I came across the following integral which would ...

**8**

votes

**0**answers

307 views

### Is the product of a discretely Lindelöf space with [0,1] discretely Lindelöf ?

A space $X$ is discretely Lindelöf iff given any discrete subset $D$ of $X$, its closure in $X$ is Lindelöf. Such spaces were introduced by Arkhangel'skii about 15 years ago (if I am not mistaken) ...

**13**

votes

**1**answer

690 views

### Idempotent measures on the free binary system?

Let $(S,*)$ be the free (non associative) binary system on one generator (so $S$ is just the set of terms in $*$ and $1$). There is an extension of $*$ to the space $P(S)$ of finitely additive ...

**3**

votes

**2**answers

375 views

### Any relationship between Viswanath's constant and the Khinchine-Lévy constant?

It is well-known that if ${\{{F_n}\}}$ is a random Fibonacci sequence then we have almost certainly $\lim \limits_{n\to\infty}\sqrt[n]{|F_n|}=\tau$ where $\tau\approx 1.554682275$ is Viswanath's ...

**0**

votes

**1**answer

85 views

### Homeomorphism between base of conifolds and spheres

Hello
Call $Y^4$ a conifold which satisfies the following condition:
$\mathfrak{Y}(z):=\sum_{\alpha=1}^{3}(z_{\alpha})^{2}=0,$
where $z_\alpha \in \mathbb{C}$. Now intersect $Y^4$ with $S^5$ to ...

**2**

votes

**0**answers

200 views

### What machine learning algorithm is appropriate for predicting one time-series from another?

I have eye-tracking data on two subjects -- a teacher, and a student. It's in the form (x, y, time), so there is a series of these for each subject. What the teacher looks at influences what the ...

**4**

votes

**0**answers

241 views

### Symmetric matrices and Hilbert's fourth problem

From the analytic viewpoint, the Busemann-Pogorelov solution of Hilbert's fourth problem is summarized in the following result:
Theorem. All straight lines are extremals of the variational problem
$$
...

**0**

votes

**0**answers

230 views

### Prime numbers characterization

When, in one endeavour, I investigated prime numbers, I came up with a formula that characterizes primes, and the job was done in essentially this way:
First i defined a function $sr$ (sum of ...

**3**

votes

**2**answers

259 views

### Constructing equivalent algebraic expressions for matrix equations

I have an expression involving matrices, of the form:
$$f(k)=x^T A_k^{-1}A x$$
where $x$ is a $1\times N$ vector, $A_k = A + k I$ and $A$ is an $N\times N$ matrix ($A_k$ is invertible for all $k$) ...