# All Questions

**18**

votes

**1**answer

359 views

### Making $\mathbb{Q}$-cohomology integral

Let $X$ be an algebraic variety (say, smooth and projective) over $\mathbb{C}$, and fix $$\alpha\in H^i(X^{\text{an}}, \mathbb{Q})$$
with $i>0$.
Does there always exist a variety $Y$ and a ...

**6**

votes

**2**answers

131 views

### Painting $n$ balls from $2n$ balls red, and guessing which ball is red, game

The game
Lucy has $2n$ distinct white colored balls numbered $1$ through $2n$. She picks $n$ different balls in any way she likes, and paint them red. She then giftwrap all the balls so that it is ...

**9**

votes

**13**answers

1k views

### Obscure Names in Mathematics [on hold]

I recently stumbled over the "Happy Ending Problem" (cf e.g. http://en.wikipedia.org/wiki/Happy_Ending_problem), which made we wonder, if there are other conjectures, theorems or problems, whose names ...

**0**

votes

**0**answers

31 views

### Is the parallelogram rule an axiom or a theorem in euclidean geometry? [on hold]

I am aware of the proof of the rule in inner product spaces. Excluding the geometry of Descartes, is it possible to prove parallelogram rule or is it an axiom?

**3**

votes

**0**answers

27 views

### Is a group uniquely determined by the sets {ab,ba} for each pair of elements a and b?

This is a cross-posted question, originally active here on math.stackexchange.
For a given group $G=(S,\cdot)$ with underlying set $S$, consider the function
$$
F_G:S\times ...

**7**

votes

**2**answers

412 views

### Decomposition of an integral operator into a composition

I've been musing about the following question for a while now. Given an integral operator $G$ defined by
$$ (Gf)(x) = \int_0^1 G(x,u) f(u)\,du $$
Is it possible to decompose this into two separate ...

**0**

votes

**1**answer

81 views

### Poisson approximation of random sub-graphs

I add the edges of $G(n)$ the complete graph on $n$ vertices one by one, at random and without replacement, and denote by $G(n,m)$ the resulting Erdos Renyi random graph process. At step $m$ in the ...

**0**

votes

**0**answers

11 views

### lower bound of a trace quadratic form

i want to find a lower bound on the following expression:
$tr(AXA^T)$ in terms of $tr(X)$
where A is real $n\times n$ matrix and $X>0$ is positive symmetric. It seems that the following bound is ...

**0**

votes

**1**answer

26 views

### Help in finding distribution of the following function of random variable

Let $X_1$ and $X_2$ be independent complex Gaussian random variables, $$X_1 \sim \mathcal{CN}(0,\sigma)$$
$$X_2 \sim \mathcal{CN}(0,\sigma)$$
If $X= aX_1 + bX_2$ where $a,b$ are constants then the ...

**0**

votes

**0**answers

60 views

### Is there any algorithm to decide whether a series with integral coefficiens is a algebraic function?

Given a series with integral coefficiens as following:
$$F(x)=\sum_0^i a_i x^i,\text{where }a_i\in \mathbb{N}\bigcup 0 $$$$\text{and there is a computable function $\psi$ such that } \forall i ...

**3**

votes

**1**answer

61 views

### Are sums of 0-1 Pareto efficient vectors Pareto efficient?

Does there exist an $m \times n$ matrix $A$ and a vector $x \in \mathbb{R}^m$ such that:
The entries of $A$ are $\in \{0, 1\}$.
For all pairs of columns $u, v$ of $A$ the entries of $u - v$ are ...

**2**

votes

**2**answers

91 views

### Are there examples of families of objects which are canonically isomorphic, but where diagrams of canonical isomorphisms don't commute?

Specifically, are there "nice" (ie, not too obscure or contrived) examples of families of objects, where say for each objects $A,B,C$ in the family, the canonical isomorphism from $A\rightarrow C$ is ...

**1**

vote

**0**answers

25 views

### Does $\mathsf{fReR}_0$ prove the existence of the cartesian product of two sets

$\mathsf{fReR}_0$ is a set-theoretical system whose axioms consist of:
(1) Axiom of extensionality: $\forall z\in x\ (z\in y)\wedge\forall z\in y\ (z\in x)\rightarrow x=y$
(2) Axiom of empty set: ...

**2**

votes

**0**answers

68 views

### Characterizing the real analytic Eisenstein series

Consider the classical real analytic Eisenstein series
$$
E(z,s)=\left(\pi^{-s}\Gamma(s)\frac{1}{2}\right)\sum_{(m,n)\neq(0,0)}\frac{y^s}{|mz+n|^{2s}},
$$
where $z=x+iy$. We think of $E(z,s)$ as a ...

**1**

vote

**0**answers

46 views

### Equations over $\mathbb{Z}[[T]]$ vs. equations over $\mathbb{Z}_p$

This question might be deemed totally unanswerable, unless there is an obvious counterexample. Answers to either effect would be welcome.
Question. Let $X$ be a finite-type scheme over ...

**2**

votes

**1**answer

74 views

### u-Invariants of p-adic function fields

In his Paper "Fields of u-invariant 9" Oleg Izhboldin points out that for a algebraic closed, finitely generated field $k$ we have $u(k)= 2^{cd(k)}$. In particular we have
...

**0**

votes

**0**answers

14 views

### Cohomological dimension of transcendental p-adic extensions

Let $k = \mathbb{Q}_p$ for any prime $p$ and set
$L = k(t_1,..,t_n)$.
The u-invariant of a field $u(k)$ is defined by $u(k):=\{ max (\mathrm{rank}(q)) $ | $ q $ is anisotropic over $k\}$.
It is ...

**6**

votes

**0**answers

133 views

### On the ratio of Gilbreath sequences

Definitions: let $n \in \mathbb{N}_{>0} \cup \{ \infty \}$ and let $E_n$ be the set of sequences $(d_i)_{i=1}^n$ such that $d_1=1$, $d_i$ is an even integer (for $i > 1$) and $0<d_i \le i$. ...

**2**

votes

**1**answer

266 views

### Beurling density and interpolation

Let $\Lambda=\{\lambda_n\}_1^\infty$ a set of points on the real line. We denote by $\bar{n}(r)$ the largest number of points in any interval $[x,x+r]$, $r>0$. Define the upper uniform density ...

**2**

votes

**1**answer

78 views

### A metric associated with a continuous surjective map $f:X\to Y$

Assume that $f:(X,d_{1})\to (Y,d_{2})$ is a continuous surjective map between compact metric spaces. We define another
metric $d_{f}$ on $Y$ With $$ d_{f}(a,b)=Hd(f^{-1}(x), f^{-1}(y))$$ ...

**3**

votes

**1**answer

120 views

### Weinstein's local classification of Lagrangian foliations

In the paper "Symplectic manifolds and their Lagrangian submanifolds", Weinstein showed that locally all the Lagrangian foliations are symplectomorhic to the fiber foliation of a cotangent bundle.
I ...

**0**

votes

**1**answer

45 views

### Nagakami behavoir

Is the sum of square Nagakami random variables Erlang distributed?
What is the distribution of euclidean norm of complex Nagakami?
Cheers!

**2**

votes

**2**answers

99 views

+100

### Projection formula for smooth representations of locally profinite groups

Let $G$ be a locally profinite (i.e., locally compact, Hausdorff, and totally disconnected) topological group, $H \le G$ a closed subgroup, $(\pi, V)$ a smooth representation of $G$, and $(\sigma, W)$ ...

**2**

votes

**0**answers

231 views

### A metric on $S^{2}$ [on hold]

Let $p:S^{3}\to S^{2}$ be the Hopf fibration $p(z,w)= (\parallel z\parallel^{2}-\parallel w\parallel^{2},\;2z\bar{w})$.
Define a metric on $S^{2}$ as follows:
$$d(x,y)=Hd(p^{-1}(x), ...

**0**

votes

**0**answers

47 views

### Zero-mean assumptions concerning r.d.'s when reading graduate-level probability texts

I'm reading through T. Taos book on random matrices (to be found here), and it is frequent to make without-loss-of-generality certain reductions when proving theorems as for example Hoeffdings ...

**10**

votes

**4**answers

2k views

### An easy proof that S(n) does not embed into A(n+1)?

Rotman's book An Introduction to the Theory of Groups (Fourth Edition) asks, on page 22, Exercise 2.8, to show that S(n) cannot be embedded in A(n+1), where S(n) = the symmetric group on n elements, ...

**8**

votes

**1**answer

984 views

### How can I have a copy of this old paper by Frobenius?

How can I have a copy of this old paper and a translation of it?
Frobenius, G. (1902). Uber primitive Gruppen des Grades n und der Klasse n - 1. S. B. Akad. Berlin 1902, 455-459.

**-5**

votes

**0**answers

43 views

### What is induction to n^(n-1) >= n! for n=9,10 [on hold]

Can somebody help me? How do I prove n^(n-1) >= n! for n=9,10..... by induction?

**1**

vote

**1**answer

146 views

### Can (how) one distinguish germs of continuous functions by a countable set of params?

Continuous functions can be distinguished by their values at say rational points of [0 1].
Germs of analytic functions can be distinguished by derivatives at a point.
So in both cases we see ...

**5**

votes

**1**answer

207 views

### Dimension of Hilbert scheme?

I have a subvariety $V \subset X$, and I want to compute the dimension of the connected component of $\textrm{Hilb}(X)$ containing $[V]$.
I can give explicit deformations of $V$ showing that the ...

**2**

votes

**1**answer

526 views

### Efficient algorithms to determine the roots of: $p(x) = r^x $ in the finite field $GF(q)$, where $r$ a primitive root of the field

I need to make sure that no efficient (i.e., polynomial time) algorithm exists for the following problem:
Exponentiating Polynomial Root Problem (EPRP)
Let $p(x)$ be a polynomial with $\deg(p) \geq ...

**5**

votes

**1**answer

209 views

### Parity of primes [duplicate]

While working on a completely different (combinatorial) problem, I ran a simple program to calculate the parity of the first ~50000 primes (number of 1s in their binary representation modulo 2). The ...

**2**

votes

**0**answers

70 views

### A homological criterion for collapsibility?

On page 256 of Kirby and Siebenmann one finds the following lemma (its proof an "elementary exercise", so they only give a hint):
Taking $A$ to be a point and iterating this collapsing lemma, this ...

**4**

votes

**1**answer

243 views

### Strong Law of Large Numbers for arrays of partly dependent random variables

Suppose $X_1$, $X_2$ are two independent real-valued random variables. Let $F$ be a continuous (unbounded) function from $\mathbb{R^2}$ to $\mathbb{R}$. Assume that the necessary measurability and ...

**7**

votes

**3**answers

1k views

### Use of Indiscernibles in Model Theory

Hi, what is the main use of Indiscernibles in model theory? reading through Chang and Keisler's Model Theory it seems that the main motivation for Indicernibles is for getting many non isomorphic ...

**0**

votes

**1**answer

117 views

### Question on the partial differential equations in complex space

As is known most of the theory now developed for partial differential equations is in the real space, especially function space like Sobolev space, BMO space, $L^p$ space, etc. However is there some ...

**2**

votes

**0**answers

114 views

### What are the minimal degrees of the real and imaginary part of an algebraic complex number? [on hold]

Let $z=a+bi\in\mathbb C$ with $b\ne0$ be an algebraic complex number of minimal degree $n$. It is obvious that $a=\dfrac {z+\bar{z}}2$ and $b=\dfrac {z-\bar{z}}{2i}$ are also algebraic. For $n=3$, it ...

**7**

votes

**2**answers

682 views

### Modular congruences related to sums of Catalan numbers

I am curious if somebody can be helpful concerning the following
experimental observation:
There exist two rational sequences $\alpha_0,\alpha_1,\dots$ and
$\beta_0,\beta_1,\dots$, both with values ...

**2**

votes

**1**answer

116 views

### Various definitions of the Bruhat decomposition of the affine Grassmannian

Let $G$ be a connected, simply connected, complex, semi-simple group with affine Grassmannian $\mathcal Gr \cong G(\mathbb C[t,t^{-1}])/G(\mathbb C[t])$. Fix a choice of maximal torus $T \subset G$ ...

**3**

votes

**1**answer

139 views

### Chances for a cosine polynomial to be positive at a point

Let $k_1,\ldots,k_n$ be distinct integers. Let $s_n(t)=\cos (k_1t)+\cdots+\cos (k_nt)$ be a trigonometric sum. Consider any interval $I\subset [-\pi,\pi)$ of length $\delta=\delta(n)$. Let $\,U$ be a ...

**3**

votes

**3**answers

1k views

### Is there a standard notation for binary relations in category theory?

In set theory, I learned that a binary relation is simply a subset of a Cartesian product. Moving on to category theory, it seems that this definition is not enough. Just as a function is no longer ...

**0**

votes

**0**answers

53 views

### Spectral sequence and HOM functor

I work with the category $A-{\rm Mod}$ of left modules over a unital ring $A$, but I could ask the same question for any abelian category with enough projectives. Let $M$ and $N$ be two $A$-modules ...

**4**

votes

**4**answers

446 views

### Consecutive Integer Squared Square

Is it possible to construct a squared square out of consecutive integer squares?
Be it 1,2,3,...n or k,k+1,k+2,...n.

**8**

votes

**2**answers

327 views

+50

### Separating points in the plane II

Let A be a set of $2m$ points on the plane so that no open set of diameter $2$ has more than m of them. Define $A+A+...+A$ ($k$ times) to be the multiset of $k$-sums from $A$. That is, we consider all ...

**-7**

votes

**0**answers

82 views

### An alternative Generalization of The Riemann Zeta Function [on hold]

If we consider the Riemann Zeta Function as follows
$$\zeta(s)=\underset{n=1}{\overset{\infty}{\sum}}\cfrac{1}{n^{s}}=\cfrac{1}{1^{s}}+\cfrac{1}{2^{s}}+\cfrac{1}{3^{s}}+\cdots.$$
could we generalize ...

**0**

votes

**0**answers

26 views

### Number of graphs with n vertices and k edges up to isomorphy [duplicate]

Is there a way to figure out the number of individual graphs with n vertices and k edges up to isomorphy?
I'm particularly looking at graphs with:
n = 25, k = 50
n = 50, k = 170
n = 100, k = 700

**1**

vote

**3**answers

373 views

### What is the extra property of this sheaf?

I have a particular mathematical structure, and I think it would be enlightening to try to place it in a categorical context.
The ...

**7**

votes

**1**answer

333 views

### What is the decomposition group at $p$ in the Galois group unramified outside $\ell?$

Let $p\ne\ell$ be two prime numbers, and let $K_{\ell}$ be the maximal extension of $\mathbb Q$ ramified only at $\ell$ and $\infty$ (i.e. it is the fixed subfield of $\overline{\mathbb Q}$ by all the ...

**0**

votes

**0**answers

27 views

### Rate of convergence in narrow convergence

Does anyone help me in the following question?
I have a sequence of probability measures $\mu_n$ and know that $\mu_n$ converges narrowly to a probability measure $\mu$. Is there any way to estimate ...

**0**

votes

**0**answers

10 views

### Bounding a function pointwise from bound on the expectation [migrated]

Suppose we have a Lipschitz continuous positive function with bounded expectation Eg < a. What can be said about the point-wise bound ?