# All Questions

**6**

votes

**0**answers

38 views

### Determinant of a determinant

Consider an $mn \times mn$ matrix over a commutative ring $A$, divided into $n \times n$ blocks that commute pairwise. One can pretend that each of the $m^2$ blocks is a number and apply the $m ...

**0**

votes

**0**answers

13 views

### Are there two-sided $\varepsilon$-expanders with independent sets of size $(1-\varepsilon)n$?

Terry Tao's notes on expander graphs has the following exercise:
Exercise 13 Let $G$ be a $k$-regular graph on $n$ vertices that is a two-sided $\epsilon$-expander for some $n > k \geq 1$ and ...

**0**

votes

**0**answers

25 views

### semi-classical Green's function

I am reading Gutzwiller's papers on the relation between Hamiltonian flows and solution to Schrodinger equations.
In the two papers, he gave a semi-classical approximation of the Green's function to ...

**1**

vote

**0**answers

36 views

### Abelian covers of compact Kahler manifolds

Let $X$ be a compact Kahler manifold and $A\subset H_1(X,\mathbb{Z})$ be a subgroup.
Corresponding to $A$ there is an abelian covering $X_A \to X$ with $Deck(X_A)=H_1(X,\mathbb{Z})/A$. For example if ...

**1**

vote

**3**answers

70 views

### Estimating a sum

Good morning everyone,
I would like to make a question about estimating a sum.
Consider the following sum
$$S_n:=\sum_{k=0}^{n-1} \frac{k^2}{(n-k)^2 (n+k)^2} $$
It is easy to see that this sum is ...

**2**

votes

**1**answer

64 views

### What is known about this series?

I recently came across the following function which intrigues me:
\begin{equation}
f(\alpha):=\sum_{i=0}^\infty \frac{\alpha^{i(i+1)/2}}{i!}.
\end{equation}
For $-1\leq \alpha\leq 1$ this function is ...

**3**

votes

**2**answers

154 views

### Difference between ZFC and ZF+GCH

I hear that the axiom of choice (AC) derives from
The generalized continuum hypothesis(GCH).
And also hear that both AC and GCH is independent of
Zermelo–Fraenkel set theory(ZF).
So, I'm just ...

**1**

vote

**0**answers

27 views

### Outer automorphisms of direct product of residually finite groups

Let $A,B$ be finite generated residually finite groups such that $\mathop{Out}(A)$ and $\mathop{Out}(B)$ are residually finite.
Is $\mathop{Out}(A\times B)$ residually finite?
If not, what is the ...

**-2**

votes

**0**answers

14 views

### Probability of having a connected network in a random graph [on hold]

I'm trying to solve this programming problem out of interest in my spare time and want to make sure my maths is correct.
"The people of Absurdistan discovered how to build roads only last year. After ...

**1**

vote

**1**answer

61 views

### Relations between characteristic classes of a group and the Stiefel-Whitney/Pontryagin classes

Let $X$ be a closed manifold and $BG$ be the classifying space of a group $G$
A map from $X$ to $BG$ induce a map from $H^*(BG,Z)$ to $H^*(X,Z)$ by pull back.
Let $GH^*(X,Z)$ be the subgroup of ...

**0**

votes

**0**answers

59 views

### Argument againts the $abcd$ conjecture with extra gcd conditions

Got an argument and numerical support againts the $abcd$
conjecture with extra gcd conditions (observe that this
is different from the $abc$ and the $abcd$ conjectures).
This thesis p. 20 defines the ...

**2**

votes

**0**answers

27 views

### Optimal planar net for catching convex shapes

Imagine you want to make a net out of string to filter and catch objects of
a certain size, minimizing the length of string employed.
(This actually arises in filtering biological impurities from ...

**-4**

votes

**0**answers

72 views

### I am doing P.H.D. in İstanbul University. My research is about Character Theory of Finite groups [on hold]

How can I define the p(t)-adic valuation and absolute value on F(t)?

**-1**

votes

**0**answers

56 views

### Algebra Constructions [on hold]

What kind of constructions of associative algebras are normally used in theory and examples besides the following ones:
group algebra
monoid algebra (such as Solomon-Tits algebra)
tensor product
...

**1**

vote

**1**answer

41 views

### locally topologically finitely generated t.d.l.c. group

I am trying to find an example of the following situation.
$G$ is a t.d.l.c. (totally disconnected locally compact) $\sigma$-compact topological group in which every compact open subgroup is ...

**-1**

votes

**0**answers

53 views

### Category of compact manifold [on hold]

Question 1: What we know about the sub-category $Met$ of $Top$ whose objects are metrizable topological spaces and whose morphisms are the isometries.
Question 2: Let $X$ be any compact manifold. We ...

**0**

votes

**0**answers

63 views

### A curious property of Ramanujan's function $\tau(n)$

As it is well known, Ramanujan's $\tau(n)$ function can be defined through the power series expansion of the modular discriminant:
$$\Delta(q)=q\prod\limits_{n=1}^\infty (1-q^n)^{24}=\sum ...

**0**

votes

**1**answer

73 views

### Sylow-subgroups of the group of units of a finite field [on hold]

Let $K$ be a finite field with $p^n$ elements. Then the group of unit is cyclic of order $p^n-1$. What are the primes which are dividing $p^n-1$? What are the Sylow-subgroups of the group of units of ...

**-3**

votes

**0**answers

90 views

### Finding closed form of : $\int_0^1 \frac{\log(1+x)}{1+x}\log\left(\log\left(\frac{1}{x}\right)\right) \ dx$

$ \newcommand{\pars}[1]{\left(\, #1 \,\right)} \newcommand{\dd}{{\rm d}}$
Finding closed form of the below:
$$\int_0^1 \frac{\log(1+x)}{1+x}\log\left(\log\left(\frac{1}{x}\right)\right) \ dx$$
This ...

**-5**

votes

**0**answers

46 views

### Legendre symbol problem [on hold]

Let $p,q$ different odd prime numbers and $a$ a positive odd number, Prove that :
$$p \equiv - q\ \ \ \ (mod \ \ 4a )\Longrightarrow \Bigg(\frac{a}{p}\Bigg) =\Bigg(\frac{a}{q}\Bigg) $$
Where ...

**4**

votes

**2**answers

101 views

### Bound on sum of complex summands involving binomial coefficients

I am trying to find the asymptotic behavior of the sum:
$$ \sum^n_{i=0} \begin{pmatrix} 2n \\ i \end{pmatrix} x^i y^{2n-i}$$
as $n\rightarrow\infty$. Here $x$, $y$ are complex numbers and I have ...

**2**

votes

**1**answer

92 views

### Canonical divisor for conic bundles

Is there a general formula for the canonical divisor $K_X$ of a smooth conic bundle $X$?
Motivation: for smooth hypersurfaces of degree $d$ in $\mathbb{P}^n$, $K_X = \mathcal{O}_X(d-n-1)$. But ...

**-2**

votes

**0**answers

42 views

### How to show this Legendre Symbol Problem [on hold]

Let $n \in \mathbb{N}$ and $p$ an odd prime number such as $p \nmid n$,
Prove that:
$\exists x, y \in \mathbb{Z};\,\, \gcd(x,y) = 1$ such as $x^{2} +ny^{2} \equiv 0\, (\mod p) \iff ...

**0**

votes

**0**answers

22 views

### Variance of sums of correlated variables when sampling without replacement?

Background
Let $X_t$ be an alternating renewal process, a stochastic process on the state space {0, 1} where 0 = 'Broken' and 1 = 'Working'. Let $U$ be the cumulative sojourn working during an ...

**14**

votes

**0**answers

142 views

### Are there periodicity phenomena in manifold topology with odd period?

The study of $n$-manifolds has some well-known periodicities in $n$ with period a power of $2$:
$n \bmod 2$ is important. Poincaré duality implies that odd-dimensional compact oriented manifolds ...

**-1**

votes

**1**answer

41 views

### extension of a continuous function [on hold]

Please is it true that if $f:K\to \mathbb{R}$ is a continuous function of a comact set $K\subset\mathbb{R}^m$ then $f$ can be extended to a continuous function of some open neighbourhood of $K$?
...

**6**

votes

**1**answer

118 views

### Is there a proof of Warning's Second Theorem using p-adic cohomology?

Let $\mathbb{F}_q$ be a finite field, $n \in \mathbb{Z}^+$, and $f_1,\ldots,f_r \in \mathbb{F}_q[t_1,\ldots,t_n]$ with $\operatorname{deg}(f_i) = d_i$. Put $d = \sum_{i=1}^n d_i$ and suppose $d< ...

**6**

votes

**1**answer

106 views

### A question about three forcings

Let $P_1$ be the finite support iteration of random forcings of length $\omega$. Let $P_2 = \text{Random} \times \text{Random}$ and $P_3 = \text{Random} \times \text{Cohen}$. Are $P_i, P_j$, for $1 ...

**3**

votes

**0**answers

50 views

### The representation-theoretic nature of an operator resolvent

Consider parameter $s$ in definition of $R(s,A)=(s I - A)^{-1}$ where $A$ is a linear operator in a vector space $X$. When $X$ is over $\mathbb{C}$, then $s$ is thought to be a complex number.
Now ...

**3**

votes

**1**answer

74 views

### expression for infinite series with powers of factorial in denominator

The series
$$\sum_{k=0}^\infty \frac{\exp(c k \beta)}{(k!)^\beta} $$
has come up when I'm trying to apply the methodology in this paper (http://www.ism.ac.jp/~eguchi/pdf/Robustify_MLE.pdf) to Poisson ...

**-2**

votes

**0**answers

56 views

### Combinatorical configuration. Proof [on hold]

Given integers $k$ and v with $1 < k < v$ show that there exists a
$$(v, \binom v k ,\binom {v-1} {k-1}, k, \binom {v-2}{k-2} ) $$ design.
Please give me a hint.
For: $(a, b, c, d,e )$
...

**4**

votes

**0**answers

105 views

### The sum of all the elements of every non empty subset of $A$ is not a multiple of $n$

Let $N=\{1,2,\ldots ,n\},n>1$. We wish to construct a set $A\subseteq N$ with the property:
The sum of all the elements of every non empty subset of $A$ is not a
multiple of $n$.
...

**11**

votes

**0**answers

238 views

### Chiral categories versus braided monoidal categories

Let $X$ be a curve over $\mathbf{C}$. As I understand from the 2008 Talbot notes, a chiral category on $X$ consists of a crystal of categories on the Ran space $\mathrm{Ran}(X)$ (see these notes of ...

**2**

votes

**0**answers

57 views

### Faithful and weakly-mixing representations of Property (T) groups in relation to left regular rep

Is it known that: Any countable Property (T) group (or more generally, a non-amenable group) has a faithful, weakly-mixing representation which is NOT weakly included in its left regular ...

**0**

votes

**0**answers

29 views

### Saddle point method for asymptotic expansion [migrated]

Any idea on how to derive an asymptotic expansion of the following integral for $x$ around zero (using Saddle point method):
$$\int_{0}^{x}\left(u+\alpha\right)^{-a - \left(b-a\right)e^{-\lambda ...

**0**

votes

**1**answer

82 views

### Probability of connected graph on torus

Let $G = (V, E)$ be a graph on n vertices constructed in the following way:
Each vertex $v \in V$ is positioned uniform randomly in $[0, 1] × [0, 1]$.
Connect two vertices $u, v \in V$ if $d(v,u) ≤ ...

**0**

votes

**1**answer

51 views

### Is it possible to define the density of the logistic map for $x<0$?

Probability density functions (PDF's) have inherent connections to the field of
Dynamical Systems.
The motivation for this question can be found in: ...

**3**

votes

**1**answer

66 views

### Estimate for Levy metric

In the Encyclopedia of Mathematics there is an inequality for Levy metric ($d_L$):
$$d_L(E,F) \leq \{\beta_r(F)\}^{r/(r+1)},$$
where $E$ is a a distribution that is degenerate at zero, $\beta_r(F)$, ...

**10**

votes

**1**answer

93 views

### Weyl group actions on 0-weight spaces

For a complex simple Lie group G with a maximal torus T, we can take a highest-weight representation V of G and look at the 0-weight space, i.e. the subspace of V of elements invariant under T. This ...

**2**

votes

**0**answers

29 views

### Fixing (non)-independency of a the subfamilies of finitely many events.

I'm would be interesting in any construction of a probability space with n events (n is given), where for every subset of these events, it is also given whether or not, the family is mutually ...

**2**

votes

**1**answer

92 views

### Lower regularity version of Moser's theorem on volume elements

A theorem of Moser, published in "On the Volume Elements of a Manifold" (Transactions of the Americal Mathematical Society 120, 1965), shows that if a $C^\infty$ compact manifold $M$ has two ...

**6**

votes

**0**answers

96 views

### Cramer's theorem in Hilbert spaces

I am interested under what conditions Cramer's theorem applies in random variables taking values in Hilbert spaces. Following these lecture notes, but using a Hilbert space:
Let $X_1,X_2,\cdots$, be ...

**12**

votes

**2**answers

270 views

### How to compute $\pi_{3}$ of $L(p,q)\# L(p',q')$?

Let $L(p,q)$ be a 3-dimensional lens space, and let $L(p',q')$ be another. Is there any known result concerning the 3rd homotopy group of the connected sum $L(p,q)\# L(p',q')$? If not, I am interested ...

**-3**

votes

**0**answers

51 views

### Mathematically compute - species diversity [on hold]

Using a simple model, and having these as paramters
numberofspecies <- 100
meaninitialpopulationsize <- 50
sdloginitialpopulationsize <- 1 #determines variation in initial population ...

**4**

votes

**1**answer

68 views

### Estimate on sum of $J_n^4$

If $J_n(x)$ is the Bessel function of order $n$, we know that for all $x$, $$\sum_{n=-\infty}^{\infty} J_n^2(x)=J_0^2(x)+2\sum_{n=1}^{\infty} J_n^2(x)=1.$$
What is known about
$$
...

**-5**

votes

**0**answers

119 views

### Mathematics Research and The Internet [on hold]

I reformulate here a question about Mathematics and The Internet. My questions are: What was the vital role of Mathematics research in the foundation of the Intranet ($\rightarrow{Internet}$) and, do ...

**1**

vote

**1**answer

85 views

### Upper bound of sum of binomial coefficients

I am looking for an upper bound - up to constant factor - for:
$\sum_{k=t}^{t+l} {n \choose k} \cdot 2^{-n}$ where:
The values of $t$ are between: $\frac{n}2+\sqrt{n} \leq t \leq \frac{9n}{10}$. ...

**3**

votes

**0**answers

44 views

### Large Deviations: Exponential decay in normed spaces

Let $(X_1,X_2,\cdots)$ be a sequence of independent and identically distributed random variables taking values in some general normed space $(V,||\cdot||)$. Denote $\mu=E[X_1]$ and ...

**-1**

votes

**0**answers

61 views

### Integration of the reciprocal of sum exponential [migrated]

Any one know the method to do the integration as
$$\int\frac{x^2\cdot \exp (-ax^2) \exp(-bx^2)}{\exp(-ax^2)+\exp(-bx^2)}dx$$
It can be simplified as
$$\int\frac{x^2}{\exp(ax^2)+\exp(bx^2)}dx$$
...

**4**

votes

**0**answers

185 views

### “Partition” of a smooth function in $\mathbb R^2$

This is a question asking for reference.
I have a proof of the following.
Let $f=f(x,y)$ be a smooth function in $\mathbb R^2$ which vanishes at the origin. Then there exist smooth functions ...