# All Questions

**0**

votes

**0**answers

5 views

### Cosets of the fixer of an action of a monoid on a finite set

Let $M$ be a monoid that acts transitively from the right on a finite set $X$.
Assume furthermore that the action of $M$ on $X$ induces for every $m \in M$ a bijection on $X \to X, x \mapsto x.m$.
Let
...

**0**

votes

**0**answers

25 views

### Self-similarity for simple algebraic structures

I'm doing this thread because I have some ideas about how to define self-similarity in algebra, but I don't know if this is known at all. Any critics, comments and references are more than welcomed. ...

**0**

votes

**0**answers

18 views

### Is there some Riemannian manifold's version of Whitney theorem?

Given any Riemannian or Semi-Riemannian manifold $(M,g)$, does there exist a Eucildean space $(E,g^\prime)$ of enough high dimension with metric $g^\prime=diag\{-1,-1,...,+1,+1,...\}$ with any n ...

**2**

votes

**0**answers

13 views

### An upper bound for a average of a function in $L_{p}([0,1))$

Suppose that $ f $ is $ 1 $-periodic and that $ f \in {L^{p}}([0,1) $, where $ p > 1 $. Let
$$
(D_{n})_{n \in \mathbb{N}_{0}} =
\left( \left\{
I^{n}_{j},~
1\leq j \leq 2^{n} \}
\right\} \right)_{n ...

**1**

vote

**0**answers

8 views

### Simultaneous vanishing of convolutions of Mertens function with itself

By a theorem of Landau, we know that all the step functions ($k\geq 1$)
$$M_k(x)=\frac{1}{2\pi i}\int^{2+i\infty}_{2-i\infty}\frac{x^sds}{\zeta^k(s)s}=\sum_{n\leq ...

**2**

votes

**0**answers

19 views

### centeredness in forcing iterations

Suppose $A$ is a complete subalgebra of a complete boolean algebra $B$, and $B$ is $\kappa$-centered. Let $G \subset A$ be a generic ultrafilter. Is $B/G$ $\kappa$-centered in $V[G]$?
Naively, we ...

**-2**

votes

**0**answers

18 views

### Treating The Differential of Independent Variables in Integral

I need some comments from real infinitessimal calculus and real analysis point of view like limit-theorems etc. regarding the following
Consider that B is constant ($\frac{dB}{dt}=0$)
There is a ...

**-4**

votes

**0**answers

59 views

### How to choose subject of pure math for PhD? [on hold]

I'm seeking about PhD in Maths. I'm really confused about choosing the area of research. I enjoyed studying and teaching Algebra & Topology, But I don't know how to start writing a research on ...

**1**

vote

**0**answers

33 views

### Decomposition of non-singular matrix

Is there any way to show that a non-singular matrix A can be partitioned as follows:
\begin{eqnarray*}
A&=&\left[
\begin{array}{cc}
\underset{\left( k\times k_{1}\right) ...

**2**

votes

**0**answers

50 views

### Faltings height in short exact sequences

Let $K$ be a number field and $0 \rightarrow A \rightarrow B \rightarrow C \rightarrow 0$ a short exact sequence of abelian varieties over $K$. Let $h(A)$ denote the logarithmic Faltings height ...

**-1**

votes

**0**answers

35 views

### Example of an operator whose domain is infinite dimensional but range is not closed [on hold]

Give me an example of an operator $T:D->R$, such that $D$ is an infinite dimensional Hilbert space and $R$ is not closed/dense.

**1**

vote

**0**answers

20 views

### Transversality of stable and unstable manifolds for geodesic flows associated to different metrics on the same manifold

Let $M$ be a closed smooth manifold carrying two negatively curved Riemannian metrics $g$ and $h$. Take a point $p \in M$ and vector $v \in T^{1}M$. Let $\gamma_{v,g}$ and $\gamma_{v,h}$ be the unique ...

**3**

votes

**1**answer

114 views

### Is a “central” extension of $\mathbb{Z}/m\mathbb{Z}$ by $\mathrm{GL}_n$ necessarily split?

Let $m \ge 1$ be an integer, let $k$ be a field of characteristic $0$, and let
$$
1 \rightarrow \mathrm{GL}_n \rightarrow E \rightarrow \mathbb{Z}/m\mathbb{Z} \rightarrow 1
$$
be an extension of ...

**-1**

votes

**0**answers

93 views

### Is $\{(\frac{3}{2})^{n}\}_{\mathbb{N}}$ is equidistributed in [0,1] (related open question) [on hold]

Is $\{(\frac{3}{2})^{n}\}_{\mathbb{N}}$ is dense in [0,1] (open question).
A more general question is: Is $\{(\frac{3}{2})^{n}\}_{\mathbb{N}}$ is equidistributed in [0,1]?If so, Then density ...

**0**

votes

**0**answers

18 views

### best MAX-SAT solver for ising spin glass

What is the best MAX-SAT solver problems for Ising spin glass? I tried Scip-Max-sat and open-wbo. While open-wbo cannot solve the instance with only 27 variable Scip-max-Sat fail to solve the one with ...

**9**

votes

**0**answers

86 views

### The real numbers object in Sh(Top)

If $X$ is a sober topological space, the real numbers object in the topos $\mathrm{Sh}(X)$ is the sheaf of continuous real-valued functions on $X$. This is proven very explicitly in Theorem VI.8.2 of ...

**4**

votes

**1**answer

83 views

### properties of formal delta functions

The formal delta function is
$\,\,\displaystyle\delta(x):=\sum_{n\in\mathbb Z}x^n.
$
If we agree that expressions $(x+y)^n$ for $n\in\mathbb Z$ are always expanded in non-negative powers of the second ...

**2**

votes

**0**answers

42 views

### Definition of Givental $J$-function of cotangent bundle of flag variety

I would like to know the definition of Givental $J$-function of cotangent bundle of flag variety. To state my question more precisely, let us briefly recall the definition of the Givental $J$-function ...

**0**

votes

**0**answers

14 views

### Eigenfunction of ergodic skew product fixed by commutator?

Background: Let $(Y, \mathcal{B},\mu,T)$ be an ergodic probability system and let $G$ be a compact metrizable group with compact subgroup $H$. Given a measurable map $\rho:Y \to G$. We may define the ...

**1**

vote

**0**answers

31 views

### Mutant pairs distinguished by [2,1]-colored HOMFLY polynomials

Morton and Cromwell showed that the famous mutant pair, Kinoshita-Terasaka and Conway knots, can be distinguished by HOMFLY polynomials colored by [2,1] Young diagram. Are there any other mutant pairs ...

**9**

votes

**1**answer

289 views

### Parity of the Prime Counting Function

I am interested in the distribution of the parity of $\pi(x)$, the prime counting function, over the natural numbers.
Let:
$\ \ E_n := \left\{ k \in \left\{1,\dots,n\right\} : \pi(k) \equiv 0 \mod 2 ...

**0**

votes

**0**answers

30 views

### Stochastic methods for solving very high-dimensional PDE

I am looking for stochastic methods for solving a very high-dimensional PDE (with one time dimension and very large number of spatial dimensions), which would reduce it to a lower-dimensional problem, ...

**0**

votes

**0**answers

27 views

### bounds on a series with binomial coefficients

I have the following series
$\sum\limits_{l=1}^n {n\choose l} \alpha^{\beta^l}$
where $\alpha > 0$and $0 \leq \beta \leq 1$.
Can anybody guide me how I can evaluate it or find some tight upper ...

**-2**

votes

**0**answers

62 views

### What is the joint distribution of sample mean and sample variance of normal distribution? [on hold]

${X_i} \sim N\left( {\mu ,{\sigma ^2}} \right)$, define $\bar X = \frac{1}{{n}}\sum\limits_{i = 1}^n {{X_i}} $, ${S^2} = \frac{1}{{n - 1}}\sum\limits_{n = 1}^n {{{\left( {{X_i} - \bar X} ...

**8**

votes

**1**answer

156 views

### Special fiber of $X(p)$ in characteritic $p$

Let $p \geq 5$ be a prime. Let $Y(p)$ be the fine moduli space representing elliptic curves + basis of the $p$-torsion over $\mathbb{Q}_p$ and let $Y_1(p)$ be the fine moduli space representing ...

**26**

votes

**0**answers

280 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 ...

**2**

votes

**0**answers

48 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 ...

**3**

votes

**1**answer

109 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 ...

**2**

votes

**0**answers

85 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

153 views

### Estimating a sum [on hold]

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 ...

**3**

votes

**2**answers

163 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 ...

**10**

votes

**4**answers

638 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 are independent of
Zermelo–Fraenkel set theory(ZF).
So, I'm just ...

**4**

votes

**0**answers

55 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

26 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 ...

**0**

votes

**1**answer

159 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 ...

**2**

votes

**0**answers

107 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 ...

**4**

votes

**0**answers

63 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

92 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)?

**-3**

votes

**0**answers

82 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

49 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 ...

**-2**

votes

**0**answers

70 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

84 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

90 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

110 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 ...

**6**

votes

**4**answers

258 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 ...

**3**

votes

**1**answer

121 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 ...

**0**

votes

**0**answers

26 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 ...

**15**

votes

**0**answers

187 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

44 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 $K$?
...

**9**

votes

**2**answers

204 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< ...