All Questions

4
votes
1answer
22 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 $E_n = \{ k \in \{1,2, .., n\} : \pi(k) \equiv 0 \mod 2 \}$ and $O_n = \{ k ...
0
votes
0answers
15 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
0answers
23 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 ...
1
vote
0answers
26 views

What is the joint distribution of sample mean and sample variance of normal distribution?

${X_i} \sim N\left( {\mu ,{\sigma ^2}} \right)$, define $\bar X = \sum\limits_{i = 1}^n {{X_i}} $, ${S^2} = \frac{1}{{n - 1}}\sum\limits_{n = 1}^n {{{\left( {{X_i} - \bar X} \right)}^2}}$. What is the ...
6
votes
0answers
68 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 ...
20
votes
0answers
167 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 ...
1
vote
0answers
33 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
1answer
59 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
0answers
58 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
3answers
126 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
1answer
102 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 ...
7
votes
3answers
348 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 ...
2
votes
0answers
35 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
0answers
18 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
1answer
85 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
0answers
89 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 ...
3
votes
0answers
36 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
0answers
82 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)?
-2
votes
0answers
72 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
1answer
44 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
0answers
61 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
0answers
72 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 ...
-1
votes
1answer
84 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
0answers
99 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
2answers
140 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
1answer
106 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
0answers
23 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
0answers
157 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
1answer
43 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 $K$? ...
8
votes
1answer
142 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
1answer
136 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 ...
4
votes
0answers
55 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
1answer
82 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
0answers
57 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
0answers
107 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
0answers
245 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 ...
3
votes
0answers
66 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
0answers
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
1answer
93 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
1answer
56 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
1answer
71 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
1answer
97 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
0answers
30 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
1answer
94 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
0answers
97 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
2answers
286 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
0answers
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
1answer
69 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 $$ ...
-6
votes
0answers
123 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
1answer
86 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: $\frac{n}2+\sqrt{n} \leq t \leq \frac{9n}{10}$. ...

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