Semi-local simple connectedness of non-collapsing Ricci limit spaces

Let $X$ be a non-collapsing Ricci limit space and let $x\in X$. We show that for any $\epsilon>0$, there is $r>0$ such that every loop in $B_t(x)$ is contractible in $B_{(1+\epsilon)t}(x)$, where $t\in(0,r]$. In particular, $X$ is semi-locally simply connected.Comment: Slightly modified the proof of Theorem 3.5 to fix a minor error on local covers. Slightly modified the proofs of Lemmas 3.2, 3.6, and 3.8 to fix a minor error on estimating $\rho(t,x)$ by a nearby point. Fixed some typo

Identities concerning Bernoulli and Euler polynomials

We establish two general identities for Bernoulli and Euler polynomials, which are of a new type and have many consequences. The most striking result in this paper is as follows: If $n$ is a positive integer, $r+s+t=n$ and $x+y+z=1$, then we have $$rF(s,t;x,y)+sF(t,r;y,z)+tF(r,s;z,x)=0$$ where $$F(s,t;x,y):=\sum_{k=0}^n(-1)^k\binom{s}{k}\binom{t}{n-k}B_{n-k}(x)B_k(y).$$ This symmetric relation implies the curious identities of Miki and Matiyasevich as well as some new identities for Bernoulli polynomials such as \sum_{k=0}^n\binom{n}{k}^2B_k(x)B_{n-k}(x)=2\sum^n\Sb k=0 k\not=n-1\endSb\binom{n}{k}\binom{n+k-1}{k}B_k(x)B_{n-k}.Comment: 21 page

Consecutive primes and Legendre symbols

Let $m$ be any positive integer and let $\delta_1,\delta_2\in\{1,-1\}$. We show that for some constanst $C_m>0$ there are infinitely many integers $n>1$ with $p_{n+m}-p_n\le C_m$ such that $$\left(\frac{p_{n+i}}{p_{n+j}}\right)=\delta_1\ \quad\text{and}\ \quad\left(\frac{p_{n+j}}{p_{n+i}}\right)=\delta_2$$ for all $0\le i<j\le m$, where $p_k$ denotes the $k$-th prime, and $(\frac {\cdot}p)$ denotes the Legendre symbol for any odd prime $p$. We also prove that under the Generalized Riemann Hypothesis there are infinitely many positive integers $n$ such that $p_{n+i}$ is a primitive root modulo $p_{n+j}$ for any distinct $i$ and $j$ among $0,1,\ldots,m$.Comment: 12 pages, final published versio

Polarization Entanglement Purification using Spatial Entanglement

Parametric down-conversion can produce photons that are entangled both in polarization and in space. Here we show how the spatial entanglement can be used to purify the polarization entanglement using only linear optical elements. Spatial entanglement as an additional resource leads to a substantial improvement in entanglement output compared to a previous scheme. Interestingly, in the present context the thermal character of down-conversion sources can be turned into an advantage. Our scheme is realizable with current technology.Comment: 5 pages, 2 figure