30,324 research outputs found
Semi-local simple connectedness of non-collapsing Ricci limit spaces
Let be a non-collapsing Ricci limit space and let . We show that
for any , there is such that every loop in is
contractible in , where . In particular,
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 by a nearby point. Fixed some typo
Consecutive primes and Legendre symbols
Let be any positive integer and let . We
show that for some constanst there are infinitely many integers
with such that
for all ,
where denotes the -th prime, and denotes the
Legendre symbol for any odd prime . We also prove that under the Generalized
Riemann Hypothesis there are infinitely many positive integers such that
is a primitive root modulo for any distinct and
among .Comment: 12 pages, final published versio
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 is a positive integer, and
, then we have where
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
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
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