58,536 research outputs found
Binomial coefficients, Catalan numbers and Lucas quotients
Let be an odd prime and let be integers with and . In this paper we determine
mod for ; for example,
where is the Jacobi symbol, and is the Lucas
sequence given by , and for
. As an application, we determine modulo for any integer , where denotes the
Catalan number . We also pose some related conjectures.Comment: 24 pages. Correct few typo
Projection Measurement of the Maximally Entangled N-Photon State for a Demonstration of N-Photon de Broglie Wavelength
We construct a projection measurement process for the maximally entangled
N-photon state (the NOON-state) with only linear optical elements and
photodetectors. This measurement process will give null result for any N-photon
state that is orthogonal to the NOON state. We examine the projection process
in more detail for N=4 by applying it to a four-photon state from type-II
parametric down-conversion. This demonstrates an orthogonal projection
measurement with a null result. This null result corresponds to a dip in a
generalized Hong-Ou-Mandel interferometer for four photons. We find that the
depth of the dip in this arrangement can be used to distinguish a genuine
entangled four-photon state from two separate pairs of photons. We next apply
the NOON state projection measurement to a four-photon superposition state from
two perpendicularly oriented type-I parametric down-conversion processes. A
successful NOON state projection is demonstrated with the appearance of the
four-photon de Broglie wavelength in the interference fringe pattern.Comment: 8 pages, 3 figures, new title, some content change, replaced Fig.
Super congruences and Euler numbers
Let be a prime. We prove that
, where E_0,E_1,E_2,... are Euler numbers. Our new approach is of
combinatorial nature. We also formulate many conjectures concerning super
congruences and relate most of them to Euler numbers or Bernoulli numbers.
Motivated by our investigation of super congruences, we also raise a conjecture
on 7 new series for , and the constant
(with (-) the Jacobi symbol), two of which are
and
\sum_{k>0}(15k-4)(-27)^{k-1}/(k^3\binom{2k}{k}^2\binom{3k}k)=K.$
Thickness dependent magnetotransport in ultra-thin manganite films
To understand the near-interface magnetism in manganites, uniform, ultra-thin
films of La_{0.67}Sr_{0.33}MnO_3 were grown epitaxially on single crystal (001)
LaAlO_3 and (110) NdGaO_3 substrates. The temperature and magnetic field
dependent film resistance is used to probe the film's structural and magnetic
properties. A surface and/or interface related dead-layer is inferred from the
thickness dependent resistance and magnetoresistance. The total thickness of
the dead layer is estimated to be for films on NdGaO_3 and for films on LaAlO_3.Comment: 11 pages, 4 figure
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