42,449 research outputs found
Semi-Finite Forms of Bilateral Basic Hypergeometric Series
We show that several classical bilateral summation and transformation
formulas have semi-finite forms. We obtain these semi-finite forms from
unilateral summation and transformation formulas. Our method can be applied to
derive Ramanujan's summation, Bailey's transformations,
and Bailey's summation.Comment: 8 pages. accepted by Proc. Amer. Math. So
The transmission or scattering of elastic waves by an inhomogeneity of simple geometry: A comparison of theories
The extended method of equivalent inclusion developed is applied to study the specific wave problems of the transmission of elastic waves in an infinite medium containing a layer of inhomogeneity, and of the scattering of elastic waves in an infinite medium containing a perfect spherical inhomogeneity. The eigenstrains are expanded as a geometric series and the method of integration for the inhomogeneous Helmholtz operator given by Fu and Mura is adopted. The results obtained by using a limited number of terms in the eigenstrain expansion are compared with exact solutions for the layer problem and for a perfect sphere. Two parameters are singled out for this comparison: the ratio of elastic moduli, and the ratio of the mass densities. General trends for three different situations are shown
On the origin and energy of oscillatory earthquake waves
A critical survey of the different modes of generation of oscillatory earthquake waves is given. The consequences of the failure of Hooke's law on the dispersion of waves is indicated. The energy of an earthquake is considered from the theory of elasticity with a discussion of Sezawa's result
On seismic rays and waves (Part One)
The equations of motion of an elastico-viscous medium in which the material constants
vary with position are deduced. These can be put into the form of a wave equation only
when the gradients of the constants are small. By the method of Sommerfeld and Runge
these equations are compared with the equation of the characteristic function, whence the
condition for the validity of the ray method is obtained. It is similar to De Broglie's criterion
in wave mechanics. Expressed in terms of measurable quantities in seismology, the condition
is applied to the data recently obtained by Gutenberg for the upper layers of the earth's
crust. The equation of the characteristic function is used in deriving the forms of the ray
paths for several particular velocity functions, following a method previously used by
Epstein
Faulhaber's Theorem on Power Sums
We observe that the classical Faulhaber's theorem on sums of odd powers also
holds for an arbitrary arithmetic progression, namely, the odd power sums of
any arithmetic progression is a polynomial in
. While this assertion can be deduced from the original
Fauhalber's theorem, we give an alternative formula in terms of the Bernoulli
polynomials. Moreover, by utilizing the central factorial numbers as in the
approach of Knuth, we derive formulas for -fold sums of powers without
resorting to the notion of -reflexive functions. We also provide formulas
for the -fold alternating sums of powers in terms of Euler polynomials.Comment: 12 pages, revised version, to appear in Discrete Mathematic
Optimal nonlocal multipartite entanglement concentration based on projection measurements
We propose an optimal nonlocal entanglement concentration protocol (ECP) for
multi-photon systems in a partially entangled pure state, resorting to the
projection measurement on an additional photon. One party in quantum
communication first performs a parity-check measurement on her photon in an
N-photon system and an additional photon, and then she projects the additional
photon into an orthogonal Hilbert space for dividing the original -photon
systems into two groups. In the first group, the N parties will obtain a subset
of -photon systems in a maximally entangled state. In the second group, they
will obtain some less-entangled N-photon systems which are the resource for the
entanglement concentration in the next round. By iterating the entanglement
concentration process several times, the present ECP has the maximal success
probability which is just equivalent to the entanglement of the partially
entangled state. That is, this ECP is an optimal one.Comment: 5 pages, 4 figure
Efficient multipartite entanglement purification with the entanglement link from a subspace
We present an efficient multipartite entanglement purification protocol
(MEPP) for N-photon systems in a Greenberger-Horne-Zeilinger state with
parity-check detectors. It contains two parts. One is the conventional MEPP
with which the parties can obtain a high-fidelity N-photon ensemble directly,
similar to the MEPP with controlled-not gates. The other is our recycling MEPP
in which the entanglement link is used to produce some -photon entangled
systems from entangled N'-photon subsystems (2 \leq N'<N) coming from the
instances which are just discarded in all existing conventional MEPPs. The
entangled N'-photon subsystems are obtained efficiently by measuring the
photons with potential bit-flip errors. With these two parts, the present MEPP
has a higher efficiency than all other conventional MEPPs.Comment: 17 pages, 9 figures, 2 tables. We correct the error in the address of
the author in the published version (Phys. Rev. A 84, 052312 (2011)
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