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 $_1\psi_1$ summation, Bailey's $_2\psi_2$ transformations, and Bailey's $_6\psi_6$ 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 $a+b, a+2b, ..., a+nb$ is a polynomial in $na+n(n+1)b/2$. 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 $r$-fold sums of powers without resorting to the notion of $r$-reflexive functions. We also provide formulas for the $r$-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 $N$-photon systems into two groups. In the first group, the N parties will obtain a subset of $N$-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 $N$-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)