Some congruences involving central q-binomial coefficients

Motivated by recent works of Sun and Tauraso, we prove some variations on the Green-Krammer identity involving central q-binomial coefficients, such as $$\sum_{k=0}^{n-1}(-1)^kq^{-{k+1\choose 2}}{2k\brack k}_q \equiv (\frac{n}{5}) q^{-\lfloor n^4/5\rfloor} \pmod{\Phi_n(q)},$$ where $\big(\frac{n}{p}\big)$ is the Legendre symbol and $\Phi_n(q)$ is the $n$th cyclotomic polynomial. As consequences, we deduce that \sum_{k=0}^{3^a m-1} q^{k}{2k\brack k}_q &\equiv 0 \pmod{(1-q^{3^a})/(1-q)}, \sum_{k=0}^{5^a m-1}(-1)^kq^{-{k+1\choose 2}}{2k\brack k}_q &\equiv 0 \pmod{(1-q^{5^a})/(1-q)}, for $a,m\geq 1$, the first one being a partial q-analogue of the Strauss-Shallit-Zagier congruence modulo powers of 3. Several related conjectures are proposed.Comment: 16 pages, detailed proofs of Theorems 4.1 and 4.3 are added, to appear in Adv. Appl. Mat

Existence of weak solutions for the generalized Navier-Stokes equations with damping

In this work we consider the generalized Navier-Stokes equations with the presence of a damping term in the momentum equation. The problem studied here derives from the set of equations which govern isothermal flows of incompressible and homogeneous non-Newtonian fluids. For the generalized Navier-Stokes problem with damping, we prove the existence of weak solutions by using regularization techniques, the theory of monotone operators and compactness arguments together with the local decomposition of the pressure and the Lipschitz-truncation method. The existence result proved here holds for any and any sigma > 1, where q is the exponent of the diffusion term and sigma is the exponent which characterizes the damping term.MCTES, Portugal [SFRH/BSAB/1058/2010]; FCT, Portugal [PTDC/MAT/110613/2010]info:eu-repo/semantics/publishedVersio

Comparative analysis of behavioral and transcriptional variation underlying CO2 sensory neuron function and development in Drosophila

Carbon dioxide is an important environmental cue for many insects, regulating many behaviors including some that have direct human impacts. To further improve our understanding of how this system varies among closely related insect species, we examined both the behavioral response to CO2 as well as the transcriptional profile of key developmental regulators of CO2 sensory neurons in the olfactory system across the Drosophila genus. We found that CO2 generally evokes repulsive behavior across most of the Drosophilids we examined, but this behavior has been lost or reduced in several lineages. Comparisons of transcriptional profiles from the developing and adult antennae for subset these species suggest that behavioral differences in some species may be due to differences in the expression of the CO2 co-receptor Gr63a. Furthermore, these differences in Gr63a expression are correlated with changes in the expression of a few genes known to be involved in the development of the CO2 circuit, namely dac, an important regulator of sensilla fate for sensilla that house CO2 ORNs, and mip120, a member of the MMB/dREAM epigenetic regulatory complex that regulates CO2 receptor expression. In contrast, most of the other known structural, molecular, and developmental components of the peripheral Drosophila CO2 olfactory system seem to be well-conserved across all examined lineages. These findings suggest that certain components of CO2 sensory ORN development may be more evolutionarily labile, and may contribute to differences in CO2-evoked behavioral responses across species

Quantum error rejection code with spontaneous parametric conversion

We propose a linear optics scheme with SPDC process to test the fault tolerance property of quantum error correction code. To transmit an unknown qubit robustly through the noisy channel, one may first encode it into a certain quantum error correction code and then transmit it. The remote party decodes it and stores it. Sending a qubit in such a way can significantly reduces the error rate compared with directly sending the qubit itself. Here we show how to realize such a scheme by linear optics.Comment: To appear in Phys. Rev. A. 18 pages, 2 figure, minor erros corrected and more explanations added to increase the readabilit

Experiment towards continuous-variable entanglement swapping: Highly correlated four-partite quantum state

We present a protocol for performing entanglement swapping with intense pulsed beams. In a first step, the generation of amplitude correlations between two systems that have never interacted directly is demonstrated. This is verified in direct detection with electronic modulation of the detected photocurrents. The measured correlations are better than expected from a classical reconstruction scheme. In the entanglement swapping process, a four--partite entangled state is generated. We prove experimentally that the amplitudes of the four optical modes are quantum correlated 3 dB below shot noise, which is due to the potential four--party entanglement.Comment: 9 pages, 10 figures, update of references 9 and 10; minor inconsistency in notation removed; format for units in the figures change

Conditional generation of arbitrary multimode entangled states of light with linear optics

We propose a universal scheme for the probabilistic generation of an arbitrary multimode entangled state of light with finite expansion in Fock basis. The suggested setup involves passive linear optics, single photon sources, strong coherent laser beams, and photodetectors with single-photon resolution. The efficiency of this setup may be greatly enhanced if, in addition, a quantum memory is available.Comment: 7 pages, 5 figure

Greenberger-Horne-Zeilinger nonlocality for continuous variable systems

As a development of our previous work, this paper is concerned with the Greenberger-Horne-Zeilinger (GHZ) nonlocality for continuous variable cases. The discussion is based on the introduction of a pseudospin operator, which has the same algebra as the Pauli operator, for each of the $N$ modes of a light field. Then the Bell-CHSH (Clauser, Horne, Shimony and Holt) inequality is presented for the $N$ modes, each of which has a continuous degree of freedom. Following Mermin's argument, it is demonstrated that for $N$-mode parity-entangled GHZ states (in an infinite-dimensional Hilbert space) of the light field, the contradictions between quantum mechanics and local realism grow exponentially with $N$, similarly to the usual $N$-spin cases.Comment: RevTEX; comments are welcomed; new version with minor change

Tunneling spectroscopy in the magnetic superconductor TmNi2B2C

We present new measurements about the tunneling conductance in the borocarbide superconductor TmNi$_2$B$_2$C. The results show a very good agreement with weak coupling BCS theory, without any lifetime broadening parameter, over the whole sample surface. We detect no particular change of the tunneling spectroscopy below 1.5K, when both the antiferromagnetic (AF) phase and the superconducting order coexist.Comment: Submitted to Phys. Rev. B, Rapid Communication

Localized surface states in HTSC: Alternative mechanism of zero-bias conductance peaks

It is shown that the quasiparticle states localized in the vicinity of surface imperfections of atomic size can be responsible for the zero-bias tunneling conductance peaks in high-Tc superconductors. The contribution from these states can be easily separated from other mechanisms using their qualitatively different response on an external magnetic field.Comment: REVTeX, 4 pages, 2 figs; to be published in PR

Optimal filtering of the LISA data

The LISA time-delay-interferometry responses to a gravitational-wave signal are rewritten in a form that accounts for the motion of the LISA constellation around the Sun; the responses are given in closed analytic forms valid for any frequency in the band accessible to LISA. We then present a complete procedure, based on the principle of maximum likelihood, to search for stellar-mass binary systems in the LISA data. We define the required optimal filters, the amplitude-maximized detection statistic (analogous to the F statistic used in pulsar searches with ground-based interferometers), and discuss the false-alarm and detection probabilities. We test the procedure in numerical simulations of gravitational-wave detection.Comment: RevTeX4, 28 pages, 9 EPS figures. Minus signs fixed in Eq. (46) and Table II. Corrected discussion of F-statistic distribution in Sec. IV