The relic gravitational wave (RGW) generated during the inflation depends on the initial condition via the amplitude, the spectral index $n_t$ and the running index $alpha_t$. CMB observations so far have only constrained the tensor-scalar ratio $r$,
but not $n_t$ nor $alpha_t$. Complementary to this, the ground-based interferometric detectors working at $sim 10^2$Hz are able to constrain the spectral indices that influence the spectrum sensitively at high frequencies. In this work we give a proper normalization of the analytical spectrum at the low frequency end, yielding a modification by a factor of $sim 1/50$ to the previous treatment. We calculate the signal-noise ratios (SNR) for various ($n_t,alpha_t$) at fixed $r=0.2$ by S6 of LIGO H-L, and obtain the observational upper limit on the running index $alpha_t<0.02093$ (i.e, at a detection rate $95%$ and a false alarm rate $5%$) at the default $(n_t=0,r=0.2)$. This is consistent with the constraint on the energy density obtained by LIGO-Virgo Collaboration. Extending to the four correlated detectors currently running, the calculated SNR improves slightly. When extending to the six correlated detectors of the second-generation in design, the calculated SNR is $sim 10^3$ times over the previous two cases, due to the high sensitivities. RGW can be directly detected by the six 2nd-generation detectors for models with $alpha_t>0.01364$.
Wave-particle duality of photons with losses in the Mach-Zehnder interferometer (MZI) is investigated experimentally and theoretically. The experiment is done with the standard MZI with the beam splitter or the beam merger being continuously varied.
The losses are deliberately introduced either inside the MZI (the two arms between the beam splitter and beam mergers) or outside the MZI (after the beam merger). It is proved that the unbalanced losses have great influence on the predictability $P$ (particle nature) and visibility $V$ (wave nature). For the former case the duality inequality holds while for the later the duality inequality is ``violated. We get $P^2+V^2>1$. This ``violation could be eliminated in principle by switching the two paths and detectors and then averaging the results. The observed results can be exactly explained theoretically. The experiment is done with coherent beam, instead of single photons, and we have proved that they are exactly equivalent in duality experiment with MZI.
How users in a dynamic system perform learning and make decision become more and more important in numerous research fields. Although there are some works in the social learning literatures regarding how to construct belief on an uncertain system sta
te, few study has been conducted on incorporating social learning with decision making. Moreover, users may have multiple concurrent decisions on different objects/resources and their decisions usually negatively influence each others utility, which makes the problem even more challenging. In this paper, we propose an Indian Buffet Game to study how users in a dynamic system learn the uncertain system state and make multiple concurrent decisions by not only considering the current myopic utility, but also taking into account the influence of subsequent users decisions. We analyze the proposed Indian Buffet Game under two different scenarios: customers request multiple dishes without budget constraint and with budget constraint. For both cases, we design recursive best response algorithms to find the subgame perfect Nash equilibrium for customers and characterize special properties of the Nash equilibrium profile under homogeneous setting. Moreover, we introduce a non-Bayesian social learning algorithm for customers to learn the system state, and theoretically prove its convergence. Finally, we conduct simulations to validate the effectiveness and efficiency of the proposed algorithms.
