A Combined Fit on the Annihilation Corrections in Bu,d,sB_{u,d,s} →\to PPPP Decays Within QCDF

Motivated by the possible large annihilation contributions implied by recent CDF and LHCb measurements on nonleptonic annihilation B meson decays, and the refined experimental measurements on hadronic B meson decays, we study the strength of annihilation contributions within QCD factorization (QCDF) in this paper. With the available measurements of two-body B_{u,d,s} -> pi pi, pi K, K K decays, a comprehensive fit on the phenomenological parameters X_A^{i,f} (or rho_A^{i,f} and phi_A^{i,f}) which are used to parameterize the endpoint singularity in annihilation amplitudes is performed with the statistical chi^2 approach. It is found that (1) flavor symmetry breaking effects are hardly to be distinguished between X_{A,s}^i and X_{A,d}^i due to the large experimental errors and theoretical uncertainties, where X_{A,s}^i and X_{A,d}^i are related to the nonfactorization annihilation contributions in B_s and B_{u,d} decays, respectively. So X_{A,s}^i = X_{A,d}^i is a good approximation by now. (2) In principle, parameter X_{A}^f which is related to the factorization annihilation contributions and independent of the initial state can be regarded as the same variable for B_{u,d,s} decays. (3) Numerically, two solutions are found, one is (rho_A^i, phi_A^i) = (2.98^+1.12_-0.86,-105^+34_-24) and (rho_A^f, phi_A^f) = (1.18^+0.20_-0.23,-40^+11_-8), the other is (rho_A^i, phi_A^i) = (2.97^+1.19_-0.90,-105^+32_-24) and (rho_A^f, phi_A^f) = (2.80^+0.25_-0.21,165^+4_-3). Obviously, nonfactorization annihilation parameter X_A^i is generally unequal to factorization annihilation parameter X_A^f, which differ from the traditional treatment. With the fitted parameters, all results for observables of B_{u,d,s} ->pi pi, pi K, K K decays are in good agreement with experimental data.Comment: 12 pages, version accepted by PL

Safety Model Checking with Complementary Approximations

Formal verification techniques such as model checking, are becoming popular in hardware design. SAT-based model checking techniques such as IC3/PDR, have gained a significant success in hardware industry. In this paper, we present a new framework for SAT-based safety model checking, named Complementary Approximate Reachability (CAR). CAR is based on standard reachability analysis, but instead of maintaining a single sequence of reachable- state sets, CAR maintains two sequences of over- and under- approximate reachable-state sets, checking safety and unsafety at the same time. To construct the two sequences, CAR uses standard Boolean-reasoning algorithms, based on satisfiability solving, one to find a satisfying cube of a satisfiable Boolean formula, and one to provide a minimal unsatisfiable core of an unsatisfiable Boolean formula. We applied CAR to 548 hardware model-checking instances, and compared its performance with IC3/PDR. Our results show that CAR is able to solve 42 instances that cannot be solved by IC3/PDR. When evaluated against a portfolio that includes IC3/PDR and other approaches, CAR is able to solve 21 instances that the other approaches cannot solve. We conclude that CAR should be considered as a valuable member of any algorithmic portfolio for safety model checking

Two Iterative algorithms for the matrix sign function based on the adaptive filtering technology

In this paper, two new efficient algorithms for calculating the sign function of the large-scale sparse matrix are proposed by combining filtering algorithm with Newton method and Newton Schultz method respectively. Through the theoretical analysis of the error diffusion in the iterative process, we designed an adaptive filtering threshold, which can ensure that the filtering has little impact on the iterative process and the calculation result. Numerical experiments are consistent with our theoretical analysis, which shows that the computational efficiency of our method is much better than that of Newton method and Newton Schultz method, and the computational error is of the same order of magnitude as that of the two methods.Comment: 18 pages,12 figure