FO-Definability of Shrub-Depth

Shrub-depth is a graph invariant often considered as an extension of tree-depth to dense graphs. We show that the model-checking problem of monadic second-order logic on a class of graphs of bounded shrub-depth can be decided by AC^0-circuits after a precomputation on the formula. This generalizes a similar result on graphs of bounded tree-depth [Y. Chen and J. Flum, 2018]. At the core of our proof is the definability in first-order logic of tree-models for graphs of bounded shrub-depth

Identifying successful sales and marketing strategies that affect customer loyalty in a coffee shop

.Most small businesses such as coffee shops are concerned with customer loyalty and satisfaction, and increasing profitability. Marketing strategy becomes important when customer loyalty is low. The aim of this research is to identify sales that affect customer loyalty in a small coffee shop and to investigate the relationship between effective marketing strategies and customer loyalty, using a survey of customers

Extracting Mass Hierarchy Information from Simple Analysis of Neutrino Mass Splitting

Based on the independent measurements on neutrino mass splitting $|\Delta m^2_{\mu\mu}|$, $|\Delta m^2_{ee}|$, $\Delta m^2_{21}$, and recent measurements by the T2K Collaboration, we carry out a simple fitting analysis on $\Delta m^2_{32}$ and $\Delta m^2_{31}$ in normal hierarchy and inverse hierarchy respectively, suggesting \Delta m^2_{32}=(2.46\pm0.07)\times10^{-3}~\mbox{eV}^2 and \Delta m^2_{31}=(2.53\pm0.07)\times10^{-3}~\mbox{eV}^2 in normal hierarchy, or $\Delta m^2_{32}$=-(2.51\pm0.07)\times10^{-3}~\mbox{eV}^2 and $\Delta m^2_{31}$=-(2.44\pm0.07)\times10^{-3}~\mbox{eV}^2 in invert hierarchy. The simple analysis indicate that both normal and inverted hierarchy are consistent with current experimental measurements on mass splitting. The p-value for normal hierarchy and that for inverted hierarchy are 62% and 55%, respectively. This reveals a slight favor for the normal hierarchy. It is suggested that further measurements on the mass splitting with higher accuracy are necessary to determine the neutrino mass hierarchy.Comment: 5 latex pages, 5 figures. Final version as publishe

The parameterized space complexity of model-checking bounded variable first-order logic

The parameterized model-checking problem for a class of first-order sentences (queries) asks to decide whether a given sentence from the class holds true in a given relational structure (database); the parameter is the length of the sentence. We study the parameterized space complexity of the model-checking problem for queries with a bounded number of variables. For each bound on the quantifier alternation rank the problem becomes complete for the corresponding level of what we call the tree hierarchy, a hierarchy of parameterized complexity classes defined via space bounded alternating machines between parameterized logarithmic space and fixed-parameter tractable time. We observe that a parameterized logarithmic space model-checker for existential bounded variable queries would allow to improve Savitch's classical simulation of nondeterministic logarithmic space in deterministic space $O(\log^2n)$. Further, we define a highly space efficient model-checker for queries with a bounded number of variables and bounded quantifier alternation rank. We study its optimality under the assumption that Savitch's Theorem is optimal

Quark-lepton complementarity and self-complementarity in different schemes

With the progress of increasingly precise measurements on the neutrino mixing angles, phenomenological relations such as quark-lepton complementarity (QLC) among mixing angles of quarks and leptons and self-complementarity (SC) among lepton mixing angles have been observed. Using the latest global fit results of the quark and lepton mixing angles in the standard Chau-Keung scheme, we calculate the mixing angles and CP-violating phases in the other eight different schemes. We check the dependence of these mixing angles on the CP-violating phases in different phase schemes. The dependence of QLC and SC relations on the CP phase in the other eight schemes is recognized and then analyzed, suggesting that measurements on CP-violating phases of the lepton sector are crucial to the explicit forms of QLC and SC in different schemes.Comment: 11 pages, 3 figures, version accepted for publication in PR