Dual of the Hopf Algebra Consisting of the Adjacency Matrices

In this article we discuss the Hopf algebras spanned by the adjacency matrices in detail. We show that there two Hopf algebraic structures concerning the adjacency matrices, one is the copy of Connes-Kreimer Hopf algebra, another one is the copy of the dual of Connes-Kreimer Hopf algebra

Hopf Algebras Consisting of Finite Sets

In this article we generalise the structure of Connes-Kreimer Hpof algebra consisting of Feynmam diagrams to the situations of abstract finite sets, matrices and star product of scalar field, where the construction for the case of finite sets is essential

Insertion and Lie Bracket Concerning Finite Sets

In this article we discuss the operations of partitions (sequence of disjoint finite subsets) which are quotient, insertion, composition and Lie bracket. Moreover, we discuss applications of those operations for Feymman diagrams and Kontesvich's graphs

From Kontsevich Graphs to Feynman graphs, a Viewpoint from the Star Products of Scalar Fields

In the present paper we construct the star products concerning scalar fields in the covariant case from a new approach. We construct the star products at three levels, which are levels of functions on Rd, fields and functionals respectively. We emphases that the star product at level of functions is essence and starting point for our setting. Firstly the star product of functions includes all algebraic and combinatorial information of the star products concerning the scalar fields and functionals almost. Secondly, a more interesting point is that the star product of functions concerns only finite dimensional issue, which is a Moyal-like star product on Rd generated by a bi-vector field with abstract coefficients. Thus the Kontsevich graphs play some roles naturally. Actually we prove that there is an ono-one correspondence between a class of Kontsevich graphs and the Feynman graphs. Additionally the Wick theorem, Wick power and the expectation of Wick-monomial are discussed in terms of the star product at level of functions. Our construction can be considered as the generalisation of the star products in perturbative algebraic quantum fields theory and twist product introduced in [1],[2]

Ground State Solutions for the Periodic Discrete Nonlinear Schrödinger Equations with Superlinear Nonlinearities

We consider the periodic discrete nonlinear Schrödinger equations with the temporal frequency belonging to a spectral gap. By using the generalized Nehari manifold approach developed by Szulkin and Weth, we prove the existence of ground state solutions of the equations. We obtain infinitely many geometrically distinct solutions of the equations when specially the nonlinearity is odd. The classical Ambrosetti-Rabinowitz superlinear condition is improved

Diurnal modulation of electron recoils from DM-nucleon scattering through the Migdal effect

Halo dark matter (DM) particles could lose energy due to the scattering off nuclei within the Earth before reaching the underground detectors of DM direct detection experiments. This Earth shielding effect can result in diurnal modulation of the DM-induced recoil event rates observed underground due to the self-rotation of the Earth. For electron recoil signals from DM-electron scatterings, the current experimental constraints are very stringent such that the diurnal modulation cannot be observed for halo DM. We propose a novel type of diurnal modulation effect: diurnal modulation in electron recoil signals induced by DM-nucleon scattering via the Migdal effect. We set so far the most stringent constraints on DM-nucleon scattering cross section via the Migdal effect for sub-GeV DM using the S2-only data of PandaX-II and PandaX-4T with improved simulations of the Earth shielding effect. Based on the updated constraints, we show that the Migdal effect induced diurnal modulation of electron events can still be significant in the low energy region, and can be probed by experiments such as PandaX-4T in the near future