Tau functions of the charged free bosons

We study bosonic tau functions in relation with the charged free bosonic fields.It is proved that up to a constant the only tau function in the Fock space M is the vacuum vector,and some tau functions are given in the completion M by using Schur functions.We also give a new proof of Borchardt’s identity and obtain several q-series identities by using the boson-boson correspondence.

MBIUS-TODA HIERARCHY AND ITS INTEGRABILITY

In this paper, we construct a new integrable equation called Mobius-Toda equation which is a generalization of q-Toda equation. Meanwhile its soliton solutions are constructed to show its integrable property. Further the Lax pairs of the Mobius-Toda equation and a whole integrable MSbius-Toda hierarchy are also constructed. To show the integrability, the bi-Hamiltonian structure and tau symmetry of the MSbius-Toda hierarchy are given and this leads to the existence of the tau function.

Abnormal energy metabolism and tau phosphorylation in the brains of middle-aged mice in response to atmospheric PM2.5 exposure

In light of the accelerated aging of the global population and the deterioration of the atmosphere pollution, we sought to clarify the potential mechanisms by which fine particulate matter（PM_（2.5）） can cause cognitive impairment and neurodegeneration through the alteration of mitochondrial structure and function. The results indicate that PM_（2.5） inhalation reduces ATP production by disrupting the aerobic tricarboxylic acid cycle and oxidative phosphorylation, thereby causing the hypophosphorylation of tau in the cortices of middle-aged mice. Furthermore, excessive reactive oxygen species generation was involved in the impairment. Interestingly, these alterations were partially reversed after exposure to PM_（2.5） ended. These findings clarify the mechanism involved in mitochondrial abnormality-related neuropathological dysfunction in response to atmospheric PM_（2.5） inhalation and provide an optimistic sight for alleviating the adverse health outcomes in polluted areas.

Symmetries and bilinear equations for the modified BKP hierarchy

The modified Kadomtsev-Petviashvili hierarchy of B type(mBKP hierarchy)and its constrained cases are investigated from the aspect of the Lax formulation.Starting from the Lax equation of the mBKP hierarchy,we firstly derive the bilinear equations and show the existence of the tau functions.Then the additional symmetries are constructed,and the corresponding generator is showed to be the squared eigenfunction symmetry.After that,the actions of the additional symmetries on the tau functions are given.At last,the Lax formulation of the constrained mBKP hierarchy is investigated and the corresponding bilinear equations are also discussed.

On the Drinfeld-Sokolov hierarchy of type E_(6)^((1))and its topological solution

In this paper we report some explicit evolutionary PDEs of the Drinfeld-Sokolov hierarchy of type E_(6)^((1)),and show how the unknown functions in these PDEs are related to the tau function.Moreover,for this hierarchy we compute its topological solution of formal series up to a certain degree,whose coefficients of monomials give the Fan-Jarvis-Ruan-Witten invariants for the E_(6) simple singularity.Based on such results we also derive several explicit evolutionary PDEs and some low-degree terms of the topological solution for the Drinfeld-Sokolov hierarchy of type F_(4)^((1)).