This paper is dedicated to provide theta function representations of algebro-geometric solutions and related crucial quantities for the two-component Hunter-Saxton (HS2) hierarchy through an initial value problem. Our main tools include polynomial recursive formalism, hyperelliptic curve with finite number genus, Baker-Akhiezer functions, meromorphic function, Dubrovin-type equations auxiliary ...

We use the hypergraph container theory of Balogh–Morris–Samotij and Saxton–Thomason to obtain general rectilinear approximations volume estimates for sequences bodies closed under certain families projections. give a number applications our results, including multicolour generalisation theorem Hatami, Janson Szegedy on entropy graph limits. Finally, we raise questions geometric analytic approac...