Narrow Results

In this paper, we construct the dispersionless coupled constrained KP hierarchy. We also introduce vital formal Laurent series ${{𝒴}}_{l,1},{{𝒴}}_{l,2}$ to construct new flows, which lead out the additional symmetries of the dispersionless coupled constrained KP hierarchy. The additional flows construct a sub-algebra of the Virasoro algebra. In addition, we give the additional flows acting on $\mu (t,\lambda ),\widehat{\mu }(t,\lambda )$ and $\rho (t,\lambda ),\widehat{\rho }(t,\lambda )$ of the dispersionless coupled constrained KP hierarchy.

The work of Helmer [Divisibility properties of integral functions, Duke Math. J. 6(2) (1940) 345–356] applied algebraic methods to the field of complex analysis when he proved the ring of entire functions on the complex plane is a Bezout domain (i.e. all finitely generated ideals are principal). This inspired the work of Henriksen [On the ideal structure of the ring of entire functions, Pacific J. Math. 2(2) (1952) 179–184. On the prime ideals of the ring of entire functions, Pacific J. Math. 3(4) (1953) 711–720] who proved a correspondence between the maximal ideals within the ring of entire functions and ultrafilters on sets of zeroes as well as a correspondence between the prime ideals and growth rates on the multiplicities of zeroes. We prove analogous results on rings of analytic functions in the non-Archimedean context: all finitely generated ideals in the ring of analytic functions on an annulus of a characteristic zero non-Archimedean field are two-generated but not guaranteed to be principal. We also prove the maximal and prime ideal structure in the non-Archimedean context is similar to that of the ordinary complex numbers; however, the methodology has to be significantly altered to account for the failure of Weierstrass factorization on balls of finite radius in fields which are not spherically complete, which was proven by Lazard [Les zeros d’une function analytique d’une variable sur un corps value complet, Publ. Math. l’IHES 14(1) (1942) 47–75].

We provide elementary proof of several congruences involving single sum and multisums of binomial coefficients.

Bateman and Duquette have initiated the study of Salem elements in positive characteristic. Later, Kerada introduced and studied j-Salem numbers ($j\ge 2$). To complete this circle of ideas we introduce 2-Salem elements over the field of formal Laurent series over a finite field ${𝔽}_q$. We extend the above-cited studies and we provide some analog results. The main result of this note is Theorem 1.2 which characterizes the set $T_2^{\prime \ast }$ of 2-Salem series when $q\ne 2^r$.