We study some specializations and extensions of the quantum version of the MacMahon Master Theorem derived by Garoufalidis, Lê and Zeilberger. In particular, we obtain a (t, q)-analogue for the Cartier-Foata noncommutative version and a semi-strong (t, q)-analogue for the contextual algebra.

Some divisor class relations for genus 0 curves are proved and used to compute the Cartier divisor class of the virtual canonical bundle for genus 0 maps to a smooth target. Many results here first appeared in [6] and [5]; our proofs use a completely different method.

We propose an extended notion of adelic R-Cartier divisors, called ℓ1-adelic which enables us to associate a Banach space each algebraic variety over the field rational numbers, and establish global continuity arithmetic volume function defined on pairs divisors R-base conditions.

we introduce a new concept of general $g$-$eta$-monotone operator generalizing the general $(h,eta)$-monotone operator cite{arvar2, arvar1}, general $h-$ monotone operator cite{xiahuang} in banach spaces, and also generalizing $g$-$eta$-monotone operator cite{zhang}, $(a, eta)$-monotone operator cite{verma2}, $a$-monotone operator cite{verma0}, $(h, eta)$-monotone operator cite{fanghuang}, $h$-...

We show that, for a pseudo-proper smooth noetherian formal scheme $\mathfrak{X}$ over positive characteristic $p$ field, its truncated De Rham complex up to the is decomposable. Moreover, if dimension of exactly $p$, then full Along way we establish Cartier isomorphism associated morphism schemes.