We study how the cohomology of a type $F_\infty$ relatively hyperbolic group pair $(G,\mathcal{P})$ changes under Dehn fillings (i.e. quotients pairs). For sufficiently long where quotient $(\bar{G},\bar{\mathcal{P}})$ is $F_\infty$, we show that there spectral sequence relating groups $H^i(G,\mathcal{P};\mathbb{Z} G)$ and $H^i\left(\bar{G},\bar{\mathcal{P}};\mathbb{Z}\bar{G}\right)$. As conseq...

Let $K$ be a compact group. For symplectic quotient $M_{\lambda}$ of Hamiltonian K\"ahler $K$-manifold, we show that the induced complex structure on is locally invariant when parameter $\lambda$ varies in $\mathrm{Lie}(K)^*$. To prove such result, take two different approaches: (i) by using geometry properties implosion construction; (ii) investigating variation GIT quotients.

Generalizing cones over projective toric varieties, we present arbitrary toric varieties as quotients of quasiaffine toric varieties. Such quotient presentations correspond to groups of Weil divisors generating the topology. Groups comprising Cartier divisors define free quotients, whereas Q-Cartier divisors define geometric quotients. Each quotient presentation yields homogeneous coordinates. ...

The concept of right (left) quotient (or residual) of an ideal η by anideal ν of an L-subring μ of a ring R is introduced. The right (left) quotients areshown to be ideals of μ . It is proved that the right quotient [η :r ν ] of an idealη by an ideal ν of an L-subring μ is the largest ideal of μ such that[η :r ν ]ν ⊆ η . Most of the results pertaining to the notion of quotients(or residual) of ...

We give a more practical variant of Shanks’ 1954 algorithm for computing the continued fraction of logb a, for integers a > b > 1, using the floor and ceiling functions and an integer parameter c > 1. The variant, when repeated for a few values of c = 10, enables one to guess if logb a is rational and to find approximately r partial quotients. 1. Shanks’ algorithm In his article [1], Shanks gav...