Divisor

About: Divisor is a research topic. Over the lifetime, 2462 publications have been published within this topic receiving 21394 citations. The topic is also known as: factor & submultiple.

Dispersionless Hirota equations and the genus 3 hyperelliptic divisor

Abstract: Equations of dispersionless Hirota type $$\begin{aligned} F(u_{x_ix_j})=0 \end{aligned}$$ have been thoroughly investigated in mathematical physics and differential geometry. It is known that the parameter space of integrable Hirota type equations in 3D is 21-dimensional, and that the action of the natural equivalence group $${\mathrm {Sp}}(6,{\mathbb {R}})$$ on the parameter space has an open orbit. However the structure of the generic equation corresponding to the open orbit remained elusive. Here we prove that the generic 3D Hirota equation is given by the remarkable formula $$\begin{aligned} \vartheta _m(\tau )=0, \qquad \tau =i\ \text {Hess}(u) \end{aligned}$$ where $$\vartheta _m$$ is any genus 3 theta constant with even characteristics and $$\text {Hess}(u)$$ is the $$3\times 3$$ Hessian matrix of a (real-valued) function $$u(x_1, x_2, x_3)$$ . Thus, generic Hirota equation coincides with the equation of the genus 3 hyperelliptic divisor (to be precise, its intersection with the imaginary part of the Siegel upper half space $${\mathfrak {H}}_3$$ ). The rich geometry of integrable Hirota type equations sheds new light on local differential geometry of the genus 3 hyperelliptic divisor, in particular, the integrability conditions can be viewed as local differential-geometric constraints that characterise the hyperelliptic divisor uniquely modulo $${\mathrm {Sp}}(6,{\mathbb {C}})$$ -equivalence.

Generalized GCD rings

Abstract: All rings are assumed to be commutative with identity. A generalized GCD ring (G-GCD ring) is a ring (zero-divisors admitted) in which the intersection of every two finitely generated (f.g.) faithful multiplication ideals is a f.g. faithful multiplication ideal. Various properties of G-GCD rings are considered. We generalize some of Jäger’s and Lüneburg’s results to f.g. faithful multiplication ideals. MSC 2000: 13A15 (primary), 13F05 (secondary)

Split remainder divider

Abstract: The invention provides computer apparatus for performing a division operation having a dividend mathematically divided by a divisor. The dividend and the divisor are split between a state machine and an array of carry save adders. The most significant bits of the dividend and the divisor are input to the state machine and the least significant bits of the dividend and the divisor are input to the carry save adder array. The state machine is fully encoded with partial remainder values and quotient digit values for all possible combinations of the most significant bits of the divisor and the dividend. The carry save adders add the respective least significant bits of the dividend and the divisor and output spillover signals to the state machine. The state machine provides partial remainders and quotient digits selected from the encoded partial remainder values and quotient digit values dependent on (i.e. as a function of) the most significant bits of the dividend, divisor and the spillover signals.

Totally invariant divisors of endomorphisms of projective spaces

Abstract: Totally invariant divisors of endomorphisms of the projective space are expected to be always unions of linear spaces. Using logarithmic differentials we establish a lower bound for the degree of the non-normal locus of a totally invariant divisor. As a consequence we prove the linearity of totally invariant divisors for \(\mathbb {P}^3\).