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.

A Novel Approximate Divider using Binomial Expansion

Abstract: This paper presents a novel approximate divider circuit using binomial expansion. The circuit is approximated using the division between A and $L = {2^{\left\lfloor {{{\log }_2}B} \right\rfloor }}$, where ⌊log 2 B⌋ is the most significant bit value of the divisor B. After that, we use the sum of binomial coefficient to approximate the values. The approximate divider is much simpler and can be implemented using shift and add operations. Moreover, the complexity of the method is $\mathcal{O}(n)$, where n is the number of bits. Experimental results show that the probability of errors is less than 0.18. The approximate circuit is useful for circuit applications contain rigorous and massive arithmetic operations such as artificial intelligence circuits.

On the Fourth Moment of the Epstein Zeta Functions and the Related Divisor Problem

Abstract: In this paper, we study the fourth moment of the Epstein zeta function $\zeta (s;Q)$ associated to a $n\times n$ positive definite symmetric matrix $Q$ ($n\geq 4$) on the line $\mathrm{Re}(s)=\frac{n-1}{2}$. We prove that the integral $\int _{0}^{T}|\zeta (\frac{n-1}{2}+it;Q)|^{4}dt$ is evaluated by $O(T(\mathrm{log}\,T)^{4})$ if $Q$ satisfies some conditions. As an application, we consider the divisor problem with respect to the coefficients of the Dirichlet series of Epstein zeta functions. Certain estimates for the error term of the sum of the Dirichlet coefficients are obtained by combining our results and Fomenko's estimates for $\zeta (\frac{n-1}{2}+it;Q)$.

On direct images of twisted pluricanonical sheaves on normal varieties

Abstract: We study the depth properties of certain direct image sheaves on normal varieties. Let $$f: Y\rightarrow X$$ be a proper morphism of relative dimension d from a smooth variety onto a normal variety such that the preimage E of the singular locus of X is a divisor. We show that for any integer $$m>0$$ , the higher direct image $$R^df_*\omega ^{\otimes m}_Y(aE)$$ modulo the torsion subsheaf is $$S_2$$ , provided that a is sufficiently large. In case f is birational, we give criteria on a for the direct image $$f_*\omega _Y(aE)$$ to coincide with $$\omega _X$$ . We also introduce an index measuring the singularities of normal varieties.

Trisection for genus 2 curves in odd characteristic

Abstract: We provide trisection (division by 3) algorithms for Jacobians of genus 2 curves over finite fields $$\mathbb {F}_q$$Fq of odd characteristic which rely on the factorization of a polynomial whose roots correspond (bijectively) to the set of trisections of the given divisor. We also construct a polynomial whose roots allow us to calculate the 3-torsion divisors. We show the relation between the rank of the 3-torsion subgroup and the factorization of this 3-torsion polynomial, and describe the factorization of the trisection polynomials in terms of the Galois structure of the 3-torsion subgroup. We also generalize these ideas for $$\ell \in \{5,7\}$$lź{5,7}.

Rational classes and divisors on curves of genus 2

Abstract: We describe a method of looking for rational divisor classes on a curve of genus 2. We have an algorithm to decide if a given class of divisors of degree 3 contains a rational divisor. It is known that the shape of the kernel of Cassel’s morphism (X − T) is related to the existence of rational classes of degree 1. Our key tool is the dual Kummer surface.