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.

Spectral Construction of Non-Holomorphic Eisenstein-Type Series and their Kronecker Limit Formula

Abstract: Let $X$ be a smooth, compact, projective Kahler variety and $D$ be a divisor of a holomorphic form $F$, and assume that $D$ is smooth up to codimension two. Let $\omega$ be a Kahler form on $X$ and $K_{X}$ the corresponding heat kernel which is associated to the Laplacian that acts on the space of smooth functions on $X$. Using various integral transforms of $K_{X}$, we will construct a meromorphic function in a complex variable $s$ whose special value at $s=0$ is the log-norm of $F$ with respect to $\mu$. In the case when $X$ is the quotient of a symmetric space, then the function we construct is a generalization of the so-called elliptic Eisenstein series which has been defined and studied for finite volume Riemann surfaces.

On a hybrid mean value of the F.Smarandache function and the divisor function

Abstract: For any positive integer n,the famous F.Smarandache function S(n) defined as the smallest positive integer m such that n|m|.That is,S(n)=min{m:n|m!,mEN}.Using the elementary meth- ods,a hybrid mean value problem involving the F.Smarandache function and the Dirichlet divisor func- tion is studied,and a sharper asymptotic formula is given for it.

Embedded system and floating-point division operation method and system thereof

Abstract: The invention provides an embedded system and a floating-point division operation method and system thereof. The method includes: firstly, acquiring a dividend and a divisor; secondly, extracting a decimal in the divisor; thirdly, acquiring a reciprocal of the divisor by using the decimal of the divisor; fourthly, multiplying a decimal of the dividend by the decimal of the divisor to obtain a decimal of a quotient; fifthly, adding an exponent of the dividend, an exponent of the divisor and an exponent of the reciprocal of the divisor to obtain an exponent of the quotient; finally acquiring a sign digit of the quotient according to a sign digit of the dividend and a sign digit of the divisor so as to obtain a division operation result of a floating point. Only the steps of addition, multiplication and the like relatively few in operation number, rather than an iterative algorithm are utilized for division operation in the embedded system and the floating-point division operation method and system thereof, the result of division operation is obtained with few steps, and the problem that operational resources of the embedded system are wasted due to the fact that an existing floating-point division operation method includes more steps can be solved.

Using the smoothness of p-1 for computing roots modulo p

Abstract: We prove, without recourse to the Extended Riemann Hypothesis, that the projection modulo $p$ of any prefixed polynomial with integer coefficients can be completely factored in deterministic polynomial time if $p-1$ has a $(\ln p)^{O(1)}$-smooth divisor exceeding $(p-1)^{{1/2}+\delta}$ for some arbitrary small $\delta$. We also address the issue of computing roots modulo $p$ in deterministic time.

Extreme positive ternary sextics

Abstract: We study nonnegative (psd) real sextic forms $q(x_0,x_1,x_2)$ that are not sums of squares (sos). Such a form has at most ten real zeros. We give a complete and explicit characterization of all sets $S\subset\mathbb{P}^2(\mathbb{R})$ with $|S|=9$ for which there is a psd non-sos sextic vanishing in $S$. Roughly, on every plane cubic $X$ with only real nodes there is a certain natural divisor class $\tau_X$ of degree~$9$, and $S$ is the real zero set of some psd non-sos sextic if, and only if, there is a unique cubic $X$ through $S$ and $S$ represents the class $\tau_X$ on $X$. If this is the case, there is a unique extreme ray $\mathbb{R}_+q_S$ of psd non-sos sextics through $S$, and we show how to find $q_S$ explicitly. The sextic $q_S$ has a tenth real zero which for generic $S$ does not lie in $S$, but which may degenerate into a higher singularity contained in $S$. We also show that for any eight points in $\mathbb{P}^2(\mathbb{R})$ in general position there exists a psd sextic that is not a sum of squares and vanishes in the given points.