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.

Efficient Algorithms for Integer Division by Constants Using Multiplication

Abstract: We present a complete analysis of the integer division of a single unsigned dividend word by a single unsigned divisor word based on double-word multiplication of the dividend by an inverse of the divisor. The well-known advantage of this method yields run-time efficiency, if the inverse of the divisor can be calculated at compile time, since multiplication is much faster than division in arithmetic units. Our analysis leads to the discovery of a limit to the straightforward application of this method in the form of a critical dividend, which fortunately associates with a minority of the possible divisors (20%) and defines only a small upper part of the available dividend space. We present two algorithms for ascertaining whether a critical dividend exists and, if so, its value along with a circumvention of this limit. For completeness, we include an algorithm for integer division of a unsigned double-word dividend by an unsigned single-word divisor in which the quotient is not limited to a single word and the remainder is an intrinsic part of the result.

The Schützenberger category of a semigroup

Abstract: In this paper we introduce the Schutzenberger category \({\mathbb {D}}(S)\) of a semigroup \(S\). It stands in relation to the Karoubi envelope (or Cauchy completion) of \(S\) in the same way that Schutzenberger groups do to maximal subgroups and that the local divisors of Diekert do to the local monoids \(eSe\) of \(S\) with \(e\in E(S)\). In particular, the objects of \({\mathbb {D}}(S)\) are the elements of \(S\), two objects of \({\mathbb {D}}(S)\) are isomorphic if and only if the corresponding semigroup elements are \({\fancyscript{D}}\)-equivalent, the endomorphism monoid at \(s\) is the local divisor in the sense of Diekert and the automorphism group at \(s\) is the Schutzenberger group of the \({\fancyscript{H}}\)-class of \(s\) in \(S\). This makes transparent many well-known properties of Green’s relations. The paper also establishes a number of technical results about the Karoubi envelope and Schutzenberger category that were used by the authors in a companion paper on syntactic invariants of flow equivalence of symbolic dynamical systems.

On the Dirichlet divisor problem in short intervals

Abstract: We present several new results involving Δ(x+U)−Δ(x), where U=o(x) and $$\varDelta(x):=\sum_{n\leq x}d(n)-x\log x-(2\gamma-1)x $$ is the error term in the classical Dirichlet divisor problem.

Lefschetz theorems for tamely ramified coverings

Abstract: As is well known, the Lefschetz theorems for the \'etale fundamental group of SGA1 do not hold. We fill a small gap in the literature showing they do for tame coverings. Let $X$ be a regular projective variety over a field $k$, and let $D\hookrightarrow X$ be a strict normal crossings divisor. Then, if $Y$ is an ample regular hyperplane intersecting $D$ transversally, the restriction functor from tame \'etale coverings of $X\setminus D$ to those of $Y\setminus D\cap Y$ is an equivalence if dimension $X \ge 3$, and fully faithful if dimension $X=2$. The method is dictated by Grothendieck-Murre ("The tame fundamental group of a formal neighbourhood of a divisor with normal crossings on a scheme", Springer LNM 208). The authors showed that one can lift tame coverings from $Y\setminus D\cap Y$ to the complement of $D\cap Y$ in the formal completion of $X$ along $Y$. One has then to further lift to $X\setminus D$.