Topic
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.
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Posted Content•
TL;DR: For a suitable irreducible base polynomial, a family of polynomials with Galois groups of order matching their degrees is constructed with the properties: as mentioned in this paper.
Abstract: For a suitable irreducible \textit{base} polynomial $f(x)\in \mathbf{Z}[x]$ of degree $k$, a family of polynomials $F_m(x)$ depending on $f(x)$ is constructed with the properties:
(i) there is exactly one irreducible factor $\Phi_{d,f}(x)$ for $F_m(x)$ for each divisor $d$ of $m$;
(ii) deg $(\Phi_{d,f}(x))=\varphi(d)\cdot\mathrm{deg} (f)$ generalizing the factorization of $x^m-1$ into cyclotomic polynomials;
(iii) when the base polynomial $f(x) = x-1$ this $F_m(x)$ coincides with $x^m-1$.
As an application, irreducible polynomials of degree 12, 24, 24 are constructed having Galois groups of order matching their degrees and isomorphic to
$S_3 \oplus C_2 , S_3 \oplus C_2\oplus C_2$
and $S_3 \oplus C_4$ respectively.
Posted Content•
TL;DR: In this article, the moments of the Dirichlet divisor problem are connected to the moments in the asymptotic formula for the mean square of the integrals.
Abstract: Let $\Delta(x)$ denote the error term in the Dirichlet divisor problem, and let $E(T)$ denote the error term in the asymptotic formula for the mean square of $|\zeta(1/2+it)|$. If $E^*(t) := E(t) - 2\pi\Delta^*(t/(2\pi))$ with $\Delta^*(x) :=
-\Delta(x) + 2\Delta(2x) - \frac{1}{2}\Delta(4x)$ and
$\int_0^T E^*(t)\,dt = \frac{3}{4}\pi T + R(T)$, then we obtain a number of results involving the moments of $|\zeta(1/2+it)|$ in short intervals, by connecting them to the moments of $E^*(T)$ and $R(T)$ in short intervals. Upper bounds and asymptotic formulas for integrals of the form $$ \int_T^{2T}\left(\int_{t-H}^{t+H}|\zeta(1/2+iu)|^2\,du\right)^k\,dt \qquad(k\in N, 1 \ll H \le T) $$ are also treated.
Patent•
17 Apr 1990
TL;DR: In this paper, a method to reduce the number of times of memory access and to accelerate a decimal division processing at an office computer level by providing a means to obtain the estimation of a quotient, an addition means, an add/subtraction means, and a correction means in case of generating a negative value in a division result was proposed.
Abstract: PURPOSE:To reduce the number of times of memory access and to accelerate a decimal division processing at an office computer level economically by providing a means to obtain the estimation of a quotient, an addition means, an addition/subtraction means, and a correction means in case of generating a negative value in a division result. CONSTITUTION:The plural high-order digits of a dividend are compared with the plural high- order digits of a divisor by using the subtracter 11 of the means 1 to obtain the estimation of the quotient, and when it is dividend < divisor, the plural high-order digits of the divisor are shifted by one digit, then, they are subtracted from the plural high-order digits of the dividend. And such operation is repeated until a subtraction result shows a negative number, then, the estimated quotient(C) of the quotient can be obtained. Next, the arithmetic operation of (divisor X quotient (C)) is performed by using an adder 21, and after that, the verification of (dividend - (divisor X quotient C)) is performed by using an adder/subtractor 31. When the negative number is obtained as a result, a subtraction result is restored to a positive number by adding the divisor, and a number(estimated number of quotient - 1) is set as a new quotient, and a true quotient and a remainder can be found by performing such arithmetic operation repeatedly. In such a way, it is possible to accelerate the decimal division processing economically at the office computer level by reducing the number of times of access.
Posted Content•
TL;DR: In this paper, the axiomatic quadratic affine Weyl group multiple Dirichlet series over rational function fields of odd characteristic admits meromorphic continuation to the interior of the corresponding complexified Tits cone.
Abstract: In this paper and its sequel \cite{DPP}, we investigate the precise relationship between the quadratic affine Weyl group multiple Dirichlet series in the sense of \cite{CG1, BD}, and those defined axiomatically by Whitehead \cite{White2} and \cite{White1}. In particular, we show that the axiomatic quadratic Weyl group multiple Dirichlet series of type $D_{\scriptscriptstyle 4}^{\scriptscriptstyle (1)}$ over rational function fields of odd characteristic admits meromorphic continuation to the interior of the corresponding complexified Tits cone. We shall also determine the polar divisor of this function, and compute the residue at each of its poles. As a consequence, we obtain an \emph{exact} formula for a weighted 4-th moment of quadratic Dirichlet $L$-functions over rational function fields; we shall also derive an asymptotic formula for this weighted moment that is expected to generalize to any global field.
Patent•
12 Dec 2017
TL;DR: In this paper, a low complexity implementation of remainder computation of any number when the divisor belongs to a type of numbers that can be represented as 2k+1 is presented.
Abstract: In some applications, such as randomization and cryptography, remainder computation for a number is required. The remainder computation is also used in modulo arithmetic. The remainder computation can be simplified when the divisor belongs to a certain class of numbers. A method and apparatus are disclosed that enable low complexity implementation of remainder computation of any number when the divisor belongs to a type of numbers that can be represented as 2k+1.