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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TL;DR: By using the Mumford’s representation of a reduced divisor D of the Jacobian J(Fq) of a hyperelliptic curve H of genus 2 with odd characteristic, a perfectly random bit string is extracted of the sum of abscissas of rational points on H in the support of D.
Abstract: Here, we proposed an improved version of the deterministic random extractors $SEJ$ and $PEJ$ proposed by R. R. Farashahi in \cite{F} in 2009. By using the Mumford's representation of a reduced divisor $D$ of the Jacobian $J(\mathbb{F}_q)$ of a hyperelliptic curve $\mathcal{H}$ of genus $2$ with odd characteristic, we extract a perfectly random bit string of the sum of abscissas of rational points on $\mathcal{H}$ in the support of $D$. By this new approach, we reduce in an elementary way the upper bound of the statistical distance of the deterministic randomness extractors defined over $\mathbb{F}_q$ where $q=p^n$, for some positive integer $n\geq 1$ and $p$ an odd prime.
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TL;DR: In this paper, the authors consider the problem of finding the greatest common divisor of a set of positive integers with a given integer parameter and an integer parameter (H) such that the divisors of H = 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 14, 15, 16, 17, 18, 20, 21, 22, 23, 24, 25, 26, 27, 28, 28
Abstract: Given a set of $n$ positive integers $\{a_1, \ldots, a_n\}$ and an integer parameter $H$ we study small additive shifts of its elements by integers $h_i$ with $|h_i| \le H$, $i =1, \ldots, n$, such that the greatest common divisor of $a_1+h_1, \ldots, a_n+h_n$ is very different from that of $a_1, \ldots, a_n$. We also consider a similar problem for the least common multiple.
Patent•
24 Jun 2016
TL;DR: In this article, a clock divider includes a pair of counters and a programmable delay line, which is configured to count using a first integer portion of the clock divisor.
Abstract: A clock divider includes, in part, a pair of counters and a programmable delay line. A first one of the counters operates at a first frequency and is configured to count using a first integer portion of the divisor. The second counter operates at a second frequency smaller than the first frequency and is configured to count using a second integer portion of the divisor. The programmable delay line includes, in part, a chain of delay elements configured to generate a multitude of delays of the output of the second counter. A multiplexer selects one of the generated delays in accordance with the fractional portion of the divisor. The second counter increases its count only when the first counter reaches a terminal count. The first and second integer portions are loaded respectively into the first and second counters when the second counter reaches its terminal count.
TL;DR: In this article, the authors considered the problem of multiple interpolation in the class of entire functions of finite order and found necessary and sufficient conditions for solvability of the problem.
Abstract: The problem of multiple interpolation in the class of entire functions of finite order was solved in [1]. The analogous problem for the class of functions of finite order in the upper half-plane C + = {z: Imz > 0} was discussed in [2], where necessary and sufficient conditions for its solvability were found. The difficulties that appear upon passage from the interpolation problem in classes of entire functions to classes of functions that are analytic in the half-plane C + are caused by the possibility of interpolation nodes accumulating at points of the real axis. The sufficient conditions found in [3] eliminate the possibility of such accumulation and make it possible to apply the methods and theory of entire functions. In the present paper we consider a problem more general than that of [2], eliminate the above-noted shortcomings, and find necessary and sufficient conditions for solvability of our problem. In its formulation, our problem is close to problems on free interpolation in H ~176 , since the values of the derivative of a function at interpolation nodes are subjected to some natural constraints. We will use ideas from [1, 2], as well as ideas due to Jones [3] for solution of problems on free interpolation in H ~176 Let Lo, ~]+ denote the class of functions that are analytic in C + and of order _ 1. Definition. A divisor D = {an, qn} (i.e., a set of different complex numbers an E C +, n = 1, 2,..., whose limit points all lie on the rea/axis, together with their multiplicities qn, qn >_ 1, is an integer) is said to be an interpolation divisor in the class Lo, oo] + if, for any sequence of numbers {bnk}, k = 1,..., q,, n = 1, 2,..., that satisfies the conditions (Irna,)~-~lb~kl In + In maxl
Posted Content•
TL;DR: In this article, it was shown that the set of all natural numbers for which there exists a divisor has a natural density, and the value of the natural density of the set is known.
Abstract: Let $n_1,\cdots,n_r$ be any finite sequence of integers and let $S$ be the set of all natural numbers $n$ for which there exists a divisor $d(x)=1+\sum_{i=1}^{deg(d)}c_ix^i$ of $x^n-1$ such that $c_i=n_i$ for $1\leq i \leq r$. In this paper we show that the set $S$ has a natural density. Furthermore, we find the value of the natural density of $S$.