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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01 Jan 2013
TL;DR: In this paper, it was shown that the set of real numbers that satisfy these conditions has full Hausdor dimension, which is the first result of W. M. Schmidt on normal numbers to dierent bases.
Abstract: Lets be an integer greater than or equal to 2. A real number is simply normal to bases if in its base-s expansion every digit 0;1;:::;s 1 occurs with the same frequency 1{s. Let S be the set of positive integers that are not perfect powers, hence S is the set t2;3;5;6;7;10;11;:::u. Let M be a function fromS to sets of positive integers such that, for each s inS, if m is in Mpsq then each divisor of m is in Mpsq and if Mpsq is infinite then it is equal to the set of all positive integers. These conditions on M are necessary for there to be a real number which is simply normal to exactly the bases s m such that s is inS and m is in Mpsq. We show these conditions are also sucient and further establish that the set of real numbers that satisfy them has full Hausdor dimension. This extends a result of W. M. Schmidt (1961/1962) on normal numbers to dierent bases.
1 citations
TL;DR: In this paper, the number of ways in which N objects may be assigned to M distinct classes is given by one can easily see that this can be true if and only if M! is a divisor of MN and this is true if or only if m = 1 or m = 2.
Abstract: Clustering may be viewed as a combinatorial optimization problem. In the above paper1, the number of ways in which N objects may be assigned to M distinct classes is given by One can easily see that this can be true if and only if M! is a divisor of MN and this is true if and only if M = 1 or M = 2.
TL;DR: The number of automorphisms of G fixing d elements of G and denote it by θ(G,d) is found, which proves a conjecture of Checco-Darling-Longfield-Wisdom.
Abstract: Let $G=\mathbf{Z}_{p} \oplus \mathbf{Z}_{p^2}$, where $p$ is a prime number. Suppose that $d$ is a divisor of the order of $G$. In this paper we find the number of automorphisms of $G$ fixing $d$ elements of $G$, and denote it by $\theta(G,d)$. As a consequence, we prove a conjecture of Checco-Darling-Longfield-Wisdom. We also find the exact number of fixed-point-free automorphisms of the group $\mathbf{Z}_{p^{a}} \oplus \mathbf{Z}_{p^{b}}$, where $a$ and $b$ are positive integers with $a
TL;DR: It is proved that the set of all C\documentclass[12pt]{minimal} -semigroups with fixed multiplicity, genus and Frobenius number is a Frobenii variety.
Abstract: A numerical semigroup S is closed under addition of its divisors (
$${\mathcal {C}}$$
-semigroup) if the following condition holds: if $$s\in S\setminus \{0\}$$
and d is a non trivial divisor of s then $$s+d\in S$$
. In this paper we prove that the set of all $${\mathcal {C}}$$
-semigroups is a Frobenius variety. As a consequence, we give algorithms to compute the set $${\mathcal {C}}$$
-semigroups with fixed multiplicity, genus and Frobenius number.
01 Jan 1993
TL;DR: The weights of C are determined by using the weights of the irreducible cyclic code of length s, generated by .
Abstract: Let s, k, integers such that s is a divisor of 2k−1 Let g(x) be a primitive divisor of xs −1 over F 2, and let π(x) be a primitive polynomial of degree k over F 2 We consider N the binary cyclic code C of length N = 2k −1, generated by For special cases, we determine the weights of C by using the weights of the irreducible cyclic code of length s, generated by