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: In this article, an effective divisor of the moduli space of stable curves was defined, which is denoted by the class of $\overline{S^{2}W}$ in the Picard group of moduli functors.
Abstract: We define an effective divisor of the moduli space of stable curves $\overline{M_g}$, which is denoted $\overline{S^{2}W}$. Writing the class of $\overline{S^{2}W}$ in the Picard group of the moduli functor Pic$_{\text{fun}}(\overline{M_{g}})\otimes \mathbb{Q}$ in terms of the so-called Harer basis $\lambda,\delta_0,\ldots,\delta_{[g/2]}$, we prove that the relations among the coefficients of $\delta_1,\ldots,\delta_{[g/2]}$ are the same relations on coefficients as the Brill-Noether divisors. We present a result on effective divisors of $\overline{M_g}$ which could be useful to get the same relations on coefficients for other divisors. We also compute the coefficient of $\lambda$.
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TL;DR: In this paper, the smallest positive integer (e(G)$ such that the map sending $g\in G$ to G:C_G(g)| is a generalized character of G is investigated.
Abstract: For a finite group $G$ we investigate the smallest positive integer $e(G)$ such that the map sending $g\in G$ to $e(G)|G:C_G(g)|$ is a generalized character of $G$. It turns out that $e(G)$ is strongly influenced by local data, but behaves irregularly for non-abelian simple groups. We interpret $e(G)$ as an elementary divisor of a certain non-negative integral matrix related to the character table of $G$. Our methods applied to Brauer characters also answers a recent question of Navarro: The $p$-Brauer character table of $G$ determines $|G|_{p'}$.
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TL;DR: In this paper, a generalization of the Mobius function is used to formulate the divisor sum and establish some identities. One such identity is a weighted sum of reciprocal of square-free numbers not exceeding $n.
Abstract: The number of tuples with positive integers pairwise relatively prime to each other with product at most $n$ is considered. A generalization of $\mu^{2}$ where $\mu$ is the Mobius function is used to formulate this divisor sum and establish some identities. One such identity is a weighted sum of reciprocal of square-free numbers not exceeding $n$. Some auxiliary number theoretic functions are introduced to formulate this sum.
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TL;DR: In this paper, the authors give a new infinite family of group homomorphisms from the braid group B_k to the symmetric group S{mk} for all k and m \geq 2.
Abstract: We give a new infinite family of group homomorphisms from the braid group B_k to the symmetric group S_{mk} for all k and m \geq 2. Most known permutation representations of braids are included in this family. We prove that the homomorphisms in this family are non-cyclic and transitive. For any divisor l of m, 1\leq l < m, we prove in particular that if \frac{m}{l} is odd then there are 1 + \frac{m}{l} non-conjugate homomorphisms included in our family.
We define a certain natural restriction on homomorphisms B_k to S_n, common to all homomorphisms in our family, which we term 'good', and of which there are two types.
We prove that all good homomorphisms B_k to S_{mk} of type 1 are included in the infinite family of homomorphisms we gave. For m=3, we prove that all good homomorphisms B_k to S_{3k} of type 2 are also included in this family.
Finally, we refute a conjecture made by Matei and Suciu regarding permutation representations of braids and give an updated conjecture.
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TL;DR: In this paper, the Laxton group of recursive sequences with arbitrary initial conditions is studied and a necessary condition is established for a prime to be a divisor of a sequence: the respective element of the ring must be a quadratic residue.
Abstract: Linear second order recursive sequences with arbitrary initial conditions
are studied. For sequences with the same parameters a ring and a group
is attached, and isomorphisms and homomorphisms are established
for related parameters. In the group, called the {\it sequence group},
sequences are identified if they differ by a scalar factor,
but not if they differ by a shift, which is the case for the Laxton group.
Prime divisors of sequences are studied with the help of the sequence group $\mod p$,
which is always cyclic of order $p\pm 1$.
Even and odd numbered subsequences are given independent status through
the introduction of one rational parameter in place of two integer
parameters. This step brings significant simplifications in the
algebra.
All elements of finite order in Laxton groups and sequence
groups are described effectively.
A necessary condition is established for a prime $p$
to be a divisor of a sequence: {\it the norm (determinant) of
the respective element of the ring must be a quadratic residue $\mod p$}.
This leads to an uppers estimate of the set of divisors
by a set of prime density $1/2$. Numerical experiments show that
the actual density is typically close to $0.35$.
A conjecture is formulated that the sets of prime divisors of the
even and odd numbered elements are independent for a large family
of parameters.