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, the authors study the $q$-bracket operator of Bloch and Okounkov and derive convolution identities for the functions $f$ and link both classes of $q-brackets through divisor sums.
Abstract: We study the $q$-bracket operator of Bloch and Okounkov when applied to $f(\lambda)=\sum_{\lambda_i \in \lambda}g(\lambda_i)$ and $f(\lambda)=\sum_{\substack{\lambda_i \in \lambda \\ \lambda_i \text{distinct} }}g(\lambda_i)$. We use these expansions to derive convolution identities for the functions $f$ and link both classes of $q$-brackets through divisor sums. As a result, we generalize Euler's classic convolution identity for the partition function and obtain an analogous identity for the totient function. As corollaries, we generalize Stanley's theorem as well as provide several new combinatorial results.
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TL;DR: In this article, it was shown that a B$\acute{e}$zout ring $R$ is an elementary divisor ring if and only if $R/J(R)$ is regular.
Abstract: We present some new conditions for a B$\acute{e}$zout ring to be an elementary divisor ring. We prove, in this note, that a B$\acute{e}$zout ring $R$ is feckly zero-adequate if and only if $R/J(R)$ is regular if and only if $R/J(R)$ is $\pi$-regular, and that every feckly zero-adequate ring is an elementary divisor ring. If $R$ has feckly adequate range 1, we prove that $R$ is an elementary divisor ring if and only if $R$ is a B$\acute{e}$zout ring. Many known results are thereby generalized to much wider class of rings, e.g. [4, Theorem 14], [5, Theorem 4], [8, Theorem 1.2.14], [10, Theorem 4] and [11, Theorem 7]. \vskip3mm {\bf Keywords:} Elementary divisor ring, B$\acute{e}$zout ring, Feckly zero-adequate ring, Feckly adequate range 1.
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TL;DR: Theorem 1.3 as mentioned in this paper states that Kohnen's result still holds when N = 4M and 8M, with M an odd and square-free integer, and then Theorem 2.4 extends the equivalence to the cases of N = 9M and N = 27M, where M is a square free integer coprime to 3.
Abstract: The aim of this paper is the generalization of the following equivalence due to Kohnen: Let f be a modular function of integral weight with respect to Gamma_0(N), N square-free. Then f has a cuspidal divisor (i.e. zeros and poles supported at the cusps) if and only if f is an eta-quotient. We first present Theorem 1.3, which states that Kohnen's result still holds when N=4M and 8M, with M an odd and square-free integer, and then Theorem 1.4, which extends the equivalence to the cases - N=9M and N=27M, with M a square-free integer coprime to 3, - N=16M and N=32M, with M an odd and square-free integer, - N=25M and N=125M, with M a square-free integer coprime to 5, by introducing a generalization of the classical Dedekind eta-function.
Patent•
23 Feb 2015
TL;DR: In this article, an address generation device for dividing, by a predetermined divisor, the sum of a first result of multiplication of an ordinal number by a first constant and a result of multiplying the square of the ordinal numbers by a second constant to generate a remainder as an address is presented.
Abstract: PROBLEM TO BE SOLVED: To reduce the circuit scale of an interleaving circuit and a de-interleaving circuit by reducing comparison/subtraction elements and thus dispensing with a large scale modulo operation circuit and also by reducing a ROM size to be stored.SOLUTION: An address generation device for dividing, by a predetermined divisor, the sum of a first result of multiplication of an ordinal number by a first constant and a result of multiplication of the square of the ordinal number by a second constant to generate a remainder as an address includes: first arithmetic means for dividing the first result by the divisor to produce a first intermediate remainder; second arithmetic means for dividing a predetermined function of the ordinal number by the divisor to produce a second intermediate remainder; and third arithmetic means for dividing the sum of the first intermediate remainder and the second intermediate remainder by the divisor to produce a final remainder. The final remainder is the address.
TL;DR: In this article, the authors studied the problem of finding the greatest divisor of a positive integer n which is coprime to a given square-free integer k by an elementary and beautiful approach.
Abstract: Denote by $$\delta _{k}(n)$$
the greatest divisor of a positive integer n which is coprime to a given $$k\ge 2$$
. In the case of $$k=p$$
(a prime) Joshi and Vaidya studied $$E_{p}(x):=\sum _{n\le x}\delta _{p}(n)-\frac{p}{2(p+1)}x^{2}$$
(as $$x\rightarrow \infty $$
) and obtained $$E_{p}(x)=\Omega _{\pm }(x)$$
by an elementary and beautiful approach. Here we study $$R_{p}^{(2)}(x):=\sum _{n\le x}\delta _{p}^{2}(n)-\frac{p^{2}}{3(p^{2}+p+1)}x^{3}+\frac{p}{6}x$$
and show $$R_{p}^{(2)}(x)=\Omega _{\pm }(x^{2})$$
. Moreover, using a method of Adhikari and Balasubramanian we consider a bound of $$|R_{k}^{(2)}(x)|/x^{2}$$
for any square-free integer k.