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, a method and apparatus for obtaining the quotient of division operations in a data processing apparatus, generate as an auxiliary value the inverse square root g of the divisor w, then multiply by the dividend v, and the intermediate result again multiplied by the auxiliary value g.
Abstract: A method and apparatus are disclosed which, for obtaining the quotient of division operations in a data processing apparatus, generate as an auxiliary value the inverse square root g of the divisor w. The auxiliary value g is then multiplied by the dividend v, and the intermediate result again multiplied by the auxiliary value g. An improvement in operation of the data processing apparatus is obtained despite introduction of the auxiliary value because the range covered by the auxiliary value is significantly smaller than that of the direct inverse of the divisor. A preferred application is the area of signal processing in communications.
4 citations
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TL;DR: In this article, the authors considered integers n for which the polynomial xn − 1 has a divisor in 1d53d;p[x] of every degree up to n, where p is a rational prime.
Abstract: In a recent paper, we considered integers n for which the polynomial xn – 1 has a divisor in ℤ[x] of every degree up to n, and we gave upper and lower bounds for their distribution. In this paper, we consider those n for which the polynomial xn – 1 has a divisor in 𝔽p[x] of every degree up to n, where p is a rational prime. Assuming the validity of the Generalized Riemann Hypothesis, we show that such integers n have asymptotic density 0.
4 citations
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TL;DR: In this article, the authors introduced the notion of hesitant walk-avoiding and proved that the associated Grossberg-Karshon twisted cube is untwisted when the character formula is a true combinatorial formula in the sense that there are no terms appearing with a minus sign.
Abstract: Let $G$ be a complex semisimple simply connected linear algebraic group. Let $\lambda$ be a dominant weight for $G$ and $\mathcal{I} = (i_1, i_2, \ldots, i_n)$ a word decomposition for an element $w = s_{i_1} s_{i_2} \cdots s_{i_n}$ of the Weyl group of $G$, where the $s_i$ are the simple reflections. In the 1990s, Grossberg and Karshon introduced a virtual lattice polytope associated to $\lambda$ and $\mathcal{I}$, which they called a twisted cube, whose lattice points encode (counted with sign according to a density function) characters of representations of $G$. In recent work, the first author and Jihyeon Yang prove that the Grossberg-Karshon twisted cube is untwisted (so the support of the density function is a closed convex polytope) precisely when a certain torus-invariant divisor on a toric variety, constructed from the data of $\lambda$ and $\mathcal{I}$, is basepoint-free. This corresponds to the situation in which the Grossberg-Karshon character formula is a true combinatorial formula in the sense that there are no terms appearing with a minus sign. In this note, we translate this toric-geometric condition to the combinatorics of $\mathcal{I}$ and $\lambda$. More precisely, we introduce the notion of hesitant $\lambda$-walks and then prove that the associated Grossberg-Karshon twisted cube is untwisted precisely when $\mathcal{I}$ is hesitant-$\lambda$-walk-avoiding.
4 citations
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TL;DR: In this paper, the authors studied the distribution of divisors of Euler's totient function φ(n) and Carmichael's function λ(n), and showed that the average size of τ(φn)) and τ(λ(n)) are each exp{ log 2 2 (log log n) for almost all n.
Abstract: Two of the most studied functions in the theory of numbers are Euler’s totient function φ(n) and Carmichael’s function λ(n), the first giving the order of the group (Z/nZ)∗ of reduced residues modulo n, and the latter giving the maximum order of any element of (Z/nZ)∗. The distribution of φ(n) and λ(n) has been investigated from a variety of perspectives. In particular, many interesting properties of these functions require knowledge of the distribution of prime factors of φ(n) and λ(n), e.g., [3], [5], [4], [6], [7], [12], [19]. The distribution of all of the divisors of φ(n) and λ(n) has thus far received little attention, perhaps due to the complicated way in which prime factors interact to form divisors. From results about the normal number of prime factors of φ(n) and λ(n) [5], one deduces immediately that τ(φ(n)) and τ(λ(n)) are each exp{ log 2 2 (log log n)} for almost all n. However, the determination of the average size of τ(φ(n)) and of τ(λ(n)) is more complex, and has been studied recently by Luca and Pomerance [13]. In this note we investigate problems about localization of divisors of φ(n) and λ(n). Our results have application to the structure of (Z/nZ)∗, since the set of divisors of λ(n) is precisely the set of orders of elements of (Z/nZ)∗. We say that a positive integer m is u-dense if whenever 1 ≤ y < m, there is a divisor of m in the interval (y, uy]. The distribution of u-dense numbers for general u has been investigated by Tenenbaum ([17], [18]) and Saias ([14], [15]). According to Theoreme 1 of [14], the number of u-dense integersm ≤ x is (x log u)/ log x, uniformly for 2 ≤ u ≤ x. In particular, the number of 2-dense integersm ≤ x is x/ log x, that is, the 2-dense integers are about as sparse as the primes. By contrast, we show that 2-dense values of φ(n) and λ(n) are very common.
4 citations
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TL;DR: In this paper, it was shown that any divisor Y of a global analytic set X subset of R(n) has a generic equation, that is, there is an analytic function vanishing on Y with multiplicity one along each irreducible component of Y.
Abstract: We prove that any divisor Y of a global analytic set X subset of R(n) has a generic equation, that is, there is an analytic function vanishing on Y with multiplicity one along each irreducible component of Y. We also prove that there are functions with arbitrary multiplicities along Y. The main result states that if X is pure dimensional, Y is locally principal, X \ Y is not connected and Y represents the zero class in H(q-1)(infinity) (X, Z(2)) then the divisor Y is globally principal.
4 citations