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 paper, the authors determine the kernel of φ for certain M and M′, as well as relate the pre-image of the Shimura subgroup under φ to the group Σ(M′)τ.
5 citations
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TL;DR: In this paper, the asymptotics of the integer moments of a random Haar-distributed unitary matrix have been studied and the central limit theorem for these functions has been derived.
Abstract: Since the seminal work of Keating and Snaith, the characteristic polynomial of a random Haar-distributed unitary matrix has seen several of its functional studied or turned into a conjecture; for instance:
$ \bullet $ its value in $1$ (Keating-Snaith theorem),
$ \bullet $ the truncation of its Fourier series up to any fraction of its degree,
$ \bullet $ the computation of the relative volume of the Birkhoff polytope,
$ \bullet $ its products and ratios taken in different points,
$ \bullet $ the product of its iterated derivatives in different points,
$ \bullet $ functionals in relation with sums of divisor functions in $ \mathbb{F}_q[X] $.
$ \bullet $ its mid-secular coefficients,
$ \bullet $ the "moments of moments", etc.
We revisit or compute for the first time the asymptotics of the integer moments of these last functionals and several others. The method we use is a very general one based on reproducing kernels, a symmetric function generalisation of some classical orthogonal polynomials interpreted as the Fourier transform of particular random variables and a local Central Limit Theorem for these random variables. We moreover provide an equivalent paradigm based on a randomisation of the mid-secular coefficients to rederive them all. These methodologies give a new and unified framework for all the considered limits and explain the apparition of Hankel determinants or Wronskians in the limiting functional.
5 citations
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TL;DR: If the natural number n has the canonical form p1a1p2a2aa2prar, then d =p1b1p 2b2b2br r for i = 1,2,r,r and σ(n) is an exponential divisor of n as discussed by the authors.
Abstract: If the natural number n has the canonical form p1a1p2a2…prar then d=p1b1p2b2…prbr is said to be an exponential divisor of n if bi|ai for i=1,2,…,r. The sum of the exponential divisors of n is denoted by σ(e)(n). n is said to be an e-perfect number if σ(e)(n)=2n; (m;n) is said to be an e-amicable pair if σ(e)(m)=m
5 citations
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IBM1
TL;DR: In this paper, an improvement in division machines for computing quotients and remainders from dividend and divisor factor data, and particularly in machines controlled by data expressed in accordance with the binary system of notation, is described.
Abstract: This invention relates to improvements in division machines for computing quotients and remainders from dividend and divisor factor data, and particularly in machines controlled by data expressed in accordance with the binary system of notation. With previous division machines, especially...
5 citations
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TL;DR: A direct comparison of cryptographic protocols using explicit formulas for real hyperelliptic curves with the corresponding protocols presented in the imaginary model is performed.
Abstract: We present a complete set of efficient explicit formulas for arithmetic
in the degree $0$ divisor class group of a genus two real hyperelliptic
curve given in affine coordinates. In addition to formulas suitable for
curves defined over an arbitrary finite field, we give simplified versions
for both the odd and the even characteristic cases. Formulas for baby
steps, inverse baby steps, divisor addition, doubling, and special
cases such as adding a degenerate divisor are provided, with
variations for divisors given in reduced and adapted basis. We
describe the improvements and the correctness together with a
comprehensive analysis of the number of field operations for each
operation. Finally, we perform a direct comparison of cryptographic
protocols using explicit formulas for real hyperelliptic curves with
the corresponding protocols presented in the imaginary model.
5 citations