Zobrazeno 1 - 10
of 6 077
pro vyhledávání: '"P. Pain"'
Autor:
Pain, Jean-Christophe, Poirier, Michel
The distributions $P(M_L,M_S)$ of the total magnetic quantum numbers $M_L$ and $M_S$ for $N$ electrons of angular momentum $\ell$, as well as the enumeration of $LS$ spectroscopic terms and spectral lines, are crucial for the calculation of atomic st
Externí odkaz:
http://arxiv.org/abs/2410.01385
We propose a relation between values of the Riemann zeta function $\zeta$ and a family of integrals. This results in an integral representation for $\zeta(2p)$, where $p$ is a positive integer, and an expression of $\zeta(2p+1)$ involving one of the
Externí odkaz:
http://arxiv.org/abs/2409.06546
Low-dose positron emission tomography (PET) image reconstruction methods have potential to significantly improve PET as an imaging modality. Deep learning provides a promising means of incorporating prior information into the image reconstruction pro
Externí odkaz:
http://arxiv.org/abs/2409.06198
A sum rule for $r$-derangements obtained from the Cauchy product of exponential generating functions
Autor:
Pain, Jean-Christophe
We propose a sum rule for $r$-derangements (meaning that the elements are restricted to be in distinct cycles in the cycle decomposition) involving binomial coefficients. The identity, obtained using the Cauchy product of two exponential generating f
Externí odkaz:
http://arxiv.org/abs/2408.15927
Autor:
Pain, Jean-Christophe
In an interesting article entitled ``A curious formula related to the Euler Gamma function'', Bakir Farhi posed the open question of whether it was possible to obtain an expression of $$ \boldsymbol{\eta}=2\int_0^1\log\Gamma(x)\,\cdot\sin(2\pi x)\,\m
Externí odkaz:
http://arxiv.org/abs/2408.14835
Reduction multigrids have recently shown good performance in hyperbolic problems without the need for Gauss-Seidel smoothers. When applied to the hyperbolic limit of the Boltzmann Transport Equation (BTE), these methods result in very close to $\math
Externí odkaz:
http://arxiv.org/abs/2408.08262
Autor:
Pain, Jean-Christophe
In this article, we propose an integral expression of the Catalan numbers, based on Malmst\'en's definite-integral representation of $\ln\left[\Gamma(x)\right]$, $\Gamma$ being the usual Gamma function. The obtained expression is likely to yield new
Externí odkaz:
http://arxiv.org/abs/2408.05130
Autor:
Pain, Jean-Christophe
In this article, we use the Touchard identity in order to obtain new integral representations for Catalan numbers. The main idea consists in combining the identity with a known integral representation and resorting to the binomial theorem. The same p
Externí odkaz:
http://arxiv.org/abs/2408.01201
Autor:
Pain, Jean-Christophe
In this article, we investigate the $p$-adic valuation $\nu_p$ of quantities such as the factorial $n!$, the hyperfactorial $H(n)$ or the superfactorial $\mathrm{sf}(n)$. In particular, we obtain simple bounds (both upper and lower) for $\nu_p$, usin
Externí odkaz:
http://arxiv.org/abs/2408.00353
Autor:
Chataignier, Louis, Pain, Michel
At high temperature, the overlap of two particles chosen independently according to the Gibbs measure of the branching Brownian motion converges to zero as time goes to infinity. We investigate the precise decay rate of the probability to obtain an o
Externí odkaz:
http://arxiv.org/abs/2407.21014