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We present an integral expression of the Catalan numbers, based on F\'eaux' integral representation of $\log\left[\Gamma(x)\right]$, $\Gamma$ being the usual Gamma function. The obtained formula may be the starting point of the derivation of new relations involving central binomial coefficients or Catalan numbers.

In this article we obtain, using an expression of the digamma function $\psi(x)$ due to Mikolas, integral representations of the zeta function of odd arguments $\zeta(2p+1)$ for any positive value of $p$. The integrand consists of the product of a polynomial by one or two elementary trigonometric functions. Examples for the first values of the argument are given. Some of them were already derived by other methods.

We present two integral representations of the logarithm of the Glaisher-Kinkelin constant. The calculations are based on definite integral expressions of $\log\Gamma(x)$, $\Gamma$ being the usual Gamma function, due respectively to F\'eaux and Kummer. The connection between the second formula and an infinite series expansion of the Glaisher-Kinkelin constant is also outlined.

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 structure and spectra, in particular for the modeling of emission or absorption properties of hot plasmas. However, no explicit formula for $P(M_L,M_S)$ is known yet. In the present work, we show that the generating function for the cumulants, which characterize the distribution, obeys a recurrence relation, similar to the Newton-Girard identities relating elementary symmetric polynomials to power sums. This enables us to provide an explicit formula for the generating function. We also analyze the possibility of representing the $P(M_L,M_S)$ distribution by a bi-variate Gram-Charlier series, which coefficients are obtained from the knowledge of the exact moments of $P(M_L,M_S)$. It is shown that a simple approximation is obtained by truncating this series to the first few terms, though it is not convergent., Comment: submitted to J. Phys. B: At. Mol. Opt. Phys

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 above mentioned integrals together with a harmonic-number sum. Simplification of the latter eventually leads to an integral representation of $\zeta(2p + 1)$., Comment: submitted to "International Journal of Number Theory"

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 problem to produce quantitatively accurate images from compromised signal. Deep learning-based methods for low-dose PET are generally poorly conditioned and perform unreliably on images with features not present in the training distribution. We present a method which explicitly models deep latent space features using a robust kernel representation, providing robust performance on previously unseen dose reduction factors. Additional constraints on the information content of deep latent features allow for tuning in-distribution accuracy and generalisability. Tests with out-of-distribution dose reduction factors ranging from $\times 10$ to $\times 1000$ and with both paired and unpaired MR, demonstrate significantly improved performance relative to conventional deep-learning methods trained using the same data. Code:https://github.com/cameronPain, Comment: 19 pages, 15 figures, 4 tables, Submitted to IEEE Transactions on Medical Imaging

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 functions, generalizes a known relation for usual derangements.

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 $$ \boldsymbol{\eta}=2\int_0^1\log\Gamma(x)\,\cdot\sin(2\pi x)\,\mathrm{d}x=0.7687478924\dots $$ in terms of the known mathematical constants as $\pi$, $\log\pi$, $\log 2$, $\Gamma(1/4)$, $e$, etc. In the present work, we show that $$ \boldsymbol{\eta}=\frac{1}{\pi}\left[\gamma+\log(2\pi)\right], $$ where $\gamma$ is the usual Euler-Mascheroni constant, and provide two different proofs, the first one involving the Glaisher-Kinkelin constant, and the second one based on the Malmst\'en integral representation of $\log\Gamma(x)$. The resulting formula can also be obtained directly from the knowledge of the Fourier series expansion of $\log\Gamma(x)$.

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 $\mathcal{O}(n)$ growth in work with problem size on unstructured grids. This scalability relies on the CF splitting producing an $A_\textrm{ff}$ block that is easy to invert. We introduce a parallel two-pass CF splitting designed to give diagonally dominant $A_\textrm{ff}$. The first pass computes a maximal independent set in the symmetrized strong connections. The second pass converts F-points to C-points based on the row-wise diagonal dominance of $A_\textrm{ff}$. We find this two-pass CF splitting outperforms common CF splittings available in hypre. Furthermore, parallelisation of reduction multigrids in hyperbolic problems is difficult as we require both long-range grid-transfer operators and slow coarsenings (with rates of $\sim$1/2 in both 2D and 3D). We find that good parallel performance in the setup and solve is dependent on several factors: repartitioning the coarse grids, reducing the number of active MPI ranks as we coarsen, truncating the multigrid hierarchy and applying a GMRES polynomial as a coarse-grid solver. We compare the performance of two different reduction multigrids, AIRG (that we developed previously) and the hypre implementation of $\ell$AIR. In the streaming limit with AIRG, we demonstrate 81\% weak scaling efficiency in the solve from 2 to 64 nodes (256 to 8196 cores) with only 8.8k unknowns per core, with solve times up to 5.9$\times$ smaller than the $\ell$AIR implementation in hypre.

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 summations involving Catalan numbers or central binomial coefficients.

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 procedure is applied to a variant of the Touchard identity proposed by Callan a few years ago. The method presented here can be generalized to derive additional integral representations from known ones, provided that the latter have a well-suited form and lend themselves to an analytical summation under the integral sign.

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 $\nu_p$, using the Legendre-de Polignac formula. Other iterated quantities such as the Berezin function, are also considered. Beyond their recreational character, such quantities, often related to very large numbers, may find applications for cryptography purposes. Finally, lower and upper bounds for the $p$-adic valuation of Stirling numbers of the first kind and Catalan numbers are briefly discussed.

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 overlap greater than $a$, for some $a>0$, in the whole subcritical phase of inverse temperatures $\beta \in [0,\beta_c)$. Moreover, we study this probability both conditionally on the branching Brownian motion and non-conditionally. Two sub-phases of inverse temperatures appear, but surprisingly the threshold is not the same in both cases., Comment: 38 pages, 4 figures

Recently, Agievich proposed an interesting upper bound on binomial coefficients in the de Moivre-Laplace form. In this article, we show that the latter bound, in the specific case of a central binomial coefficient, is larger than the one proposed by Sasvari and obtained using the Binet formula for the Gamma function. In addition, we provide the expression of the next-order bound and apply it to Catalan numbers $C_n$. The bounds are very close to the exact value, the difference decreasing with $n$ and with the order of the upper bound.

We present electron transport calculations of shocked argon based on an average-atom modeling of the plasma, and compare them with measurements, involving both incident and reflected shock waves. Since the corresponding experiments are subject to a 5 T magnetic field, the impact of the latter on the Rankine-Hugoniot equations is taken into account, starting from the magneto-resistive hydrodynamics, and the resistivity tensor is deduced from the Boltzmann equation. The resistivity tensor yields the electrical and Hall resistivities. Our average-atom code Paradisio provides the quantities required for the calculation of electrical resistivity within the Ziman-Evans formalism, as well as for the Hall resistivity. We obtain a good agreement between calculated conductivities and experimental values, both for the incident and reflected shocks. Our values of the Hall constant are compared to experimental values derived from Hall voltage measurements, as well as to theoretical ones based on the quantum statistical linear-relaxation-time approach., Comment: submitted to Phys. Rev. E

The many-body localised phase of quantum systems is an unusual dynamical phase wherein the system fails to thermalise and yet, entanglement grows unboundedly albeit very slowly in time. We present a microscopic theory of this ultraslow growth of entanglement in terms of dynamical eigenstate correlations of strongly disordered, interacting quantum systems in the many-body localised regime. These correlations involve sets of four or more eigenstates and hence, go beyond correlations involving pairs of eigenstates which are usually studied in the context of eigenstate thermalisation or lack thereof. We consider the minimal case, namely the second R\'enyi entropy of entanglement, of an initial product state as well as that of the time-evolution operator, wherein the correlations involve quartets of four eigenstates. We identify that the dynamics of the entanglement entropy is dominated by the spectral correlations within certain special quartets of eigenstates. We uncover the spatial structure of these special quartets and the ensuing statistics of the spectral correlations amongst the eigenstates therein, which reveals a hierarchy of timescales or equivalently, energyscales. We show that the hierarchy of these timescales along with their non-trivial distributions conspire to produce the logarithmic in time growth of entanglement, characteristic of the many-body localised regime. The underlying spatial structures in the set of special quartets also provides a microscopic understanding of the spacetime picture of the entanglement growth. The theory therefore provides a much richer perspective on entanglement growth in strongly disordered systems compared to the commonly employed phenomenological approach based on the $\ell$-bit picture., Comment: 19 pages, 12 figures

In this note, the first-order Dickson polynomials are introduced through a particular case of the expression of the trace of the $n^{th}$ power of a matrix in terms of powers of the trace and determinant of the matrix itself. The technique relies on the Cayley-Hamilton theorem and its application to the derivation of formulas due to Carlitz and to second-order Dickson polynomials is straightforward. Finally, generalization of Dickson polynomials over finite fields and multivariate Dickson polynomials are evoked as potential avenues of investigation in the same framework.

The intriguing law of anomalous numbers, also named Benford's law, states that the significant digits of data follow a logarithmic distribution favoring the smallest values. In this work, we test the compliance with this law of the atomic databases developed at the National Institute of Standards and Technology (NIST) focusing on line energies, oscillator strengths, Einstein coefficients and radiative opacities. The considered databases are the Atomic Spectra Database (ASD) and the NIST-LANL (Los Alamos National Laboratory) Lanthanide/Actinide Opacity Database. The present study is not limited to the first digit and the case of multipole lines is also considered. The fact that the law is verified with a high accuracy means that the occurrence of digits reflects the constraints induced, in a given angular-momentum coupling, by the selection rules for atomic processes. As a consequence, Benford's law may be of great interest to detect inconsistencies in atomic databases., Comment: submitted to "Journal of Quantitative Spectroscopy and Radiative Transfer"

Reactive transport in porous media plays a pivotal role in subsurface reservoir processes, influencing fluid properties and geochemical characteristics. However, coupling fluid flow and transport with geochemical reactions is computationally intensive, requiring geochemical calculations at each grid cell and each time step within a discretized simulation domain. Although recent advancements have integrated machine learning techniques as surrogates for geochemical simulations, ensuring computational efficiency and accuracy remains a challenge. This chapter investigates machine learning models as replacements for a geochemical module in a reactive transport in porous media simulation. We test this approach on a well-documented cation exchange problem. While the surrogate models excel in isolated predictions, they fall short in rollout predictions over successive time steps. By introducing modifications, including physics-based constraints and tailored dataset generation strategies, we show that machine learning surrogates can achieve accurate rollout predictions. Our findings emphasize that, when judiciously designed, machine learning surrogates can substantially expedite the cation exchange problem without compromising accuracy, offering significant potential for a range of reactive transport applications.

In the Super-Transition-Array statistical method for the computation of radiative opacity of hot dense matter, the moments of the absorption or emission features involve partition functions with reduced degeneracies, occurring through the calculation of averages of products of subshell populations. In the present work, we discuss several aspects of the computation of such peculiar partition functions, insisting on the precautions that must be taken in order to avoid numerical difficulties. In a previous work, we derived a formula for supershell partition functions, which takes the form of a functional of the distribution of energies within the supershell and allows for fast and accurate computations, truncating the number of terms in the expansion. The latter involves coefficients for which we obtained a recursion relation and an explicit formula. We show that such an expansion can be combined with the recurrence relation for shifted partition functions. We also propose, neglecting the effect of fine structure as a first step, a positive-definite formula for the Super-Transition-Array moments of any order, providing an insight into the asymmetry and sharpness of the latter. The corresponding formulas are free of alternating sums. Several ways to speed up the calculations are also presented., Comment: submitted to "High Energy Density Physics"

We present two integral representations of the logarithm of the Glaisher-Kinkelin constant. Both are based on a definite integral of $\ln[\Gamma(x + 1)]$, $\Gamma$ being the usual Gamma function. The first one relies on an integral representation of $\ln[\Gamma(x + 1)]$ due to Binet, and the second one results from the so-called Malmst\'en formula. The numerical evaluation is easier with the latter expression than with the former.

We propose a semi-empirical formula for the cross section of ionization by electron impact. The formula involves adjustable parameters which are determined by comparison with measured or numerically calculated cross sections. In the latter case, the ions are perturbed by their environment which is a high-density plasma. As a consequence, the cross section is significantly modified. We investigate Be-like carbon, nitrogen and oxygen as well as aluminum ions. We also show that the formula is well-suited for interpolation and extrapolation. Knowing the cross section, we calculate the rate coefficient within the Boltzmann and Fermi-Dirac statistics. In the first case, the rate can be calculated analytically. In the second one, it can be expressed in terms of special functions, but the numerical evaluation is more convenient while providing accurate results. Our results are compared to experiment and to other calculations., Comment: submitted to J. Phys. B: At. Mol. Opt. Phys

In comparison with Derrida's REM, we investigate the influence of the so-called decoration processes arising in the limiting extremal processes of numerous log-correlated Gaussian fields. In particular, we focus on the branching Brownian motion and two specific quantities from statistical physics in the vicinity of the critical temperature. The first one is the two-temperature overlap, whose behavior at criticality is smoothened by the decoration process - unlike the one-temperature overlap which is identical - and the second one is the temperature susceptibility, as introduced by Sales and Bouchaud, which is strictly larger in the presence of decorations and diverges, close to the critical temperature, at the same speed as for the REM but with a different multiplicative constant. We also study some general decorated cases in order to highlight the fact that the BBM has a critical behavior in some sense to be made precise., Comment: 51 pages. The main difference with the first version is Example 4.6, which now exhibits more general behaviors

The polynomials introduced by Sister Celine cover different usual orthogonal polynomials as special cases. Among them, the Jacobi and discrete Hahn polynomials are of particular interest for the quantum theory of angular momentum. In this note, we show that characters of irreducible representations of the rotation group as well as Wigner rotation "d" matrices, can be expressed as Sister Celine's polynomials. Since many relations were proposed for the latter polynomials, such connections could lead to new identities for quantities important in quantum mechanics and atomic physics.

Recently, there has been a huge effort focused on developing highly efficient open source libraries to perform Artificial Intelligence (AI) related computations on different computer architectures (for example, CPUs, GPUs and new AI processors). This has not only made the algorithms based on these libraries highly efficient and portable between different architectures, but also has substantially simplified the entry barrier to develop methods using AI. Here, we present a novel methodology to bring the power of both AI software and hardware into the field of numerical modelling by repurposing AI methods, such as Convolutional Neural Networks (CNNs), for the standard operations required in the field of the numerical solution of Partial Differential Equations (PDEs). The aim of this work is to bring the high performance, architecture agnosticism and ease of use into the field of the numerical solution of PDEs. We use the proposed methodology to solve the advection-diffusion equation, the non-linear Burgers equation and incompressible flow past a bluff body. For the latter, a convolutional neural network is used as a multigrid solver in order to enforce the incompressibility constraint. We show that the presented methodology can solve all these problems using repurposed AI libraries in an efficient way, and presents a new avenue to explore in the development of methods to solve PDEs and Computational Fluid Dynamics problems with implicit methods., Comment: 24 pages, 6 figures

A particular case of degenerate Clebsch-Gordan coefficient can be expressed with three binomial coefficients. Such a formula, which may be obtained using the standard ladder operator procedure, can also be derived from the Racah-Shimpuku formula or from expressions of Clebsch-Gordan coefficients in terms of $_3F_2$ hypergeometric functions. The O'Hara interesting interpretation of this Clebsch-Gordan coefficient by binomial random variables can also be related to hypergeometric functions ($_2F_1$), in the case where one of the parameters tends to infinity. This emphasizes the links between Clebsch-Gordan coefficients, hypergeometric functions and, what has been less exploited until now, the notion of probability within the framework of the quantum theory of angular momentum.

In this article, we would like to pay tribute to Gabor Kalman, outlining his contribution to a model widely used in dense plasma physics: the high-temperature Thomas-Fermi model. The approach of Ruoxian Ying and Kalman relies on the separation of the bound and free electrons, a physically reasonable definition of the bound electrons, a description of the source density in the Poisson equation through the electron-ion and ion-ion pair correlation functions and a determination of the degree of ionization from the minimization of the total free energy. We also report on different approximations of the function $\Phi$, which is a cornerstone of the original Thomas-Femi model., Comment: submitted to "Contributions to Plasma Physics"

This paper solves the discretised multiphase flow equations using tools and methods from machine-learning libraries. The idea comes from the observation that convolutional layers can be used to express a discretisation as a neural network whose weights are determined by the numerical method, rather than by training, and hence, we refer to this approach as Neural Networks for PDEs (NN4PDEs). To solve the discretised multiphase flow equations, a multigrid solver is implemented through a convolutional neural network with a U-Net architecture. Immiscible two-phase flow is modelled by the 3D incompressible Navier-Stokes equations with surface tension and advection of a volume fraction field, which describes the interface between the fluids. A new compressive algebraic volume-of-fluids method is introduced, based on a residual formulation using Petrov-Galerkin for accuracy and designed with NN4PDEs in mind. High-order finite-element based schemes are chosen to model a collapsing water column and a rising bubble. Results compare well with experimental data and other numerical results from the literature, demonstrating that, for the first time, finite element discretisations of multiphase flows can be solved using an approach based on (untrained) convolutional neural networks. A benefit of expressing numerical discretisations as neural networks is that the code can run, without modification, on CPUs, GPUs or the latest accelerators designed especially to run AI codes., Comment: 34 pages, 18 figures, 4 tables

The low-lying structure of $^{13}$Be has remained an enigma for decades. Despite numerous experimental and theoretical studies, large inconsistencies remain. Being both unbound, and one neutron away from $^{14}$Be, the heaviest bound beryllium nucleus, $^{13}$Be is difficult to study through simple reactions with weak radioactive ion beams or more complex reactions with stable-ion beams. Here, we present the results of a study using the $^{12}$Be(d,p)$^{13}$Be reaction in inverse kinematics using a 9.5~MeV per nucleon $^{12}$Be beam from the ISAC-II facility. The solid deuteron target of IRIS was used to achieve an increased areal thickness compared to conventional deuterated polyethylene targets. The Q-value spectrum below -4.4~MeV was analyzed using a Bayesian method with GEANT4 simulations. A three-point angular distribution with the same Q-value gate was fit with a mixture of $s$- and $p$-wave, $s$- and $d$-wave, or pure $p$-wave transfer. The Q-value spectrum was also compared with GEANT simulations obtained using the energies and widths of states reported in four previous works. It was found that our results are incompatible with works that revealed a wide $5/2^+$ resonance but shows better agreement with ones that reported a narrower width., Comment: 10 pages, 5 figures

We calculate the cross-section of ionization by free-electron impacts in high or moderate density plasmas. We show that the so-called ionization potential depression (IPD) strongly affects the magnitude of the cross-section in the high-density domain. We use the well-known IPD formulas of Stewart-Pyatt and Ecker-Kr\"oll. A more recent approach based on classical molecular dynamics simulation is also investigated. The latter provides an alternative way to calculate IPD values. At near-solid densities the effects of the free-electron degeneracy should be investigated. The rates are then calculated within the Fermi-Dirac statistics. We first use the semi-empirical formula of Lotz for ionization cross-section. The results may differ significantly from measured cross-sections or calculations with reliable atomic codes. Then, in a second step, we propose a new formula that combines the Lotz formula and a polynomial expansion in terms of the ratio of the energy of the incident electron and the ionization energy. The coefficients of the polynomial expansion are adjusted to fit the cross-section provided by robust atomic codes. A great advantage of the new formula is that it allows a fully analytical calculation of the ionization rate. Our results are compared to experiments measuring IPDs, cross-sections and rate coefficients on aluminum at high and moderate densities and on Be-like CNO ions., Comment: submitted to J. Phys. B: At. Mol. Opt. Phys

We present new results on the ionization by electron impacts in a dense plasma. We are interested in the density effect known as the ionization potential depression and in its role in atomic structure. Rather than using the well-known Stewart-Pyatt or Ecker-Kr\"oll formulas for the ionization potential depression, we consider a distribution function of the ionization energy, which involves the plasma fluctuations due to ion dynamics. This distribution is calculated within classical molecular dynamics. The removal of the noise yields a new distribution which is composed of a small set of Gaussian peaks among which one peak is selected by considering the signal-to-noise ratio. This approach provides an ionization potential depression in good agreement with experimental results obtained at the Linac Coherent Light Source facility. Our results are also compared to other calculations. In a second part, we investigate the effects of the ionization potential depression and the fluctuations on ionization by electron impacts. We propose a new expression of the cross section, based on an average over the ionization energy distribution. This cross section can be calculated analytically. The main strength of our work is to account for the fluctuations due to ion dynamics., Comment: submitted to Phys. Rev. E

We present calculations of electrical resistivity for expanded boron, aluminum, titanium and copper plasmas using the Ziman formulation in the framework of the average-atom model. Our results are compared to experimental data, as well as with other theoretical calculations, relying on the Ziman and Kubo-Greenwood formulations, and based on average-atom models or quantum-molecular-dynamics simulations. The impact of the definition of ionization, paying a particular attention to the consistency between the latter and the perfect free electron gas assumption made in the formalism, is discussed. We propose a definition of the mean ionization generalizing to expanded plasmas the idea initially put forward for dense plasmas, consisting in dropping the contribution of quasi-bound states from the ionization due to continuum ones. It is shown that our recommendation for the calculation of the quasi-bound density of states provides the best agreement with measurements., Comment: submitted to Phys. Rev. E

The variations of exchange Slater integrals with respect to their order $k$ are not well known. While direct Slater integrals $F^k$ are positive and decreasing when the order increases, this is not stricto sensu the case for exchange integrals $G^k$. However, two inequalities were published by Racah in his seminal article "Theory of complex spectra. II". In this article, we show that the technique used by Racah can be generalized, albeit with cumbersome calculations, to derive further relations, and provide two of them, involving respectively three and four exchange integrals. Such relations can prove useful to detect regularities in complex atomic spectra and classify energy levels., Comment: submitted to J. Phys. B: At. Mol. Opt. Phys

In this note, we propose an integral representation for $\zeta(4)$, where $\zeta$ is the Riemann zeta function. The corresponding expression is obtained using relations for polylogarithms. A possible generalization to any even argument of the zeta function is considered.