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The study on the generating function approach to entropy become popular as it generates several well-known entropy measures discussed in the literature. In this work, we define the weighted cumulative residual entropy generating function (WCREGF) and study its properties. We then introduce the dynamic weighted cumulative residual entropy generating function (DWCREGF). It is shown that the DWCREGF determines the distribution uniquely. We study some characterization results using the relationship between the DWCREGF and the hazard rate and/or the mean residual life function. Using a characterization based on DWCREGF, we develop a new goodness fit test for Rayleigh distribution. A Monte Carlo simulation study is conducted to evaluate the proposed test. Finally, the test is illustrated using two real data sets.

An important aim of this paper is to convey some basics of mathematical logic to the legal community working with Artificial Intelligence. After analysing what AI is, we decide to delimit ourselves to rule-based AI leaving Neural Networks and Machine Learning aside. Rule based AI allows for Formal methods which are described in a rudimentary form. We will then see how mathematical logic interacts with legal rule-based AI practice. We shall see how mathematical logic imposes limitations and complications to AI applications. We classify the limitations and interactions between mathematical logic and legal AI in three categories: logical, computational and mathematical. The examples to showcase the interactions will largely come from European traffic regulations. The paper closes off with some reflections on how and where AI could be used and on basic mechanisms that shape society.

We provide a new theoretical framework for the variable-step deferred correction (DC) methods based on the well-known BDF2 formula. By using the discrete orthogonal convolution kernels, some high-order BDF2-DC methods are proven to be stable on arbitrary time grids according to the recent definition of stability (SINUM, 60: 2253-2272). It significantly relaxes the existing step-ratio restrictions for the BDF2-DC methods (BIT, 62: 1789-1822). The associated sharp error estimates are established by taking the numerical effects of the starting approximations into account, and they suggest that the BDF2-DC methods have no aftereffect, that is, the lower-order starting scheme for the BDF2 scheme will not cause a loss in the accuracy of the high-order BDF2-DC methods. Extensive tests on the graded and random time meshes are presented to support the new theory.

This article presents a new algorithm to compute all the roots of two families of polynomials that are of interest for the Mandelbrot set $\mathcal{M}$ : the roots of those polynomials are respectively the parameters $c\in\mathcal{M}$ associated with periodic critical dynamics for $f_c(z)=z^2+c$ (hyperbolic centers) or with pre-periodic dynamics (Misiurewicz-Thurston parameters). The algorithm is based on the computation of discrete level lines that provide excellent starting points for the Newton method. In practice, we observe that these polynomials can be split in linear time of the degree. This article is paired with a code library \citelib{MLib} that implements this algorithm. Using this library and about 723 000 core-hours on the HPC center \textit{Rom\'eo} (Reims), we have successfully found all hyperbolic centers of period $\leq 41$ and all Misiurewicz-Thurston parameters whose period and pre-period sum to $\leq 35$. Concretely, this task involves splitting a tera-polynomial, i.e. a polynomial of degree $\sim10^{12}$, which is orders of magnitude ahead of the previous state of the art. It also involves dealing with the certifiability of our numerical results, which is an issue that we address in detail, both mathematically and along the production chain. The certified database is available to the scientific community. For the smaller periods that can be represented using only hardware arithmetic (floating points FP80), the implementation of our algorithm can split the corresponding polynomials of degree $\sim10^{9}$ in less than one day-core. We complement these benchmarks with a statistical analysis of the separation of the roots, which confirms that no other polynomial in these families can be split without using higher precision arithmetic.

We have introduced a q-deformation, i.e., a polynomial in q with natural coefficients, of the binomial coefficient of two finite words u and v counting the number of occurrences of v as a subword of u. In this paper, we examine the q-deformation of Parikh matrices as introduced by E\u{g}ecio\u{g}lu in 2004. Many classical results concerning Parikh matrices generalize to this new framework: Our first important observation is that the elements of such a matrix are in fact q-deformations of binomial coefficients of words. We also study their inverses and as an application, we obtain new identities about q-binomials. For a finite word z and for the sequence $(p_n)_{n\ge 0}$ of prefixes of an infinite word, we show that the polynomial sequence $\binom{p_n}{z}_q$ converges to a formal series. We present links with additive number theory and k-regular sequences. In the case of a periodic word $u^\omega$, we generalize a result of Salomaa: the sequence $\binom{u^n}{z}_q$ satisfies a linear recurrence relation with polynomial coefficients. Related to the theory of integer partition, we describe the growth and the zero set of the coefficients of the series associated with $u^\omega$. Finally, we show that the minors of a q-Parikh matrix are polynomials with natural coefficients and consider a generalization of Cauchy's inequality. We also compare q-Parikh matrices associated with an arbitrary word with those associated with a canonical word $12\cdots k$ made of pairwise distinct symbols.

Classical Krylov subspace projection methods for the solution of linear problem $Ax = b$ output an approximate solution $\widetilde{x}\simeq x$. Recently, it has been recognized that projection methods can be understood from a statistical perspective. These probabilistic projection methods return a distribution $p(\widetilde{x})$ in place of a point estimate $\widetilde{x}$. The resulting uncertainty, codified as a distribution, can, in theory, be meaningfully combined with other uncertainties, can be propagated through computational pipelines, and can be used in the framework of probabilistic decision theory. The problem we address is that the current probabilistic projection methods lead to the poorly calibrated posterior distribution. We improve the covariance matrix from previous works in a way that it does not contain such undesirable objects as $A^{-1}$ or $A^{-1}A^{-T}$, results in nontrivial uncertainty, and reproduces an arbitrary projection method as a mean of the posterior distribution. We also propose a variant that is numerically inexpensive in the case the uncertainty is calibrated a priori. Since it usually is not, we put forward a practical way to calibrate uncertainty that performs reasonably well, albeit at the expense of roughly doubling the numerical cost of the underlying projection method.

We propose a generalization of nonlinear stability of numerical one-step integrators to Riemannian manifolds in the spirit of Butcher's notion of B-stability. Taking inspiration from Simpson-Porco and Bullo, we introduce non-expansive systems on such manifolds and define B-stability of integrators. In this first exposition, we provide concrete results for a geodesic version of the Implicit Euler (GIE) scheme. We prove that the GIE method is B-stable on Riemannian manifolds with non-positive sectional curvature. We show through numerical examples that the GIE method is expansive when applied to a certain non-expansive vector field on the 2-sphere, and that the GIE method does not necessarily possess a unique solution for large enough step sizes. Finally, we derive a new improved global error estimate for general Lie group integrators.

We study the problem of sequentially predicting properties of a probabilistic model and its next outcome over an infinite horizon, with the goal of ensuring that the predictions incur only finitely many errors with probability 1. We introduce a general framework that models such prediction problems, provide general characterizations for the existence of successful prediction rules, and demonstrate the application of these characterizations through several concrete problem setups, including hypothesis testing, online learning, and risk domination. In particular, our characterizations allow us to recover the findings of Dembo and Peres (1994) with simple and elementary proofs, and offer a partial resolution to an open problem posed therein.

Researchers would often like to leverage data from a collection of sources (e.g., primary studies in a meta-analysis) to estimate causal effects in a target population of interest. However, traditional meta-analytic methods do not produce causally interpretable estimates for a well-defined target population. In this paper, we present the CausalMetaR R package, which implements efficient and robust methods to estimate causal effects in a given internal or external target population using multi-source data. The package includes estimators of average and subgroup treatment effects for the entire target population. To produce efficient and robust estimates of causal effects, the package implements doubly robust and non-parametric efficient estimators and supports using flexible data-adaptive (e.g., machine learning techniques) methods and cross-fitting techniques to estimate the nuisance models (e.g., the treatment model, the outcome model). We describe the key features of the package and demonstrate how to use the package through an example.

In this note, we apply some techniques developed in [1]-[3] to give a particular construction of bivariate Abelian Codes from cyclic codes, multiplying their dimension and preserving their apparent distance. We show that, in the case of cyclic codes whose maximum BCH bound equals its minimum distance the obtained abelian code verifies the same property; that is, the strong apparent distance and the minimum distance coincide. We finally use this construction to multiply Reed-Solomon codes to abelian codes