Search Results (131)

In this work, we use the Riemann–Liouville (R-L) fractional derivative of order $\alpha \in (0,1)$ to study the oscillation criteria for damped matrix fractional differential equations and determine sufficient conditions under which all prepared solutions of the system show oscillatory behaviour. The criteria are novel even for the linear undamped case and extend conventional oscillation results for integer-order matrix differential systems to the fractional setting. The goal of the current effort is to better understand the relationships between solutions and their derivatives. Using the matrix-valued Riccati transformation converts the system into a Riccati-type inequality, and the oscillation conditions are then derived by integrating against a weighted kernel via the operator L. Both results generalise the integer-order oscillation criteria to the fractional matrix setting, extending their applicability to fractional-order control systems, viscoelastic structural models, and anomalous diffusion processes. This work develops new conditions and analytical techniques that deepen insight and provide useful results for analysing oscillatory behaviour and asymptotic stability of of the considered systems. To illustrate the significance of the obtained oscillation results, we give two examples. Full article

Poisson mixed-effects models are essential for analyzing repeated count data, relying on latent random effects to account for unobserved heterogeneity and longitudinal dependence. However, the validity of likelihood-based inference in these models is highly sensitive to the specification of both the fixed-effects structure and the distributional assumptions of the random effects. While diagnostics based on the information matrix equality (IME) provide a theoretical framework for detecting misspecification, their high dimensionality and reliance on second-order derivatives often result in numerical instability and poor finite-sample performance in nonlinear settings. Here we introduce the Contrast of Information by Volume (CIV) test, a low-dimensional information-based diagnostic test for Poisson generalized linear mixed models (GLMMs). By integrating the scalar CIV statistics with novel graphical diagnostics, our approach facilitates the interpretation of specification errors in the random-effects structure. We derive the asymptotic behaviour of the CIV statistics under local misspecification and evaluate their properties through Monte Carlo simulations. To ensure robust inference in moderate samples, a parametric bootstrap procedure is employed for size calibration. Simulation results demonstrate that the CIV diagnostics maintain accurate Type I error control and achieve competitive power against common misspecification, including heteroskedasticity, correlation, and heavy-tailed random-effect distributions. Compared to traditional IME diagnostics, estimator-comparison tests, and GMM-based procedures, the CIV approach offers a superior balance between finite-sample stability and detection power. Finally, an empirical application illustrates the utility of the CIV framework in diagnosing latent misspecification and guiding the selection of random-effects covariance structures in applied research. Full article

In this paper, we introduce and study a Szász-type family of positive linear operators generated by Euler polynomials of negative order on $[0,\infty )$. The construction is based on an explicit finite representation of these polynomials with non-negative terms, which ensures the positivity of the corresponding kernel. We prove the basic properties of the operators and show that they can be represented as finite convex combinations of shifted classical Szász operators. We also provide a probabilistic representation of the kernel as a finite mixture of Poisson distributions, which clarifies the role of the parameter k and the resulting moment structure. The corresponding algebraic and central moment identities are derived and used to establish convergence on compact intervals and to obtain quantitative estimates in terms of the modulus of continuity, Lipschitz-type classes, and Peetre’s K-functional. Furthermore, Voronovskaya-type asymptotic results are obtained, including a quantitative form and a second-order asymptotic formula. Numerical tables and a graphical illustration are presented for selected test functions and parameter values, and the results are consistent with the theoretical convergence behaviour. The paper shows that Euler polynomials of negative order provide a positive and structurally tractable framework for constructing Szász-type approximation operators on the positive real axis. Full article

This article introduces and develops a comprehensive theory of the Mixed Degenerate Gould–Hopper–Appell Type Polynomials $MDGHA$-$TPs$, constructed by embedding an Appell factor into the framework of degenerate Gould–Hopper generating functions. Beginning with the generating function formulation, we derive explicit series representations, monomial-type operational identities, recurrence relations, and a determinantal form that encodes the algebraic structure of the family. Summation identities expressed via Stirling numbers of the first kind and addition-type formulas are established. A detailed numerical investigation of the zero distributions of these polynomials is then carried out, with graphical illustrations revealing symmetry patterns and geometric arrangements in the complex plane. Connections with classical sequences of Appell, Hermite, and Gould–Hopper are explored throughout. The article concludes with remarks on open problems including the orthogonality of the $MDGHA$-$TPs$ with respect to suitable weight functions, the asymptotic behaviour of their zeros as the degree tends to infinity, and potential applications to boundary-value problems in heat diffusion, perturbation expansions in quantum mechanics, and signal processing in non-homogeneous media. Full article

The purpose of the paper is to give some sequences that converge quickly to a generalization of Ioachimescu’s constant, i.e., the limit of the sequence ${\left({\displaystyle \frac{1}{a^s}}+{\displaystyle \frac{1}{{(a+1)}^s}}+\cdots +{\displaystyle \frac{1}{{(a+n-1)}^s}}-{\displaystyle \frac{1}{1-s}}\left({(a+n-1)}^{1-s}-a^{1-s}\right)\right)}_{n\in \mathbb{N}},$ where $a\in (0,+\infty )$ and $s\in (0,1)$. Full article

This research paper examines the spatial and time variation of demographic dependency in Europe in a 30-year horizon of the evolution of the demographic dividend regarding the economic dependency ratio (ADR1). We used the Curve Fit Forecast tool to estimate the trends of ADR1 in each of the EU Member States using data on Eurostat projections and a sophisticated geostatistical analysis tool developed in ArcGIS Pro 3.2.2. The findings indicate that the dependency in all countries has increased significantly in a statistically significant manner as the Gompertz function has appeared as the best curve in a third of the cases. It is an S-shaped asymptotic behaviour of this function that effectively describes the nonlinear patterns of acceleration and saturation of demographic ageing. As indicated in the analysis, the European regions are increasingly moving apart, with the southern and eastern nations such as Romania demonstrating the most alarming decline in ADR1. These trends highlight the need to reform labour market policies and social protection mechanisms to an ageing population. The paper combines the curve-fitting, descriptive statistics (median, skewness, interquartile range (IQR)) with time clustering (value, correlation, and Fourier) to provide an effective, replicable approach to early warning and policy prioritisation. Overall, the results highlight the importance of integrating predictive spatial modelling and demographic economics to support anticipatory and evidence-based policy decisions. The proposed approach proves to be a robust and transferable framework, applicable to a wide range of socio-economic phenomena characterised by inertia and structural change. Future research should extend the analysis to subnational levels, incorporate additional explanatory variables, and develop scenario-based simulations, including multivariate Gompertz-type models, to further enhance both predictive accuracy and policy relevance in the context of emerging structural labour scarcity. Full article

Modern reliability experiments frequently face operational constraints that require balancing test duration, precision, and removal strategies, rendering classical censoring schemes inadequate for contemporary multidisciplinary applications. This study introduces a novel multi-progressive generalized Type-II censoring (MP-GC-T2) framework that unifies and extends existing progressive and generalized censoring structures through the integration of staged failure-proportion controls, dual temporal termination thresholds, and adaptive withdrawal of surviving units. The proposed mechanism provides enhanced flexibility in experiment design while retaining analytical tractability for statistical inference. Assuming Weibull lifetimes, we develop a complete inferential framework including maximum likelihood estimation, asymptotic interval construction, and Bayesian estimation via hybrid Metropolis–Hastings–Gibbs sampling with informative gamma priors, together with multiple interval estimation strategies for reliability characteristics. Extensive Monte Carlo investigations assess estimator bias, precision, coverage behaviour, and interval efficiency across diverse censoring configurations, demonstrating robustness and inferential gains relative to conventional schemes. Furthermore, optimal progressive-removal planning criteria are explored to guide practitioners in selecting censoring patterns that maximize inferential accuracy under practical constraints. The versatility and practical relevance of the MP-GC-T2 design are illustrated through applications to heterogeneous real datasets arising from clinical, chemical, geological, physical, and petroleum sciences, confirming its adaptability to distinct reliability structures and data-generation mechanisms. Collectively, the proposed methodology contributes a unified experimental and inferential platform that advances censoring design, reliability estimation, and cross-disciplinary statistical modelling. Full article

This research establishes an innovative analytical framework for examining ${\mathcal{C}}^0$-solutions of pseudo-almost automorphic type in evolution systems governed by nonlocal initial conditions within Banach spaces. Our methodological approach is founded upon the compactness properties of the resolvent operator ${(I-\mathcal{A})}^{-1}$ and the semigroup generated by A. By integrating semigroup theory with fixed-point methodologies, we develop a unified strategy that successfully overcomes challenges arising from nonlocal initial specifications. The practical applicability of our theoretical framework is verified through its implementation on a transport equation with nonlocal initial history, thereby demonstrating both the existence of solutions and their distinctive character. Full article

The dynamical processes in the atmosphere are coupled with the chemistry of the atmosphere. Internal gravity waves influence the distribution of chemical constituents in the atmosphere through their effects on the background wind or mean flow. We examine a coupled system of equations comprising a nonlinear transport equation of Fisher type for the distribution of the chemical species, along with nonlinear Boussinesq equations for internal gravity waves in a vertically stratified and vertically sheared fluid flow in a two-dimensional region. In our model, a horizontally localized gravity-wave packet is generated and propagates upward into a localized region where the chemical species is present. Numerical solutions show that the wave-induced mean flow resulting from nonlinear gravity-wave interactions in the vicinity of a critical level leads to modifications in the distribution of the chemical. An asymptotic analysis of a related qualitatively similar problem gives us information on the dominant behaviour of the chemical concentration perturbation. We conclude that nonlinearity and vertical shear play a vital role in the interplay between gravity-wave dynamics and chemical distributions in the atmosphere. Full article

We study the Sneddon $\mathcal{R}$-transform and its inverse in the setting of Lebesgue spaces. Generated by the mixed trigonometric kernel $xcos\left(xt\right)+hsin\left(xt\right)$, the $\mathcal{R}$-transform acts as a unifying operator for sine- and cosine-type integral transforms. Boundedness, continuity, and weighted $L^p$-estimates are established in an appropriate Banach space framework, together with Parseval–Goldstein type identities. Initial and final value theorems are derived for generalized functions in Zemanian-type spaces, yielding precise asymptotic behaviour at the origin and at infinity. A finite-interval theory is also developed, leading to polynomial growth estimates and final value theorems for the finite $\mathcal{R}$-transform. Full article

A transformation method was applied to the biconfluent (BHE), doubly confluent (DHE), and triconfluent (THE) Heun equations to generate and classify exactly solvable quantum mechanical potentials derived from them. With this, the range of potentials solvable in terms of the confluent hypergeometric function can be extended. The resulting potentials contained five independently tunable terms and two terms originating from the Schwartzian derivative that depended only on the parameters of the $z\left(x\right)$ transformation function. The polynomial solutions of these potentials contain expansion coefficients obtained from three-term (BHE and DHE) and four-term (THE) recurrence relations. For the simplest $z\left(x\right)$ transformation functions, the Lemieux–Bose potentials have been recovered for the BHE and DHE. The coupling parameters of these potentials and also of five potentials derived from the THE have been expressed in terms of the parameters of the respective differential equations. The present scheme offers a general framework into which a number of earlier results can be integrated in a systematic way. These include special cases of potentials obtained from less general versions of the Heun-type equations and individual solvable potentials obtained from various methods that do not necessarily refer to the Heun-type equations considered here. Several potentials derived here were found to coincide with or reduce to potentials found earlier from the quasi-exactly solvable (QES) formalism. Based on their mathematical form, their physically relevant features (domain of definition, asymptotic behaviour, single- or multi-well structure) were discussed, and possible fields of applications were pointed out. Full article

We prove a probabilistic limit theorem for the Epstein zeta-function $\zeta (s;Q)$ in the interval $[N,N+M]$ as $N\to \infty$, using discrete shifts $\zeta (\sigma +ikh;Q)$, where $h>0$ and $\sigma >{\displaystyle \frac{n-1}{2}}$ are fixed. Here, Q is a positive-definite $n\times n$ matrix, and the interval length M satisfies $h^{-1}{\left(Nh\right)}^{27/82}⩽M⩽h^{-1}{\left(Nh\right)}^{1/2}$. The limit measure is given explicitly. This theorem is the first result in short intervals for $\zeta (s;Q)$. The obtained theorem improves the known results established for the interval of length N. Since the considered probability measures are defined in terms of frequency, theorems in short intervals have a certain advantage in the detection of $\zeta (\sigma +ikh;Q)$ with a given property, as well as in the characterization of the asymptotic behaviour of $\zeta (s;Q)$ in general. Full article

Corporate Social Responsibility (CSR) actively enhances social, economic, and environmental well-being, increasingly impacting society. It plays a vital role in building a trustworthy and transparent image for the banking system’s relationship with the community. In this context, the paper aims to analyse the effects of delayed adaptation by the banking system to reporting requirements, as well as the reasons that may cause oscillating behaviour on their part. Accordingly, three scenarios are developed to describe the behaviour of banks that experience regular fluctuations in the level of external sustainability reporting requirements, meaning the pressure to comply with these requirements may vary over time. The research method employed involves a dynamic analysis, utilising a mathematical model described by a nonlinear system with time delay. The goal of the research is to identify the equilibrium point of the system and analyse its asymptotic stability. Moreover, the critical time delay is provided, beyond which banks’ responses become oscillatory rather than stable. Numerical simulations illustrate the theoretical findings and reveal a critical delay value under which banks can stabilise their resources to meet sustainability requirements. Full article

The arithmetic average of the first n primes, ${\overline{p}}_n={\displaystyle \frac{1}{n}}{\sum }_{i=1}^n{p}_i$, exhibits very many interesting and subtle properties. Since the transformation from $p_n\to {\overline{p}}_n$ is extremely easy to invert, $p_n=n{\overline{p}}_n-(n-1){\overline{p}}_{n-1}$, it is clear that these two sequences $p_n⟷{\overline{p}}_n$ must ultimately carry exactly the same information. But the averaged sequence ${\overline{p}}_n$, while very closely correlated with the primes, (${\overline{p}}_n\sim {\displaystyle \frac{1}{2}}{p}_n$), is much “smoother” and much better behaved. Using extensions of various standard results, I shall demonstrate that the prime-averaged sequence ${\overline{p}}_n$ satisfies prime-averaged analogues of the Cramer, Andrica, Legendre, Oppermann, Brocard, Fourges, Firoozbakht, Nicholson, and Farhadian conjectures. (So these prime-averaged analogues are not conjectures; they are theorems). The crucial key to enabling this pleasant behaviour is the “smoothing” process inherent in averaging. While the asymptotic behaviour of the two sequences is very closely correlated, the local fluctuations are quite different. Full article

In this article, we introduce a distribution that is an extension of the Power Maxwell (PM) distribution, which is based on the quotient of two independent random variables. These are the PM and a gamma distribution, respectively. In this way, the result is a model with greater kurtosis than the PM distribution. We study its probability density function and some properties, such as moments, asymmetry and kurtosis coefficient. An EM algorithm is proposed to estimate the parameters via the maximum likelihood method. A simulation study is carried out to study the asymptotic behaviour of our estimators. An application to a real dataset is also included. Full article