Alcohol consumption dynamics may involve behavioural memory, relapse latency, and random environmental influences beyond classical deterministic models. This study develops a fractional stochastic delay model that integrates these mechanisms. The population comprises potential drinkers, current drinkers, temporary quitters, and permanent quitters. Behavioural memory is represented by a Caputo fractional derivative, relapse latency by a discrete delay, and random variability by multiplicative stochastic perturbations. We derive the basic reproduction number, characterise the drinking-free and drinking-present equilibria, and investigate local stability of the drinking-present equilibrium. The characteristic equation assesses the influence of relapse delay on stability, while a stochastic Volterra formulation describes random perturbations. Simulations show that fractional memory changes transient relaxation, relapse delay can strengthen persistence and oscillations, and stochastic perturbations generate variability between realisations. Comparisons with the classical deterministic model distinguish the dynamical roles of memory, delay, and stochasticity, providing a basis for future data-driven studies of alcohol consumption dynamics.
We investigate numerically a family of two-dimensional discrete-time dynamical systems. The family includes the Arnold Cat Map and the Cerbelli-Giona map, and is formed from compositions of piecewise-linear shear maps. Although many dynamical properties of the family can be understood analytically, details of multifractal behaviour require numerical computation. The piecewise-linearity of the maps allows us to accurately construct global stable and unstable manifolds, and to establish, for the first time for this family, good estimates for various multifractal properties, such as generalised fractal dimension, [Formula: see text] spectrum and cancellation exponents. We also discuss the relevance of these results to applications such as fluid mixing.
In this paper, we introduce and systematically study two new families of generalised Genocchi polynomials, namely the r-Whitney Genocchi polynomials and the r-Dowling Genocchi polynomials. Our approach is to multiply the Genocchi polynomials by the r-Whitney numbers and r-Dowling polynomials as weights, and then investigate the new generating functions and their properties that arise from this. We derive various identities, including explicit connections with Whitney numbers, recurrence relations, and summation formulas. In addition, by associating these polynomials with a random variable Y and defining them in a probabilistic version, we provide a probabilistic perspective on these polynomials.
We define an [Formula: see text]-dimensional degenerate Sumudu transform associated with the degenerate exponential kernel. A logarithmic change of variables conjugates this transform to the standard [Formula: see text]-dimensional Laplace transform. We develop the corresponding anisotropic parameter domains, weighted [Formula: see text] spaces, degenerate convolution, and Banach-algebra structure. Under explicit hypotheses, classical Laplace results yield existence, analyticity, uniqueness, Bromwich inversion, and operational rules for the degenerate derivatives [Formula: see text]. Applications include a one-dimensional initial value problem, a two-dimensional separable problem, and numerical examples.
In this paper, we introduce the poly-derangement polynomials and unipoly-derangement polynomials, and investigate their fundamental properties. In addition, we explore the relationships between these polynomials and numbers and the Stirling numbers of the first and second kinds, Bernoulli numbers, derangement numbers, Bell polynomials, and Fubini polynomials. Furthermore, for certain parameter values, we compute the numerical roots of these polynomials, examine how the roots behave as the degree increases, and present the numerical results using Mathematica.
