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This music score was submitted for the Kaleidoscope 2020 Call for Scores, an open access collaboration with the UCLA Music Library.

We develop a full randomization of the classical hyper‐logistic growth model by obtaining closed‐form expressions for relevant quantities of interest, such as the first probability density function of its solution, the time until a given fixed population is reached, and the population at the inflection point. These results are obtained under very general hypotheses on the distributions of the random model parameters by taking extensive advantage of the so‐called random variable transformation method. To illustrate the practical implications of our findings, we apply them to model the growth of multicellular tumor spheroids using empirical data. In this context, we explore two methodologies—the Bayesian approach and the random least mean square method—aimed at effectively addressing the challenge of assigning appropriate distributions to model parameters. This ensures that probabilistic fits accurately capture the inherent uncertainties of tumor growth dynamics. Finally, we notably show that the results obtained using both approaches in the randomized hyper‐logistic model align closely with each other, surpassing those yielded by the randomized logistic model. [ABSTRACT FROM AUTHOR]

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We study, from a probabilistic standpoint, a full randomization of nonhomogeneous second‐order linear difference equations assuming that its data (initial conditions, coefficients, and forcing term) are random variables. Our analysis consists of computing the so‐called first probability density function of the solution, which is a stochastic process, and then analyzing the stability of the solution assuming that all data have an arbitrary joint probability density function. To achieve these goals, we take extensive advantage of the so‐called random variable transformation technique. The theoretical results extend their deterministic counterpart, and then, they have many applications in real‐world problems where uncertainty plays a key role. Our findings are first illustrated by means of several numerical examples, where different simulations are carried out, and, second, by means of a model belonging to biomathematics. [ABSTRACT FROM AUTHOR]

This paper presents a methodology to quantify computationally the uncertainty in a class of differential equations often met in Mathematical Physics, namely random non-autonomous second-order linear differential equations, via adaptive generalized Polynomial Chaos (gPC) and the stochastic Galerkin projection technique. Unlike the random Fröbenius method, which can only deal with particular random linear differential equations and needs the random inputs (coefficients and forcing term) to be analytic, adaptive gPC allows approximating the expectation and covariance of the solution stochastic process to general random second-order linear differential equations. The random inputs are allowed to functionally depend on random variables that may be independent or dependent, both absolutely continuous or discrete with infinitely many point masses. These hypotheses include a wide variety of particular differential equations, which might not be solvable via the random Fröbenius method, in which the random input coefficients may be expressed via a Karhunen-Loève expansion.