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Myriad random phenomena in nature possess fractal and Hurst characteristics. Random processes/fields, such as those with Cauchy or Dagum correlations, enable modeling such stochastic structures in time and space. In the first place, this paper provides a compact review of these models, including their spectral properties, for wide ranges of the fractal dimension and Hurst parameter. The Cauchy and Dagum models can be used to determine stochastic responses of dynamical systems and/or spatial problems in 1d, 2d, or 3d in the presence of fractal and Hurst characteristics. The paper surveys various examples ranging from vibration problems, rods and beams with random properties under random loadings, waves and wavefronts, fracture, homogenization of random media, and statistical turbulence, to stochastically evolving spontaneous violations of the entropy inequality in granular flows. The latter case shows the route to examine whether a mechanical system gives rise to stochastics with such intriguing features. Common features brought out in this survey show what can and how can be achieved with Cauchy and Dagum-type models and related constructs in mechanics.

The paper extends the analysis of the entropies of the Poisson distribution with parameter λ. It demonstrates that the Tsallis and Sharma–Mittal entropies exhibit monotonic behavior with respect to λ, whereas two generalized forms of the Rényi entropy may exhibit “anomalous” (non-monotonic) behavior. Additionally, we examine the asymptotic behavior of the entropies as λ→∞ and provide both lower and upper bounds for them.

Many phenomena in physics are described by a vector-valued homogeneous and isotropic random function in two or three space variables, or a random field. On the one hand, such a field can be uniquely represented as a sum of two components: a solenoidal one without divergence, and an irrotational one without curl. On the other hand, it may be represented as a sum of a longitudinal part, parallel to a fixed space direction, and a lateral part, orthogonal to that direction. If the longitudinal part of the solenoidal component has certain fractal and memory properties, what are the corresponding properties of its lateral part? Similarly, if the lateral part of the irrotational component has certain fractal and memory properties, what are the corresponding properties of its longitudinal part? We give an answer to those questions using well-known tools of classical real analysis. © 2025 Society for Industrial and Applied Mathematics.

Gaussian random fields are completely characterised by their mean value and covariance function. Random fields on hyperbolic spaces have been studied to a limited extent only, namely for the case of scalar-valued fields that are not evolving over time. This paper challenges the problem of the second-order characteristics of multivariate (vector-valued) random fields that evolve temporally over hyperbolic spaces. Specifically, we characterise the continuous space–time covariance functions that are isotropic (radially symmetric) over space (the hyperbolic space) and stationary over time (the real line). Our finding is the analogue of recent findings that have been shown for the case where the space is either the n-dimensional sphere or more generally a two-point homogeneous space. Our main result can be read as a spectral representation theorem, and we also detail the main result for the subcase of covariance functions having a spectrum that is absolutely continuous with respect to the Lebesgue measure (technical details are reported below).

Combining research methods from various areas of mathematics and physics, Probabilistic Models of Cosmic Backgrounds describes the isotropic random sections of certain fiber bundles and their applications to creating rigorous mathematical models of both discovered and hypothetical cosmic backgrounds. Previously scattered and hard-to-find mathematical and physical theories have been assembled from numerous textbooks, monographs, and research papers, and explained from different or even unexpected points of view. This consists of both classical and newly discovered results necessary for understanding a sophisticated problem of modelling cosmic backgrounds. The book contains a comprehensive description of mathematical and physical aspects of cosmic backgrounds with a clear focus on examples and explicit calculations. Its reader will bridge the gap of misunderstanding between the specialists in various theoretical and applied areas who speak different scientific languages. The audience of the book consists of scholars, students, and professional researchers. A scholar will find basic material for starting their own research. A student will use the book as supplementary material for various courses and modules. A professional mathematician will find a description of several physical phenomena at the rigorous mathematical level. A professional physicist will discover mathematical foundations for well-known physical theories.

We consider two types of entropy, namely, Shannon and R & eacute;nyi entropies of the Poisson distribution, and establish their properties as the functions of intensity parameter. More precisely, we prove that both entropies increase with intensity. While for Shannon entropy the proof is comparatively simple, for R & eacute;nyi entropy, which depends on additional parameter alpha > 0, we can characterize it as nontrivial. The proof is based on application of Karamata’s inequality to the terms of Poisson distribution.

This paper is devoted to the study of the properties of entropy as a function of the Hurst index, which corresponds to the fractional Gaussian noise. Since the entropy of the Gaussian vector depends on the determinant of the covariance matrix, and the behavior of this determinant as a function of the Hurst index is rather difficult to study analytically at high dimensions, we also consider simple alternative entropy functionals, whose behavior, on the one hand, mimics the behavior of entropy and, on the other hand, is not difficult to study. Asymptotic behavior of the normalized entropy (so called entropy rate) is also studied for the entropy and for the alternative functionals.

The goal of the 2019 conference on Stochastic Processes and Algebraic Structures held in SPAS2019, Västerås, Sweden, from September 30th to October 2nd 2019 was to showcase the frontiers of research in several important topics of mathematics, mathematical statistics, and its applications. The conference has been organized along the following tracks: 

1. Stochastic processes and modern statistical methods in theory and practice, 

2. Engineering Mathematics, 

3. Algebraic Structures and applications.  

This book highlights the latest advances in algebraic structures and applications focused on mathematical notions, methods, structures, concepts, problems, algorithms, and computational methods for the natural sciences, engineering, and modern technology. In particular, the book features mathematical methods and models from non-commutative and non-associative algebras and rings associated to generalizations of differential calculus, quantum deformations of algebras, Lie algebras, Lie superalgebras, color Lie algebras, Hom-algebras and their n-ary generalizations, semigroups and group algebras, non-commutative and non-associative algebras and computational algebra interplay with q-special functions and q-analysis, topology, dynamical systems, representation theory, operator theory and functional analysis, applications of algebraic structures in coding theory, information analysis, geometry and probability theory. 

The book gathers selected, high-quality contributed chapters from several large research communities working on modern algebraic structures and their applications. The chapters cover both theory and applications, and are illustrated with a wealth of ideas, theorems, notions, proofs, examples, open problems, and results on the interplay of algebraic structures with other parts of Mathematics. The applications help readers grasp the material, and encourage them to develop new mathematical methods and concepts in their future research. Presenting new methods and results, reviews of cutting-edge research, open problems, and directions for future research, this book will serve as a source of inspiration for a broad range of researchers and students.