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Normality equations describe Newtonian dynamical systems admitting normal shift of hypersurfaces. They were first derived in Euclidean geometry, then in Riemannian geometry. Recently they were rederived in more general case, when geometry…
A condensed review is presented for two basic topics in the theory of pattern formation in nonlinear dissipative media: (i) domain walls (DWs, alias grain boundaries), which appear as transient layers between different states occupying…
The theory of quaternionic slice regular functions was introduced in 2006 and successfully developed for about a decade over symmetric slice domains, which appeared to be the natural setting for their study. Some recent articles paved the…
We present a differential algebra of generalized functions over a field of generalized scalars by means of several axioms in terms of general algebra and topology. Our differential algebra is of Colombeau type in the sense that it contains…
We extend the construction of [19] by introducing spaces of generalized tensor fields on smooth manifolds that possess optimal embedding and consistency properties with spaces of tensor distributions in the sense of L. Schwartz. We thereby…
In the early 1960's, Moreau and Rockafellar introduced a concept of called \emph{subgradient} for convex functions, initiating the developments of theoretical and applied convex analysis. The needs of going beyond convexity motivated the…
The modeling of probability distributions, specifically generative modeling and density estimation, has become an immensely popular subject in recent years by virtue of its outstanding performance on sophisticated data such as images and…
Nonlinear field theories can be used to study both standard physics questions, or to study questions such as the emergence of order and complexity. These theories are generally derived from the symmetries of a given problem and the…
Ultrafunctions are a particular class of functions defined on a Non Archimedean field R^{*}\supset R. They have been introduced and studied in some previous works ([1],[2],[3]). In this paper we introduce a modified notion of ultrafunction…
By means of several examples, we motivate that universal properties are the simplest way to solve a given mathematical problem, explaining in this way why they appear everywhere in mathematics. In particular, we present the co-universal…
The universal approximation theorem is generalised to uniform convergence on the (noncompact) input space $\mathbb{R}^n$. All continuous functions that vanish at infinity can be uniformly approximated by neural networks with one hidden…
Many real-world problems encountered in several disciplines deal with the modeling of time-series containing different underlying dynamical regimes, for which probabilistic approaches are very often employed. In this paper we describe…
Probabilistic diffusion models enjoy increasing popularity in the deep learning community. They generate convincing samples from a learned distribution of input images with a wide field of practical applications. Originally, these…
The authors lay the foundations for the study of normal families of holomorphic functions and mappings on an infinite-dimensional normed linear space. Characterizations of normal families, in terms of value distribution, spherical…
This paper is concerned with the distribution of normalized zero-sets of random entire functions. The normalization of the zero-set is performed in the same way as that of the counting function for an entire function in Nevanlinna theory.…
Normalizing flows have emerged as an important family of deep neural networks for modelling complex probability distributions. In this note, we revisit their coupling and autoregressive transformation layers as probabilistic graphical…
Within the framework of mappings between affine spaces, the notion of $n$-th polarization of a function will lead to an intrinsic characterization of polynomial functions. We prove that the characteristic features of derivations, such as…
The recent significant enrichment of the Order Completion Method for nonlinear Systems of PDEs resulted in the global existence of generalized solutions to a large class of such equations. In this paper we investigate the existence and…
We characterize microlocal regularity of Colombeau generalized functions by an appropriate extension of the classical notion of micro-ellipticity to pseudodifferential operators with slow scale generalized symbols. Thus we obtain an…
Over the centuries mathematicians have been challenged by the partial differential equations (PDEs) that describe the motion of fluids in many physical contexts. Important and beautiful results were obtained in the past one hundred years,…