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It is well-known that many diffusion equations can be recast as Wasserstein gradient flows. Moreover, in recent years, by modifying the Wasserstein distance appropriately, this technique has been transferred to further evolution equations…
Methods from additive number theory are applied to construct families of finitely generated linear semigroups with intermediate growth.
Reaction-diffusion equations are widely used as the governing evolution equations for modeling many physical, chemical, and biological processes. Here we derive reaction-diffusion equations to model transport with reactions on a…
The contraction semigroup $S(t)={\rm e}^{t\mathbb{A}}$ generated by the abstract linear dissipative evolution equation $$ \ddot u + A u + f(A) \dot u=0 $$ is analyzed, where $A$ is a strictly positive selfadjoint operator and $f$ is an…
Normalizing flows are exact-likelihood generative neural networks which approximately transform samples from a simple prior distribution to samples of the probability distribution of interest. Recent work showed that such generative models…
The concepts and methods used for the study of disordered systems have proven useful in the analysis of the evolution equations of quantum chromodynamics in the high-energy regime: Indeed, parton branching in the semi-classical…
We provide an overview of the diffusion model as a method to generate new samples. Generative models have been recently adopted for tasks such as art generation (Stable Diffusion, Dall-E) and text generation (ChatGPT). Diffusion models in…
We study diffusion-type equations supported on structures that are randomly varying in time. After settling the issue of well-posedness, we focus on the asymptotic behavior of solutions: our main result gives sufficient conditions for…
Evolution of systems in which Hamiltonians are generators of gauge transformations is a notion that requires more structure than the canonical theory provides. We identify and study this additional structure in the framework of relational…
Evolution of genus density is calculated from Gaussian initial conditions using Zel'dovich approximation. A new approach is introduced which formulates the desired quantity in a rotationally invariant manner. It is shown that normalized…
In this work, we study good semigroups of $\mathbb{N}^n$ introducing the definition of length and genus for these objects. We show how to count the local good semigroup with a fixed genus. Furthermore, we study the relationships of these…
We propose a general algebraic analytic scheme for the spectral transform of solutions of nonlinear evolution equations. This allows us to give the general integrable evolution corresponding to an arbitrary time and space dependence of the…
A transitive permutation group is semiprimitive if each of its normal subgroups is transitive or semiregular. Interest in this class of groups is motivated by two sources: problems arising in universal algebra related to collapsing monoids…
We use semigroup theory to describe the group of automorphisms of some semigroups of interest in holomorphic dynamical systems. We show, with some examples, that representation theory of semigroups is related to usual constructions in…
The gradient flow is the evolution of fields and physical quantities along a dimensionful parameter~$t$, the flow time. We give a simple argument that relates this gradient flow and the Wilsonian renormalization group (RG) flow. We then…
Recent studies suggest utilizing generative models instead of traditional auto-regressive algorithms for time series forecasting (TSF) tasks. These non-auto-regressive approaches involving different generative methods, including GAN,…
In this work, we propose a new family of generative flows on an augmented data space, with an aim to improve expressivity without drastically increasing the computational cost of sampling and evaluation of a lower bound on the likelihood.…
The evolution equations for the generalized microscopic phase densities are introduced. The evolution equations of average values of microscopic phase densities are derived and a solution of the initial-value problem of the obtained…
A simple way to define the flow rules of plasticity models is the assumption of generalized normality associated with a suitable pseudo-potential function. This approach, however, is not usually employed to formulate endochronic theory and…
We survey recent generalizations and improvements of the linear programming method that involve semidefinite programming. A general framework using group representations and tools from graph theory is provided.