For certain Sheffer sequences (sn)n=0∞ on C, Grabiner (1988) proved that, for each α∈0,1, the corresponding Sheffer operator zn↦sn(z) extends to a linear self-homeomorphism of Eminα(C), the Fréchet ...topological space of entire functions of order at most α and minimal type (when the order is equal to α>0). In particular, every function f∈Eminα(C) admits a unique decomposition f(z)=∑n=0∞cnsn(z), and the series converges in the topology of Eminα(C). Within the context of a complex nuclear space Φ and its dual space Φ′, in this work we generalize Grabiner's result to the case of Sheffer operators corresponding to Sheffer sequences on Φ′. In particular, for Φ=Φ′=Cn with n≥2, we obtain the multivariate extension of Grabiner's theorem. Furthermore, for an Appell sequence on a general co-nuclear space Φ′, we find a sufficient condition for the corresponding Sheffer operator to extend to a linear self-homeomorphism of Eminα(Φ′) when α>1. The latter result is new even in the one-dimensional case.
We analyze an interacting particle system with a Markov evolution of birth-and-death type. We have shown that a local competition mechanism (realized via a density dependent mortality) leads to a ...globally regular behavior of the population in the course of the stochastic evolution.
We describe a general derivation scheme for the Vlasov-type equations for Markov evolutions of particle systems in continuum. This scheme is based on a proper scaling of corresponding Markov ...generators and has an algorithmic realization in terms of related hierarchical chains of correlation functions equations. Several examples of realization of the proposed approach in particular models are presented.
Theoretical and applied cancer studies that use individual-based models (IBMs) have been limited by the lack of a mathematical formulation that enables rigorous analysis of these models. However, ...spatial cumulant models (SCMs), which have arisen from theoretical ecology, describe population dynamics generated by a specific family of IBMs, namely spatio-temporal point processes (STPPs). SCMs are spatially resolved population models formulated by a system of differential equations that approximate the dynamics of two STPP-generated summary statistics: first-order spatial cumulants (densities), and second-order spatial cumulants (spatial covariances). We exemplify how SCMs can be used in mathematical oncology by modelling theoretical cancer cell populations comprising interacting growth factor-producing and non-producing cells. To formulate model equations, we use computational tools that enable the generation of STPPs, SCMs and mean-field population models (MFPMs) from user-defined model descriptions (Cornell et al. Nat Commun 10:4716, 2019). To calculate and compare STPP, SCM and MFPM-generated summary statistics, we develop an application-agnostic computational pipeline. Our results demonstrate that SCMs can capture STPP-generated population density dynamics, even when MFPMs fail to do so. From both MFPM and SCM equations, we derive treatment-induced death rates required to achieve non-growing cell populations. When testing these treatment strategies in STPP-generated cell populations, our results demonstrate that SCM-informed strategies outperform MFPM-informed strategies in terms of inhibiting population growths. We thus demonstrate that SCMs provide a new framework in which to study cell-cell interactions, and can be used to describe and perturb STPP-generated cell population dynamics. We, therefore, argue that SCMs can be used to increase IBMs’ applicability in cancer research.
The aim of this paper is to analyze different regulation mechanisms in spatial continuous stochastic development models. We describe the density behavior for models with global mortality and local ...establishment rates. We prove that the local self-regulation via a competition mechanism (density dependent mortality) may suppress a unbounded growth of the averaged density if the competition kernel is superstable.
Letters to the editor Helgason, Sigurdur; Finkelshtein, Dmitri; Gogolev, Andrey ...
Notices of the American Mathematical Society,
2021, Letnik:
68, Številka:
9
Journal Article
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