Related papers: Continuous selections of multivalued mappings
This survey provides a comprehensive exploration of applications of Topological Data Analysis (TDA) within neural network analysis. Using TDA tools such as persistent homology and Mapper, we delve into the intricate structures and behaviors…
The intersection set of Bayesian and nonparametric statistics was almost empty until about 1973, but now is growing at a healthy rate. This chapter, for the {\it Highly Structured Stochastic Systems} book (Oxford University Press, 2003)…
The study of temporal networks is motivated by the simple and important observation that just as network structure can affect dynamics, so can structure in time. Just as network topology can teach us about the system in question, so can its…
We describe a new methodology for studying persistence of topological features across a family of spaces or point-cloud data sets, called zigzag persistence. Building on classical results about quiver representations, zigzag persistence…
We study Borel systems and continuous systems of measures, with a focus on mapping properties: compositions, liftings, fibred products and disintegration. Parts of the theory we develop can be derived from known work in the literature, and…
There has been much recent work on multiwinner voting systems. However, sometimes a committee is highly structured, and if we want to vote for such a committee, our voting method should be more structured as well. We consider committees…
This paper is a collection of recent results on discrete-time and continuous-time branching random walks. Some results are new and others are known. Many aspects of this theory are considered: local, global and strong local survival, the…
This paper seeks to advance the theory of nonexpansive mappings by introducing and exploring a novel class of nonexpansive type mappings, which we aptly designate as perimetric nonexpansive mappings. We establish that the collection of…
A substantial progress in development of new and efficient tensor factorization techniques has led to an extensive research of their applicability in recommender systems field. Tensor-based recommender models push the boundaries of…
Each complex network (or class of networks) presents specific topological features which characterize its connectivity and highly influence the dynamics of processes executed on the network. The analysis, discrimination, and synthesis of…
Let $I=[0,1]$ and consider disjoint closed regions $G_{1},....,G_{n}$ in $% I\times I$ and subintervals $I_{1},......,I_{n},$ such that $G_{i}$ projects onto $I_{i.}$ We define the lower and upper maps $\tau_{1},$ $\tau_{2}$ by the lower…
High dimensional statistical problems arise from diverse fields of scientific research and technological development. Variable selection plays a pivotal role in contemporary statistical learning and scientific discoveries. The traditional…
Convection is a well-studied topic in fluid dynamics, yet it is less understood in the context of networks flows. Here, we incorporate techniques from topological data analysis (namely, persistent homology) to automate the detection and…
Spaces of quasi-invariant measures supplied with different topologies are studied. Their embeddings, projective decompositions, conditions for their metrizability are investigated. Theorems about convergence of nets of quasi-invariant…
The theory of multidimensional persistence captures the topology of a multifiltration -- a multiparameter family of increasing spaces. Multifiltrations arise naturally in the topological analysis of scientific data. In this paper, we give a…
Modern recommender systems operate in uniquely dynamic settings: user interests, item pools, and popularity trends shift continuously, and models must adapt in real time without forgetting past preferences. While existing tutorials on…
The paper contains a very simple proof of the classical Hasumi's theorem that each usco mapping defined on an extremally disconnected space has a continuous selection. The paper also contains a very simple proof of a recent result about…
A class of measure-valued processes which model multilevel populations undergoing mutation, selection, genetic drift and spatial migration is considered. We investigate the qualitative behaviour of models with multilevel selection and the…
This sequel to our previous paper [MS11b] continues the study of topological contact dynamics and applications to contact dynamics and topological dynamics. We provide further evidence that the topological automorphism groups of a contact…
We believe three ingredients are needed for further progress in persistence and its use: invariants not relying on decomposition theorems to go beyond 1-dimension, outcomes suitable for statistical analysis and a setup adopted for…