Related papers: Sheaf and duality methods for analyzing multi-mode…
Understanding and interacting with everyday physical scenes requires rich knowledge about the structure of the world, represented either implicitly in a value or policy function, or explicitly in a transition model. Here we introduce a new…
Through the subsequent discussion we consider a certain particular sort of (topological) algebras, which may substitute the `` structure sheaf algebras'' in many--in point of fact, in all--the situations of a geometrical character that…
Large scale models of physical phenomena demand the development of new statistical and computational tools in order to be effective. Many such models are `sloppy', i.e., exhibit behavior controlled by a relatively small number of parameter…
The initial motivation of this work was to give a topological interpretation of two-periodic twisted de-Rham cohomology which is generalizable to arbitrary coefficients. To this end we develop a sheaf theory in the context of locally…
Modellers of complex biological or social systems are often faced with an invidious choice: to use simple models with few mechanisms that can be fully analysed, or to construct complicated models that include all the features which are…
Language is contextual and sheaf theory provides a high level mathematical framework to model contextuality. We show how sheaf theory can model the contextual nature of natural language and how gluing can be used to provide a global…
In naturalistic learning problems, a model's input contains a wide range of features, some useful for the task at hand, and others not. Of the useful features, which ones does the model use? Of the task-irrelevant features, which ones does…
We introduce and study the notion of a locally proper map between topological spaces. We show that fundamental constructions of sheaf theory, more precisely proper base change, projection formula, and Verdier duality, can be extended from…
Hierarchy is one of the most conspicuous features of numerous natural, technological and social systems. The underlying structures are typically complex and their most relevant organizational principle is the ordering of the ties among the…
If a Quillen model category can be specified using a certain logical syntax (intuitively, ``is algebraic/combinatorial enough''), so that it can be defined in any category of sheaves, then the satisfaction of Quillen's axioms over any site…
This paper makes contributions to ``pure'' sheaf model theory, the part of model theory in which the models are sheaves over a complete Heyting algebra. We start by outlining the theory in a way we hope is readable for the non-specialist.…
Networks have in recent years emerged as an invaluable tool for describing and quantifying complex systems in many branches of science. Recent studies suggest that networks often exhibit hierarchical organization, where vertices divide into…
This paper outlines a program in what one might call spectral sheaf theory --- an extension of spectral graph theory to cellular sheaves. By lifting the combinatorial graph Laplacian to the Hodge Laplacian on a cellular sheaf of vector…
Many real-world systems studied are governed by complex, nonlinear dynamics. By modeling these dynamics, we can gain insight into how these systems work, make predictions about how they will behave, and develop strategies for controlling…
The topology of any complex system is key to understanding its structure and function. Fundamentally, algebraic topology guarantees that any system represented by a network can be understood through its closed paths. The length of each path…
Complex systems are characterized by many interacting units that give rise to emergent behavior. A particularly advantageous way to study these systems is through the analysis of the networks that encode the interactions among the system's…
Several approaches to cognition and intelligence research rely on statistics-based models testing, namely factor analysis. In the present work we exploit the emerging dynamical systems perspective putting the focus on the role of the…
In the present paper, we show how to construct an algebraic sheaf by means of the topological generalized group defined by Molaei in [16] by considering both homotopy and sheaf theory.
The fundamental concept of applying the system methodology to network analysis declares that network architecture should take into account services and applications which this network provides and supports. This work introduces a formal…
Mathematical models come in many forms across biological applications. In the case of complex, spatial dynamics and pattern formation, stochastic models also face two main challenges: pattern data is largely qualitative, and model…