Related papers: Interval-sphere model structures
Rigidity transitions in simple models of confluent cells have been a powerful organizing principle in understanding the dynamics and mechanics of dense biological tissue. In this work we explore the interplay between geometry and rigidity…
We define two notions. The first one is a $rank\ compression\ system$ $\xi$ for a finite poset $\mathbf{P}$ that assigns each interval subposet $I$ to an order-preserving map $\xi_I \colon I^{\xi} \to \mathbf{P}$ satisfying some conditions,…
Different cell types aggregate and sort into hierarchical architectures during the formation of animal tissues. The resulting spatial organization depends (in part) on the strength of adhesion of one cell type to itself relative to other…
In this paper, a unified framework for representing uncertain information based on the notion of an interval structure is proposed. It is shown that the lower and upper approximations of the rough-set model, the lower and upper bounds of…
The persistence diagram is an increasingly useful tool from Topological Data Analysis, but its use alongside typical machine learning techniques requires mathematical finesse. The most success to date has come from methods that map…
We introduce an algebraic system which can be used as a model for spaces with geodesic paths between any two of their points. This new algebraic structure is based on the notion of mobility algebra which has recently been introduced as a…
We show that a persistence module (for a totally ordered indexing set) consisting of finite-dimensional vector spaces is a direct sum of interval modules. The result extends to persistence modules with the descending chain condition on…
We discuss how the kernel convolution approach can be used to accurately approximate the spatial covariance model on a sphere using spherical distances between points. A detailed derivation of the required formulas is provided. The proposed…
These are notes from a multi-year learning seminar on the algebraic approach to Constraint Satisfaction Problems (CSPs). The main topics covered are the theory of algebraic structures with few subpowers, the theory of absorbing subalgebras…
We study the process of compactification as a topology change. It is shown how the mediating spacetime topology, or cobordism, may be simplified through surgery. Within the causal Lorentzian approach to quantum gravity, it is shown that any…
Although the preservation of shape continuity and physiological anatomy is a natural assumption in the segmentation of medical images, it is often neglected by deep learning methods that mostly aim for the statistical modeling of input data…
Multiparameter persistent homology has emerged as a powerful generalization of topological data analysis, capable of encoding multivariate filtrations. However, the algebraic complexity of multiparameter persistence modules, marked by wild…
We study the injection of air in a granular suspension. We use a linear Hele-Saw cell filled with a suspension which is displaced by air, leading to a Saffman-Taylor (fingering) instability. For the suspension, we use an iso-dense mixture…
This paper treats the problem of the merging of formations, where the underlying model of a formation is graphical. We first analyze the rigidity and persistence of meta-formations, which are formations obtained by connecting several rigid…
We apply poset cocalculus, a functor calculus framework for functors out of a poset, to study the problem of decomposing multipersistence modules into simpler components. We both prove new results in this topic and offer a new perspective…
A method to apply and visualize persistent homology of time series is proposed. The method captures persistent features in space and time, in contrast to the existing procedures, where one usually chooses one while keeping the other fixed.…
We construct constraints from consistency between estimated parameters from different spheres, termed inter-sphere consistency. It facilitates more flexible calibration using only two spheres, which has been considered a challenging and not…
In this short note, we construct a class of models of an extension of homotopy type theory, which we call homotopy type theory with an interval type.
We establish a regular sampling theory in the range of the analysis operator of a continuous frame having a unitary structure. The unitary structure is related with a unitary representation of a locally compact abelian group on a separable…
In Topological Data Analysis, filtered chain complexes enter the persistence pipeline between the initial filtering of data and the final persistence invariants extraction. It is known that they admit a tame class of indecomposables, called…