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Morse complexes and Morse-Smale complexes are topological descriptors popular in topology-based visualization. Comparing these complexes plays an important role in their applications in feature correspondences, feature tracking, symmetry…
We present an optimal mass transport framework on the space of Gaussian mixture models, which are widely used in statistical inference. Our method leads to a natural way to compare, interpolate and average Gaussian mixture models.…
Let X and Y be curves over a finite field. In this article we explore methods to determine whether there is a rational map from Y to X by considering L-functions of certain covers of X and Y and propose a specific family of covers to…
In this paper, by reviewing the concept of subcovering and semicovering maps, we extend the notion of subcovering map to subsemicovering map. We present some necessary or sufficient conditions for a local homeomorphism to be a…
We develop a differential geometric framework for parallel transport over path spaces and a corresponding discrete theory, an integrated version of the continuum theory, using a category-theoretic framework.
Several central problems in quantum information theory (such as measurement compatibility and quantum steering) can be rephrased as membership in the minimal matrix convex set corresponding to special polytopes (such as the hypercube or its…
Optimal transport and its related problems, including optimal partial transport, have proven to be valuable tools in machine learning for computing meaningful distances between probability or positive measures. This success has led to a…
We survey research on the homotopy theory of the space map(X, Y) consisting of all continuous functions between two topological spaces. We summarize progress on various classification problems for the homotopy types represented by the…
Two discrete dynamical systems are discussed and analyzed whose trajectories encode significant explicit information about a number of problems in combinatorial probability, including graphical enumeration on Riemann surfaces and random…
Convex polytopes are convex hulls of point sets in the $n$-dimensional space $\E^n$ that generalize 2-dimensional convex polygons and 3-dimensional convex polyhedra. We concentrate on the class of $n$-dimensional polytopes in $\E^n$ called…
We show the existence of polynomial maps which have a regular bifurcation value, while over a neighbourhood of this value the fibres are connected and diffeomorphic.
Lower and upper probabilities, also known as Choquet capacities, are widely used as a convenient representation for sets of probability distributions. This paper presents a graphical decomposition and exact propagation algorithm for…
We investigate some combinatorial properties of convex polytopes simple in edges. For polytopes whose nonsimple vertices are located sufficiently far one from another, we prove an analog of the Hard Lefschetz theorem. It implies Stanley's…
In this paper, we study information transport in multiplex networks comprised of two coupled subnetworks. The upper subnetwork, called the logical layer, employs the shortest paths protocol to determine the logical paths for packets…
The probability that the commutator of two group elements is equal to a given element has been introduced in literature few years ago. Several authors have investigated this notion with methods of the representation theory and with…
In this paper we will study the representations of isomorphisms between bases of topological spaces. It turns out that the perfect setting for this study is that of regular open subsets of complete metric spaces, but we have achieved some…
We extend the usual notion of parallel transport along a path to triangulated surfaces. A homotopy of paths is lifted into a fibered category with connection and this defines a functor between the fibers above the boundary paths. These…
The displacement and deviation vectors in spaces (manifolds), the tangent bundle of which is endowed with a transport along paths, are introduced. In case these spaces are equipped with a linear connection, the deviation equations (between…
Optimal transportation distances are valuable for comparing and analyzing probability distributions, but larger-scale computational techniques for the theoretically favorable quadratic case are limited to smooth domains or regularized…
A neighborhood homotopy is an equivalence relation on spatial graphs which is generated by crossing changes on the same component and neighborhood equivalence. We give a complete classification of all 2-component spatial graphs up to…