Related papers: Shape dimension of maps
Deep learning models have seen significant successes in numerous applications, but their inner workings remain elusive. The purpose of this work is to quantify the learning process of deep neural networks through the lens of a novel…
In previous work, we introduced a method for modeling a configuration of objects in 2D and 3D images using a mathematical "medial/skeletal linking structure." In this paper, we show how these structures allow us to capture positional…
We provide upper bounds for the cardinality of the value set of a polynomial map in several variables over a finite field. These bounds generalize earlier bounds for univariate polynomials.
"An invariant of metric spaces under bornologous equivalences" gives an invariant and "A coarse invariant" extends the invariant to coarse equivalences. In both papers the invariant is defined for a class of metric spaces called sigma…
Metric mean dimension is a metric-depedent quantity to characterize the topological complexity of systems with infinite topological entropy. In this paper, we investigate metric mean dimension of factor maps. (1) We introduce three types of…
In this paper we investigate a problem of approximation of continuous mappings by smooth mappings with nonnegative Jacobian.
The topological properties of a material depend on its symmetries, parameters, and spatial dimension. Changes in these properties due to parameter and symmetry variations can be understood by computing the corresponding topological…
This paper describes a novel framework for computing geodesic paths in shape spaces of spherical surfaces under an elastic Riemannian metric. The novelty lies in defining this Riemannian metric directly on the quotient (shape) space, rather…
Among Thurston maps (orientation-preserving, postcritically finite branched coverings of the 2-sphere to itself), those that arise as subdivision maps of a finite subdivision rule form a special family. For such maps, we investigate…
We explore the notion of sectional number of a group homomorphism, leading to a generalization of the covering number of a group, and present several characterizations when the sectional number is finite, providing estimates for computing…
A topological invariant of the geodesic laminations on a modular surface is constructed. The invariant has a continuous part (the tail of a continued fraction) and a combinatorial part (the singularity data). It is shown, that the invariant…
The main aim of the present paper is to introduce new classes of functions called $ \alpha $ $^m $ continuous maps and $ \alpha $ $^m $ irresolute maps. We obtain some characterizations of these classes and properties are studied.
Dimension four provides a peculiarly idiosyncratic setting for the interplay between scalar curvature and differential topology. Here we will explain some of the peculiarities of the four-dimensional realm via a careful discussion of the…
Quantification of the number of variables needed to locally explain complex data is often the first step to better understanding it. Existing techniques from intrinsic dimension estimation leverage statistical models to glean this…
Shapes do not define a linear space. This paper explores the linear structure of deformations as a representation of shapes. This transforms shape optimization to a variant of optimal control. The numerical challenges of this point of view…
We study the topological dynamics by iterations of a piecewise continuous, non linear and locally contractive map in a real finite dimensional compact ball. We consider those maps satisfying the "separation property": different continuity…
This paper deals with projective shape analysis, which is a study of finite configurations of points modulo projective transformations. The topic has various applications in machine vision. We introduce a convenient projective shape space,…
The paper studies the deformation theory of a holomorphic surjective map from a normal compact complex space to a compact Kaehler manifold and describes the component of the space of holomorphic maps, generalizing results in the projective…
We describe differential invariants of infinite-dimensional algebras being equivalence algebras of some classes of PDE and study structure of these algebras.
In this paper we consider dynamical properties of set-valued mappings and their implications on the associated inverse limit space. Specifically, we define the specification property and topological entropy for set-valued functions and…