arXiv research

Extends Popularity Bias Memorization theorem to new conditions.

Study of digital topology concepts like hyperspaces and function graphs.

Defines finite type Multivalued Shape using hyperspaces.

Let n be a natural number equal or greater than 2. In this paper we study the topological structure of certain hyperspaces of convex subsets of constant width, equipped with the Hausdorff metric topology. We focus our attention on the hyperspace cw_D(R^n) of all compact convex subsets with constant width d\in D, where …

We determine the homeomorphism type of the hyperspace of positively curved $C^\infty$ convex bodies in $\mathbb R^n$, and derive various properties of its quotient by the group of Euclidean isometries. We make a systematic study of hyperspaces of convex bodies that are at least $C^1$. We show how to destroy the symmetr…

We characterize metric spaces $X$ whose hyperspaces $2^X$ or $Bd(X)$ of non-empty closed (bounded) subsets, endowed with the Hausdorff metric, are absolute [neighborhood] retracts.

The paper studies the topology of hyperspaces of k-dimensional convex sets.

It is shown that the hyperspace of all nonempty closed subsets $\Cld_{AW}(X)$ of a separable metric space $X$ endowed with the Attouch-Wets topology is homeomorphic to a separable Hilbert space if and only if the completion of $X$ is proper, locally connected and contains no bounded connected component, $X$ is topologi…

The notion of max-plus convex subset of Euclidean space can be naturally extended to other linear spaces. The aim of this paper is to describe the topology of hyperspaces of max-plus convex subsets of Tychonov powers $\mathbb R^τ$ of the real line. We show that the corresponding spaces are AR's if and only if $τ\leω_1$…

We study topological properties of the Gromov-Hausdorff metric on the set of isometry classes of nonnegatively curved $2$-spheres.

In this work we define a new pseudometric in $\mathcal K^n_*$, the hyperspace of all non-degenerated compact convex sets of $\mathbb R^n$, which is invariant under similarities. We will prove that the quotient space generated by this pseudometric (which is the orbit space generated by the natural action of the group of…

Recently we have obtained the Cartan connection for the Finsler space whose metric is given by an exponential change with an h-vector. In this paper, we discuss certain geometric properties of a Finslerian hyperspace subjected to an h-exponential change of metric.

Study on the topology of tensorial bodies, showing they are homeomorphic to a product space.

The family of Wilder continua in cubes of dimension > 2 and its two subfamilies-of continuum-wise Wilder continua and of hereditarily arcwise connected continua-are recognized as coanalytic absorbers in the hyperspace of subcontinua of the cubes. In particular, each of them is homeomorphic to the set of all nonempty co…

In a preceding work it is determined when a centrally symmetric convex body in $\mathbb{R}^d,$ $d=d_1\cdots d_l,$ is the closed unit ball of a reasonable crossnorm on $\mathbb{R}^{d_1}\otimes\cdots\otimes\mathbb{R}^{d_l}.$ Consequently, the class of tensorial bodies is introduced, an associated tensorial Banach-Mazur d…

We present an alternative proof of the following fact: the hyperspace of compact closed subsets of constant width in $\mathbb R^n$ is a contractible Hilbert cube manifold. The proof also works for certain subspaces of compact convex sets of constant width as well as for the pairs of compact convex sets of constant rela…

Bing and Moise proved, independently, that any Peano continuum admits a length metric d. We treat non-degenerate Peano continua with a length metric as evolution systems instead of stationary objects. For any compact length space (X, d) we consider a semiflow in the hyperspace $2^X$ of all non-empty closed sets in X. T…

The paper proves rigidity and vanishing theorems for translating solitons.

The implementation of artificial neural networks in hardware substrates is a major interdisciplinary enterprise. Well suited candidates for physical implementations must combine nonlinear neurons with dedicated and efficient hardware solutions for both connectivity and training. Reservoir computing addresses the proble…

Let us denote by $\mathcal K_n$ the hyperspace of all convex bodies of $\mathbb R^n$ equipped with the Hausdorff distance topology. An affine invariant point $p$ is a continuous and Aff(n)-equivariant map $p:\mathcal K_n\to \mathbb R^n$, where Aff(n) denotes the group of all nonsingular affine maps of $\mathbb R^n$. Fo…

The redundancy is widely recognized in Convolutional Neural Networks (CNNs), which enables to remove unimportant filters from convolutional layers so as to slim the network with acceptable performance drop. Inspired by the linear and combinational properties of convolution, we seek to make some filters increasingly clo…

Study evaluates margin parameter effects on knowledge embedding quality.

Proposes a new method for time series classification and clustering.

Under the framework of spectral clustering, the key of subspace clustering is building a similarity graph which describes the neighborhood relations among data points. Some recent works build the graph using sparse, low-rank, and $\ell_2$-norm-based representation, and have achieved state-of-the-art performance. Howeve…

Čech cohomology $H^n(X)$ of a separable metrizable space $X$ is defined in terms of cohomology of its nerves (or ANR neighborhoods) $P_β$ whereas Steenrod-Sitnikov homology $H_n(X)$ is defined in terms of homology of compact subsets $K_α\subset X$. We show that one can also go vice versa: in a sense, $H^n(X)$ can be re…

Optimizes bounds for threefold singularity volumes.

Study describes singularities of height functions on specific singular surfaces.

The paper extends affine connection results to singular warped and twisted products.

New singularity concept in GR: volume singularities.

The study shows stability of neckpinch singularities in mean curvature flows.

Study describes singularities of distance squared functions on singular surfaces.

In this paper, we define the set of singular grid diagrams $\mathcal{SG}$ which provides a unified description for singular links, singular Legendrian links, singular transverse links, and singular braids. We also classify the complete set of all equivalence relations on $\mathcal{SG}$ which induce the bijection onto e…

Proves positive mass theorem for AF spin manifolds with conical singularities.

The study examines singularities and geometric properties of surfaces derived from frontals with specific singular points.

Rectifies flat singular points of area-minimizing currents with singularity degree > 1.

Paper generalizes a theorem for real analytic singularities.

The paper studies singularities of pedal curves of hyperbolic frontals.

The paper extends deformation theory to Calabi-Yau varieties with isolated log canonical singularities.

Maxfaces can have cuspidal edges near certain singularities.

New invariant distinguishes singular knots and links.

Study on singular twisted links and virtual braids, extending knot theory concepts.

Geodesics near singularities either hit or wind around, with winding number dependent on singularity type.

Virtual singular braids are generalizations of singular braids and virtual braids. We define the virtual singular braid monoid via generators and relations, and prove Alexander- and Markov-type theorems for virtual singular links. We also show that the virtual singular braid monoid has another presentation with fewer g…

We study Legendrian singular links up to contact isotopy. Using a special property of the singular points, we define the singular connected sum of Legendrian singular links. This concept is a generalization of the connected sum and can be interpreted as a tangle replacement, which provides a way to classify Legendrian …

Study neck pinches in Lagrangian flows, proving stability and introducing new singularities.

Classifies neighborhoods around specific leaf structures.

We study the local symplectic algebra of curves. We use the method of algebraic restrictions to classify symplectic $T_7$ singularities. We define discrete symplectic invariants - the Lagrangian tangency orders. We use these invariants to distinguish symplectic singularities of classical $A-D-E$ singularities of planar…

The article investigates conditions for isomorphism of singular tangent bundles.