Various concepts and constructions in homotopy theory have been defined in the digital setting. Although there have been several attempts at a definition of a fibration in the digital setting, robust examples of these digital fibrations are few and far between. In this paper, we develop a digital Hopf fibration within the category of tolerance spaces. By widening our category to that of tolerance spaces, we are able to give a construction of this digital Hopf fibration which mimics the smooth setting.

Gregory Lupton John Oprea Nicholas A. Scoville

We define a fundamental group for digital images. Namely, we construct a functor from digital images to groups, which closely resembles the ordinary fundamental group from algebraic topology. Our construction differs in several basic ways from previously established versions of a fundamental group in the digital setting. Our development gives a prominent role to subdivision of digital images. We show that our fundamental group is preserved by subdivision.

Gregory Lupton John Oprea Nicholas Scoville

We generalize results from topological robotics on the topological complexity (TC) of aspherical spaces to sectional categories of fibrations inducing subgroup inclusions on the level of fundamental groups. In doing so, we establish new lower bounds on sequential TCs of aspherical spaces as well as the parametrized TC of epimorphisms. Moreover, we generalize the Costa-Farber canonical class for TC to classes for sequential TCs and explore their properties. We combine them with the results on sequential TCs of aspherical spaces to obtain results on spaces that are not necessarily aspherical.

Arturo Espinosa Baro Michael Farber Stephan Mescher John Oprea

We show that in the presence of a geometric condition such as non-negative Ricci curvature, the distributional category of a manifold may be used to bound invariants, such as the first Betti number and macroscopic dimension, from above. Moreover, \`a la Bochner, when the bound is an equality, special constraints are imposed on the manifold. We show that the distributional category of a space also bounds the rank of the Gottlieb group, with equality imposing constraints on the fundamental group. These bounds are refined in the setting of cohomologically symplectic manifolds, enabling us to get specific computations for the distributional category and LS-category.

Ekansh Jauhari John Oprea

Interaction in virtual reality (VR) environments is essential to achieve a pleasant and immersive experience. Most of the currently existing VR applications, lack of robust object grasping and manipulation, which are the cornerstone of interactive systems. Therefore, we propose a realistic, flexible and robust grasping system that enables rich and real-time interactions in virtual environments. It is visually realistic because it is completely user-controlled, flexible because it can be used for different hand configurations, and robust because it allows the manipulation of objects regardless their geometry, i.e. hand is automatically fitted to the object shape. In order to validate our proposal, an exhaustive qualitative and quantitative performance analysis has been carried out. On the one hand, qualitative evaluation was used in the assessment of the abstract aspects such as: hand movement realism, interaction realism and motor control. On the other hand, for the quantitative evaluation a novel error metric has been proposed to visually analyze the performed grips. This metric is based on the computation of the distance from the finger phalanges to the nearest contact point on the object surface. These contact points can be used with different application purposes, mainly in the field of robotics. As a conclusion, system evaluation reports a similar performance between users with previous experience in virtual reality applications and inexperienced users, referring to a steep learning curve.

Sergiu Oprea Pablo Martinez-Gonzalez Alberto Garcia-Garcia John Alejandro Castro-Vargas Sergio Orts-Escolano Jose Garcia-Rodriguez

The ability to predict, anticipate and reason about future outcomes is a key component of intelligent decision-making systems. In light of the success of deep learning in computer vision, deep-learning-based video prediction emerged as a promising research direction. Defined as a self-supervised learning task, video prediction represents a suitable framework for representation learning, as it demonstrated potential capabilities for extracting meaningful representations of the underlying patterns in natural videos. Motivated by the increasing interest in this task, we provide a review on the deep learning methods for prediction in video sequences. We firstly define the video prediction fundamentals, as well as mandatory background concepts and the most used datasets. Next, we carefully analyze existing video prediction models organized according to a proposed taxonomy, highlighting their contributions and their significance in the field. The summary of the datasets and methods is accompanied with experimental results that facilitate the assessment of the state of the art on a quantitative basis. The paper is summarized by drawing some general conclusions, identifying open research challenges and by pointing out future research directions.

Sergiu Oprea Pablo Martinez-Gonzalez Alberto Garcia-Garcia John Alejandro Castro-Vargas Sergio Orts-Escolano Jose Garcia-Rodriguez Antonis Argyros

We give a brief exposition of the 2d TQFT that captures the structure of the GL Verlinde numbers, following Witten.

Alina Marian Dragos Oprea

A new method for the determination of environmental Gd by neutron transmission (NT) experiments is proposed. The NT method is based on the measurements of neutron spectra passing through a target. From the attenuation neutron spectra new data as concentration, width, resonance energies and cross section have been obtained.

Cristiana Oprea Ioan Alexandru Oprea Alexandru Mihul

We prove that the Lusternik-Schnirelmann category $cat(M)$ of a closed symplectic manifold $(M, \omega)$ equals the dimension $dim(M)$ provided that the symplectic cohomology class vanishes on the image of the Hurewicz homomorphism. This holds, in particular, when $\pi_2(M)=0$. The Arnold conjecture asserts that the number of fixed points of a Hamiltonian symplectomorphism of $M$ is greater than or equal to the number of critical points of some function on $M$. A modified form of the conjecture, replacing the latter quantity (via Lusternik-Schnirelmann theory) by $cup(M) + 1$, has been proved recently by various authors using techniques of Floer. The first author has also recently shown that the original form of the conjecture holds when $cat(M) =dim(M)$. Thus, this paper completes the proof of the original Arnold conjecture for closed symplectic manifolds with, for example, $\pi_2(M)=0$.

Yuli B. Rudyak John Oprea

By the work of Li, a compact co-K\"ahler manifold $M$ is a mapping torus $K_\varphi$, where $K$ is a K\"ahler manifold and $\varphi$ is a Hermitian isometry. We show here that there is always a finite cyclic cover $\bar M$ of the form $\bar M \cong K \times S^1$, where $\cong$ is equivariant diffeomorphism with respect to an action of $S^1$ on $M$ and the action of $S^1$ on $K \times S^1$ by translation on the second factor. Furthermore, the covering transformations act diagonally on $S^1$, $K$ and are translations on the $S^1$ factor. In this way, we see that, up to a finite cover, all compact co-K\"ahler manifolds arise as the product of a K\"ahler manifold and a circle.

Giovanni Bazzoni John Oprea