Real-time, accurate prediction of human steering behaviors has wide applications, from developing intelligent traffic systems to deploying autonomous driving systems in both real and simulated worlds. In this paper, we present ContextVAE, a context-aware approach for multi-modal vehicle trajectory prediction. Built upon the backbone architecture of a timewise variational autoencoder, ContextVAE observation encoding employs a dual attention mechanism that accounts for the environmental context and the dynamic agents' states, in a unified way. By utilizing features extracted from semantic maps during agent state encoding, our approach takes into account both the social features exhibited by agents on the scene and the physical environment constraints to generate map-compliant and socially-aware trajectories. We perform extensive testing on the nuScenes prediction challenge, Lyft Level 5 dataset and Waymo Open Motion Dataset to show the effectiveness of our approach and its state-of-the-art performance. In all tested datasets, ContextVAE models are fast to train and provide high-quality multi-modal predictions in real-time. Our code is available at: this https URL.

Predicting pedestrian movement is critical for human behavior analysis and also for safe and efficient human-agent interactions. However, despite significant advancements, it is still challenging for existing approaches to capture the uncertainty and multimodality of human navigation decision making. In this paper, we propose SocialVAE, a novel approach for human trajectory prediction. The core of SocialVAE is a timewise variational autoencoder architecture that exploits stochastic recurrent neural networks to perform prediction, combined with a social attention mechanism and a backward posterior approximation to allow for better extraction of pedestrian navigation strategies. We show that SocialVAE improves current state-of-the-art performance on several pedestrian trajectory prediction benchmarks, including the ETH/UCY benchmark, Stanford Drone Dataset, and SportVU NBA movement dataset. Code is available at: this https URL.

Topological Data Analysis (TDA) is a modern approach to Data Analysis focusing on the topological features of data; it has been widely studied in recent years and used extensively in Biology, Physics, and many other areas. However, financial markets have been studied slightly through TDA. Here we present a quick review of some recent applications of TDA on financial markets, including applications in the early detection of turbulence periods in financial markets and how TDA can help to get new insights while investing. Also, we propose a new turbulence index based on persistent homology -- the fundamental tool for TDA -- that seems to capture critical transitions in financial data; we tested our index with different financial time series (S&P500, Russel 2000, S&P/BMV IPC and Nikkei 225) and crash events (Black Monday crash, dot-com crash, 2007-08 crash and COVID-19 crash). Furthermore, we include an introduction to persistent homology so the reader can understand this paper without knowing TDA.

In this work, we propose the use of a Natural User Interface (NUI) through body gestures using the open source library OpenPose, looking for a more dynamic and intuitive way to control a drone. For the implementation, we use the Robotic Operative System (ROS) to control and manage the different components of the project. Wrapped inside ROS, OpenPose (OP) processes the video obtained in real-time by a commercial drone, allowing to obtain the user's pose. Finally, the keypoints from OpenPose are obtained and translated, using geometric constraints, to specify high-level commands to the drone. Real-time experiments validate the full strategy.

We show that for at most three closed geodesics with linearly independent directions, the homeomorphism type of its complement in the 3-torus is determine by the orbit of their direction vectors subspaces under the action of $PSL_3(\mathbb{Z}).$ Moreover, we provide asymptotically sharp volume bounds for a family of closed geodesics complements. The bounds depend only on the distance in the Farey graph.

A novel unification for the problem of search of optimal clusters under a well pair potential function is presented. My formulation introduces appropriate sets and lattices from where efficient methods can address this problem. First, as results of my propositions a discrete set is depicted such that the solution of a continuous and discrete search of an optimal cluster is the same. Then, this discrete set is approximated by a special lattice IF. IF stands for a lattice that combines lattices IC and FC together. In fact, two lattices IF with 9483 and 1739 particles are obtained with the property that they include all putative optimal clusters from 2 trough 1000 particles, even the difficult optimal Lennard-Jones clusters, C*38, C*98, and the Ino's decahedrons. C*98 is the only cluster where its initial configuration has a different geometry than the putative optimal cluster in term of the adjacency matrix stated by Hoare. My paper is not a benchmark, I develop a theory and a numerical experiment for the state of the art of the optimal Lennard-Jones clusters and even I found new optimal Lennard-Jones clusters with a greedy search method called Modified Peeling Method. The paper includes all the necessary data to allow the researchers reproduce the state of the art of the optimal Lennard-Jones clusters at April 8, 2005. This novel formulation unifies the geometrical motifs of the optimal Lennard-Jones clusters and gives new insight towards the understanding of the complexity of the NP problems.

In this paper, we show that Virtual Reality (VR) sickness is associated with a reduction in attention, which was detected with the P3b Event-Related Potential (ERP) component from electroencephalography (EEG) measurements collected in a dual-task paradigm. We hypothesized that sickness symptoms such as nausea, eyestrain, and fatigue would reduce the users' capacity to pay attention to tasks completed in a virtual environment, and that this reduction in attention would be dynamically reflected in a decrease of the P3b amplitude while VR sickness was experienced. In a user study, participants were taken on a tour through a museum in VR along paths with varying amounts of rotation, shown previously to cause different levels of VR sickness. While paying attention to the virtual museum (the primary task), participants were asked to silently count tones of a different frequency (the secondary task). Control measurements for comparison against the VR sickness conditions were taken when the users were not wearing the Head-Mounted Display (HMD) and while they were immersed in VR but not moving through the environment. This exploratory study shows, across multiple analyses, that the effect mean amplitude of the P3b collected during the task is associated with both sickness severity measured after the task with a questionnaire (SSQ) and with the number of counting errors on the secondary task. Thus, VR sickness may impair attention and task performance, and these changes in attention can be tracked with ERP measures as they happen, without asking participants to assess their sickness symptoms in the moment.

OGTT is a common test, frequently used to diagnose insulin resistance or diabetes, in which a patient's blood sugar is measured at various times over the course of a few hours. Recent developments in the study of OGTT results have framed it as an inverse problem which has been the subject of Bayesian inference. This is a powerful new tool for analyzing the results of an OGTT test,and the question arises as to whether the test itself can be improved. It is of particular interest to discover whether the times at which a patient's glucose is measured can be changed to improve the effectiveness of the test. The purpose of this paper is to explore the possibility of finding a better experimental design, that is, a set of times to perform the test. We review the theory of Bayesian experimental design and propose an estimator for the expected utility of a design. We then study the properties of this estimator and propose a new method for quantifying the uncertainty in comparisons between designs. We implement this method to find a new design and the proposed design is compared favorably to the usual testing scheme.

Collaborative competitions have gained popularity in the scientific and technological fields. These competitions involve defining tasks, selecting evaluation scores, and devising result verification methods. In the standard scenario, participants receive a training set and are expected to provide a solution for a held-out dataset kept by organizers. An essential challenge for organizers arises when comparing algorithms' performance, assessing multiple participants, and ranking them. Statistical tools are often used for this purpose; however, traditional statistical methods often fail to capture decisive differences between systems' performance. This manuscript describes an evaluation methodology for statistically analyzing competition results and competition. The methodology is designed to be universally applicable; however, it is illustrated using eight natural language competitions as case studies involving classification and regression problems. The proposed methodology offers several advantages, including off-the-shell comparisons with correction mechanisms and the inclusion of confidence intervals. Furthermore, we introduce metrics that allow organizers to assess the difficulty of competitions. Our analysis shows the potential usefulness of our methodology for effectively evaluating competition results.

In linear models it is common to have situations where several regression coefficients are zero. In these situations a common tool to perform regression is a variable selection operator. One of the most common such operators is the LASSO operator, which promotes point estimates which are zero. The LASSO operator and similar approaches, however, give little in terms of easily interpretable parameters to determine the degree of variable selectivity. In this paper we propose a new family of selection operators which builds on the geometry of LASSO but which yield an easily interpretable way to tune selectivity. These operators correspond to Bayesian prior densities and hence are suitable for Bayesian inference. We present some examples using simulated and real data, with promising results.

We study the problem of existence of F-structures on compact complex surfaces, giving a complete classification modulo the gap in the classification of surfaces of class VII. We then use these results to study the minimal entropy problem for compact complex surfaces. For instance we prove that compact Kahler surfaces of Kodaira number different from 2 have minimal entropy 0, and such a surface admits a metric with entropy 0 if and only if it is the complex projective space, a ruled surface of genus 0 or 1, a complex torus or a hyperelliptic surface. The key result we use to prove this, is a new topological obstruction to the existence of metrics with vanishing topological entropy. Finally we show that these results fit perfectly into Wall's study of geometric structures on compact complex surfaces.

This paper continues our previous work done in math.AG/0008207 and is an attempt to establish a conceptual framework which generalizes the work of Manin on the relation between non-linear second order ODEs of type Painleve VI and integrable systems. The principle behind everything is a strong interaction between K-theory and Picard-Fuchs type differential equations via Abel-Jacobi maps. Our main result is an extension of a theorem of Donagi and Markman to our setup.

It is known that the so-called Bercovici-Pata bijection can be explained in terms of certain Hermitian random matrix ensembles $(M_{d})_{d\geq1}$ whose asymptotic spectral distributions are free infinitely divisible. We investigate Hermitian L\'{e}vy processes with jumps of rank one associated to these random matrix ensembles introduced in [6] and [10]. A sample path approximation by covariation processes for these matrix L\'{e}vy processes is obtained. As a general result we prove that any $d\times d$ complex matrix subordinator with jumps of rank one is the quadratic variation of an $\mathbb{C}^{d}$-valued L\'{e}vy process. In particular, we have the corresponding result for matrix subordinators with jumps of rank one associated to the random matrix ensembles $(M_{d})_{d\geq1}$

These are the lecture notes from my short course of the same title at the CIMPA Research School on Associative and Nonassociative Algebras and Dialgebras: Theory and Algorithms - In Honour of Jean-Louis Loday (1946-2012), held at CIMAT, Guanajuato, Mexico, February 17 to March 2, 2013. The underlying motivation is to apply the theory of noncommutative Grobner bases in free associative algebras to the construction of universal associative envelopes for nonassociative structures defined by multilinear operations. Trilinear operations were classified by the author and Peresi in 2007. In her Ph.D. thesis of 2012, Elgendy studied the universal associative envelopes of nonassociative triple systems obtained by applying these trilinear operations to the 2-dimensional simple associative triple system. In these notes I use computer algebra to extend some aspects of her work to the 4-dimensional and 6-dimensional simple associative triple systems.

We study a simple model of bicycle motion: a segment of fixed length in multi-dimensional Euclidean space, moving so that the velocity of the rear end is always aligned with the segment. If the front track is prescribed, the trajectory of the rear wheel is uniquely determined via a certain first order differential equation -- the bicycle equation. The same model, in dimension two, describes another mechanical device, the hatchet planimeter. Here is a sampler of our results. We express the linearized flow of the bicycle equation in terms of the geometry of the rear track; in dimension three, for closed front and rear tracks, this is a version of the Berry phase formula. We show that in all dimensions a sufficiently long bicycle also serves as a planimeter: it measures, approximately, the area bivector defined by the closed front track. We prove that the bicycle equation also describes rolling, without slipping and twisting, of hyperbolic space along Euclidean space. We relate the bicycle problem with two completely integrable systems: the AKNS (Ablowitz, Kaup, Newell and Segur) system and the vortex filament equation. We show that "bicycle correspondence" of space curves (front tracks sharing a common back track) is a special case of a Darboux transformation associated with the AKNS system. We show that the filament hierarchy, encoded as a single generating equation, describes a 3-dimensional bike of imaginary length. We show that a series of examples of "ambiguous" closed bicycle curves (front tracks admitting self bicycle correspondence), found recently F. Wegner, are buckled rings, or solitons of the planar filament equation. As a case study, we give a detailed analysis of such curves, arising from bicycle correspondence with multiply traversed circles.

The Tait-Kneser theorem, first demonstrated by Peter G. Tait in 1896, states that the osculating circles along a plane curve with monotone non-vanishing curvature are pairwise disjoint and nested. This note contains a proof of this theorem using the Lorentzian geometry of the space of circles. We show how a similar proof applies to two variations on the theorem, concerning the osculating Hooke and Kepler conics along a plane curve. We also prove a version of the 4-vertex theorem for Kepler conics.

Given two polynomials $p(x), q(x)$ of degree $d$, we give a combinatorial formula for the finite free cumulants of $p(x)\boxtimes_d q(x)$. We show that this formula admits a topological expansion in terms of non-crossing multi-annular permutations on surfaces of different genera. This topological expansion, on the one hand, deepens the connection between the theories of finite free probability and free probability, and in particular proves that $\boxtimes_d$ converges to $\boxtimes$ as $d$ goes to infinity. On the other hand, borrowing tools from the theory of second order freeness, we use our expansion to study the infinitesimal distribution of certain families of polynomials which include Hermite and Laguerre, and draw some connections with the theory of infinitesimal distributions for real random matrices. Finally, building off our results we give a new short and conceptual proof of a recent result [Steinerberger (2020), Hoskins and Kabluchko (2020)] that connects root distributions of polynomial derivatives with free fractional convolution powers.

A program searching for symmetry structures behind some features of the standard Model is launched. After addressing known no-go theorems, we construct a novel symmetry mixing gauge and Higgs fields which is a Lorentz symmetry extension involving gauge symmetries. We construct a field theory model with such a local symmetry, which is in turn invariant under local Lorentz transformations. We address the action of extended symmetries on the geometric frame, and models with extra dimensions. This features the novelty of diverse covariant derivatives associated to non--commuting differential operators, and their associated curvatures. Mass--like terms arise accompanying non--commuting differential operators. A novel formalism for constructing an invariant Lagrangian in direct correspondence to obtained casimir operators --embodying a long sought paradigm-- is developed. It recovers both gauge as well as some sort of Higgs-like potential. Equations of motion and conserved currents are found. Some exploration is devoted to possible invariant measures hinting possible approaches to address anomalies.

We propose and analyze a stochastic model to investigate epigenetic mutations, i.e., modifications of the genetic information that control gene expression patterns in a cell but do not alter the DNA sequence. Epigenetic mutations are related to environmental fluctuations, which leads us to consider (additive) noise as the driving element for such mutations. We focus on two applications: firstly, cancer immunotherapy involving macrophages' epigenetic modifications that we call tumor microenvironment noise-induced polarizations, and secondly, cell fate determination and mutation of the flower Arabidopsis thaliana. Due to the technicalities involving cancer biology for the first case, we present only a general review of this topic and show the details in a separate manuscript since our principal concerns here are the mathematical results that are important to validate our system as an appropriate epigenetic model; for such results, we rely on the theory of Stochastic PDE, theory of large deviations, and ergodic theory. Moreover, since epigenetic mutations are reversible, a fact currently exploited to develop so-called epi-drugs to treat diseases like cancer, we also investigate an optimal control problem for our system to study the reversal of epigenetic mutations.