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We present a theorem of resolution of singularities for real analytic constrained differential systems $A(x)\dot{x} = F(x)$ defined on a 2-manifold with corners having impasse set $\{x; \det A(x) = 0\}$. This result can be seen as a generalization of the classical one for 2-dimensional real analytic vector fields. Our proof is based on weighted blow-ups and the Newton polygon., Comment: 35 pages, 12 figures

This paper covers the modeling and nonlinear control of the Cubli, a cube with three reaction wheels mounted on orthogonal faces that becomes a reaction wheel-based 1D/3D inverted pendulum when positioned in one of its edges (1D) or vertices (3D). Instead of angles, unit complex numbers are used as control states for the 1D configuration. This approach is useful not only to get rid of trigonometric functions, but mainly because it is a specific case of the 3D configuration, that utilizes unit ultra-complex numbers (quaternions) as system states, and therefore facilitates its understanding. The derived nonlinear control law is equivalent to a linear one and is characterized by only three straightforward tuning parameters. Experiment results are presented to validate modeling and control., Comment: Paper submitted to ACC 2021

Multidimensional Projection is a fundamental tool for high-dimensional data analytics and visualization. With very few exceptions, projection techniques are designed to map data from a high-dimensional space to a visual space so as to preserve some dissimilarity (similarity) measure, such as the Euclidean distance for example. In fact, although adopting distinct mathematical formulations designed to favor different aspects of the data, most multidimensional projection methods strive to preserve dissimilarity measures that encapsulate geometric properties such as distances or the proximity relation between data objects. However, geometric relations are not the only interesting property to be preserved in a projection. For instance, the analysis of particular structures such as clusters and outliers could be more reliably performed if the mapping process gives some guarantee as to topological invariants such as connected components and loops. This paper introduces TopoMap, a novel projection technique which provides topological guarantees during the mapping process. In particular, the proposed method performs the mapping from a high-dimensional space to a visual space, while preserving the 0-dimensional persistence diagram of the Rips filtration of the high-dimensional data, ensuring that the filtrations generate the same connected components when applied to the original as well as projected data. The presented case studies show that the topological guarantee provided by TopoMap not only brings confidence to the visual analytic process but also can be used to assist in the assessment of other projection methods.

In this paper we characterize the phase portrait of the Riccati quadratic polynomial differential systems $$\dot{x}= \alpha_2(x),\quad\dot{y} = ky^2+\beta_1(x) y + \gamma_2(x), $$ with $(x,y)\in\mathbb{R}^2$, $\gamma_2(x)$ non-zero (otherwise the system is a Bernoulli differential system), $k\neq0$ (otherwise the system is a Lienard differential system), $\beta_ 1(x)$ a polynomial of degree at most $1$, $\alpha_ 2(x)$ and $\gamma_ 2(x)$ polynomials of degree at most 2, and the maximum of the degrees of $ \alpha_2(x)$ and $k y^2+\beta_1(x) y + \gamma_2(x)$ is 2. We give the complete description of their phase portraits in the Poincare disk, Comment: 22 pages

Starting from the lattice Landau gauge gluon and ghost propagator data we use a sequence of Pad\'e approximants, identify the poles and zeros for each approximant and map them into the analytic structure of the propagators. For the Landau gauge gluon propagator the Pad\'e analysis identifies a pair of complex conjugate poles and a branch cut along the negative real axis of the Euclidean $p^2$ momenta. For the Landau gauge ghost propagator the Pad\'e analysis shows a single pole at $p^2 = 0$ and a branch cut also along the negative real axis of the Euclidean $p^2$ momenta. The method gives precise estimates for the gluon complex poles, that agree well with other estimates found in the literature. For the branch cut the Pad\'e analysis gives, at least, a rough estimate of the corresponding branch point.

Given the large adoption and economical impact of permissionless blockchains, the complexity of the underlying systems and the adversarial environment in which they operate, it is fundamental to properly study and understand the emergent behavior and properties of these systems. We describe our experience on a detailed, one-month study of the Ethereum network from several geographically dispersed observation points. We leverage multiple geographic vantage points to assess the key pillars of Ethereum, namely geographical dispersion, network efficiency, blockchain efficiency and security, and the impact of mining pools. Among other new findings, we identify previously undocumented forms of selfish behavior and show that the prevalence of powerful mining pools exacerbates the geographical impact on block propagation delays. Furthermore, we provide a set of open measurement and processing tools, as well as the data set of the collected measurements, in order to promote further research on understanding permissionless blockchains.

We study flows of smooth vector fields $X$ over invariant surfaces $M$ which are levels of rational first integrals. It leads us to study constrained systems, that is, systems with impasses. We identify a subset $\mathcal{I} \subset M$ which we call "pseudo-impasse" set and analyze the flow of X by points of $\mathcal{I}$. Systems well known in the literature exemplify our results: Lorenz, Chen, Falkner-Skan and Fisher-Kolmogorov. We also study 1-parameter families of integrable systems and unfolding of minimal sets. Our main tool is the geometric singular perturbation theory.