Quantum isometries and noncommutative spheres

We introduce and study two new examples of noncommutative spheres: the half-liberated sphere, and the free sphere. Together with the usual sphere, these two spheres have the property that the corresponding quantum isometry group is "easy", in the representation theory sense. We present as well some general comments on the axiomatization problem, and on the "untwisted" and "non-easy" case.Comment: 16 page

Evolution of Polygonal Lines by the Binormal Flow

The aim of this paper is threefold. First we display solutions of the cubic nonlinear Schr ̈odinger equation on R in link with initial data a sum of Dirac masses. Secondly we show a Talbot effect for the same equation. Finally we prove the existence of a unique solution of the binormal flow with datum a polygonal line. This equation is used as a model for the vortex filaments dynamics in 3-D fluids and superfluids. We also construct solutions of the binormal flow that present an intermittency phenomena. Finally, the solution we construct for the binormal flow is continued for negative times, yielding a geometric way to approach the continuation after blow-up for the 1-D cubic nonlinear Schr ̈odinger equation

On the energy of critical solutions of the binormal flow

The binormal flow is a model for the dynamics of a vortex filament in a 3-D inviscid incompressible fluid. The flow is also related with the classical continuous Heisen- berg model in ferromagnetism, and the 1-D cubic Schr ̈odinger equation. We consider a class of solutions at the critical level of regularity that generate singularities in finite time. One of our main results is to prove the existence of a natural energy associated to these solutions. This energy remains constant except at the time of the formation of the singularity when it has a jump discontinuity. When interpreting this conservation law in the framework of fluid mechanics, it involves the amplitude of the Fourier modes of the variation of the direction of the vorticity

Riemann's non-differentiable function and the binormal curvature flow

We make a connection between a famous analytical object introduced in the 1860s by Riemann, as well as some variants of it, and a nonlinear geometric PDE, the binormal curvature flow. As a consequence this analytical object has a non-obvious nonlinear geometric interpretation. We recall that the binormal flow is a standard model for the evolution of vortex filaments. We prove the existence of solutions of the binormal flow with smooth trajectories that are as close as desired to curves with a multifractal behavior. Finally, we show that this behavior falls within the multifractal formalism of Frisch and Parisi, which is conjectured to govern turbulent fluids

Unbounded growth of the energy density associated to the Schrödinger map and the binormal flow

We consider the binormal flow equation, which is a model for the dynamics of vortex filaments in Euler equations. Geometrically it is a flow of curves in three dimensions, explicitly connected to the 1-D Schr¨odinger map with values on the 2-D sphere, and to the 1-D cubic Schr¨odinger equation. Although these equations are completely integrable we show the existence of an unbounded growth of the energy density. The density is given by the amplitude of the high frequencies of the derivative of the tangent vectors of the curves, thus giving information of the oscillation at small scales. In the setting of vortex filaments the variation of the tangent vectors is related to the derivative of the direction of the vorticity, that according to the Constantin-Fefferman-Majda criterion plays a relevant role in the possible development of singularities for Euler equations

A maximality result for orthogonal quantum groups

We prove that the quantum group inclusion $O_n \subset O_n^*$ is "maximal", where $O_n$ is the usual orthogonal group and $O_n^*$ is the half-liberated orthogonal quantum group, in the sense that there is no intermediate compact quantum group $O_n\subset G\subset O_n^*$. In order to prove this result, we use: (1) the isomorphism of projective versions $PO_n^*\simeq PU_n$, (2) some maximality results for classical groups, obtained by using Lie algebras and some matrix tricks, and (3) a short five lemma for cosemisimple Hopf algebras.Comment: 10 page

When IoT Meets DevOps: Fostering Business Opportunities

The Internet of Things (IoT) is the new digital revolution for the near-future society, the second after the creation of the Internet itself. The software industry is converging towards the large-scale deployment of IoT devices and services, and there’s broad support from the business environment for this engineering vision. The Development and Operations (DevOps) project management methodology, with continuous delivery and integration, is the preferred approach for achieving and deploying applications to all levels of the IoT architecture. In this paper we also discuss the promising trend of associating devices with microservices, which are further encapsulated into functional packages called containers. Docker is considered the market leader in container-based service delivery, though other important software companies are promoting this concept as part of the technology solution for their IoT customers. In the experimental section we propose a three-layer IoT model, business-oriented, and distributed over multiple cloud environments, comprising the Physical, Fog/Edge, and Application layers.     Keywords: Internet-of-Things, software technologies, project management, business environment Heading

Quantum Isometries of the finite noncommutative geometry of the Standard Model

We compute the quantum isometry group of the finite noncommutative geometry F describing the internal degrees of freedom in the Standard Model of particle physics. We show that this provides genuine quantum symmetries of the spectral triple corresponding to M x F where M is a compact spin manifold. We also prove that the bosonic and fermionic part of the spectral action are preserved by these symmetries.Comment: 29 pages, no figures v3: minor change

Enhancing Fashion Sustainability Through a Data Systemic Approach

Today everyday life is characterized by the interaction with an ever-increasing flow of digital data. The research aims to analyze the fashion industry as a data-driven enterprise in which the correlation of data characterized by greater information power and higher quality gives the chance to make a more informed decision making that lead to undertaking better and more sustainable actions in all the value chain. Data, in this focus, could have the power of increasing the efficiency of the system and reducing its impact at the same time, creating a new model that is not only able to improve environmental, economic and social sustainability but also communicative, enabling a more human-centered products and services designing. This research highlights the importance of giving an integrated and holistic perspective through a data systemic approach to deal with a complex and fragmented sustainable problem, proposing an information flow strategy that makes accessible information improving transparency and traceability. This paper presents several case studies that show how data-oriented projects can contribute some benefits to a fashion system that has environmental sustainability as its priority, but also that the lack of correlation of all these strategies is not yet able to generate and lead to a systemic change