Heteroclinic synchronization: Ultrasubharmonic locking

According to the traditional view of synchronization, a weak periodic input is able to lock a nonlinear oscillator at a frequency close to that of the input (1∶1 zone). If the forcing increases, it is possible to achieve synchronization at subharmonic bands also. Using a competitive dynamical system we show the inverse phenomenon: with a weak signal the 1∶1 zone is narrow, but the synchronization of ultrasubharmonics is dominant. In the system’s phase space, there exists a heteroclinic contour in the autonomous regime, which is the image of sequential dynamics. Under the action of a weak periodic forcing, in the vicinity of the contour a stable limit cycle with long period appears. This results in the locking of very low-frequency oscillations with the finite frequency of the forcing. We hypothesize that this phenomenon can be the origin for the synchronization of slow and fast brain rhythms.This work was supported by National Institute of Neurological Disorders and Stroke Grant No. 7R01-NS-38022, National Science Foundation Grant No. EIA-0130708, Spanish MEC BFI2003-07276, and Fundación BBVA

Mapping the Topological Phase Diagram of Multiband Semiconductors with Supercurrents

We show that Josephson junctions made of multiband semiconductors with strong spin-orbit coupling carry a critical supercurrent $I_c$ that contains information about the non-trivial topology of the system. In particular, we find that the emergence and annihilation of Majorana bound states in the junction is reflected in strong even-odd effects in $I_c$ at small junction transparency. This effect allows for a mapping between $I_c$ and the topological phase diagram of the junction, thus providing a dc measurement of its topology.Comment: 5 pages 3 figures. Published versio

SNS junctions in nanowires with spin-orbit coupling: role of confinement and helicity on the sub-gap spectrum

We study normal transport and the sub-gap spectrum of superconductor-normal-superconductor (SNS) junctions made of semiconducting nanowires with strong Rashba spin-orbit coupling. We focus, in particular, on the role of confinement effects in long ballistic junctions. In the normal regime, scattering at the two contacts gives rise to two distinct features in conductance, Fabry-Perot resonances and Fano dips. The latter arise in the presence of a strong Zeeman field $B$ that removes a spin sector in the leads (\emph{helical} leads), but not in the central region. Conversely, a helical central region between non-helical leads exhibits helical gaps of half-quantum conductance, with superimposed helical Fabry-Perot oscillations. These normal features translate into distinct subgap states when the leads become superconducting. In particular, Fabry-Perot resonances within the helical gap become parity-protected zero-energy states (parity crossings), well below the critical field $B_c$ at which the superconducting leads become topological. As a function of Zeeman field or Fermi energy, these zero-modes oscillate around zero energy, forming characteristic loops, which evolve continuously into Majorana bound states as $B$ exceeds $B_c$. The relation with the physics of parity crossings of Yu-Shiba-Rusinov bound states is discussed.Comment: 12 pages main article, 14 figures + 5 pages supplementary material, 5 figures. Added new appendix. Other minor changes. Published versio

Multiple Andreev reflection and critical current in topological superconducting nanowire junctions

We study transport in a voltage biased superconductor-normal-superconductor (SNS) junction made of semiconducting nanowires with strong spin-orbit coupling, as it transitions into a topological superconducting phase for increasing Zeeman field. Despite the absence of a fractional steady-state ac Josephson current in the topological phase, the dissipative multiple Andreev reflection (MAR) current I_dc at different junction transparencies is particularly revealing. It exhibits unique features related to topology, such as the gap inversion, the formation of Majorana bound states, and fermion-parity conservation. Moreover, the critical current I_c, which remarkably does not vanish at the critical point where the system becomes gapless, provides direct evidence of the topological transition.Comment: Published version, 21 pages, 7 figures, 3 appendice

Quantifying wave-function overlaps in inhomogeneous Majorana nanowires

A key property of Majorana zero modes is their protection against local perturbations. In the standard picture, this protection is guaranteed by a high degree of spatial nonlocality of the Majoranas, namely a suppressed wave-function overlap, in the topological phase. However, a careful characterization of resilience to local noise goes beyond mere spatial separation and must also take into account the projection of wave-function spin. By considering the susceptibility of a given zero mode to different local perturbations, we find the relevant forms of spin-resolved wave-function overlaps that measure its resilience. We quantify these overlaps and study their dependence with nanowire parameters in several classes of experimentally relevant configurations. These include nanowires with inhomogeneous depletion and induced pairing, barriers, and quantum dots. Smooth inhomogeneities have been shown to produce near-zero modes, so-called pseudo-Majoranas, below the critical Zeeman field in the bulk. Surprisingly, their resilience is found to be comparable or better than that of topological Majoranas in realistic systems. We further study how accurately their overlaps can be estimated using a purely local measurement on one end of the nanowire, accessible through conventional transport experiments. In uniform nanowires, this local estimator is remarkably accurate. In inhomogeneous cases, it is less accurate but can still provide reasonable estimates for potential inhomogeneities of the order of the superconducting gap. We further analyze the zero-mode wave-function structure, spin texture, and spectral features associated with each type of inhomogeneity. All our results highlight the strong connection between internal wave-function degrees of freedom, nonlocality, and protection in smoothly inhomogeneous nanowiresWe acknowledge financial support from the Spanish Ministry of Economy and Competitiveness through Grants No. FIS2015-65706-P, No. FIS2015-64654-P, and No. FIS2016-80434-P (AEI/FEDER, EU), the Ramón y Cajal programme, Grants No. RYC-2011-09345 and No. RYC-2013-14645, and the “María de Maeztu” Programme for Units of Excellence in Research and Development (MDM-2014-0377

Medidas de riesgo y teorías de elección

Treballs Finals de Grau de Matemàtiques, Facultat de Matemàtiques, Universitat de Barcelona, Any: 2014, Director: José Manuel Corcuera ValverdeThe purpose of this undergraduate thesis is to study and explain different kinds of risk measures, in terms of their axiomatic definitions and of the economic theories of choice that they can derive from. The theory of choice under risk has historically been one of the recurrent problems in the economy and financial world. It has been a challenge developing the necessary mechanisms to allow the modeling of an economic agent behaviour, when it comes to choosing amongst a number of options with uncertain future. Within these models, the concept of risk always emerge, and each theory will analise and measure it in its own way. In the first part we will study different theories formulated throughout history, particularly the expected utility theory (von Neuman and Morgenstern, 1947), the dual theory of choice (Yaari, 1987) and the generalised expected utility theory (Quiggin, 1993), which derives from the lesser ones. All that by explaining the motivations that led to their development, as well as their main advantages and inconveniences (including important paradoxes that contributed to the revision of the theories). In the second part of the essay, we will explain the nature of risk measuring as well as the different ways of approaching it depending on the theory of choice. Amongst them, we will particularly make a point on that derived from the generalised expected utility theory, which we will name distortion-exponential principle

"Si él lo necesita": Gypsy fairness in Vallecas

"Si él lo necesita" (if he really needs it) was the most common argument given by the subjects who accepted the zero offer in the ultimatum game during experiments were conducted among illiterate (adult) gypsies in Vallecas, Madrid. Interestingly the acceptance of the zero offer was not a rare case but, in contrast, it was the modal value. This is even more remarkable if we consider that the 97% of the subjects proposed the equal split.Gypsies, fairness, social welfare, strategy method ultimatum game, bargaining.

Transient cognitive dynamics, metastability, and decision making

Transient Cognitive Dynamics, Metastability, and Decision Making. Rabinovich et al. PLoS Computational Biology. 2008. 4(5) doi:10.1371/journal.pcbi.1000072The idea that cognitive activity can be understood using nonlinear dynamics has been intensively discussed at length for the last 15 years. One of the popular points of view is that metastable states play a key role in the execution of cognitive functions. Experimental and modeling studies suggest that most of these functions are the result of transient activity of large-scale brain networks in the presence of noise. Such transients may consist of a sequential switching between different metastable cognitive states. The main problem faced when using dynamical theory to describe transient cognitive processes is the fundamental contradiction between reproducibility and flexibility of transient behavior. In this paper, we propose a theoretical description of transient cognitive dynamics based on the interaction of functionally dependent metastable cognitive states. The mathematical image of such transient activity is a stable heteroclinic channel, i.e., a set of trajectories in the vicinity of a heteroclinic skeleton that consists of saddles and unstable separatrices that connect their surroundings. We suggest a basic mathematical model, a strongly dissipative dynamical system, and formulate the conditions for the robustness and reproducibility of cognitive transients that satisfy the competing requirements for stability and flexibility. Based on this approach, we describe here an effective solution for the problem of sequential decision making, represented as a fixed time game: a player takes sequential actions in a changing noisy environment so as to maximize a cumulative reward. As we predict and verify in computer simulations, noise plays an important role in optimizing the gain.This work was supported by ONR N00014-07-1-0741. PV acknowledges support from Spanish BFU2006-07902/BFI and CAM S-SEM-0255-2006

Prognosticating Physique: Machine Learning for Future Body Shape Estimations in Weight Loss

This research presents the development of a predictive model to forecast morphological changes in individuals undergoing weight loss treatment. The initiative, Tech4Diet, draws from the public health imperative to address the global obesity crisis and utilized 3D body scans and supplementary medical data to enhance adherence to treatment. An extensive review of the current literature on 3D human body model representation forms the foundation of this work, leading to the selection of the Skinned Multi-Person Linear Model (SMPL) model for encoding body scans. Long Short-Term Memory (LSTM) networks are employed to analyze these encoded datasets and predict potential body changes before the treatment concludes. The process includes a comprehensive analysis of collected data, body model representation, neural network design, model training, and evaluation. The resulting model successfully generates 3D meshes of predicted body transformations, offering a novel approach to visualizing weight loss progress. Further chapters detail the data acquisition, model design, training process, and results