The topological classification of one-dimensional symmetric quantum walks

We give a topological classification of quantum walks on an infinite 1D lattice, which obey one of the discrete symmetry groups of the tenfold way, have a gap around some eigenvalues at symmetry protected points, and satisfy a mild locality condition. No translation invariance is assumed. The classification is parameterized by three indices, taking values in a group, which is either trivial, the group of integers, or the group of integers modulo 2, depending on the type of symmetry. The classification is complete in the sense that two walks have the same indices if and only if they can be connected by a norm continuous path along which all the mentioned properties remain valid. Of the three indices, two are related to the asymptotic behaviour far to the right and far to the left, respectively. These are also stable under compact perturbations. The third index is sensitive to those compact perturbations which cannot be contracted to a trivial one. The results apply to the Hamiltonian case as well. In this case all compact perturbations can be contracted, so the third index is not defined. Our classification extends the one known in the translation invariant case, where the asymptotic right and left indices add up to zero, and the third one vanishes, leaving effectively only one independent index. When two translationally invariant bulks with distinct indices are joined, the left and right asymptotic indices of the joined walk are thereby fixed, and there must be eigenvalues at $1$ or $-1$ (bulk-boundary correspondence). Their location is governed by the third index. We also discuss how the theory applies to finite lattices, with suitable homogeneity assumptions.Comment: 36 pages, 7 figure

Two species coagulation approach to consensus by group level interactions

We explore the self-organization dynamics of a set of entities by considering the interactions that affect the different subgroups conforming the whole. To this end, we employ the widespread example of coagulation kinetics, and characterize which interaction types lead to consensus formation and which do not, as well as the corresponding different macroscopic patterns. The crucial technical point is extending the usual one species coagulation dynamics to the two species one. This is achieved by means of introducing explicitly solvable kernels which have a clear physical meaning. The corresponding solutions are calculated in the long time limit, in which consensus may or may not be reached. The lack of consensus is characterized by means of scaling limits of the solutions. The possible applications of our results to some topics in which consensus reaching is fundamental, like collective animal motion and opinion spreading dynamics, are also outlined

Multiple peak aggregations for the Keller-Segel system

In this paper we derive matched asymptotic expansions for a solution of the Keller-Segel system in two space dimensions for which the amount of mass aggregation is $8\pi N$, where $N=1,2,3,...$ Previously available asymptotics had been computed only for the case in which N=1

Shrinkers, expanders, and the unique continuation beyond generic blowup in the heat flow for harmonic maps between spheres

Using mixed analytical and numerical methods we investigate the development of singularities in the heat flow for corotational harmonic maps from the $d$-dimensional sphere to itself for $3\leq d\leq 6$. By gluing together shrinking and expanding asymptotically self-similar solutions we construct global weak solutions which are smooth everywhere except for a sequence of times $T_1<T_2<...<T_k<\infty$ at which there occurs the type I blow-up at one of the poles of the sphere. We show that in the generic case the continuation beyond blow-up is unique, the topological degree of the map changes by one at each blow-up time $T_i$, and eventually the solution comes to rest at the zero energy constant map.Comment: 24 pages, 8 figures, minor corrections, matches published versio

The varying burden of depressive symptoms across adulthood : Results from six NHANES cohorts

Background: Depressive symptoms differ from each other in the degree of functional impairment they cause. The incidence of depression varies across the adult lifespan. We examined whether age moderates the impairment caused by depressive symptoms. Methods: The study sample (n = 21,056) was adults drawn from six multistage probability samples from the National Health and Nutrition Examination Survey series (NHANES, years 2005-2016) conducted in the United States using cross-sectional, representative cohorts. Depressive symptoms were assessed with the nine-item Patient Health Questionnaire (PHQ-9). We used regression models to predict high functional impairment, while controlling for sociodemographic variables and physical disorders. Results: Age moderated the association between depressive symptoms and functional impairment: middle-aged adults perceived moderate and severe symptoms as more impairing than did others. Older adults reported slightly higher impairment due to mild symptoms. The individual symptoms of low mood, feelings of worthlessness and guilt, and concentration difficulties were more strongly related to high impairment in mid-adulthood as compared to early and late adulthood. Limitations: Cross-sectional data allows only between-person comparisons. The PHQ-9 is brief and joins compound symptoms into single items. There was no information available concerning comorbid mental disorders. Co-occurring physical disorders were self-reported. Conclusions: Symptoms of depression may imply varying levels of impairment at different ages. The results suggest a need for age adjustments when estimating the functional impact of depression in the general population. Additionally, they show a need for more accurate assessments of depression-related impairment at older ages. Evidence-based programs may generally benefit from symptom- and age-specific findings.Peer reviewe

Utilising the Cross Industry Standard Process for Data Mining to reduce uncertainty in the Measurement and Verification of energy savings

This paper investigates the application of Data Mining (DM) to predict baseline energy consumption for the improvement of energy savings estimation accuracy in Measurement and Verification (M&V). M&V is a requirement of a certified energy management system (EnMS). A critical stage of the M&V process is the normalisation of data post Energy Conservation Measure (ECM) to pre-ECM conditions. Traditional M&V approaches utilise simplistic modelling techniques, which dilute the power of the available data. DM enables the true power of the available energy data to be harnessed with complex modelling techniques. The methodology proposed incorporates DM into the M&V process to improve prediction accuracy. The application of multi-variate regression and artificial neural networks to predict compressed air energy consumption in a manufacturing facility is presented. Predictions made using DM were consistently more accurate than those found using traditional approaches when the training period was greater than two months

Dynamic clustering of time series with Echo State Networks

In this paper we introduce a novel methodology for unsupervised analysis of time series, based upon the iterative implementation of a clustering algorithm embedded into the evolution of a recurrent Echo State Network. The main features of the temporal data are captured by the dynamical evolution of the network states, which are then subject to a clustering procedure. We apply the proposed algorithm to time series coming from records of eye movements, called saccades, which are recorded for diagnosis of a neurodegenerative form of ataxia. This is a hard classification problem, since saccades from patients at an early stage of the disease are practically indistinguishable from those coming from healthy subjects. The unsupervised clustering algorithm implanted within the recurrent network produces more compact clusters, compared to conventional clustering of static data, and provides a source of information that could aid diagnosis and assessment of the disease.Universidad de Málaga. Campus de Excelencia Internacional Andalucía Tec

Particle approximation of the one dimensional Keller-Segel equation, stability and rigidity of the blow-up

We investigate a particle system which is a discrete and deterministic approximation of the one-dimensional Keller-Segel equation with a logarithmic potential. The particle system is derived from the gradient flow of the homogeneous free energy written in Lagrangian coordinates. We focus on the description of the blow-up of the particle system, namely: the number of particles involved in the first aggregate, and the limiting profile of the rescaled system. We exhibit basins of stability for which the number of particles is critical, and we prove a weak rigidity result concerning the rescaled dynamics. This work is complemented with a detailed analysis of the case where only three particles interact