Models of quantum walks which admit continuous time and continuous spacetime limits have recently led to quantum simulation schemes for simulating fermions in relativistic and non relativistic regimes (Di Molfetta and Arrighi, 2020). This work continues the study of relationships between discrete time quantum walks (DTQW) and their ostensive continuum counterparts by developing a more general framework than was done in (Di Molfetta and Arrighi, 2020) to evaluate the continuous time limit of these discrete quantum systems. Under this framework, we prove two constructive theorems concerning which internal discrete transitions ("coins") admit nontrivial continuum limits in 1D+1. We additionally prove that the continuous space limit of the continuous time limit of the DTQW can only yield massless states which obey the Dirac equation. We also demonstrate that for general coins the continuous time limit of the DTQW can be identified with the canonical continuous time quantum walk (CTQW) when the coin is allowed to transition through the continuum limit process. Finally, we introduce the Plastic Quantum Walk, or a quantum walk which admits both continuous time and continuous spacetime limits and, as a novel result, we use our 1D+1 results to obtain necessary and sufficient conditions concerning which DTQWs admit plasticity in 2D+1, showing the resulting Hamiltonians. We consider coin operators as general 4 parameter unitary matrices, with parameters which are functions of the lattice step size 𝜖. This dependence on 𝜖 encapsulates all functions of 𝜖 for which a Taylor series expansion in 𝜖 is well defined, making our results very general.

Background: Physical functioning and mobility of older populations are of increasing interest when populations are aging. Lower body functioning such as walking is a fundamental part of many actions in daily life. Limitations in mobility threaten independent living as well as quality of life in old age. In this study we examine differences in physical aging and convert those differences into the everyday measure of single years of age. Methods: We use the English Longitudinal Study of Ageing, which was collected biennially between 2002 and 2012. Data on physical performance, health as well as information on economics and demographics of participants were collected. Lower body performance was assessed with two timed walks at normal pace each of 8 ft (2.4 m) of survey participants aged at least 60 years. We employed growth curve models to study differences in physical aging and followed the characteristic-based age approach to illustrate those differences in single years of age. Results: First, we examined walking speed of about 11,700 English individuals, and identified differences in aging trajectories by sex and other characteristics (e.g. education, occupation, regional wealth). Interestingly, higher educated and non-manual workers outperformed their counterparts for both men and women. Moreover, we transformed the differences between subpopulations into single years of age to demonstrate the magnitude of those gaps, which appear particularly high at early older ages. Conclusions: This paper expands research on aging and physical performance. In conclusion, higher education provides an advantage in walking of up to 15 years for men and 10 years for women. Thus, enhancements in higher education have the potential to ensure better mobility and independent living in old age for a longer period. (author's Abstract)

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The present work aims to contribute to the study of networks by mapping the temporal evolution of the degree to a random walk in degree space. We analyzed how and when the degree approximates a pre-established value through a parallel with the first-passage problem of random walks. The mean time for the first-passage was calculated for the dynamical versions the Watts-Strogatz and Erdos-Renyi models. We also analyzed the degree variance for the random recursive tree and Barabasi-Albert models / O presente trabalho visa contribuir com a pesquisa na área de redes através do mapeamento da evolução temporal do grau com um passeio aleatório no espaço do mesmo. Para tanto, foi feita uma análise de quando e como a quantidade de ligações do vértice se aproxima de um valor pré-estabelecido, mediante um paralelo com o problema da primeira passagem de passeios aleatórios. O tempo médio para a primeira passagem para as versões dinâmicas dos modelos Watts-Strogatz e Erdos-Rényi foram calculados. Além disso, foi realizado um estudo da variância do grau para os modelos da árvore recursiva aleatória e Barabási-Albert

Considere um meio desordenado formado por $N$ pontos cujas coordenadas são geradas aleatoriamente com probabilidade uniforme ao longo das arestas unitárias de um hipercubo de $d$ dimensões. Um caminhante, partindo de um ponto qualquer desse meio, se desloca seguindo a regra determinista de dirigir-se sempre ao ponto mais próximo que não tenha sido visitado nos últimos $\\mu$ passos. Esta dinâmica de movimentação, denominada caminhada determinista do turista, leva a trajetórias formadas por uma parte inicial transiente de $t$ pontos, e uma parte final cíclica de $p$ pontos. A exploração do meio se limita aos $t+p$ pontos percorridos na trajetória. O sucesso da exploração depende do valor da memória $\\mu$ do viajante. Para valores pequenos de $\\mu$ a exploração é altamente localizada e o sistema não é satisfatoriamente explorado. Já para $\\mu$ da ordem de $N$, aparecem ciclos longos, permitindo a exploração global do meio. O objetivo deste estudo é determinar o valor de memória $\\mu_1$ para o qual ocorre uma transição abrupta no comportamento exploratório do turista em meios unidimensionais. Procuramos também entender a distribuição da posição final do turista após atingir um estado estacionário que é atingido quando o turista fica aprisionado nos ciclos. Os resultados obtidos por simulações numéricas e por um tratamento analítico mostram que $\\mu_1 = \\log_2 N$. O estudo também mostrou a existência de uma região de transição com largura $\\varepsilon = e/ \\ln 2$ constante, caracterizando uma transição aguda de fase no comportamento exploratório do turista em uma dimensão. A análise do estado estacionário da caminhada em função da memória mostrou que, para $\\mu$ distante de $\\mu_1$, a dinâmica de exploração ocorre como um processo difusivo tradicional (distribuição gaussiana). Já para $\\mu$ próximo de $\\mu_1$ (região de transição), essa dinâmica segue um processo superdifusivo não-linear, caracterizado por distribuições $q$-gaussianas e distribuições $\\alpha$-estáveis de Lévy. Neste processo, o parâmetro $q$ funciona como parâmetro de ordem da transição. / Consider a disordered medium formed by $N$ point whose coordinates are randomly generated with uniform probability along the unitary edges of a $d$-dimensional hypercube. A walker, starting to walk from any point of that medium, moves following the deterministic rule of always going to the nearest point that has not been visited in the last $\\mu$ steps. This dynamic of moving, called deterministic tourist walk, leads to trajectories formed by a initial transient part of $t$ points and a final cycle of $p$ points. The exploration of the medium is limited to the $t+p$ points covered. The success of the exploration depends on the traveler\'s memory value $\\mu$. For small values of $\\mu$, the exploration is highly localized and the whole system remains unexplored. For values of $\\mu$ of the order of $N$, however, long cycles appear, allowing global exploration of the medium. The objective of this study is to determine the memory value $\\mu_1$ for which a sharp transition in the exploratory behavior of the tourist in one-dimensional media occurs. We also want to understand the distribution of the final position of the tourist after reaches a steady state in exploring the medium. That steady state is reached when the tourist is trapped in cycles. The results achieved by numerical simulations and analytical treatment has shown that $\\mu_1 = \\log_2 N$. The study has also shown the existence of a transition region, with a constant width of $\\varepsilon = e/ \\ln 2$, characterizing a phase transition in the exploratory behavior of the tourist in one dimension. The analysis of the walk steady state as a function of the memory has shown that for $\\mu$ far from $\\mu_1$, the exploratory dynamic follows a traditional diffusion process (with gaussian distribution). In the other hand, for $\\mu$ near $\\mu_1$ (transition region), the dynamic follows a non-linear superdiffusion process, characterized by $q$-gaussian distributions and Lèvy $\\alpha$-stable distributions. In this process, the parameter $q$ plays the role of a transition order parameter.

The present work aims to contribute to the study of networks by mapping the temporal evolution of the degree to a random walk in degree space. We analyzed how and when the degree approximates a pre-established value through a parallel with the first-passage problem of random walks. The mean time for the first-passage was calculated for the dynamical versions the Watts-Strogatz and Erdos-Renyi models. We also analyzed the degree variance for the random recursive tree and Barabasi-Albert models / O presente trabalho visa contribuir com a pesquisa na área de redes através do mapeamento da evolução temporal do grau com um passeio aleatório no espaço do mesmo. Para tanto, foi feita uma análise de quando e como a quantidade de ligações do vértice se aproxima de um valor pré-estabelecido, mediante um paralelo com o problema da primeira passagem de passeios aleatórios. O tempo médio para a primeira passagem para as versões dinâmicas dos modelos Watts-Strogatz e Erdos-Rényi foram calculados. Além disso, foi realizado um estudo da variância do grau para os modelos da árvore recursiva aleatória e Barabási-Albert

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