Exponential dichotomies of evolution operators in Banach spaces

This paper considers three dichotomy concepts (exponential dichotomy, uniform exponential dichotomy and strong exponential dichotomy) in the general context of non-invertible evolution operators in Banach spaces. Connections between these concepts are illustrated. Using the notion of Green function, we give necessary conditions and sufficient ones for strong exponential dichotomy. Some illustrative examples are presented to prove that the converse of some implication type theorems are not valid

Recurrence spectrum in smooth dynamical systems

We prove that for conformal expanding maps the return time does have constant multifractal spectrum. This is the counterpart of the result by Feng and Wu in the symbolic setting

Generalized nonuniform dichotomies and local stable manifolds

We establish the existence of local stable manifolds for semiflows generated by nonlinear perturbations of nonautonomous ordinary linear differential equations in Banach spaces, assuming the existence of a general type of nonuniform dichotomy for the evolution operator that contains the nonuniform exponential and polynomial dichotomies as a very particular case. The family of dichotomies considered allow situations for which the classical Lyapunov exponents are zero. Additionally, we give new examples of application of our stable manifold theorem and study the behavior of the dynamics under perturbations.Comment: 18 pages. New version with minor corrections and an additional theorem and an additional exampl

Analytical models for CO2 emissions and travel time for short-to-medium-haul flights considering available seats

Recently, there has been much interest in measuring the environmental impact of short-to-medium-haul flights. Emissions of CO2 are usually measured to consider the environmental footprint, and CO2 calculators are available using different types of approximations. We propose analytical models calculating gate-to-gate CO2 emissions and travel time based on the flight distance and on the number of available seats. The accuracy of the numerical results were in line with other CO2 calculators, and when applying an analytical fitting, the error of interpolation was low. The models presented the advantage with respect to other calculators of being sensitive to the number of available seats, a parameter generally not explicitly considered. Its applicability was shown in two practical examples where emissions and travel time per kilometre were calculated for several European routes in a simple and efficient manner. The model enabled the identification of routes where rail would be a viable alternative both from the emissions and total travel time perspectives

Hausdorff Dimension in Convex Bornological Spaces

AbstractFor non-metrizable spaces the classical Hausdorff dimension is meaningless. We extend the notion of Hausdorff dimension to arbitrary locally convex linear topological spaces and thus to a large class of non-metrizable spaces. This involves a limiting procedure using the canonical bornological structure. In the case of normed spaces the new notion of Hausdorff dimension is equivalent to the classical notion

The Impact of COVID-19 on Physical Activity and Sedentary Behavior in Children: A Pilot Study

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Topological pressure of simultaneous level sets

Multifractal analysis studies level sets of asymptotically defined quantities in a topological dynamical system. We consider the topological pressure function on such level sets, relating it both to the pressure on the entire phase space and to a conditional variational principle. We use this to recover information on the topological entropy and Hausdorff dimension of the level sets. Our approach is thermodynamic in nature, requiring only existence and uniqueness of equilibrium states for a dense subspace of potential functions. Using an idea of Hofbauer, we obtain results for all continuous potentials by approximating them with functions from this subspace. This technique allows us to extend a number of previous multifractal results from the $C^{1+\epsilon}$ case to the $C^1$ case. We consider ergodic ratios $S_n \phi/S_n \psi$ where the function $\psi$ need not be uniformly positive, which lets us study dimension spectra for non-uniformly expanding maps. Our results also cover coarse spectra and level sets corresponding to more general limiting behaviour.Comment: 32 pages, minor changes based on referee's comment

Assessment of oil contamination in the bay of Porto Grande (Cape Verde) using the mullet Chelon bispinosus

Polycyclic aromatic hydrocarbons (PAHs) are a group of persistent organic pollutants, some of which are mutagenic and carcinogenic so PAH concentrations in fish used for human consumption are crucial to assess impact to human health. Total PAH concentrations in muscle and liver of mullets Chelon bispinosus from the Bay of Porto Grande (Cape Verde) (four sites in the bay and a control) ranged from 112.7 to 779.5 and 291.5 to 7548.7 ng/g d. w., respectively. Two and three ring PAHs were the most frequent (72.8 to 90.8% in the muscle and 75.9 to 98.3% in the liver), but levels of carcinogenic PAHs (mainly Dibenzo (a,h) antracene) in certain sites (CN and PG) are of concern. Results reflect a chronic PAH pollution in the bay and sources are a mixture of anthropogenic (petrogenic and pyrolytic) and natural sources, making their identification extremely complex. Although, BaP levels were below the threshold established by Cape Verde and the European Union, BaPEs levels in muscle ranged from 0.28 to 3.66 ng/g w. w. and BAPEs and TPAHs exposure for the average adult was 0.02 to 0.26 and 1.6 to 11.2 μg/day, respectively. Further knowledge of PAH concentrations in other species are necessary for a proper environmental risk assessment policy.Key words: Bay of Porto Grande, Cape Verde, Chelon bispinosus, mullets, PAHs, BaPEs, daily intake

Time-metric equivalence and dimension change under time reparameterizations

We study the behavior of dynamical systems under time reparameterizations, which is important not only to characterize chaos in relativistic systems but also to probe the invariance of dynamical quantities. We first show that time transformations are locally equivalent to metric transformations, a result that leads to a transformation rule for all Lyapunov exponents on arbitrary Riemannian phase spaces. We then show that time transformations preserve the spectrum of generalized dimensions D_q except for the information dimension D_1, which, interestingly, transforms in a nontrivial way despite previous assertions of invariance. The discontinuous behavior at q=1 can be used to constrain and extend the formulation of the Kaplan-Yorke conjecture

Modelling the complex nature of the tumor microenvironment: 3D tumor spheroids as an evolving tool

Cancer remains a serious burden in society and while the pace in the development of novel and more effective therapeutics is increasing, testing platforms that faithfully mimic the tumor microenvironment are lacking. With a clear shift from animal models to more complex in vitro 3D systems, spheroids emerge as strong options in this regard. Years of development have allowed spheroid-based models to better reproduce the biomechanical cues that are observed in the tumor-associated extracellular matrix (ECM) and cellular interactions that occur in both a cellâ cell and cell-ECM manner. Here, we summarize some of the key cellular interactions that drive tumor development, progression and invasion, and how successfully are these interactions recapitulated in 3D spheroid models currently in use in the field. We finish by speculating on future advancements in the field and on how these can shape the relevance of spherical 3D models for tumor modelling.Authors would like to acknowledge the fnancial support from the European Research Council through the Starting Grant “CapBed” (ERC-2018-STG-805411) and FCT/MCTES (Fundação para a Ciência e a Tecnologia/ Ministério da Ciência, Tecnologia, e Ensino Superior) through the grant SFRH/BD/119756/2016 (D.B.R.). The authors would additionally like to thank the contributions to this work from the project “TERM RES Hub—Scientifc Infrastructure for Tissue Engineering and Regenerative Medicine”, reference PINFRA/22190/2016 (Norte-01-0145-FEDER-022190), funded by the Portuguese National Science Foundation (FCT) in cooperation with the Northern Portugal Regional Coordination and Development Commission (CCDR-N). Ultimately, we would like to equally acknowledge fnancial support fromhttps://doi.org/10.54499/ UIDB/50026/2020);https://doi.org/10.54499/UIDP/50026/2020) andhttps://doi. org/10.54499/LA/P/0050/2020)