On embeddings and dimensions of global attractors associated with dissipative partial differential equations

Hunt and Kaloshin (1999) proved that it is possible to embed a compact subset X of a Hilbert space with upper box-counting dimension d into RN for any N > 2d+1, using a linear map L whose inverse is Hölder continuous with exponent α < (N - 2d)/N(1 + τ(X)/2), where τ(X) is the 'thickness exponent' of X. More recently, Ott et al. (2006) conjectured that "many of the attractors associated with the evolution equations of mathematical physics have thickness exponent zero". In Chapter 2 we study orthogonal sequences in a Hilbert space H, whose elements tend to zero, and similar sequences in the space c0 of null sequences. These examples are used to show that Hunt and Kaloshin's result, and a related result due to Robinson (2009) for subsets of Banach spaces, are asymptotically sharp. An analogous argument shows that the embedding theorems proved by Robinson (2010), in terms of the Assouad dimension, for the Hilbert and Banach space case are asymptotically sharp. In Chapter 3 we introduce a variant of the thickness exponent, the Lipschitz deviation dev(X). We show that Hunt and Kaloshin's result and Corollary 3.9 in Ott et al. (2006) remain true with the thickness replaced by the Lipschitz deviation. We then prove that dev(X) = 0 for the attractors of a wide class of semilinear parabolic equations, thus providing a partial answer to the conjecture of Ott, Hunt, & Kaloshin. In Chapter 4 we study the regularity of the vector field on the global attractor associated with parabolic equations. We show that certain dissipative equations possess a linear term that is log-Lipschitz continuous on the attractor. We then prove that this property implies that the associated global attractor A lies within a small neighbourhood of a smooth manifold, given as a Lipschitz graph over a finite number of Fourier modes. This provides an alternative proof that the global attractor A has zero Lipschitz deviation. In Chapter 5 we use shape theory and the concept of cellularity to show that if A is the global attractor associated with a dissipative partial differential equation in a real Hilbert space H and the set A - A has finite Assouad dimension d, then there is an ordinary differential equation in Rm+1, with m > d, that has unique solutions and reproduces the dynamics on A. Moreover, the dynamical system generated by this new ordinary differential equation has a global attractor X arbitrarily close to LA, where L is a homeomorphism from A into Rm+1

Embedding of global attractors and their dynamics

Using shape theory and the concept of cellularity, we show that if $A$ is the global attractor associated with a dissipative partial differential equation in a real Hilbert space $H$ and the set $A-A$ has finite Assouad dimension $d$, then there is an ordinary differential equation in ${\mathbb R}^{m+1}$, with $m >d$, that has unique solutions and reproduces the dynamics on $A$. Moreover, the dynamical system generated by this new ordinary differential equation has a global attractor $X$ arbitrarily close to $LA$, where $L$ is a homeomorphism from $A$ into ${\mathbb R}^{m+1}$

On embeddings and dimensions of global attractors associated with dissipative partial differential equations

Hunt and Kaloshin (1999) proved that it is possible to embed a compact subset X of a Hilbert space with upper box-counting dimension d into RN for any N > 2d+1, using a linear map L whose inverse is Hölder continuous with exponent α d, that has unique solutions and reproduces the dynamics on A. Moreover, the dynamical system generated by this new ordinary differential equation has a global attractor X arbitrarily close to LA, where L is a homeomorphism from A into Rm+1.EThOS - Electronic Theses Online ServiceBrazil. Coordenação do Aperfeiçoamento de Pessoal de Nível Superior (CAPES)GBUnited Kingdo

Height and body-mass index trajectories of school-aged children and adolescents from 1985 to 2019 in 200 countries and territories: a pooled analysis of 2181 population-based studies with 65 million participants

Summary Background Comparable global data on health and nutrition of school-aged children and adolescents are scarce. We aimed to estimate age trajectories and time trends in mean height and mean body-mass index (BMI), which measures weight gain beyond what is expected from height gain, for school-aged children and adolescents. Methods For this pooled analysis, we used a database of cardiometabolic risk factors collated by the Non-Communicable Disease Risk Factor Collaboration. We applied a Bayesian hierarchical model to estimate trends from 1985 to 2019 in mean height and mean BMI in 1-year age groups for ages 5–19 years. The model allowed for non-linear changes over time in mean height and mean BMI and for non-linear changes with age of children and adolescents, including periods of rapid growth during adolescence. Findings We pooled data from 2181 population-based studies, with measurements of height and weight in 65 million participants in 200 countries and territories. In 2019, we estimated a difference of 20 cm or higher in mean height of 19-year-old adolescents between countries with the tallest populations (the Netherlands, Montenegro, Estonia, and Bosnia and Herzegovina for boys; and the Netherlands, Montenegro, Denmark, and Iceland for girls) and those with the shortest populations (Timor-Leste, Laos, Solomon Islands, and Papua New Guinea for boys; and Guatemala, Bangladesh, Nepal, and Timor-Leste for girls). In the same year, the difference between the highest mean BMI (in Pacific island countries, Kuwait, Bahrain, The Bahamas, Chile, the USA, and New Zealand for both boys and girls and in South Africa for girls) and lowest mean BMI (in India, Bangladesh, Timor-Leste, Ethiopia, and Chad for boys and girls; and in Japan and Romania for girls) was approximately 9–10 kg/m2. In some countries, children aged 5 years started with healthier height or BMI than the global median and, in some cases, as healthy as the best performing countries, but they became progressively less healthy compared with their comparators as they grew older by not growing as tall (eg, boys in Austria and Barbados, and girls in Belgium and Puerto Rico) or gaining too much weight for their height (eg, girls and boys in Kuwait, Bahrain, Fiji, Jamaica, and Mexico; and girls in South Africa and New Zealand). In other countries, growing children overtook the height of their comparators (eg, Latvia, Czech Republic, Morocco, and Iran) or curbed their weight gain (eg, Italy, France, and Croatia) in late childhood and adolescence. When changes in both height and BMI were considered, girls in South Korea, Vietnam, Saudi Arabia, Turkey, and some central Asian countries (eg, Armenia and Azerbaijan), and boys in central and western Europe (eg, Portugal, Denmark, Poland, and Montenegro) had the healthiest changes in anthropometric status over the past 3·5 decades because, compared with children and adolescents in other countries, they had a much larger gain in height than they did in BMI. The unhealthiest changes—gaining too little height, too much weight for their height compared with children in other countries, or both—occurred in many countries in sub-Saharan Africa, New Zealand, and the USA for boys and girls; in Malaysia and some Pacific island nations for boys; and in Mexico for girls. Interpretation The height and BMI trajectories over age and time of school-aged children and adolescents are highly variable across countries, which indicates heterogeneous nutritional quality and lifelong health advantages and risks

Heterogeneous contributions of change in population distribution of body mass index to change in obesity and underweight NCD Risk Factor Collaboration (NCD-RisC)

From 1985 to 2016, the prevalence of underweight decreased, and that of obesity and severe obesity increased, in most regions, with significant variation in the magnitude of these changes across regions. We investigated how much change in mean body mass index (BMI) explains changes in the prevalence of underweight, obesity, and severe obesity in different regions using data from 2896 population-based studies with 187 million participants. Changes in the prevalence of underweight and total obesity, and to a lesser extent severe obesity, are largely driven by shifts in the distribution of BMI, with smaller contributions from changes in the shape of the distribution. In East and Southeast Asia and sub-Saharan Africa, the underweight tail of the BMI distribution was left behind as the distribution shifted. There is a need for policies that address all forms of malnutrition by making healthy foods accessible and affordable, while restricting unhealthy foods through fiscal and regulatory restrictions

Lipschitz deviation and embeddings of global attractors

Hunt and Kaloshin (1999 Nonlinearity 12 1263-75) proved that it is possible to embed a compact subset X of a Hilbert space with upper box-counting dimension d 2k + 1, using a linear map L whose inverse is Holder continuous with exponent alpha R-N, d(H) (L(X)) >= min(N, d(H)(X)/(1 + tau(X)/2)). They also conjectured that 'many of the attractors associated with the evolution equations of mathematical physics have thickness exponent zero'. In this paper we introduce a variant of the thickness exponent, the Lipschitz deviation dev(X): we show that in both of the above results this can be used in place of the thickness exponent, and-appealing to results from the theory of approximate inertial manifolds-we prove that dev(X) = 0 for the attractors of a wide class of semilinear parabolic equations, thus providing a partial answer to the conjecture of Ott, Hunt and Kaloshin. In particular, dev(X) = 0 for the attractor of the 2D Navier-Stokes equations with forcing f is an element of L-2, while current results only guarantee that tau(X) = 0, when f is an element of C-infinity

Embedding of global attractors and their dynamics

Suppose that is the global attractor associated with a dissipative dynamical system on a Hilbert space . If the set has finite Assouad dimension , then for any there are linear homeomorphisms such that is a cellular subset of and is log-Lipschitz (i.e. Lipschitz to within logarithmic corrections). We give a relatively simple proof that a compact subset of is the global attractor of some smooth ordinary differential equation on if and only if it is cellular, and hence we obtain a dynamical system on for which is the global attractor. However, consists entirely of stationary points. In order for the dynamics on to reproduce those on we need to make an additional assumption, namely that the dynamics restricted to are generated by a log-Lipschitz continuous vector field (this appears overly restrictive when is infinite-dimensional, but is clearly satisfied when the initial dynamical system is generated by a Lipschitz ordinary differential equation on ). Given this we can construct an ordinary differential equation in some (where is determined by and ) that has unique solutions and reproduces the dynamics on . Moreover, the dynamical system generated by this new ordinary differential equation has a global attractor arbitrarily close to

Proceedings Of The 23Rd Paediatric Rheumatology European Society Congress: Part Two

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