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New bounds are proven on the mean vertical convective heat transport, ⟨wT⟩¯¯¯¯¯¯¯¯¯¯¯, for uniform internally heated (IH) convection in the limit of infinite Prandtl number. For fluid in a horizontally-periodic layer between isothermal boundaries, we show that ⟨wT⟩¯¯¯¯¯¯¯¯¯¯¯≤12−cR−2, where R is a nondimensional `flux' Rayleigh number quantifying the strength of internal heating and c=216. Then, ⟨wT⟩¯¯¯¯¯¯¯¯¯¯¯=0 corresponds to vertical heat transport by conduction alone, while ⟨wT⟩¯¯¯¯¯¯¯¯¯¯¯>0 represents the enhancement of vertical heat transport upwards due to convective motion. If, instead, the lower boundary is a thermal insulator, then we obtain ⟨wT⟩¯¯¯¯¯¯¯¯¯¯¯≤12−cR−4, with c≈0.0107. This result implies that the Nusselt number Nu, defined as the ratio of the total-to-conductive heat transport, satisfies Nu≲R4. Both bounds are obtained by combining the background method with a minimum principle for the fluid's temperature and with Hardy--Rellich inequalities to exploit the link between the vertical velocity and temperature. In both cases, power-law dependence on R improves the previously best-known bounds, which, although valid at both infinite and finite Prandtl numbers, approach the uniform bound exponentially with R.

The background method is a widely used technique to bound mean properties of turbulent flows rigorously. This work reviews recent advances in the theoretical formulation and numerical implementation of the method. First, we describe how the background method can be formulated systematically within a broader "auxiliary function" framework for bounding mean quantities, and explain how symmetries of the flow and constraints such as maximum principles can be exploited. All ideas are presented in a general setting and are illustrated on Rayleigh-Bénard convection between stress-free isothermal plates. Second, we review a semidefinite programming approach and a timestepping approach to optimizing bounds computationally, revealing that they are related to each other through convex duality and low-rank matrix factorization. Open questions and promising directions for further numerical analysis of the background method are also outlined.

We prove decomposition theorems for sparse positive (semi)definite polynomial matrices that can be viewed as sparsity-exploiting versions of the Hilbert--Artin, Reznick, Putinar, and Putinar--Vasilescu Positivstellensätze. First, we establish that a polynomial matrix $P(x)$ with chordal sparsity is positive semidefinite for all $x\in \mathbb{R}^n$ if and only if there exists a sum-of-squares (SOS) polynomial $\sigma(x)$ such that $\sigma P$ is a sum of sparse SOS matrices. Second, we show that setting $\sigma(x)=(x_1^2 + \cdots + x_n^2)^\nu$ for some integer $\nu$ suffices if $P$ is homogeneous and positive definite globally. Third, we prove that if $P$ is positive definite on a compact semialgebraic set $\mathcal{K}=\{x:g_1(x)\geq 0,\ldots,g_m(x)\geq 0\}$ satisfying the Archimedean condition, then $P(x) = S_0(x) + g_1(x)S_1(x) + \cdots + g_m(x)S_m(x)$ for matrices $S_i(x)$ that are sums of sparse SOS matrices. Finally, if $\mathcal{K}$ is not compact or does not satisfy the Archimedean condition, we obtain a similar decomposition for $(x_1^2 + \cdots + x_n^2)^\nu P(x)$ with some integer $\nu\geq 0$ when $P$ and $g_1,\ldots,g_m$ are homogeneous of even degree. Using these results, we find sparse SOS representation theorems for polynomials that are quadratic and correlatively sparse in a subset of variables, and we construct new convergent hierarchies of sparsity-exploiting SOS reformulations for convex optimization problems with large and sparse polynomial matrix inequalities. Numerical examples demonstrate that these hierarchies can have a significantly lower computational complexity than traditional ones.

We present a method for finding lower bounds on the global infima of integral variational problems, wherein $\int_\Omega f(x,u(x),\nabla u(x)){\rm d}x$ is minimized over functions $u\colon\Omega\subset\mathbb{R}^n\to\mathbb{R}^m$ satisfying given equality or inequality constraints. Each constraint may be imposed over $\Omega$ or its boundary, either pointwise or in an integral sense. These global minimizations are generally non-convex and intractable. We formulate a particular convex maximization, here called the pointwise dual relaxation (PDR), whose supremum is a lower bound on the infimum of the original problem. The PDR can be derived by dualizing and relaxing the original problem; its constraints are pointwise equalities or inequalities over finite-dimensional sets, rather than over infinite-dimensional function spaces. When the original minimization can be specified by polynomial functions of $(x,u,\nabla u)$, the PDR can be further relaxed by replacing pointwise inequalities with polynomial sum-of-squares (SOS) conditions. The resulting SOS program is computationally tractable when the dimensions $m,n$ and number of constraints are not too large. The framework presented here generalizes an approach of Valmorbida, Ahmadi, and Papachristodoulou (IEEE Trans. Automat. Contr., 61:1649--1654, 2016). We prove that the optimal lower bound given by the PDR is sharp for several classes of problems, whose special cases include leading eigenvalues of Sturm-Liouville problems and optimal constants of Poincar\'e inequalities. For these same classes, we prove that SOS relaxations of the PDR converge to the sharp lower bound as polynomial degrees are increased. Convergence of SOS computations in practice is illustrated for several examples.

We prove a new rigorous bound for the mean vertical heat transport ⟨wT⟩ in internally heated convection with an insulating lower boundary and an upper boundary with fixed heat flux that balances the total internal heating. An analytical application of the background method, based on the construction of a quadratic auxiliary function, yields ⟨wT⟩≤12(12+13√)−1.6899R−1/3, where R is the Rayleigh number. Numerical optimisation of the auxiliary function suggests that the asymptotic value of this bound and the −1/3 exponent are optimal within our bounding framework. This new result halves the best existing (uniform) bound and the Rayleigh number dependence is consistent with previous conjectures and heuristic scaling arguments.

We establish rigorous upper bounds on the time-averaged heat transport for a model of rotating Rayleigh-Benard convection between no-slip boundaries at infinite Prandtl number and with Ekman pumping. The analysis is based on the asymptotically reduced equations derived for rotationally constrained dynamics with no-slip boundaries, and hence includes a lower order correction that accounts for the Ekman layer and corresponding Ekman pumping into the bulk. Using the auxiliary functional method we find that, to leading order, the temporally averaged heat transport is bounded above as a function of the Rayleigh and Ekman numbers Ra and Ek according to $Nu \leq 0.3704 Ra^2 Ek^2$. Dependent on the relative values of the thermal forcing represented by $Ra$ and the effects of rotation represented by $Ek$, this bound is both an improvement on earlier rigorous upper bounds, and provides a partial explanation of recent numerical and experimental results that were consistent yet surprising relative to the previously derived upper bound of $Nu \lesssim Ra^3 k^4$.

We prove weak duality between two recent convex relaxation methods for bounding the optimal value of a constrained variational problem in which the objective is an integral functional. The first approach, proposed by Valmorbida et al. (IEEE Trans. Automat. Control 61(6):1649--1654, 2016), replaces the variational problem with a convex program over sufficiently smooth functions, subject to pointwise non-negativity constraints. The second approach, discussed by Korda et al. (arXiv:1804.07565v1 [math.OC]), relaxes the variational problem into a convex program over scaled probability measures. We also prove that the duality between these infinite-dimensional convex programs is strong, meaning that their optimal values coincide, when the range and gradients of admissible functions in the variational problem are constrained to bounded sets. For variational problems with polynomial data, the optimal values of each convex relaxation can be approximated by solving weakly dual hierarchies of finite-dimensional semidefinite programs (SDPs). These are strongly dual under standard constraint qualification conditions irrespective of whether strong duality holds at the infinite-dimensional level. Thus, the two relaxation approaches are equivalent for the purposes of computations.

RESUMEN Antecedentes: En nuestro país la prevalencia de diabetes tipo 2 (DT2) y de factores de riesgo cardiovascular (FRCV) aumenta continuamente. Aunque el fenómeno se acompaña de adopción de estilos de vida no saludable que facilitan dicho crecimiento, es escasa la implementación de estrategias que puedan modificar la situación. Objetivo: Revisar la evidencia disponible sobre la magnitud del problema de la diabetes y los FRCV en nuestro país, su posible relación con la práctica de actividad física y potencial mecanismo de acción. Metodología: Evaluación de datos de la tercera Encuesta Nacional de Factores de Riesgo (ENFR) e información referida a factores que contribuyen al crecimiento de la prevalencia de DT2. Igualmente estrategias exitosas utilizadas a nivel mundial para su prevención. Resultados: El índice de masa corporal registrado en la población estudiada muestra un aumento del porcentaje de personas con sobrepeso/obesidad inverso a la práctica de actividad física. Igualmente los resultados de las pruebas de tolerancia a la glucosa oral muestran que sus alteraciones (prediabetes/diabetes) son menores entre quienes realizaban actividad física. El porcentaje de personas con valores de presión arterial dentro del rango normal al igual que de colesterol circulante (según valores meta de guías internacionales), es también significativamente menor entre quienes practicaban actividad física. Conclusión: La evidencia presentada demuestra objetivamente la necesidad/ventajas de implementar un programa de prevención primaria de diabetes a gran escala a nivel nacional para disminuir su crecimiento y la pertinencia de incluir la práctica de actividad física como estrategia de prevención tal como propone el PPDBA. ABSTRACT Background: In our country, the prevalence of type 2 diabetes (DT2) and cardiovascular risk factors (CVRF) increases continuously. Although the phenomenon is accompanied by the adoption of unhealthy lifestyles that facilitate such growth, there is little implementation of strategies that can modify the situation. Objective: To review the available evidence on the magnitude of the problem of diabetes and CVRF in our country, its possible relationship with the practice of physical activity and potential mechanism of action. Methodology: Evaluation of data from the Third National Survey of Risk Factors (ENFR) and information referred to factors that promote the prevalence growth of T2D. Additionally, successful strategies have been used worldwide for its prevention. Results: The body mass index registered in the studied population shows an increase in the percentage of people with overweight/obesity inverse to the practice of physical activity. Likewise, the results of the oral glucose tolerance tests show that their alterations (prediabetes/ diabetes) are lower among those who performed physical activity. The percentage of people with blood pressure values within the normal range as well as circulating cholesterol (according to target values of international guidelines), is also significantly lower among those who practiced physical activity. Conclusion: The presented evidence objectively demonstrates the need/advantages of implementing a large-scale diabetes primary prevention program at the national level to.

We prove the first rigorous bound on the heat transfer for three-dimensional Rayleigh-Bénard convection of finite-Prandtl-number fluids between free-slip boundaries with an imposed heat flux. Using the auxiliary functional method with a quadratic functional, which is equivalent to the background method, we prove that the Nusselt number $Nu$ is bounded by $Nu \leq 0.5999 R^{1/3}$ uniformly in the Prandtl number, where $R$ is the Rayleigh number based on the imposed heat flux. In terms of the Rayleigh number based on the mean vertical temperature drop, $Ra$, we obtain $Nu \leq 0.4646 Ra^{1/2}$. The scaling with Rayleigh number is the same as that of bounds obtained with no-slip isothermal, free-slip isothermal, and no-slip fixed flux boundaries, and numerical optimisation of the bound suggests that it cannot be improved within our bounding framework. Contrary to the two-dimensional case, therefore, the $Ra$-dependence of rigorous upper bounds on the heat transfer obtained with the background method for three-dimensional Rayleigh-Bénard convection is insensitive to both the thermal and the velocity boundary conditions.

The vertical heat transfer in Bénard–Marangoni convection of a fluid layer with infinite Prandtl number is studied by means of upper bounds on the Nusselt number 𝑁𝑢 as a function of the Marangoni number 𝑀𝑎 . Using the background method for the temperature field, it has recently been proved by Hagstrom & Doering (Phys. Rev. E, vol. 81, 2010, art. 047301) that 𝑁𝑢⩽0.838𝑀𝑎2/7 . In this work we extend previous background method analysis to include balance parameters and derive a variational principle for the bound on 𝑁𝑢 , expressed in terms of a scaled background field, that yields a better bound than Hagstrom & Doering’s formulation at a given 𝑀𝑎 . Using a piecewise-linear, monotonically decreasing profile we then show that 𝑁𝑢⩽0.803𝑀𝑎2/7 , lowering the previous prefactor by 4.2 %. However, we also demonstrate that optimisation of the balance parameters does not affect the asymptotic scaling of the optimal bound achievable with Hagstrom & Doering’s original formulation. We subsequently utilise convex optimisation to optimise the bound on 𝑁𝑢 over all admissible background fields, as well as over two smaller families of profiles constrained by monotonicity and convexity. The results show that 𝑁𝑢⩽𝑂(𝑀𝑎2/7(ln𝑀𝑎)−1/2) when the background field has a non-monotonic boundary layer near the surface, while a power-law bound with exponent 2/7 is optimal within the class of monotonic background fields. Further analysis of our upper-bounding principle reveals the role of non-monotonicity, and how it may be exploited in a rigorous mathematical argument.

We propose an efficient first-order method, based on the alternating direction method of multipliers (ADMM), to solve the homogeneous self-dual embedding problem for a primal-dual pair of semidefinite programs (SDPs) with chordal sparsity. Using a series of block eliminations, the per-iteration cost of our method is the same as applying a splitting method to the primal or dual alone. Moreover, our approach is more efficient than other first-order methods for generic sparse conic programs since we work with smaller semidefinite cones. In contrast to previous first-order methods that exploit chordal sparsity, our algorithm returns both primal and dual solutions when available, and it provides a certificate of infeasibility otherwise. Our techniques are implemented in the open-source MATLAB solver CDCS. Numerical experiments on three sets of benchmark problems from the library SDPLIB show speed-ups compared to some common state-of-the-art software packages.

The Erice Charter was unanimously approved at the conclusion of the 47th Residential Course "Adapted Physical Activity in Sport, Wellness and Fitness: New Challenges for Prevention and Health Promotion", held on 20-24 April 2015 in Erice, Italy, at the "Ettore Majorana" Foundation and Centre for Scientific Culture, and promoted by the International School of Epidemiology and Preventive Medicine "G. D'Alessandro" and the Study Group on Movement Sciences for Health of the Italian Society of Hygiene, Preventive Medicine and Public Health. After an intense discussion the participants identified the main points associated with the relevance of physical activity for Public Health, claiming the pivotal role of the Department of Prevention in coordinating and managing preventive actions. The participants underlined the importance of the physicians specialized in Hygiene, Preventive Medicine and Public Health. The contribution of other operators such as physicians specialized in Sport Medicine was stressed. Further, the holders of the new degree in Human Movement and Sport Sciences were considered fundamental contributors for the performance of physical activity and their presence was seen as a promising opportunity for the Departments of Prevention. Primary prevention based on recreational physical activities should become easily accessible for the population, avoiding obstacles such as certification steps or complex bureaucracy. The Sport Doctor is recognized as the principal referent for preliminary physical evaluation and clinical monitoring in secondary and tertiary prevention actions based on adapted physical activities. Developing research in the field is essential as well as implementing higher education on physical activity management in Schools of Public Health.

Obiettivi: evidenziare limiti e punti di forza dell’insegnamento di Educazione Fisica (EF) nell’attuale sistema scolastico italiano sulla base delle opinioni degli studenti. Metodi: ad un campione di matricole provenienti da 14 università italiane è stato somministrato un questionario anonimo da autocompilare riguardante esperienze e giudizi relativi all’insegnamento di EF ricevuto nell’ultimo biennio di scuola. Sono state anche analizzate eventuali differenze tra Nord, Centro e Sud. Risultati: su 7.033 partecipanti, il 42,2% riportava come obiettivo principale perseguito durante le lezioni di EF quello “ricreativo”. Più della metà degli studenti (62,2%) ha riferito nessuna o modesta influenza delle lezioni di EF sulla pratica corrente di Attività Fisica (AF). Il 67,2% del campione si ritiene soddisfatto degli insegnamenti di educazione fisica ricevuti, soprattutto nelle regioni del Nord. Discussione e conclusioni: l’analisi condotta ha evidenziato la necessità di migliorare l’offerta di tale insegnamento nella scuola italiana, sia incrementando il tempo dedicato alle lezioni, sia impiegando insegnanti adeguatamente preparati.

We introduce a new technique to optimize a linear cost function subject to a one-dimensional affine homogeneous quadratic integral inequality, i.e., the requirement that a homogeneous quadratic integral functional, affine in the optimization variables, is non-negative over a space of functions defined by homogeneous boundary conditions. Such problems arise in stability analysis, input-to-state/output analysis, and control of many systems governed by partial differential equations (PDEs), in particular fluid dynamical systems. First, we derive outer approximations for the feasible set of a homogeneous quadratic integral inequality in terms of linear matrix inequalities (LMIs), and show that under mild assumptions a convergent, non-decreasing sequence of lower bounds for the optimal cost can be computed with a sequence of semidefinite programs (SDPs). Second, we obtain inner approximations in terms of LMIs and sum-of-squares constraints, so upper bounds for the optimal cost and strictly feasible points for the integral inequality can also be computed with SDPs. To aid the formulation and solution of our SDP relaxations, we implement our techniques in QUINOPT, an open-source add-on to YALMIP. We demonstrate our techniques by solving problems arising from the stability analysis of PDEs., 31 pages, 3 figures. Codes available from https://github.com/aeroimperial-optimization/QUINOPT/. Latest version: improved exposition, fixed proof of Theorem 1, added comparison to other method, added discussion on scalability, corrected typos

L’edizione italiana di questo volume rappresenta uno strumento molto utile per chi si interessa degli aspetti sanitari collegati agli ambienti acquatici nei quali si svolge attività ricreativa: viene quindi messo a disposizione del lettore il testo di riferimento più autorevole sugli aspetti igienico-sanitari propri della frequentazione di piscine e di ambienti simili. I contenuti derivano dalla revisione completa della letteratura scientifica sull’argomento, alla quale abbiamo contribuito con numerosi studi svolti sul nostro territorio nazionale, e permettono utili confronti tra diverse realtà a livello internazionale. È comunque da rilevare come si tratti di un documento senza alcuna valenza di tipo normativo o impositivo: vengono presentate infatti indicazioni, vere e proprie “linee-guida”, che possono essere adattate alle diverse realtà locali, con l’intento di contribuire alla salute pubblica in una prospettiva di autocontrollo che prevede una stretta collaborazione tra l’autorità sanitaria, i gestori e gli utenti. Ed è grazie alla coerenza tra la natura propositiva di queste linee-guida e l’interesse alla promozione della salute pubblica da parte di un gruppo di studiosi della SItI che ne è stata predisposta la traduzione italiana: il gruppo che ha svolto il lavoro è costituito da docenti universitari afferenti all’area dello sport (Scienze Motorie e Sportive), spesso molto vicini nelle proprie sedi regionali sia al mondo delle Federazioni sia a quello delle Aziende Unità Sanitarie Locali, e dai loro studenti, appassionati utilizzatori delle piscine. Queste linee-guida hanno quindi un primo obiettivo: quello di fornire un riferimento di base, già condiviso da esperti di molti paesi, sulle caratteristiche di qualità degli ambienti acquatici ad uso ricreativo, che possono variare in funzione dei diversi tipi di struttura. Quanto riportato è stato tratto dai numerosi studi scientifici che hanno riguardato, tra gli altri temi, la valutazione dei rischi per la salute e l’efficacia delle misure di prevenzione e di controllo, unitamente ad una descrizione completa delle possibilità di monitoraggio e di valutazione di salubrità in un’ottica di tutela della salute pubblica. L’adozione di queste linee-guida fornirà quindi un suggerimento per il metodo di lavoro che può essere adottato dai gestori e dai controllori locali, oltre che rappresentare un riferimento autorevole per i progettisti di impianti di piscina e per le industrie che operano nel settore.