Exercise sheet 2

Exercises and solutions in LaTex

Calorimetría indirecta versus Harris-Benedict para determinar gasto energético basal en pacientes ventilados

El inadecuado aporte nutricional en los pacientes con enfermedades criticas, ha llevado al desarrollo de complicaciones que incrementan la mortalidad y los costos de la atención en salud. Muchos factores están involucrados en el consumo de los nutrientes por el organismo, como: los traslados, las intervenciones quirúrgicas, el uso de vasopresores, la ventilación mecánica, entre otros. Si se presenta imprecisión en la de-terminación del gasto energético, puede conllevar a un sobre aporte alimenticio en el paciente el cual puede afectar la evolución y pronosti-co del individuo. Es de conocimiento universal que los costos en salud cada día se in-crementan, en especial cuando se presenta requerimiento del manejo de un paciente en la unidad de cuidados intensivos. Para predecir el gasto energético basal de los paciente en las UCI se cuentan con herramientas de evaluación sencillas, de fácil uso y económicas, como la ecuación de Harris-Benedict, o herramientas complejas y de difícil manejo como la Calorimetría Indirecta. El incremento en la demanda de servicios con mayor tecnología en el tratamiento de los pacientes, enfrenta al personal de salud para ser más crítico en el uso de la nueva tecnología, por tal motivo, se evaluó la presencia de correlación entre las ecuaciones de Harris-Benedict y Calorimetría Indirecta, encontrando que se presenta una buena correlación entre las ecuaciones, con un valor de Pearson de 0,700 y una p = 0.002. Por lo que se puede concluir que las ecuaciones pueden ser utilizadas para estimar el gasto energético basal de los pacientes en la UCI.

Alteraciones respiratorias durante el sueño en niños con Síndrome Down a la altura de Bogotá

Objetivo: determinar la frecuencia de las diferentes alteraciones respiratorias durante el sueño (ARS) e hipertensión pulmonar (HTP) y establecer la saturación de oxígeno (SpO2) en vigilia, sueño y durante los eventos en niños con Síndrome Down (SD) a la altura de Bogotá (2640m) de acuerdo a grupos de edad e IMC. Métodos: estudio descriptivo de corte transversal, se incluyeron todos los niños con SD con sospecha de ARS remitidos a polisonograma (PSG) de octubre de 2011 a enero de 2013 a la Fundación Neumológica Colombina (FNC). Se dividieron en 3 grupos: apnea obstructiva, apnea obstructiva y central, sin apneas. Resultados: 74 niños, el 36,5% mujeres, edad media 4 años. 47,3% presento apnea obstructiva, más frecuente en >2 años, 35,1% apnea obstructiva y central, más frecuente en < 2 años y 17,6 % sin apnea. SpO2 promedio en apnea obstructiva 84,63%, apnea obstructiva y central: 81,8% y sin apnea: 86,85% (p 0,058). 23% presento obesidad, 16% con apnea obstructiva. 53 pacientes tenían ecocardiograma: 28% HTP, 53,3% tuvo apnea obstructiva y 26,7 apnea obstructiva y central, no diferencias significativas. SpO2 promedio en HTP 88,3% vigilia, 86,2% sueño REM, 85,7 % sueño no REM, no diferencia significativa comparada con pacientes sin HTP. Conclusiones: Las ARS son frecuentes en los niños con SD, La desaturación está presente en los niños con SD independiente del tipo de apnea. A todos los niños SD se les debe practicar un PSG en el primer año de vida.

Ecuaciones diferenciales estocásticas con condición final y soluciones de viscosidad de EDPS semilineales de segundo orden

El objetivo de este documento es recopilar algunos resultados clasicos sobre existencia y unicidad ´ de soluciones de ecuaciones diferenciales estocasticas (EDEs) con condici ´ on final (en ingl ´ es´ Backward stochastic differential equations) con particular enfasis en el caso de coeficientes mon ´ otonos, y su cone- ´ xion con soluciones de viscosidad de sistemas de ecuaciones diferenciales parciales (EDPs) parab ´ olicas ´ y el´ıpticas semilineales de segundo orden.

Crop yield reduction in the tropics under climate change: Processes and uncertainties

Many modelling studies examine the impacts of climate change on crop yield, but few explore either the underlying bio-physical processes, or the uncertainty inherent in the parameterisation of crop growth and development. We used a perturbed-parameter crop modelling method together with a regional climate model (PRECIS) driven by the 2071-2100 SRES A2 emissions scenario in order to examine processes and uncertainties in yield simulation. Crop simulations used the groundnut (i.e. peanut; Arachis hypogaea L.) version of the General Large-Area Model for annual crops (GLAM). Two sets of GLAM simulations were carried out: control simulations and fixed-duration simulations, where the impact of mean temperature on crop development rate was removed. Model results were compared to sensitivity tests using two other crop models of differing levels of complexity: CROPGRO, and the groundnut model of Hammer et al. [Hammer, G.L., Sinclair, T.R., Boote, K.J., Wright, G.C., Meinke, H., and Bell, M.J., 1995, A peanut simulation model: I. Model development and testing. Agron. J. 87, 1085-1093]. GLAM simulations were particularly sensitive to two processes. First, elevated vapour pressure deficit (VPD) consistently reduced yield. The same result was seen in some simulations using both other crop models. Second, GLAM crop duration was longer, and yield greater, when the optimal temperature for the rate of development was exceeded. Yield increases were also seen in one other crop model. Overall, the models differed in their response to super-optimal temperatures, and that difference increased with mean temperature; percentage changes in yield between current and future climates were as diverse as -50% and over +30% for the same input data. The first process has been observed in many crop experiments, whilst the second has not. Thus, we conclude that there is a need for: (i) more process-based modelling studies of the impact of VPD on assimilation, and (ii) more experimental studies at super-optimal temperatures. Using the GLAM results, central values and uncertainty ranges were projected for mean 2071-2100 crop yields in India. In the fixed-duration simulations, ensemble mean yields mostly rose by 10-30%. The full ensemble range was greater than this mean change (20-60% over most of India). In the control simulations, yield stimulation by elevated CO2 was more than offset by other processes-principally accelerated crop development rates at elevated, but sub-optimal, mean temperatures. Hence, the quantification of uncertainty can facilitate relatively robust indications of the likely sign of crop yield changes in future climates. (C) 2007 Elsevier B.V. All rights reserved.

On the energetics of stratified turbulent mixing, irreversible thermodynamics, Boussinesq models and the ocean heat engine controversy

In this paper, the available potential energy (APE) framework of Winters et al. (J. Fluid Mech., vol. 289, 1995, p. 115) is extended to the fully compressible Navier– Stokes equations, with the aims of clarifying (i) the nature of the energy conversions taking place in turbulent thermally stratified fluids; and (ii) the role of surface buoyancy fluxes in the Munk & Wunsch (Deep-Sea Res., vol. 45, 1998, p. 1977) constraint on the mechanical energy sources of stirring required to maintain diapycnal mixing in the oceans. The new framework reveals that the observed turbulent rate of increase in the background gravitational potential energy GPEr , commonly thought to occur at the expense of the diffusively dissipated APE, actually occurs at the expense of internal energy, as in the laminar case. The APE dissipated by molecular diffusion, on the other hand, is found to be converted into internal energy (IE), similar to the viscously dissipated kinetic energy KE. Turbulent stirring, therefore, does not introduce a new APE/GPEr mechanical-to-mechanical energy conversion, but simply enhances the existing IE/GPEr conversion rate, in addition to enhancing the viscous dissipation and the entropy production rates. This, in turn, implies that molecular diffusion contributes to the dissipation of the available mechanical energy ME =APE +KE, along with viscous dissipation. This result has important implications for the interpretation of the concepts of mixing efficiency γmixing and flux Richardson number Rf , for which new physically based definitions are proposed and contrasted with previous definitions. The new framework allows for a more rigorous and general re-derivation from the first principles of Munk & Wunsch (1998, hereafter MW98)’s constraint, also valid for a non-Boussinesq ocean: G(KE) ≈ 1 − ξ Rf ξ Rf Wr, forcing = 1 + (1 − ξ )γmixing ξ γmixing Wr, forcing , where G(KE) is the work rate done by the mechanical forcing, Wr, forcing is the rate of loss of GPEr due to high-latitude cooling and ξ is a nonlinearity parameter such that ξ =1 for a linear equation of state (as considered by MW98), but ξ <1 otherwise. The most important result is that G(APE), the work rate done by the surface buoyancy fluxes, must be numerically as large as Wr, forcing and, therefore, as important as the mechanical forcing in stirring and driving the oceans. As a consequence, the overall mixing efficiency of the oceans is likely to be larger than the value γmixing =0.2 presently used, thereby possibly eliminating the apparent shortfall in mechanical stirring energy that results from using γmixing =0.2 in the above formula.

Alternative methodologies - stretching the in vitro box

Current gas-based in vitro evaluation systems are extremely powerful research techniques. However they have the potential to generate a great deal more than simple fermentation dynamics. Details from four experiments are presented in which adaptation, and novel application, of an in vitro system allowed widely differing objectives to be examined. In the first two studies, complement methodologies were utilised. In such assays, an activity or outcome is inferred through the occurrence of a secondary event rather than by direct observation. Using an N-deficient incubation medium, the increase in starch fermentation, when supplemented with individual amino acids (i.e., known level of N) relative to that of urea (i.e., known quantity and N availability), provided an estimate of their microbial utilisation. Due to the low level of response observed with some arnino acids (notably methionine and lysine), it was concluded, that they may not need to be offered in a rumen-inert form to escape rumen microbial degradation. In another experiment, the extent to which degradation of plant cell wall components was inhibited by lipid supplementation was evaluated using fermentation gas release profiles of washed hay. The different responses due to lipid source and level of inclusion suggested that the degree of rumen protection required to ameliorate this depression was supplement dependent. That in vitro inocula differ in their microbial composition is of little interest per se, as long as the outcome is the same (i.e., that similar substrates are degraded at comparable rates and end-product release is equivalent). However where a microbial population is deficient in a particular activity, increasing the level of inoculation will have no benefit. Estimates of hydrolytic activity were obtained by examining fermentation kinetics of specific substrates. A number of studies identified a fundamental difference between rumen fluid and faecal inocula, with the latter having a lower fibrolytic activity, which could not be completely attributed to microbial numbers. The majority of forage maize is offered as an ensiled feed, however most of the information on which decisions such as choice of variety, crop management and harvesting date are made is based on fresh crop measurements. As such, an attempt was made to estimate ensiled maize quality from an in vitro analysis of the fresh crop. Fermentation profiles and chemical analysis confirmed changes in crop composition over the growing season, and loss of labile carbohydrates during ensiling. In addition, examination of degradation residues allowed metabolizable energy (ME) contents to be estimated. Due to difficulties associated with starch analysis, the observation that this parameter could be predicted by difference (together with an assumed degradability), allowed an estimate of ensiled maize ME to be developed from fresh material. In addition, the contribution of the main carbohydrates towards ME showed the importance of delaying harvest until maximum starch content has been achieved. (c) 2005 Elsevier B.V. All rights reserved.

A Riemann solver for compressible flow of a real gas in a Lagrangian frame

An approximate Riemann solver is presented for the compressible flow equations with a general (convex) equation of state in a Lagrangian frame of reference.

Cylindrically symmetric flows in a Lagrangian frame

An approximate Riemann solver, in a Lagrangian frame of reference, is presented for the compressible flow equations with cylindrical and spherical symmetry, including flow in a duct of variable cross section. The scheme is applied to a cylindrically symmetric problem involving the interaction of shocks.

Retrieval of global atmospheric electrical activity at a polluted urban site

The atmospheric electrical Potential Gradient (PG) arises from global thunderstorm activity, but surface measurements of the atmospheric Potential Gradient (PG) are influenced by global thunderstorms and local aerosol concentration changes. The local aerosol change can be monitored independently, and in some cases the concentration changes are closely related to PG changes. For these circumstances, a general theory to remove the local aerosol influence on PG measurements has been developed. Continuous measurements of PG and aerosol mass concentration were made during 24–31 Dec, 2005 within an urban environment at Reading, UK. The average diurnal variation of PG showed a double diurnal cycle, with maxima in the early morning and evening hours. The aerosol concentration has similar double maxima. Removing the aerosol using from the PG and aerosol correlation returns a single diurnal cycle, suggestive of the more global PG diurnal cycle.

Generalised Dirichelt-to-Neumann map in time dependent domains

We study the heat, linear Schrodinger and linear KdV equations in the domain l(t) < x < ∞, 0 < t < T, with prescribed initial and boundary conditions and with l(t) a given differentiable function. For the first two equations, we show that the unknown Neumann or Dirichlet boundary value can be computed as the solution of a linear Volterra integral equation with an explicit weakly singular kernel. This integral equation can be derived from the formal Fourier integral representation of the solution. For the linear KdV equation we show that the two unknown boundary values can be computed as the solution of a system of linear Volterra integral equations with explicit weakly singular kernels. The derivation in this case makes crucial use of analyticity and certain invariance properties in the complex spectral plane. The above Volterra equations are shown to admit a unique solution.

Drag produced by trapped lee waves and propagating mountain waves in a two-layer atmosphere

The surface drag force produced by trapped lee waves and upward propagating waves in non-hydrostatic stratified flow over a mountain ridge is explicitly calculated using linear theory for a two-layer atmosphere with piecewise-constant static stability and wind speed profiles. The behaviour of the drag normalized by its hydrostatic single-layer reference value is investigated as a function of the ratio of the Scorer parameters in the two layers l_2/l_1 and of the corresponding dimensionless interface height l_1 H, for selected values of the dimensionless ridge width l_1 a and ratio of wind speeds in the two layers. When l_2/l_1 → 1, the propagating wave drag approaches 1 in approximately hydrostatic conditions, and the trapped lee wave drag vanishes. As l_2/l_1 decreases, the propagating wave drag progressively displays an oscillatory behaviour with l_1 H, with maxima of increasing magnitude due to constructive interference of reflected waves in the lower layer. The trapped lee wave drag shows localized maxima associated with each resonant trapped lee wave mode, occurring for small l_2/l_1 and slightly higher values of l_1 H than the propagating wave drag maxima. As l1a decreases, i.e. the flow becomes more non-hydrostatic, the propagating wave drag decreases and the regions of non-zero trapped lee wave drag extend to higher l_2/l_1. These results are confirmed by numerical simulations for l_2/l_1 = 0.2. In parameter ranges of meteorological relevance, the trapped lee wave drag may have a magnitude comparable to that of propagating wave drag, and be larger than the reference single-layer drag. This may have implications for drag parametrization in global climate and weather-prediction models.

Limitations of small x resummation methods from F2data

We discuss several methods of calculating the DIS structure functions F2(x,Q2) based on BFKL-type small x resummations. Taking into account new HERA data ranging down to small xand low Q2, the pure leading order BFKL-based approach is excluded. Other methods based on high energy factorization are closer to conventional renormalization group equations. Despite several difficulties and ambiguities in combining the renormalization group equations with small x resummed terms, we find that a fit to the current data is hardly feasible, since the data in the low Q2 region are not as steep as the BFKL formalism predicts. Thus we conclude that deviations from the (successful) renormalization group approach towards summing up logarithms in 1/x are disfavoured by experiment.

Pulsating equilibria: stability through migration

This paper is to present a model of spatial equilibrium using a nonlinear generalization of Markov-chain type model, and to show the dynamic stability of a unique equilibrium. Even at an equilibrium, people continue to migrate among regions as well as among agent-types, and yet their overall distribution remain unchanged. The model is also adapted to suggest a theory of traffic distribution in a city.

A moving mesh approach for modelling avascular tumour growth

A key step in many numerical schemes for time-dependent partial differential equations with moving boundaries is to rescale the problem to a fixed numerical mesh. An alternative approach is to use a moving mesh that can be adapted to focus on specific features of the model. In this paper we present and discuss two different velocity-based moving mesh methods applied to a two-phase model of avascular tumour growth formulated by Breward et al. (2002) J. Math. Biol. 45(2), 125-152. Each method has one moving node which tracks the moving boundary. The first moving mesh method uses a mesh velocity proportional to the boundary velocity. The second moving mesh method uses local conservation of volume fraction of cells (masses). Our results demonstrate that these moving mesh methods produce accurate results, offering higher resolution where desired whilst preserving the balance of fluxes and sources in the governing equations.