Integration of chaos theory and mathematical models in building simulation: Part I: Literature review

Current mathematical models in building research have been limited in most studies to linear dynamics systems. A literature review of past studies investigating chaos theory approaches in building simulation models suggests that as a basis chaos model is valid and can handle the increasingly complexity of building systems that have dynamic interactions among all the distributed and hierarchical systems on the one hand, and the environment and occupants on the other. The review also identifies the paucity of literature and the need for a suitable methodology of linking chaos theory to mathematical models in building design and management studies. This study is broadly divided into two parts and presented in two companion papers. Part (I) reviews the current state of the chaos theory models as a starting point for establishing theories that can be effectively applied to building simulation models. Part (II) develops conceptual frameworks that approach current model methodologies from the theoretical perspective provided by chaos theory, with a focus on the key concepts and their potential to help to better understand the nonlinear dynamic nature of built environment systems. Case studies are also presented which demonstrate the potential usefulness of chaos theory driven models in a wide variety of leading areas of building research. This study distills the fundamental properties and the most relevant characteristics of chaos theory essential to building simulation scientists, initiates a dialogue and builds bridges between scientists and engineers, and stimulates future research about a wide range of issues on building environmental systems.

Integration of chaos theory and mathematical models in building simulation: Part II: Conceptual frameworks

Current mathematical models in building research have been limited in most studies to linear dynamics systems. A literature review of past studies investigating chaos theory approaches in building simulation models suggests that as a basis chaos model is valid and can handle the increasing complexity of building systems that have dynamic interactions among all the distributed and hierarchical systems on the one hand, and the environment and occupants on the other. The review also identifies the paucity of literature and the need for a suitable methodology of linking chaos theory to mathematical models in building design and management studies. This study is broadly divided into two parts and presented in two companion papers. Part (I), published in the previous issue, reviews the current state of the chaos theory models as a starting point for establishing theories that can be effectively applied to building simulation models. Part (II) develop conceptual frameworks that approach current model methodologies from the theoretical perspective provided by chaos theory, with a focus on the key concepts and their potential to help to better understand the nonlinear dynamic nature of built environment systems. Case studies are also presented which demonstrate the potential usefulness of chaos theory driven models in a wide variety of leading areas of building research. This study distills the fundamental properties and the most relevant characteristics of chaos theory essential to (1) building simulation scientists and designers (2) initiating a dialogue between scientists and engineers, and (3) stimulating future research on a wide range of issues involved in designing and managing building environmental systems.

Sensitivity analysis of a cyanobacterial growth and movement model under two different flow regimes

Bloom-forming and toxin-producing cyanobacteria remain a persistent nuisance across the world. Modelling cyanobacterial behaviour in freshwaters is an important tool for understanding their population dynamics and predicting the location and timing of the bloom events in lakes, reservoirs and rivers. A new deterministic–mathematical model was developed, which simulates the growth and movement of cyanobacterial blooms in river systems. The model focuses on the mathematical description of the bloom formation, vertical migration and lateral transport of colonies within river environments by taking into account the major factors that affect the cyanobacterial bloom formation in rivers including light, nutrients and temperature. A parameter sensitivity analysis using a one-at-a-time approach was carried out. There were two objectives of the sensitivity analysis presented in this paper: to identify the key parameters controlling the growth and movement patterns of cyanobacteria and to provide a means for model validation. The result of the analysis suggested that maximum growth rate and day length period were the most significant parameters in determining the population growth and colony depth, respectively.

A simple relationship between volcanic sulfate aerosol optical depth and surface temperature change simulated in an atmosphere-ocean general circulation model.

In this study we quantify the relationship between the aerosol optical depth increase from a volcanic eruption and the severity of the subsequent surface temperature decrease. This investigation is made by simulating 10 different sizes of eruption in a global circulation model (GCM) by changing stratospheric sulfate aerosol optical depth at each time step. The sizes of the simulated eruptions range from Pinatubo‐sized up to the magnitude of supervolcanic eruptions around 100 times the size of Pinatubo. From these simulations we find that there is a smooth monotonic relationship between the global mean maximum aerosol optical depth anomaly and the global mean temperature anomaly and we derive a simple mathematical expression which fits this relationship well. We also construct similar relationships between global mean aerosol optical depth and the temperature anomaly at every individual model grid box to produce global maps of best‐fit coefficients and fit residuals. These maps are used with caution to find the eruption size at which a local temperature anomaly is clearly distinct from the local natural variability and to approximate the temperature anomalies which the model may simulate following a Tambora‐sized eruption. To our knowledge, this is the first study which quantifies the relationship between aerosol optical depth and resulting temperature anomalies in a simple way, using the wealth of data that is available from GCM simulations.

A titration model: an analysis and solution

The Newton‐Raphson method is proposed for the solution of the nonlinear equation arising from a theoretical model of an acid/base titration. It is shown that it is necessary to modify the form of the equation in order that the iteration is guaranteed to converge. A particular example is considered to illustrate the analysis and method, and a BASIC program is included that can be used to predict the pH of any weak acid/weak base titration.

The DigiBog model of peatland development 1: rationale, conceptual model, and hydrological basis

Using a literature review, we argue that new models of peatland development are needed. Many existing models do not account for potentially important ecohydrological feedbacks, and/or ignore spatial structure and heterogeneity. Existing models, including those that simulate a near total loss of the northern peatland carbon store under a warming climate, may produce misleading results because they rely upon oversimplified representations of ecological and hydrological processes. In this, the first of a pair of papers, we present the conceptual framework for a model of peatland development, DigiBog, which considers peatlands as complex adaptive systems. DigiBog accounts for the interactions between the processes which govern litter production and peat decay, peat soil hydraulic properties, and peatland water-table behaviour, in a novel and genuinely ecohydrological manner. DigiBog consists of a number of interacting submodels, each representing a different aspect of peatland ecohydrology. Here we present in detail the mathematical and computational basis, as well as the implementation and testing, of the hydrological submodel. Remaining submodels are described and analysed in the accompanying paper. Tests of the hydrological submodel against analytical solutions for simple aquifers were highly successful: the greatest deviation between DigiBog and the analytical solutions was 2·83%. We also applied the hydrological submodel to irregularly shaped aquifers with heterogeneous hydraulic properties—situations for which no analytical solutions exist—and found the model's outputs to be plausible.

Determination of the degradation kinetics of anthocyanins in a model juice system using isothermal and non-isothermal methods

The effect of temperature on the degradation of blackcurrant anthocyanins in a model juice system was determined over a temperature range of 4–140 °C. The thermal degradation of anthocyanins followed pseudo first-order kinetics. From 4–100 °C an isothermal method was used to determine the kinetic parameters. In order to mimic the temperature profile in retort systems, a non-isothermal method was applied to determine the kinetic parameters in the model juice over the temperature range 110–140 °C. The results from both isothermal and non-isothermal methods fit well together, indicating that the non-isothermal procedure is a reliable mathematical method to determine the kinetics of anthocyanin degradation. The reaction rate constant (k) increased from 0.16 (±0.01) × 10−3 to 9.954 (±0.004) h−1 at 4 and 140 °C, respectively. The temperature dependence of the rate of anthocyanin degradation was modelled by an extension of the Arrhenius equation, which showed a linear increase in the activation energy with temperature.

Survival of Lactobacillus plantarum in model solutions and fruit juices

The aim of the work was to study the survival of Lactobacillus plantarum NCIMB 8826 in model solutions and develop a mathematical model describing its dependence on pH, citric acid and ascorbic acid. A Central Composite Design (CCD) was developed studying each of the three factors at five levels within the following ranges, i.e., pH (3.0-4.2), citric acid (6-40 g/L), and ascorbic acid (100-1000 mg/L). In total, 17 experimental runs were carried out. The initial cell concentration in the model solutions was approximately 1 × 10(8)CFU/mL; the solutions were stored at 4°C for 6 weeks. Analysis of variance (ANOVA) of the stepwise regression demonstrated that a second order polynomial model fits well the data. The results demonstrated that high pH and citric acid concentration enhanced cell survival; one the other hand, ascorbic acid did not have an effect. Cell survival during storage was also investigated in various types of juices, including orange, grapefruit, blackcurrant, pineapple, pomegranate, cranberry and lemon juice. The model predicted well the cell survival in orange, blackcurrant and pineapple, however it failed to predict cell survival in grapefruit and pomegranate, indicating the influence of additional factors, besides pH and citric acid, on cell survival. Very good cell survival (less than 0.4 log decrease) was observed after 6 weeks of storage in orange, blackcurrant and pineapple juice, all of which had a pH of about 3.8. Cell survival in cranberry and pomegranate decreased very quickly, whereas in the case of lemon juice, the cell concentration decreased approximately 1.1 logs after 6 weeks of storage, albeit the fact that lemon juice had the lowest pH (pH~2.5) among all the juices tested. Taking into account the results from the compositional analysis of the juices and the model, it was deduced that in certain juices, other compounds seemed to protect the cells during storage; these were likely to be proteins and dietary fibre In contrast, in certain juices, such as pomegranate, cell survival was much lower than expected; this could be due to the presence of antimicrobial compounds, such as phenolic compounds.

Investigation of the factors influencing the survival of Bifidobacterium longum in model acidic solutions and fruit juices

The survival of Bifidobacterium longum NCIMB 8809 was studied during refrigerated storage for 6 weeks in model solutions, based on which a mathematical model was constructed describing cell survival as a function of pH, citric acid, protein and dietary fibre. A Central Composite Design (CCD) was developed studying the influence of four factors at three levels, i.e., pH (3.2–4), citric acid (2–15 g/l), protein (0–10 g/l), and dietary fibre (0–8 g/l). In total, 31 experimental runs were carried out. Analysis of variance (ANOVA) of the regression model demonstrated that the model fitted well the data. From the regression coefficients it was deduced that all four factors had a statistically significant (P < 0.05) negative effect on the log decrease [log10N0 week−log10N6 week], with the pH and citric acid being the most influential ones. Cell survival during storage was also investigated in various types of juices, including orange, grapefruit, blackcurrant, pineapple, pomegranate and strawberry. The highest cell survival (less than 0.4 log decrease) after 6 weeks of storage was observed in orange and pineapple, both of which had a pH of about 3.8. Although the pH of grapefruit and blackcurrant was similar (pH ∼3.2), the log decrease of the former was ∼0.5 log, whereas of the latter was ∼0.7 log. One reason for this could be the fact that grapefruit contained a high amount of citric acid (15.3 g/l). The log decrease in pomegranate and strawberry juices was extremely high (∼8 logs). The mathematical model was able to predict adequately the cell survival in orange, grapefruit, blackcurrant, and pineapple juices. However, the model failed to predict the cell survival in pomegranate and strawberry, most likely due to the very high levels of phenolic compounds in these two juices.

A stress relaxation model for the viscoelastic solids based on the steady-state creep equation

Creep and stress relaxation are inherent mechanical behaviors of viscoelastic materials. It is considered that both are different performances of one identical physical phenomenon. The relationship between the decay stress and time during stress relaxation has been derived from the power law equation of the steady-state creep. The model was used to analyse the stress relaxation curves of various different viscoelastic materials (such as pure polycrystalline ice, polymers, foods, bones, metal, animal tissues, etc.). The calculated results using the theoretical model agree with the experimental data very well. Here we show that the new mathematical formula is not only simple but its parameters have the clear physical meanings. It is suitable to materials with a very broad scope and has a strong predictive ability.

A kinetic model for the formation of acrylamide during the finish-frying of commercial French fries

Acrylamide is formed from reducing sugars and asparagine during the preparation of French fries. The commercial preparation of French fries is a multi-stage process involving the preparation of frozen, par-fried potato strips for distribution to catering outlets where they are finish fried. The initial blanching, treatment in glucose solution and par-frying steps are crucial since they determine the levels of precursors present at the beginning of the finish frying process. In order to minimize the quantities of acrylamide in cooked fries, it is important to understand the impact of each stage on the formation of acrylamide. Acrylamide, amino acids, sugars, moisture, fat and color were monitored at time intervals during the frying of potato strips which had been dipped in varying concentrations of glucose and fructose during a typical pretreatment. A mathematical model of the finish-frying was developed based on the fundamental chemical reaction pathways, incorporating moisture and temperature gradients in the fries. This showed the contribution of both glucose and fructose to the generation of acrylamide, and accurately predicted the acrylamide content of the final fries.

Kinetic model for the formation of acrylamide during the finish-frying of commercial French fries

Acrylamide is formed from reducing sugars and asparagine during the preparation of French fries. The commercial preparation of French fries is a multistage process involving the preparation of frozen, par-fried potato strips for distribution to catering outlets, where they are finish-fried. The initial blanching, treatment in glucose solution, and par-frying steps are crucial because they determine the levels of precursors present at the beginning of the finish-frying process. To minimize the quantities of acrylamide in cooked fries, it is important to understand the impact of each stage on the formation of acrylamide. Acrylamide, amino acids, sugars, moisture, fat, and color were monitored at time intervals during the frying of potato strips that had been dipped in various concentrations of glucose and fructose during a typical pretreatment. A mathematical model based on the fundamental chemical reaction pathways of the finish-frying was developed, incorporating moisture and temperature gradients in the fries. This showed the contribution of both glucose and fructose to the generation of acrylamide and accurately predicted the acrylamide content of the final fries.

Molecular and phenotypic characterization of a mouse model of oculopharyngeal muscular dystrophy reveals severe muscular atrophy restricted to fast glycolytic fibres

Oculopharyngeal muscular dystrophy (OPMD) is an adult-onset disorder characterized by ptosis, dysphagia and proximal limb weakness. Autosomal-dominant OPMD is caused by a short (GCG)8–13 expansions within the first exon of the poly(A)-binding protein nuclear 1 gene (PABPN1), leading to an expanded polyalanine tract in the mutated protein. Expanded PABPN1 forms insoluble aggregates in the nuclei of skeletal muscle fibres. In order to gain insight into the different physiological processes affected in OPMD muscles, we have used a transgenic mouse model of OPMD (A17.1) and performed transcriptomic studies combined with a detailed phenotypic characterization of this model at three time points. The transcriptomic analysis revealed a massive gene deregulation in the A17.1 mice, among which we identified a significant deregulation of pathways associated with muscle atrophy. Using a mathematical model for progression, we have identified that one-third of the progressive genes were also associated with muscle atrophy. Functional and histological analysis of the skeletal muscle of this mouse model confirmed a severe and progressive muscular atrophy associated with a reduction in muscle strength. Moreover, muscle atrophy in the A17.1 mice was restricted to fast glycolytic fibres, containing a large number of intranuclear inclusions (INIs). The soleus muscle and, in particular, oxidative fibres were spared, even though they contained INIs albeit to a lesser degree. These results demonstrate a fibre-type specificity of muscle atrophy in this OPMD model. This study improves our understanding of the biological pathways modified in OPMD to identify potential biomarkers and new therapeutic targets.

Towards a model-based integration of co-registered electroencephalography/functional magnetic resonance imaging data with realistic neural population meshes

Brain activity can be measured with several non-invasive neuroimaging modalities, but each modality has inherent limitations with respect to resolution, contrast and interpretability. It is hoped that multimodal integration will address these limitations by using the complementary features of already available data. However, purely statistical integration can prove problematic owing to the disparate signal sources. As an alternative, we propose here an advanced neural population model implemented on an anatomically sound cortical mesh with freely adjustable connectivity, which features proper signal expression through a realistic head model for the electroencephalogram (EEG), as well as a haemodynamic model for functional magnetic resonance imaging based on blood oxygen level dependent contrast (fMRI BOLD). It hence allows simultaneous and realistic predictions of EEG and fMRI BOLD from the same underlying model of neural activity. As proof of principle, we investigate here the influence on simulated brain activity of strengthening visual connectivity. In the future we plan to fit multimodal data with this neural population model. This promises novel, model-based insights into the brain's activity in sleep, rest and task conditions.

A Hamiltonian weak-wave model for shallow-water flow

A reduced dynamical model is derived which describes the interaction of weak inertia–gravity waves with nonlinear vortical motion in the context of rotating shallow–water flow. The formal scaling assumptions are (i) that there is a separation in timescales between the vortical motion and the inertia–gravity waves, and (ii) that the divergence is weak compared to the vorticity. The model is Hamiltonian, and possesses conservation laws analogous to those in the shallow–water equations. Unlike the shallow–water equations, the energy invariant is quadratic. Nonlinear stability theorems are derived for this system, and its linear eigenvalue properties are investigated in the context of some simple basic flows.