Mathematical modelling of patient risks in a hospital environment, using multiple regression analysis

The aim of this research work was primarily to examine the relevance of patient parameters, ward structures, procedures and practices, in respect of the potential hazards of wound cross-infection and nasal colonisation with multiple resistant strains of Staphylococcus aureus, which it is thought might provide a useful indication of a patient's general susceptibility to wound infection. Information from a large cross-sectional survey involving 12,000 patients from some 41 hospitals and 375 wards was collected over a five-year period from 1967-72, and its validity checked before any subsequent analysis was carried out. Many environmental factors and procedures which had previously been thought (but never conclusively proved) to have an influence on wound infection or nasal colonisation rates, were assessed, and subsequently dismissed as not being significant, provided that the standard of the current range of practices and procedures is maintained and not allowed to deteriorate. Retrospective analysis revealed that the probability of wound infection was influenced by the patient's age, duration of pre-operative hospitalisation, sex, type of wound, presence and type of drain, number of patients in ward, and other special risk factors, whilst nasal colonisation was found to be influenced by the patient's age, total duration of hospitalisation, sex, antibiotics, proportion of occupied beds in the ward, average distance between bed centres and special risk factors. A multi-variate regression analysis technique was used to develop statistical models, consisting of variable patient and environmental factors which were found to have a significant influence on the risks pertaining to wound infection and nasal colonisation. A relationship between wound infection and nasal colonisation was then established and this led to the development of a more advanced model for predicting wound infections, taking advantage of the additional knowledge of the patient's state of nasal colonisation prior to operation.

Adaptive sending rate over wireless mesh networks using SNR

Wireless Mesh Networks (WMNs) have emerged as a key technology for the next generation of wireless networking. Instead of being another type of ad-hoc networking, WMNs diversify the capabilities of ad-hoc networks. Several protocols that work over WMNs include IEEE 802.11a/b/g, 802.15, 802.16 and LTE-Advanced. To bring about a high throughput under varying conditions, these protocols have to adapt their transmission rate. In this paper, we have proposed a scheme to improve channel conditions by performing rate adaptation along with multiple packet transmission using packet loss and physical layer condition. Dynamic monitoring, multiple packet transmission and adaptation to changes in channel quality by adjusting the packet transmission rates according to certain optimization criteria provided greater throughput. The key feature of the proposed method is the combination of the following two factors: 1) detection of intrinsic channel conditions by measuring the fluctuation of noise to signal ratio via the standard deviation, and 2) the detection of packet loss induced through congestion. We have shown that the use of such techniques in a WMN can significantly improve performance in terms of the packet sending rate. The effectiveness of the proposed method was demonstrated in a simulated wireless network testbed via packet-level simulation.

Pichia pastoris regulates its gene-specific response to different carbon sources at the transcriptional, rather than the translational, level

Background: The methylotrophic, Crabtree-negative yeast Pichia pastoris is widely used as a heterologous protein production host. Strong inducible promoters derived from methanol utilization genes or constitutive glycolytic promoters are typically used to drive gene expression. Notably, genes involved in methanol utilization are not only repressed by the presence of glucose, but also by glycerol. This unusual regulatory behavior prompted us to study the regulation of carbon substrate utilization in different bioprocess conditions on a genome wide scale. Results: We performed microarray analysis on the total mRNA population as well as mRNA that had been fractionated according to ribosome occupancy. Translationally quiescent mRNAs were defined as being associated with single ribosomes (monosomes) and highly-translated mRNAs with multiple ribosomes (polysomes). We found that despite their lower growth rates, global translation was most active in methanol-grown P. pastoris cells, followed by excess glycerol- or glucose-grown cells. Transcript-specific translational responses were found to be minimal, while extensive transcriptional regulation was observed for cells grown on different carbon sources. Due to their respiratory metabolism, cells grown in excess glucose or glycerol had very similar expression profiles. Genes subject to glucose repression were mainly involved in the metabolism of alternative carbon sources including the control of glycerol uptake and metabolism. Peroxisomal and methanol utilization genes were confirmed to be subject to carbon substrate repression in excess glucose or glycerol, but were found to be strongly de-repressed in limiting glucose-conditions (as are often applied in fed batch cultivations) in addition to induction by methanol. Conclusions: P. pastoris cells grown in excess glycerol or glucose have similar transcript profiles in contrast to S. cerevisiae cells, in which the transcriptional response to these carbon sources is very different. The main response to different growth conditions in P. pastoris is transcriptional; translational regulation was not transcript-specific. The high proportion of mRNAs associated with polysomes in methanol-grown cells is a major finding of this study; it reveals that high productivity during methanol induction is directly linked to the growth condition and not only to promoter strength.

A multi-modular second life hybrid battery energy storage system for utility grid applications

The modern grid system or the smart grid is likely to be populated with multiple distributed energy sources, e.g. wind power, PV power, Plug-in Electric Vehicle (PEV). It will also include a variety of linear and nonlinear loads. The intermittent nature of renewable energies like PV, wind turbine and increased penetration of Electric Vehicle (EV) makes the stable operation of utility grid system challenging. In order to ensure a stable operation of the utility grid system and to support smart grid functionalities such as, fault ride-through, frequency response, reactive power support, and mitigation of power quality issues, an energy storage system (ESS) could play an important role. A fast acting bidirectional energy storage system which can rapidly provide and absorb power and/or VARs for a sufficient time is a potentially valuable tool to support this functionality. Battery energy storage systems (BESS) are one of a range suitable energy storage system because it can provide and absorb power for sufficient time as well as able to respond reasonably fast. Conventional BESS already exist on the grid system are made up primarily of new batteries. The cost of these batteries can be high which makes most BESS an expensive solution. In order to assist moving towards a low carbon economy and to reduce battery cost this work aims to research the opportunities for the re-use of batteries after their primary use in low and ultra-low carbon vehicles (EV/HEV) on the electricity grid system. This research aims to develop a new generation of second life battery energy storage systems (SLBESS) which could interface to the low/medium voltage network to provide necessary grid support in a reliable and in cost-effective manner. The reliability/performance of these batteries is not clear, but is almost certainly worse than a new battery. Manufacturers indicate that a mixture of gradual degradation and sudden failure are both possible and failure mechanisms are likely to be related to how hard the batteries were driven inside the vehicle. There are several figures from a number of sources including the DECC (Department of Energy and Climate Control) and Arup and Cenex reports indicate anything from 70,000 to 2.6 million electric and hybrid vehicles on the road by 2020. Once the vehicle battery has degraded to around 70-80% of its capacity it is considered to be at the end of its first life application. This leaves capacity available for a second life at a much cheaper cost than a new BESS Assuming a battery capability of around 5-18kWhr (MHEV 5kWh - BEV 18kWh battery) and approximate 10 year life span, this equates to a projection of battery storage capability available for second life of >1GWhrs by 2025. Moreover, each vehicle manufacturer has different specifications for battery chemistry, number and arrangement of battery cells, capacity, voltage, size etc. To enable research and investment in this area and to maximize the remaining life of these batteries, one of the design challenges is to combine these hybrid batteries into a grid-tie converter where their different performance characteristics, and parameter variation can be catered for and a hot swapping mechanism is available so that as a battery ends it second life, it can be replaced without affecting the overall system operation. This integration of either single types of batteries with vastly different performance capability or a hybrid battery system to a grid-tie 3 energy storage system is different to currently existing work on battery energy storage systems (BESS) which deals with a single type of battery with common characteristics. This thesis addresses and solves the power electronic design challenges in integrating second life hybrid batteries into a grid-tie energy storage unit for the first time. This study details a suitable multi-modular power electronic converter and its various switching strategies which can integrate widely different batteries to a grid-tie inverter irrespective of their characteristics, voltage levels and reliability. The proposed converter provides a high efficiency, enhanced control flexibility and has the capability to operate in different operational modes from the input to output. Designing an appropriate control system for this kind of hybrid battery storage system is also important because of the variation of battery types, differences in characteristics and different levels of degradations. This thesis proposes a generalised distributed power sharing strategy based on weighting function aims to optimally use a set of hybrid batteries according to their relative characteristics while providing the necessary grid support by distributing the power between the batteries. The strategy is adaptive in nature and varies as the individual battery characteristics change in real time as a result of degradation for example. A suitable bidirectional distributed control strategy or a module independent control technique has been developed corresponding to each mode of operation of the proposed modular converter. Stability is an important consideration in control of all power converters and as such this thesis investigates the control stability of the multi-modular converter in detailed. Many controllers use PI/PID based techniques with fixed control parameters. However, this is not found to be suitable from a stability point-of-view. Issues of control stability using this controller type under one of the operating modes has led to the development of an alternative adaptive and nonlinear Lyapunov based control for the modular power converter. Finally, a detailed simulation and experimental validation of the proposed power converter operation, power sharing strategy, proposed control structures and control stability issue have been undertaken using a grid connected laboratory based multi-modular hybrid battery energy storage system prototype. The experimental validation has demonstrated the feasibility of this new energy storage system operation for use in future grid applications.

Numerical Analysis of Chemical Reactions by Monte Carlo Simulation

Добри Данков, Владимир Русинов, Мария Велинова, Жасмина Петрова - Изследвана е химическа реакция чрез два начина за моделиране на вероятността за химическа реакция използвайки Direct Simulation Monte Carlo метод. Изследван е порядъка на разликите при температурите и концентрациите чрез тези начини. Когато активността на химическата реакция намалява, намаляват и разликите между концентрациите и температурите получени по двата начина. Ключови думи: Механика на флуидите, Кинетична теория, Разреден газ, DSMC

Adaptive sending rate over wireless mesh networks using SNR

Wireless Mesh Networks (WMNs) have emerged as a key technology for the next generation of wireless networking. Instead of being another type of ad-hoc networking, WMNs diversify the capabilities of ad-hoc networks. Several protocols that work over WMNs include IEEE 802.11a/b/g, 802.15, 802.16 and LTE-Advanced. To bring about a high throughput under varying conditions, these protocols have to adapt their transmission rate. This paper proposes a scheme to improve channel conditions by performing rate adaptation along with multiple packet transmission using packet loss and physical layer condition. Dynamic monitoring, multiple packet transmission and adaptation to changes in channel quality by adjusting the packet transmission rates according to certain optimization criteria provided greater throughput. The key feature of the proposed method is the combination of the following two factors: 1) detection of intrinsic channel conditions by measuring the fluctuation of noise to signal ratio via the standard deviation, and 2) the detection of packet loss induced through congestion. The authors show that the use of such techniques in a WMN can significantly improve performance in terms of the packet sending rate. The effectiveness of the proposed method was demonstrated in a simulated wireless network testbed via packet-level simulation.

Increased risk of diabetes mellitus among persons with psychotic symptoms:results from the WHO World Health Survey

OBJECTIVE: To analyze with a symptom-based approach the relationship between psychosis and diabetes mellitus in the general population. METHOD: Nationally representative samples from the World Health Organization (WHO) World Health Survey, totaling 224,743 randomly selected adults 18 years and older from 52 countries worldwide, were interviewed to establish the presence of psychotic symptoms and diabetes mellitus. Presence of psychotic symptoms was established using questions pertaining to positive symptoms from the psychosis screening module of the Composite International Diagnostic Interview. Presence of diabetes was established with a response of "yes" to the question, "Have you ever been diagnosed with diabetes (high blood sugar)?" The World Health Survey was conducted between 2002 and 2004. RESULTS: An increasing number of psychotic symptoms was related to increasing likelihood of diabetes mellitus (OR = 1.27; 95% CI, 1.24-1.30). As compared to no symptoms, at least 1 psychotic symptom substantially elevated the risk (OR = 1.71; 95% CI, 1.61-1.81). In people with a lifetime diagnosis of schizophrenia or psychosis, the prevalence of diabetes was higher in those with current psychotic symptoms (7.3% vs 5.2%; OR = 1.65; 95% CI, 1.21-2.26), suggesting that the persistence of symptoms over time could play a central role. After controlling for different potential confounders, there was a clear increase in the probability of having diabetes as the number of psychotic symptoms increased. The relationship between psychotic symptoms and diabetes was tested with multiple mediation models and path analyses for categorical outcomes. Only body mass index appeared as a relevant mediator in a model with a good fit (ie, χ21 = 3.2, P = .0742; comparative fit index = 0.999). CONCLUSIONS: Psychotic symptoms are related to increased rates of diabetes mellitus in nonclinical samples, independent of several potential confounders-including a clinical diagnosis of psychosis or schizophrenia, previous antipsychotic treatment, depression, lifestyle, and individual or country socioeconomic status. The findings highlight the worldwide relevance of the problem and the importance of identifying the specific paths of this association.

The integration of ProxiMAX randomisation with CIS display for the production of novel peptides

Saturation mutagenesis is a powerful tool in modern protein engineering, which permits key residues within a protein to be targeted in order to potentially enhance specific functionalities. However, the creation of large libraries using conventional saturation mutagenesis with degenerate codons (NNN or NNK/S) has inherent redundancy and consequent disparities in codon representation. Therefore, both chemical (trinucleotide phosphoramidites) and biological methods (sequential, enzymatic single codon additions) of non-degenerate saturation mutagenesis have been developed in order to combat these issues and so improve library quality. Large libraries with multiple saturated positions can be limited by the method used to screen them. Although the traditional screening method of choice, cell-dependent methods, such as phage display, are limited by the need for transformation. A number of cell-free screening methods, such as CIS display, which link the screened phenotype with the encoded genotype, have the capability of screening libraries with up to 1014 members. This thesis describes the further development of ProxiMAX technology to reduce library codon bias and its integration with CIS display to screen the resulting library. Synthetic MAX oligonucleotides are ligated to an acceptor base sequence, amplified, and digested, subsequently adding a randomised codon to the acceptor, which forms an iterative cycle using the digested product of the previous cycle as the base sequence for the next. Initial use of ProxiMAX highlighted areas of the process where changes could be implemented in order to improve the codon representation in the final library. The refined process was used to construct a monomeric anti-NGF peptide library, based on two proprietary dimeric peptides (Isogenica) that bind NGF. The resulting library showed greatly improved codon representation that equated to a theoretical diversity of ~69%. The library was subsequently screened using CIS display and the discovered peptides assessed for NGF-TrkA inhibition by ELISA. Despite binding to TrkA, these peptides showed lower levels of inhibition of the NGF-TrkA interaction than the parental dimeric peptides, highlighting the importance of dimerization for inhibition of NGF-TrkA binding.

Patterns of acculturative stress: Continuity, discontinuity, and adjustment among Latino youth

This study evaluated inter- and intra-individual changes in acculturation, acculturative stress, and adaptation experiences, as well as their associations with adjustment outcomes among a group of Latino adolescents in South Florida. Specifically, the current study investigated the incidence, changes, and effects of stressors that arise from acculturation experiences (e.g., related to culture, discrimination, language difficulties) among Latino youth by employing a person-centered approach and a longitudinal research design. Four separate groups of analyses were conducted to investigate (a) within-group differences in levels of reported acculturative stress, (b) patterns of continuity and discontinuity in levels of acculturative stress across time, (c) adjustment outcomes associated with distinct patterns of acculturative stress within each measurement occasion, and (d) predictive relations between longitudinal acculturative stress trajectories in early adolescence and psychosocial adjustment outcomes in young adulthood. ^ Results from the multivariate analyses indicated great within group heterogeneity in acculturative stress among Latino youth during early adolescence, as well as significant continuity and discontinuity in the patterns of shifts among acculturative stress profiles between contiguous measurement occasions. Within each developmental period, membership in acculturative stress clusters was significantly and differentially associated with multiple adjustment outcomes, suggesting that maladaptive outcomes are more likely to occur among Latino adolescents experiencing high levels of psychological distress across multiple acculturative domains. In general, Latino youth acculturation is best understood as multi-dimensional, to be variable across time, and to be fluid and responsive to multiple factors and influences. Implications for preventive strategies are discussed with regard to acculturation and developmental psychology research literatures. ^

Towards simulation and emulation of large-scale computer networks

Developing analytical models that can accurately describe behaviors of Internet-scale networks is difficult. This is due, in part, to the heterogeneous structure, immense size and rapidly changing properties of today's networks. The lack of analytical models makes large-scale network simulation an indispensable tool for studying immense networks. However, large-scale network simulation has not been commonly used to study networks of Internet-scale. This can be attributed to three factors: 1) current large-scale network simulators are geared towards simulation research and not network research, 2) the memory required to execute an Internet-scale model is exorbitant, and 3) large-scale network models are difficult to validate. This dissertation tackles each of these problems. ^ First, this work presents a method for automatically enabling real-time interaction, monitoring, and control of large-scale network models. Network researchers need tools that allow them to focus on creating realistic models and conducting experiments. However, this should not increase the complexity of developing a large-scale network simulator. This work presents a systematic approach to separating the concerns of running large-scale network models on parallel computers and the user facing concerns of configuring and interacting with large-scale network models. ^ Second, this work deals with reducing memory consumption of network models. As network models become larger, so does the amount of memory needed to simulate them. This work presents a comprehensive approach to exploiting structural duplications in network models to dramatically reduce the memory required to execute large-scale network experiments. ^ Lastly, this work addresses the issue of validating large-scale simulations by integrating real protocols and applications into the simulation. With an emulation extension, a network simulator operating in real-time can run together with real-world distributed applications and services. As such, real-time network simulation not only alleviates the burden of developing separate models for applications in simulation, but as real systems are included in the network model, it also increases the confidence level of network simulation. This work presents a scalable and flexible framework to integrate real-world applications with real-time simulation.^

Combination of 16S rRNA variable regions provides a detailed analysis of bacterial community dynamics in the lungs of cystic fibrosis patients

Chronic bronchopulmonary bacterial infections remain the most common cause of morbidity and mortality among patients with cystic fibrosis (CF). Recent community sequencing work has now shown that the bacterial community in the CF lung is polymicrobial. Identifying bacteria in the CF lung through sequencing can be costly and is not practical for many laboratories. Molecular techniques such as terminal restriction fragment length polymorphism or amplicon length heterogeneity-polymerase chain reaction (LH-PCR) can provide many laboratories with the ability to study CF bacterial communities without costly sequencing. The aim of this study was to determine if the use of LH-PCR with multiple hypervariable regions of the 16S rRNA gene could be used to identify organisms found in sputum DNA. This work also determined if LH-PCR could be used to observe the dynamics of lung infections over a period of time. Nineteen samples were analysed with the V1 and the V1_V2 region of the 16S rRNA gene. Based on the amplicon size present in the V1_V2 region, Pseudomonas aeruginosa was confirmed to be in all 19 samples obtained from the patients. The V1 region provided a higher power of discrimination between bacterial profiles of patients. Both regions were able to identify trends in the bacterial population over a period of time. LH profiles showed that the CF lung community is dynamic and that changes in the community may in part be driven by the patient's antibiotic treatment. LH-PCR is a tool that is well suited for studying bacterial communities and their dynamics.

Dynamic image precompensation for improving visual performance of computer users with ocular aberrations

With the progress of computer technology, computers are expected to be more intelligent in the interaction with humans, presenting information according to the user's psychological and physiological characteristics. However, computer users with visual problems may encounter difficulties on the perception of icons, menus, and other graphical information displayed on the screen, limiting the efficiency of their interaction with computers. In this dissertation, a personalized and dynamic image precompensation method was developed to improve the visual performance of the computer users with ocular aberrations. The precompensation was applied on the graphical targets before presenting them on the screen, aiming to counteract the visual blurring caused by the ocular aberration of the user's eye. A complete and systematic modeling approach to describe the retinal image formation of the computer user was presented, taking advantage of modeling tools, such as Zernike polynomials, wavefront aberration, Point Spread Function and Modulation Transfer Function. The ocular aberration of the computer user was originally measured by a wavefront aberrometer, as a reference for the precompensation model. The dynamic precompensation was generated based on the resized aberration, with the real-time pupil diameter monitored. The potential visual benefit of the dynamic precompensation method was explored through software simulation, with the aberration data from a real human subject. An "artificial eye'' experiment was conducted by simulating the human eye with a high-definition camera, providing objective evaluation to the image quality after precompensation. In addition, an empirical evaluation with 20 human participants was also designed and implemented, involving image recognition tests performed under a more realistic viewing environment of computer use. The statistical analysis results of the empirical experiment confirmed the effectiveness of the dynamic precompensation method, by showing significant improvement on the recognition accuracy. The merit and necessity of the dynamic precompensation were also substantiated by comparing it with the static precompensation. The visual benefit of the dynamic precompensation was further confirmed by the subjective assessments collected from the evaluation participants.

Trait Contributions to Fish Community Assembly Emerge From Trophic Interactions in an Individual-Based Model

Community ecology seeks to understand and predict the characteristics of communities that can develop under different environmental conditions, but most theory has been built on analytical models that are limited in the diversity of species traits that can be considered simultaneously. We address that limitation with an individual-based model to simulate assembly of fish communities characterized by life history and trophic interactions with multiple physiological tradeoffs as constraints on species performance. Simulation experiments were carried out to evaluate the distribution of 6 life history and 4 feeding traits along gradients of resource productivity and prey accessibility. These experiments revealed that traits differ greatly in importance for species sorting along the gradients. Body growth rate emerged as a key factor distinguishing community types and defining patterns of community stability and coexistence, followed by egg size and maximum body size. Dominance by fast-growing, relatively large, and fecund species occurred more frequently in cases where functional responses were saturated (i.e. high productivity and/or prey accessibility). Such dominance was associated with large biomass fluctuations and priority effects, which prevented richness from increasing with productivity and may have limited selection on secondary traits, such as spawning strategies and relative size at maturation. Our results illustrate that the distribution of species traits and the consequences for community dynamics are intimately linked and strictly dependent on how the benefits and costs of these traits are balanced across different conditions.

Presenting SAPUSS: solving aerosol problem by using synergistic strategies in Barcelona, Spain

This paper presents the summary of the key objectives, instrumentation and logistic details, goals, and initial scientific findings of the European Marie Curie Action SAPUSS project carried out in the western Mediterranean Basin (WMB) during September-October in autumn 2010. The key SAPUSS objective is to deduce aerosol source characteristics and to understand the atmospheric processes responsible for their generations and transformations - both horizontally and vertically in the Mediterranean urban environment. In order to achieve so, the unique approach of SAPUSS is the concurrent measurements of aerosols with multiple techniques occurring simultaneously in six monitoring sites around the city of Barcelona (NE Spain): a main road traffic site, two urban background sites, a regional background site and two urban tower sites (150 m and 545 m above sea level, 150 m and 80 m above ground, respectively). SAPUSS allows us to advance our knowledge sensibly of the atmospheric chemistry and physics of the urban Mediterranean environment. This is well achieved only because of both the three dimensional spatial scale and the high sampling time resolution used. During SAPUSS different meteorological regimes were encountered, including warm Saharan, cold Atlantic, wet European and stagnant regional ones. The different meteorology of such regimes is herein described. Additionally, we report the trends of the parameters regulated by air quality purposes (both gaseous and aerosol mass concentrations); and we also compare the six monitoring sites. High levels of traffic-related gaseous pollutants were measured at the urban ground level monitoring sites, whereas layers of tropospheric ozone were recorded at tower levels. Particularly, tower level night-time average ozone concentrations (80 +/- 25 mu g m(-3)) were up to double compared to ground level ones. The examination of the vertical profiles clearly shows the predominant influence of NOx on ozone concentrations, and a source of ozone aloft. Analysis of the particulate matter (PM) mass concentrations shows an enhancement of coarse particles (PM2.5-10) at the urban ground level (+64 %, average 11.7 mu g m(-3)) but of fine ones (PM1) at urban tower level (+28 %, average 14.4 mu g m(-3)). These results show complex dynamics of the size-resolved PM mass at both horizontal and vertical levels of the study area. Preliminary modelling findings reveal an underestimation of the fine accumulation aerosols. In summary, this paper lays the foundation of SAPUSS, an integrated study of relevance to many other similar urban Mediterranean coastal environment sites.

Optimal Capital Requirements in Financial Networks with Fire Sales

I explore and analyze a problem of finding the socially optimal capital requirements for financial institutions considering two distinct channels of contagion: direct exposures among the institutions, as represented by a network and fire sales externalities, which reflect the negative price impact of massive liquidation of assets.These two channels amplify shocks from individual financial institutions to the financial system as a whole and thus increase the risk of joint defaults amongst the interconnected financial institutions; this is often referred to as systemic risk. In the model, there is a trade-off between reducing systemic risk and raising the capital requirements of the financial institutions. The policymaker considers this trade-off and determines the optimal capital requirements for individual financial institutions. I provide a method for finding and analyzing the optimal capital requirements that can be applied to arbitrary network structures and arbitrary distributions of investment returns.

In particular, I first consider a network model consisting only of direct exposures and show that the optimal capital requirements can be found by solving a stochastic linear programming problem. I then extend the analysis to financial networks with default costs and show the optimal capital requirements can be found by solving a stochastic mixed integer programming problem. The computational complexity of this problem poses a challenge, and I develop an iterative algorithm that can be efficiently executed. I show that the iterative algorithm leads to solutions that are nearly optimal by comparing it with lower bounds based on a dual approach. I also show that the iterative algorithm converges to the optimal solution.

Finally, I incorporate fire sales externalities into the model. In particular, I am able to extend the analysis of systemic risk and the optimal capital requirements with a single illiquid asset to a model with multiple illiquid assets. The model with multiple illiquid assets incorporates liquidation rules used by the banks. I provide an optimization formulation whose solution provides the equilibrium payments for a given liquidation rule.

I further show that the socially optimal capital problem using the ``socially optimal liquidation" and prioritized liquidation rules can be formulated as a convex and convex mixed integer problem, respectively. Finally, I illustrate the results of the methodology on numerical examples and

discuss some implications for capital regulation policy and stress testing.