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Within the likes of any highly contagious and unpredictable disease, lies a predictable and attainable growth rate that researchers can find in order to make logistical conclusions about that particular disease and its affected regions' counterparts. The foundation that researchers pull from when studying a particular disease and looking for its growth rate is the Susceptible-Infected-Removed (SIR) model, presented by a series of differential equations. The issue with the SIR model lies not in its complexity, but actually its simplicity and lack of a potentially high-finite amount of factors; the limit being bounded by the amount of data available for that particular factor. Our research involves the application of multiple regressions to pinpoint and identify our Covid lockdown periods, followed by the modification of the SIR model. This involved creating new model approximations such as the time-delayed SIR model and the reinfected SIR model in order to take into account factors such as incubation and reinfection, and get the lowest error discrepancy as possible for our infection rate. We were able to conclude that the more factors that we took into account, our error rate became lower and our results became more accurate. We could also identify outlier Metros and draw certain conclusions on performance level and the reasons behind them. We then moved on to find correlations, if any, between the infection rates and outside factors. We looked at demographic and weather data to demonstrate whether correlations appeared. We found that there are a few factors with high correlations, including graduate education and low temperatures.

In social systems, the evolution of interpersonal appraisals and individual opinions are not independent processes but intertwine with each other. Despite extensive studies on both opinion dynamics and appraisal dynamics separately, no previous work has ever combined these two processes together. In this paper, we propose a novel and intuitive model on the interplay between homophily-based appraisal dynamics and influence-based opinion dynamics. We assume that individuals' opinions are updated via the influence network constructed from their interpersonal appraisals, which are in turn updated based on the individual opinions via the homophily mechanism. By theoretical analysis, we characterize the set of equilibria and some transient behavior of our model. Moreover, we establish the equivalence among the convergence of the appraisal network to social balance, the modulus consensus of individual opinions, and the non-vanishing appraisals. Monte Carlo validations further show that the non-vanishing appraisals condition holds for generic initial conditions. Compared with previous works that explain the emergence of social balance via person-to-person homophily mechanism, our model provides an alternative explanation in terms of the person-to-entity homophily mechanism. In addition, our model also describes how individuals' opinions on multiple irrelevant issues become correlated and converge to modulus consensus over time-varying influence networks.

Time-delay systems are an important class of dynamical systems which provide a solid mathematical framework to deal with many application domains of interest ranging from biology, chemical, electrical, and mechanical engineering, to economics. However, the inherent complexity of such systems poses serious difficulties to control design, when control objectives depart from the standard ones investigated in the current literature, e.g. stabilization, regulation, and etc. In this paper we propose one approach to control design, which is based on the construction of symbolic models, where each symbolic state and each symbolic label correspond to an aggregate of continuous states and to an aggregate of input signals in the original system. The use of symbolic models offers a systematic methodology for control design in which constraints coming from software and hardware, interacting with the physical world, can be integrated. The main contribution of this paper is in showing that incrementally input-to-state stable time-delay systems do admit symbolic models that are approximately bisimilar to the original system, with a precision that can be rendered as small as desired. An algorithm is also presented which computes the proposed symbolic models. When the state and input spaces of time-delay systems are bounded, which is the case in many realistic situations, the proposed algorithm is shown to terminate in a finite number of steps., Comment: 12 pages

Switched systems constitute an important modeling paradigm faithfully describing many engineering systems in which software interacts with the physical world. Despite considerable progress on stability and stabilization of switched systems, the constant evolution of technology demands that we make similar progress with respect to different, and perhaps more complex, objectives. This paper describes one particular approach to address these different objectives based on the construction of approximately equivalent (bisimilar) symbolic models for switched systems. The main contribution of this paper consists in showing that under standard assumptions ensuring incremental stability of a switched system (i.e. existence of a common Lyapunov function, or multiple Lyapunov functions with dwell time), it is possible to construct a finite symbolic model that is approximately bisimilar to the original switched system with a precision that can be chosen a priori. To support the computational merits of the proposed approach, we use symbolic models to synthesize controllers for two examples of switched systems, including the boost DC-DC converter., Comment: 17 pages

Some twenty years ago we introduced a nonstandard matrix Riccati equation to solve the partial stochastic realization problem. In this paper we provide a new derivation of this equation in the context of system identification. This allows us to show that the nonstandard matrix Riccati equation is universal in the sense that it can be used to solve more general analytic interpolation problems by only changing certain parameters. Such interpolation problems are ubiquitous in systems and control. In this context we also discuss a question posed by R.E. Kalman in beginning of the 1970s.

The decision by the United Kingdom (UK) government in 2007 that the country should build a 33GW wind fleet, capable of generating 25 percent of the UK electricity requirement, was controversial. Proponents argued that it was the most attractive means of lowering the UK greenhouse gas emissions, whereas opponents noted that it would result in an unnecessary and burdensome additional expense to industry and households. Subsequently there have been calls for the wind fleet target to be further increased to perhaps 50 percent of demand. Although the National Grid has had little difficulty in accommodating the current output of about 10 percent of the total demand on the grid, this will not be the case for a substantially larger wind fleet. When the wind blows strongly, turbines shed wind which is surplus to demand, leading to significant reductions in generating efficiency. The purpose of the research described in this paper has been to develop a method for investigating the likely performance of future large wind fleets. The method relies on the use of mathematical models based on National Grid records for 2013 to 2015, each year being separately analysed. It was found that the incremental load factor of the wind fleet will be reduced to 63 percent of its current level should the wind fleet increase from its current size of 14GW to 35GW, assuming a base load of 15GW, Comment: 24 pages, 15 figures, 4 tables

Familiar laws of physics are applied to study human relations, modelled by their world lines (worldlines, WLs) combined with social networks. We focus upon the simplest, basic element of any society: a married couple, stable due to the dynamic balance between attraction and repulsion. By building worldlines/worldsheets, we arrive at a two-level coordinate systems: one describing the behaviour of a string-like binary system (here, a married couple), the other one, external, corresponding to the motion of this couple in the medium, in which the worldline is embedded, sweeping there a string-like sheet or brane. The approach is illustrated by simple examples (semi-quantitative toy models) of worldlines/sheets, open to further extension, perfections and generalization. World lines (WLs) are combined with social networks (SN). Our innovation is in the application of basic physical laws, attraction and repulsion to human behaviour. Simple illustrative examples with empirical inputs taken from intuition and/or observation are appended. This is an initial attempt, open to unlimited applications., Comment: 16 pages, 11 figures

It is anticipated that penetration of renewable energy sources (RESs) in power systems will increase further in the next decades mainly due to environmental issues. In the long term of several decades, which we refer to in terms of the future grid (FG), balancing between supply and demand will become dependent on demand actions including demand response (DR) and energy storage. So far, FG feasibility studies have not considered these new demand-side developments for modelling future demand. In Australia, installed rooftop photovoltaic (PV) generation has been increasing significantly in recent years, and this is increasingly influence the nett load profile for FG feasibility studies. This paper proposes an aggregate nett load or demand model to be used at higher voltage levels considering the effect of rooftop PV, energy storage and DR for FG scenarios, which is inspired by the smart home concept. The proposed model is formulated as an optimization problem aiming at minimizing the electricity cost. As a case study, the effect of the load model is studied on the load profile, balancing and loadability of the Australian National Electricity Market in 2020 with the increased penetration of wind and solar generation., Comment: 11 pages, 6 figures, 3 Tables, submitted to IET Generation, Transmission & Distribution

There has been vigorous debate on how different countries responded to the COVID-19 pandemic. To secure public safety, South Korea actively used personal information at the risk of personal privacy whereas France encouraged voluntary cooperation at the risk of public safety. In this article, after a brief comparison of contextual differences with France, we focus on South Korea's approaches to epidemiological investigations. To evaluate the issues pertaining to personal privacy and public health, we examine the usage patterns of original data, de-identification data, and encrypted data. Our specific proposal discusses the COVID index, which considers collective infection, outbreak intensity, availability of medical infrastructure, and the death rate. Finally, we summarize the findings and lessons for future research and the policy implications., Comment: 11pages

In this report a lumped transfer function model for High Pressure Natural Gas Pipelines is derived. Starting with a partial nonlinear differential equation (PDE) model a high order continuous state space (SS) linear model is obtained using a finite difference method. Next, from the SS representation an infinite order transfer function (TF) model is calculated. In the end, this TF is approximated by a compact non-rational function., Comment: report with 41 pages, 8 figures, that contains all the proofs of a paper submitted to the CDC2010.

In this paper we introduce a new mathematical tool to solve fractional equations representing models of fractional systems : The Ultradistributions. Ultradistributions permit us to unify the notion of integral and derivative in one only operation. Several examples of application of the results obtained are given., Comment: 17 pages. No figures

The objective of this paper is to solve the controller synthesis problem for bisimulation equivalence in a wide variety of scenarios including discrete-event systems, nonlinear control systems, behavioral systems, hybrid systems and many others. This will be accomplished by showing that the arguments underlying proofs of existence and methods for the construction of controllers are extraneous to the particular class of systems being considered and thus can be presented in greater generality., Comment: Added references, comments and examples

Control systems are usually modeled by differential equations describing how physical phenomena can be influenced by certain control parameters or inputs. Although these models are very powerful when dealing with physical phenomena, they are less suitable to describe software and hardware interfacing the physical world. For this reason there is a growing interest in describing control systems through symbolic models that are abstract descriptions of the continuous dynamics, where each "symbol" corresponds to an "aggregate" of states in the continuous model. Since these symbolic models are of the same nature of the models used in computer science to describe software and hardware, they provide a unified language to study problems of control in which software and hardware interact with the physical world. Furthermore the use of symbolic models enables one to leverage techniques from supervisory control and algorithms from game theory for controller synthesis purposes. In this paper we show that every incrementally globally asymptotically stable nonlinear control system is approximately equivalent (bisimilar) to a symbolic model. The approximation error is a design parameter in the construction of the symbolic model and can be rendered as small as desired. Furthermore if the state space of the control system is bounded the obtained symbolic model is finite. For digital control systems, and under the stronger assumption of incremental input-to-state stability, symbolic models can be constructed through a suitable quantization of the inputs., Comment: Corrected typos

Electrical networks, and physical systems in general, are known to satisfy a power balance equation which states that the rate of change of the energy in time equals the power at the port of the network minus the power dissipated. However, when complex power is considered, there does not seem to exist a similar statement for the imaginary power, either in the time-domain or the frequency-domain. Recently, in the context of electromagnetic fields, it has been shown by complexifying the time to t+js and interpreting s as reactive time, that it is possible to set up an imaginary power balance in terms of the rate of change of reactive energy in reactive time. Here these ideas are specialized to linear and time-invariant RLC networks. For non-sinusoidal waveforms it is shown that the rate of change of reactive energy in reactive time contains all the essential properties and features of the commonly accepted definition of reactive power under sinusoidal conditions. We believe that this provides an unambiguous and physically motivated resolution to the longstanding debate on how to generalize reactive power to non-sinusoidal waveforms., Comment: 6 pages and 2 figures

Proper heat transfer management is important to key electronic components in microelectronic applications. Pulsating heat pipes (PHP) can be an efficient solution to such heat transfer problems. However, mathematical modelling of a PHP system is still very challenging, due to the complexity and multiphysics nature of the system. In this work, we present a simplified, two-phase heat transfer model, and our analysis shows that it can make good predictions about startup characteristics. Furthermore, by considering parameter estimation as a nonlinear constrained optimization problem, we have used the firefly algorithm to find parameter estimates efficiently. We have also demonstrated that it is possible to obtain good estimates of key parameters using very limited experimental data.

We consider integrated circuits with semiconductors modeled by modified nodal analysis and drift-diffusion equations. The drift-diffusion equations are discretized in space using mixed finite element method. This discretization yields a high dimensional differential-algebraic equation. We show how proper orthogonal decomposition (POD) can be used to reduce the dimension of the model. We compare reduced and fine models and give numerical results for a basic network with one diode. Furthermore we discuss an adaptive approach to construct POD models which are valid over certain parameter ranges. Finally, numerical investigations for the reduction of a 4-diode rectifier network are presented, which clearly indicate that POD model reduction delivers surrogate models for the diodes involved, which depend on the position of the semiconductor in the network.

Terrorist organizations change over time because of processes such as recruitment and training as well as counter-terrorism (CT) measures, but the effects of these processes are typically studied qualitatively and in separation from each other. Seeking a more quantitative and integrated understanding, we constructed a simple dynamic model where equations describe how these processes change an organization's membership. Analysis of the model yields a number of intuitive as well as novel findings. Most importantly it becomes possible to predict whether counter-terrorism measures would be sufficient to defeat the organization. Furthermore, we can prove in general that an organization would collapse if its strength and its pool of foot soldiers decline simultaneously. In contrast, a simultaneous decline in its strength and its pool of leaders is often insufficient and short-termed. These results and other like them demonstrate the great potential of dynamic models for informing terrorism scholarship and counter-terrorism policy making., Comment: To appear as Springer Lecture Notes in Computer Science v2: vectorized 4 figures, fixed two typos, more detailed bibliography

We define observability and detectability for linear switching systems as the possibility of reconstructing and respectively of asymptotically reconstructing the hybrid state of the system from the knowledge of the output for a suitable choice of the control input. We derive a necessary and sufficient condition for observability that can be verified computationally. A characterization of control inputs ensuring observability of switching systems is given. Moreover, we prove that checking detectability of a linear switching system is equivalent to checking asymptotic stability of a suitable switching system with guards extracted from it, thus providing interesting links to Kalman decomposition and the theory of stability of hybrid systems.

A big-isotropic structure $E$ is an isotropic subbundle of $TM\oplus T^*M$, endowed with the metric defined by pairing. The structure $E$ is said to be integrable if the Courant bracket $[\mathcal{X},\mathcal{Y}]\in\Gamma E$, $\forall\mathcal{X},\mathcal{Y}\in\Gamma E$. Then, necessarily, one also has $[\mathcal{X},\mathcal{Z}]\in\Gamma E^\perp$, $\forall\mathcal{Z}\in\Gamma E^\perp$ \cite{V-iso}. A weak-Hamiltonian dynamical system is a vector field $X_H$ such that $(X_H,dH)\in E^\perp$ $(H\in C^\infty(M))$. We obtain the explicit expression of $X_H$ and of the integrability conditions of $E$ under the regularity condition $dim(pr_{T^*M}E)=const.$ We show that the port-controlled, Hamiltonian systems (in particular, constrained mechanics) \cite{{BR},{DS}} may be interpreted as weak-Hamiltonian systems. Finally, we give reduction theorems for weak-Hamiltonian systems and a corresponding corollary for constrained mechanical systems., Comment: 19 pages, minor improvements

It is shown that in the multivariate case the orders p, of the AR part, and q, of the MA part, are not invariants of the time series. Thus, it is concluded that it only makes sense to define the class of ARMA(p,p)- irreducible models, where p is the biggest of the system's Kronecker indices. This class is shown not to be a differentiable manifold, but to contain one, which is a generic subset of systems with all Kronecker indices equal to p. A formula which gives the metric tensor for riemannian manifolds of linear systems as a line integral in the complex plane is introduced for deterministic and stochastic cases and some tensors are obtained with it., Comment: 27 pages, english

A formula for the Riemannian metric tensor of differentiable manifolds of linear dynamical systems of same McMillan degree is presented in terms of their transfer function matrices. The necessary calculations for its application to ARMA and state space overlapping parametrizations are drafted. The importance of this approach for systems identification and multiple time series analysis and forecasting is explained., Comment: 3 pages in portuguese, reference links to author's detailed proofs in english

We describe in a rudimentary fashion how SVM(support vector machine) plays the role of classifier in a mathematical setting. We then discuss its application in the study of multiple SNP(single nucleotide polymorphism) variations. Also presented is a set of preliminary test results with clinical data., Comment: Presentations accepted to DIMACS workshop and International Conference on Mathematical and Theoretical Biology And Annual Meeting of the Society for Mathematical Biology Joint with Japanese Association for Mathematical Biology