Some Practical Aspects of the Verification and Validation of Simulation Models

This paper reviews the philosophical questions of model validity and describes the need for practical tests of the adequacy and representativeness of computerized simulation models. The distinction between the concepts of the verification and the validation of such general models is made by means of a delineation of test procedures, primarily those of a statistical nature, which are applicable to the determination of a simulation model's adequacy.     

The Mathematics of Quantum Chemistry

A brief historical account of the use of wave mechanics in the field of theoretical chemistry leads to a consideration of three parts of mathematics that have been found particularly important. These are:

The use of Rayleigh's principle to obtain approximate eigenvalues and eigenfunctions.

The use of group theory, particularly the more familiar point groups, to classify electronic states of a molecule, and indicate which transitions are allowed.

The evaluation of the very complicated many‐centre integrals that are inescapable if acceptable numerical validity is desired. Finally reference is made to three other directions in which additional mathematical power would be of advantage to the theoretical chemist.     

Issues arising on the use of hand‐held calculators in schools†

This paper considers the use of hand‐held calculators (HHCs) in schools from the pedagogical and sociological viewpoints. Using arguments based on the observed pattern of mathematical education, we discuss the effect of the use of HHCs, from the learning viewpoint and from the viewpoint of providing students with a ready competence in arithmetical manipulation. We shall also discuss the effects of the adoption of HHCs in undeveloped countries and for deprived minorities.

The topics are developed from philosophical/psychopedagogical reasoning and from developmental policy. Strategies designed to make mathematical education more immediately useful are discussed. An example is provided, and the relationship between Polya's problem‐solving approach and the use of HHCs is examined.     

A method for reduction of human ventricular action potential model

ABSTRACT

Mathematical modelling and computer simulations are important tools in the field of cardiac electrophysiology. High computational costs of complex models make them difficult to apply in large-scale simulations like tissue. Therefore, model reduction are of particular importance in heart studies. In this paper, we introduce a technique for simplification of ventricular cell(VC) complex models. By using this technique, starting with a complex model of human VC including 17state variables, we reduce the number of state variables to two. Our simplified model is compared with the original one via several electrophysiological features and computational efficiency. Results show that the reduced model has acceptable behaviours in single cell and one-dimensional simulation, moreover, is 55 times faster than the original one. As the presented method does not depend on the reference model, it may be applied to every cardiac cell models or each complex excitable dynamical systems with the same dynamics as VC.     

Modeling of Galactic Wind Formation from Supernovae Using High-Performance Computations

Fundamental processes in the dynamics of the interstellar medium, namely, galactic wind, i.e., ejections of interstellar matter from central regions of galaxies, which are presumably caused by the formation of supernovae, are mathematically modeled in detail on high-performance parallel computer systems. The mathematical simulation is based on a kinetically consistent gasdynamic approach developed for such class of problems in astrophysics. A kinetically consistent algorithm is well adapted to the architecture of high-performance computer systems with massive parallelism, so that complex large-scale astrophysical phenomena can be efficiently studied with a high resolution. The approach, method, and algorithms are described, and numerical results are presented.

     

FIRESCAP: A computer simulation model of reaction to a fire alarm

We develop a theoretical model and its computer‐simulation translation of evacuation response to a fire alarm, based on past research on fire emergencies and on crowd behavior as well as on our past and continuing investigation of the Beverly Hills fire disaster. The simulation model presents graphically the status of individuals and socially tied pairs in a room as they initiate evacuation—typically in response to ambiguous information—and move toward alternative exits. Individual and pair status—including choice of exit—can change during evacuation in response to changing physical and social cues; statuses include pace of movement and cooperativeness‐competitiveness. We also present some experimental results on the sensitivity of the model's outcomes to variation in several parameters.     

System dynamics and microworlds for policymakers

In the past ten years, system dynamics has become more accessible to policymakers and to the academic community. The paper reviews four major developments in the subject that have brought about this change. There have been improvements in the symbols and software used to map and model system structure. New ideas have been adopted from behavioural decision theory which help to transfer policymakers' knowledge into computer models. There have been improvements in methods of simulation analysis that enable modelers and model users to gain better insight into dynamic behaviour. Greater emphasis has been placed on small transparent models, on games and on dialogue between ‘mental models’ and computer simulations. Together these developments allow modelers to create computer-based learning environments (or microworlds) for policymakers to ‘play-with’ their knowledge of business and social systems and to debate policy and strategy change. The paper concludes with some thoughts on future research.     

Case-based modeling and the SACS Toolkit: a mathematical outline

Researchers in the social sciences currently employ a variety of mathematical/computational models for studying complex systems. Despite the diversity of these models, the majority can be grouped into one of three types: agent (rule-based) modeling, dynamical (equation-based) modeling and statistical (aggregate-based) modeling. The purpose of the current paper is to offer a fourth type: case-based modeling. To do so, we review the SACS Toolkit: a new method for quantitatively modeling complex social systems, based on a case-based, computational approach to data analysis. The SACS Toolkit is comprised of three main components: a theoretical blueprint of the major components of a complex system (social complexity theory); a set of case-based instructions for modeling complex systems from the ground up (assemblage); and a recommended list of case-friendly computational modeling techniques (case-based toolset). Developed as a variation on Byrne (in Sage Handbook of Case-Based Methods, pp.?260?C268, 2009), the SACS Toolkit models a complex system as a set of k-dimensional vectors (cases), which it compares and contrasts, and then condenses and clusters to create a low-dimensional model (map) of a complex system??s structure and dynamics over time/space. The assembled nature of the SACS Toolkit is its primary strength. While grounded in a defined mathematical framework, the SACS Toolkit is methodologically open-ended and therefore adaptable and amenable, allowing researchers to employ and bring together a wide variety of modeling techniques. Researchers can even develop and modify the SACS Toolkit for their own purposes. The other strength of the SACS Toolkit, which makes it a very effective technique for modeling large databases, is its ability to compress data matrices while preserving the most important aspects of a complex system??s structure and dynamics across time/space. To date, while the SACS Toolkit has been used to study several topics, a mathematical outline of its case-based approach to quantitative analysis (along with a case study) has yet to be written?Chence the purpose of the current paper.     

Dynamical Analysis and Control Strategies of Rumor Spreading Models in Both Homogeneous and Heterogeneous Networks

In recent years, rumor propagation in social networks attracts more researchers’ attention. In this paper, we have established I2S2R rumor spreading models in both homogeneous networks and heterogeneous networks considering the effect of time delay. In the homogeneous network model, we obtain the basic reproduction number by means of the next-generation matrix. Besides, the local stability and the global stability of the equilibrium points are discussed by linearization approach of nonlinear systems and Lyapunov function. In the heterogeneous network model, we calculate the basic reproduction number through algebraic method. In addition, Lyapunov functional method and Lasalle invariance principle are applied to study the stability of equilibrium points in the complex network model. Further, we put forward some useful strategies to control the spreading of rumor based on the complex network theory. Finally, we take advantage of numerical simulations to verify the theory above and come up with necessary conclusions.

     

Understandings of Solutions to Differential Equations Through Contexts,Web‐Based Simulations,and Student Discussion

As in the case of elementary mathematics, the instruction of high‐level mathematical concepts can often be sacrificed at the expense of a focus on algorithmic procedures. Computer‐based simulations can expand an undergraduate mathematics instructor's opportunity to explore high‐level mathematical concepts in an applied environment. This study describes one instructor's approach to incorporating simulations and classroom discussions in a differential equations course and the subsequent effects on student learning attitudes and outcomes. Students made modest gains in the area of conceptualizing and applying ideas regarding solutions to differential equations in this learning environment. Implications of the study include the identification of specific gains relative to computer‐mediated learning environments and recommendations for using simulations to support conceptual development.     

N‐person Prisoner's Dilemma†

Multi‐person versions of Prisoner's Dilemma are widely applicable in the social sciences. Examination of two important classes of real‐world situations reveals that although both can appropriately be called Prisoner's Dilemma, they have incompatible payoff structures. Thus Prisoner's Dilemma games constitute an important but apparently ambiguous set of models.

We therefore undertake a taxonomy of multi‐person Prisoner's Dilemma. Some aspects of the well‐studied two‐person case provide a useful beginning for the task. In the general multi‐person form, however, some properties of the two‐person game are found incompatible with others and so are dropped. Additional properties are suggested by strategic considerations and the associated social phenomena. We demonstrate interdependencies among the various properties and relate some of them to a simple graphical representation.     

Hybrid synthesis of the optimal discrete nonlinear filter

It has been known for many years that an optimal discrete nonlinear filter may be synthesized for systems whose plant dynamics, sensor characteristics and signal statistics are known by applying Bayes' Rule to sequentially update the conditional probability density function from the latest data. However, it was not until 1969 that a digital computer algorithm implementing the theory for a one-state variable one-step predictor appeared in the literature. This delay and the continuing scarcity of multidimensional nonlinear filters result from the overwhelming computational task which leads to unrealistic data processing times. For many nonlinear filtering problems analog and digital computers (a hybrid computation) combine to yield a higher data rate than can be obtained by con¬ventional digital methods. This paper describes an implementation of the theory by means of a hybrid computer algorithm for the optimal nonlinear one-step predictor.

The hybrid computer algorithm presented reduces the overall solution time per prediction because:

1) Many large computations of identical form are executed on the analog computer in parallel.

2) The discrete running variable in the digital algorithm may be replaced with a continuous analog computer variable in one or more dimensions leading to increased computational speed and finer resolution of the exponential transformation.

3) The modern analog computer is well suited to generate functions such as the expo¬nential at high speed with modest equipment.

4) The arithmetic, storage, and control functions performed rapidly by the digital computer are utilized without introducing extensive auxiliary calculations.

To illustrate pertinent aspects of the algorithm developed, the scalar cubed sensor problem previously described by Bucy is treated extensively. The hybrid algorithm is described. Problems associated with partitioning of equations between analog and digital computers, machine representations of variables, setting of initial conditions and floating of grid base are discussed. The effects of analog component bandwidths, digital-to-analog and analog-to-digital conversion times, analog computer mode switching times and digital computer I/O data rates on overall processing time are examined. The effect of limited analog computer dynamic range on accuracy is discussed. Results from a simulation of this optimal predictor using MOBSSL, a continuous system simulation language, are given. Timing estimates are presented and compared against similar estimates for the all digital algorithm.

For example, given a four-state variable optimal 1-step predictor utilizing 7 discrete points in each dimension, the hybrid algorithm can be used to generate predictions accurate to 2 decimal places once every 10 seconds. An analog computer complement of 250 integra¬tors and multipliers and a high-speed 3rd generation digital computer such as the CDC 6600 or IBM 360/85 are required. This compares with a lower bound of about 3 seconds per all digital prediction which would require 49 CDC 6600's operating in parallel. Analytical and simulation work quantifying errors in one state variable filters is presented. Finally, the use of an interactive graphic system for real time display and for filter evaluation is described.     

Teaching Children to be Mathematicians Versus Teaching About Mathematics

The important difference between the work of a child in an elementary mathematics class and that of a mathematician is not in the subject matter (old fashioned numbers versus groups or categories or whatever) but in the fact that the mathematician is creatively engaged in the pursuit of a personally meaningful project. In this respect a child's work in an art class is often close to that of a grown‐up artist. The paper presents the results of some mathematical research guided by the goal of producing mathematical concepts and topics to close this gap. The prime example used here is ‘Turtle Geometry’, which is concerned with programming a moving point to generate geometric forms. By embodying the moving point as a ‘cybernetic turtle’ controlled by an actual computer, the constructive aspects of the theory come out sufficiently to capture the minds and imaginations of almost all the elementary school children with whom we have worked—including some at the lowest levels of previous mathematical performance.

     

Developing a mathematical theory of computability which speaks the language of levels

In this paper, we use the modern systems theory to retrace the history of some of the important and interesting philosophical problems. And, based on the discussion, it is shown that the multilevel structure of the nature can be approached by making use of the general systems theory approach started by Mesarovic in the early 1960s. Two difficulties appearing in modern physics are listed, and studied in terms of a new mathematical theory. This theory reflects the characteristic of multilevels of the nature. Some elementary properties of the new theory are listed. Some important and fruitful leads for future research are posed in the final section.     

Dynamical behavior of fractional‐order energy‐saving and emission‐reduction system and its discretization

This paper attempts to explore dynamical behavior and mathematical properties of the three‐dimensional fractional‐order energy‐saving and emission‐reduction system. Theoretically, the conditions of local stability of fractional‐order system's equilibrium points are obtained. Numerical investigations on the dynamics of this system are carried out, and the existence of the asymptotically stable attractor is found. Combined with the fractional‐order subsystem, we discuss the relationship between energy‐saving and emission‐reduction and economic growth, and carbon emissions and economic growth. Furthermore, we discretize the fractional‐order system and give necessary and sufficient conditions of its stabilization. It is shown that the stability of the discretization system is impacted by the system's fractional parameter. Numerical simulations show the richer dynamical behavior of the fractional‐order system and verify the theoretical results. Recommendations for Resource Managers

• The impact of carbon emissions on economic growth is one of the main reasons for energy‐saving and emission‐reduction.
• Control measures on people's low‐carbon life through government intervention are required to protect the natural environment.
• New energy‐saving and emission‐reduction technologies should be implemented to achieve sustainable social and economic development.

     

Affect control theory: An assessment*

This paper reviews affect control theory's major strengths, the contributions of recent work to its growth, and the most promising avenues for future work. Affect control theory's strengths include (1) the precision of its mathematical statement and empirical base (especially when compared with earlier interpretive sociologies), (2) its ability to link the internal processing that generates social action to the socio‐cultural system upon which that action is based, and (3) the generality that allows a parsimonious explanation of a wide range of processes and previous research findings. Recent advances provide (1) new, more accurate impression‐change formulas, (2) the expansion of the theory to encompass settings, emotions, and traits, (3) new dictionaries of evaluation, potency and acitivity meanings and (4) tests of the theory using likelihood judgments, verbal scenarios and actual behavior of naive experimental subjects. Further work must include links to cognitive structures that will further delineate definition of situation and behavior selection processes. In addition, integration of affect control theory with new sociological work on the development of shared social knowledge and on institutionalized production systems expand the theory in useful ways. Finally, new work must find innovative and convincing ways to test simulation outcomes using both verbal accounts and behavior.     

Disambiguating verbal comments in social interaction: A computer model of meaning*

This paper describes a computer program for disambiguating the meaning of verbal acts in social interaction. The program was implemented using artificial intelligence techniques, representing utterances by frames, designing separate agents employing procedural rules to infer values of each coding dimension, and sharing information through a blackboard. The computer algorithms are discussed and illustrated with examples, then results of an empirical test of the program are reported. A training‐set of verbal interactions among health care practitioners and patients was examined to build the program, with an independent test‐set of data used to assess program performance. On a practical level, this program offers a promising approach for computer‐assisted or even automatic coding of interaction processes, reducing coding costs and improving reliability and validity. On a theoretical level, these algorithms offer a model of how individuals disambiguate the meaning of verbal comments in social interaction, providing substantive insights into the mechanisms of social interaction.     

On Farkas' lemma and related propositions in BISH

In this paper we analyse in the framework of constructive mathematics (BISH) the validity of Farkas' lemma and related propositions, namely the Fredholm alternative for solvability of systems of linear equations, optimality criteria in linear programming, Stiemke's lemma and the Superhedging Duality from mathematical finance, and von Neumann's minimax theorem with application to constructive game theory.