This chapter describes the recognition of patterns. Recognition can be regarded as a basic attribute of living organisms. A pattern is a description of an object. Recognition of patterns is a basic activity of all living organisms. The ability to recognize patterns is a necessary part of survival. In the animal kingdom, one's survival depends upon his ability to recognize a friend and a foe. An infant learns to recognize its mother's face and voice at an early age. One can recognize people from an analysis of their hand-writing, fingerprints, and voice prints. Acts of recognition can be divided into two major types: recognition of concrete items and recognition of abstract items. Recognition of spatial and temporal patterns using one's visual and aural sensory apparatus belongs to the former type. Examples of spatial patterns are alphanumeric characters, fingerprints, and pictures. Temporal patterns include speech waveforms, electrocardiograms, time series, and target signatures. Recognition of conceptual patterns, such as the proof of a theorem, belongs to the latter type. The subject of pattern recognition spans a number of disciplines.

This chapter presents an analysis of competitive situations. An important class of problems in the domain of modern mathematics and system theory involves selecting a combination of variables that maximizes or minimizes some predefined variable or function of the problem. This problem, called the optimization problem, becomes formidable when a number of combinations in which the variables can be combined are large. With the development of modern computers, the computational difficulties are no longer insurmountable. The chapter describes linear programming. The general problem of finding the maximum or minimum value of a function of one or many variables—when the function might be linear or non-linear, discrete valued or continuous, constrained or unconstrained—is a difficult problem to solve. The twin concepts maximum and minimum have been distinguished. Mathematically, the minimum value that is attained by a function f(x) is the same as the maximum value attained by the function -f(x). It is possible to talk about maximization or minimization, interchangeably, without a loss of generality.

This chapter presents the modeling process. A model is a representation of a system in a convenient fashion so that the conjectures that are made about the performance of the system can be readily tested. The term “model” implies a manifestation of the interpretation that a scientist gives to observed facts. Facts remain unchanged, but models change. The term “model” covers a variety of configurations. It can be a symbolic model, such as a set of mathematical or logical equations; an iconic model, such as a map or a scale model; an analog model, such as an electrical or mechanical device; or a logical model, such as a flow chart or a computer program. Models can be categorized into different classes and sub-classes, and this classification can be done in different ways. A distinction can be made between descriptive models and prescriptive models. A mathematical model is a set of equations that characterizes a real-life system, the prototype system, in the sense that some of the excitation response relationships of the prototype system are correctly represented.

This chapter provides an overview of the primitive models. One of the methods of developing a model is to follow the Leibnizian approach, that is, to invoke the well-established laws of nature and develop mathematical relations from them. One can follow the Lockean approach. When a well-tested law is not readily apparent, this is a powerful approach to follow. The chapter discusses the interdependence of model building and data gathering phases in the study of complex phenomena. The example used demonstrates a synthesis of several ideas. First, the time element has been introduced. Second, the methodology of using intuition, heuristic arguments, graphical procedures, and systematic computational algorithms has been demonstrated. The example illustrates how the need arises to refine and modify a model after it is initially determined. Another example illustrates the methods of formalizing the ever-exciting problems, such as puzzles, on the one hand, and the equally challenging problems of building models of social systems on the other hand.

This chapter provides an overview of forecasting. Forecasting, or prediction of the probable future course of events, is the most pervasive and important element in efficient planning. The ultimate effect of a decision depends on the outcome of factors that cannot be foreseen at the time a decision is made. The role of forecasting cuts across many fields of endeavor management, such as, operations research, economics, and systems analysis. In the majority of day to day decisions, forecasting is done using a heuristic and intuitive approach that is based on subjective evaluations. While dealing with complex systems involving large investments of money and manpower, mathematical, and statistical methods would be useful in imparting quantitative attributes to the quality of forecasts made. As no forecast can be made in a complete vacuum, an essential prerequisite for a good forecast is a good set of data.

This chapter discusses simple growth processes using discrete time or difference equation models. It focuses on continuous growth processes described by ordinary differential equations. Models representing growth processes are non-linear, the non-linearity representing the limiting nature of growth. Therefore, growth processes are characterized by non-linear differential equations. Growth and decay processes are important in modeling complex systems. Growth of population and pollution, growth of cancer cells, growth of poverty, and unrest are the problems of interest. Whenever and wherever there is a change, it can be regarded as a growth or decay process. Any growth or decay process can be treated as discrete or continuous. The choice depends upon one's viewpoint, the time duration involved, and the resolution level used. Some of the simplest growth processes can be found in everyday experience.

This chapter provides an overview of the system that is one of the most widely used concepts in scientific investigations. Different types of systems are familiar to us from everyday experience: mechanical systems, such as clocks; electrical systems, such as radios; industrial systems, such as factories; educational systems, such as universities; information processing systems, such as computers; medical systems, such as hospitals; organizational systems; environmental systems; and cybernetic systems. The behavior of systems is not always exemplary. Economic systems are subject to inflation, recession, and depression; biological systems are subject to disease and decay; educational systems are subject to obsolescence; ecological systems are subject to pests and pollution; and hydrosystems are subject to floods and droughts. Different ways help to improve the quality of performance of systems. One can build a new system and discard the old one. This is a common phenomenon in some political systems and can be done with simple mechanical systems. In several cases, replacement is not at all a feasible solution. Examples belonging to this category include human bodies and environment.

This chapter discusses the language of system theory. The concept of “system” is as old as civilization. A system is a collection of the objects that are united by some form of interaction or interdependence. A system might be regarded as a set of interrelated elements. The nature of elements that are involved and the relations that might exist among the elements depend upon the type of system under consideration. The fact that a system can be regarded as a set is useful because this allows one to draw freely from the edifice of knowledge that is available in set theory, topology, and function theory. The concept of a system can be traced back to antiquity; the “systems approach” concept appears to be of a recent origin. The phrase “systems approach” is associated with a cadre of disciplines that have arisen since World War II, the most familiar being operations research, management science, and systems analysis.

This chapter describes the linear dynamical structure. One of the goals in modeling is to learn about a system so as to gain control of it and to make it operate in a more desirable way. The goal of a control problem is to force one or more of the system variables to follow a prescribed pattern of behavior. In a biological system, such as a human being, the goal of the control problem might be to reduce pain and prolong life. In an economic system, the factors of interest might be economic growth, inflation, and expenditure of resources. Analogous problems abound in a broad spectrum of processes including electrical, mechanical, industrial, biological, economical, and societal systems. The chapter highlights the Pontryagin's maximum principle. Pontryagin's maximum principle is a method based on the calculus of variations and is not unlike the Lagrange multiplier method. The maximum principle provides only necessary conditions, but not sufficient conditions.

This chapter describes the static equilibrium models. The complexity of structure of many systems is revealed by a graphical approach. The graphs that have been presented in the chapter are simple geometric figures consisting of points and lines that connect some of these points. These graphs are also called “linear graphs.” As graphs are pictorial in nature, they convey and carry the information that is most suitable for human processing. As the directed lines represent relations, collections of directed lines, or directed graphs, can be used to represent the interactions among various entities of a system. While modeling large systems, a graphical representation of the various activities is a useful first step. If the set of edges in a linear graph is finite, it is called a “finite graph.” A subset of a graph is called a “sub-graph.” If the edges of a graph can be ordered such that each edge has one node in common with the preceding edge and one with the succeeding edge, then the ordered sequence of edges is called a “path.”