transactional analysis: see group psychotherapy.

complex variable analysis, branch of mathematics that deals with the calculus of functions of a complex variable, i.e., a variable of the form z=x+iy, where x and y are real and i=-1 (see number). A function w=f(z) of a complex variable z is separable into two parts, w = g₁(x,y) + ig₂(x,y), where g₁ and g₂ are real-valued functions of the real variables x and y. The theory of functions of a complex variable is concerned mainly with functions that have a derivative at every point of a given domain of values for z; such functions are called analytic, regular, or holomorphic. If a function is analytic in a given domain, then it also has continuous derivatives of higher order and can be expanded in an infinite series in terms of these derivatives (i.e., a Taylor's series). The function can also be expressed in the infinite serieswhere z₀ is a point in the domain. Also of interest in complex variable analysis are the points in a domain, called singular points, where a function fails to have a derivative. The theory of functions of a complex variable was developed during the 19th cent. by A. L. Cauchy, C. F. Gauss, B. Riemann, K. T. Weierstrass, and others.

chemical analysis, the study of the chemical composition and structure of substances. More broadly, it may be considered the corpus of all techniques whereby any exact chemical information is obtained. There are two branches in analytical chemistry: qualitative analysis and quantitative analysis. Qualitative analysis is the determination of those elements and compounds that are present in a sample of unknown material. Quantitative analysis is the determination of the amount by weight of each element or compound present. The procedures by which these aims may be achieved include testing for the chemical reaction of a putative constituent with an admixed reagent or for some well-defined physical property of the putative constituent. Classical methods include use of the analytical balance, gas manometer, buret, and visual inspection of color change. Gas and paper chromatography are particularly important modern methods. Physical techniques such as use of the mass spectrometer are also employed. For samples in the gaseous state, optical spectroscopy provides the best technique for determining which atomic and molecular species are present.

analysis, chemical: see chemical analysis.

analysis situs: see topology.

analysis, branch of mathematics that utilizes the concepts and methods of the calculus. It includes not only basic calculus, but also advanced calculus, in which such underlying concepts as that of a limit are subjected to rigorous examination; differential and integral equations, in which the unknowns are functions rather than numbers, as in algebraic equations; complex variable analysis, in which the variables are of the form z=x+iy, where i is the imaginary unit; vector analysis and tensor analysis; differential geometry; and many other fields.

Any of a group of chemical analysis methods that depend on measurement of the wavelength and intensity of electromagnetic radiation. It is used chiefly to determine the arrangement of atoms and electrons in molecules on the basis of the amounts of energy absorbed during changes in their structure or motion. In more common usage, it usually refers to ultraviolet (UV) and visible emission spectroscopy or to UV, visible, and infrared (IR) absorption spectrophotometry.

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Branch of analysis devoted to identifying elements and compounds and elucidating atomic and molecular structure by measuring the radiant energy absorbed or emitted by a substance at characteristic wavelengths of the electromagnetic spectrum (including gamma ray, X-ray, ultraviolet, visible light, infrared, microwave, and radio-frequency radiation) on excitation by an external energy source. The instruments used are spectroscopes (for direct visual observation) or spectrographs (for recording spectra). Experiments involve a light source, a prism or grating to form the spectrum, detectors (visual, photoelectric, radiometric, or photographic) for observing or recording its details, devices for measuring wavelengths and intensities, and interpretation of the measured quantities to identify chemicals or give clues to the structure of atoms and molecules. Helium, cesium, and rubidium were discovered in the mid-19th century by spectroscopy of the Sun's spectrum. Specialized techniques include Raman spectroscopy (see Chandrasekhara Venkata Raman), nuclear magnetic resonance (NMR), nuclear quadrupole resonance (NQR), dynamic reflectance spectroscopy, microwave and gamma ray spectroscopy, and electron spin resonance (ESR). Spectroscopy now also includes the study of particles (e.g., electrons, ions) that have been sorted or otherwise differentiated into a spectrum as a function of some property (such as energy or mass). Seealso mass spectrometry; spectrometer; spectrophotometry.

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Branch of applied mathematics that studies methods for solving complicated equations using arithmetic operations, often so complex that they require a computer, to approximate the processes of analysis (i.e., calculus). The arithmetic model for such an approximation is called an algorithm, the set of procedures the computer executes is called a program, and the commands that carry out the procedures are called code. An example is an algorithm for deriving π by calculating the perimeter of a regular polygon as its number of sides becomes very large. Numerical analysis is concerned not just with the numerical result of such a process but with determining whether the error at any stage is within acceptable bounds.

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Economic analysis developed by Wassily Leontief, in which the interdependence of an economy's various productive sectors is observed by viewing the product of each industry both as a commodity for consumption and as a factor in the production of itself and other goods. For example, input-output analysis will break down a nation's total production of trucks, showing that some trucks are used in the production of more trucks, some in farming, some in the production of houses, and so on. An input-output analysis is usually summarized in a gridlike table showing what various industries buy from and sell to one another.

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Technique used in the physical sciences and engineering to reduce physical properties such as acceleration, viscosity, energy, and others to their fundamental dimensions of length, mass, and time. This technique facilitates the study of interrelationships of systems (or models of systems) and their properties. Acceleration, for example, is expressed as length per unit of time squared; whether the units of length are in the English or metric system is immaterial. Dimensional analysis is often the basis of mathematical models of real situations.

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In statistics and related subfields of philosophy, the theory and method of formulating and solving general decision problems. Such a problem is specified by a set of possible states of the environment or possible initial conditions; a set of available experiments and a set of possible outcomes for each experiment, giving information about the state of affairs preparatory to making a decision; a set of available acts depending on the experiments made and their consequences; and a set of possible consequences of the acts, in which each possible act assigns to each possible initial state some particular consequence. The problem is dealt with by assessing probabilities of consequences conditional on different choices of experiments and acts and by assigning a utility function to the set of consequences according to some scheme of value or preference of the decision maker. An optimal solution consists of an optimal decision function, which assigns to each possible experiment an optimal act that maximizes the utility, or value, and a choice of an optimal experiment. See also cost-benefit analysis, game theory.

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In governmental planning and budgeting, the attempt to measure the social benefits of a proposed project in monetary terms and compare them with its costs. The procedure was first proposed in 1844 by Arsène-Jules-Étienne-Juvénal Dupuit (1804–66). It was not seriously applied until the 1936 U.S. Flood Control Act, which required that the benefits of flood-control projects exceed their costs. A cost-benefit ratio is determined by dividing the projected benefits of a program by the projected costs. A wide range of variables, including nonquantitative ones such as quality of life, are often considered because the value of the benefits may be indirect or projected far into the future.

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Laboratory examination of the physical and chemical properties and components of a sample of blood. Analysis includes number of red and white blood cells (erythrocytes and leukocytes); red cell volume, sedimentation (settling) rate, and hemoglobin concentration; blood typing; cell shape and structure; hemoglobin and other protein structure; enzyme activity; and chemistry. Special tests detect substances characteristic of specific infections.

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Field of mathematics that incorporates the methods of algebra and calculus—specifically of limits, continuity, and infinite series—to analyze classes of functions and equations having general properties (e.g., differentiability). Analysis builds on the work of G.W. Leibniz and Isaac Newton by exploring the applications of the derivative and the integral. Several distinct but related subfields have developed, including the calculus of variations, differential equations, Fourier analysis (see Fourier transform), complex analysis, vector and tensor analysis, real analysis, and functional analysis. Seealso numerical analysis.

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