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# analysis

[uh-nal-uh-sis]
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.
analysis, chemical: see chemical analysis.

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.

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.

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.

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.

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.

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.

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.

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.

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.

Analysis (from Greek ἀνάλυσις, "a breaking up") is the process of breaking a complex topic or substance into smaller parts to gain a better understanding of it. The technique has been applied in the study of mathematics and logic since before Aristotle, though analysis as a formal concept is a relatively recent development.

As a formal concept, the method has variously been ascribed by Ibn al-Haytham, Descartes (Discourse on the Method), Galileo, and Isaac Newton, as a practical method of physical discovery.

## Use in specific fields

### Chemistry

The field of chemistry uses analysis to break down chemical processes and examine chemical reactions between elements of matter. For example, analysis of the concentration of elements is important in managing a nuclear reactor, so nuclear scientists will analyze neutron activation to develop discrete measurements within vast samples. A matrix can have a considerable effect on the way a chemical analysis is conducted and the quality of its results. Analysis can be done manually or with a device. Chemical analysis is an important element of national security among the major world powers with Materials Measurement and Signature Intelligence (MASINT) capabilities.

#### Isotopes

Chemists can use isotopes to assist analysts with issues in anthropology, archeology, food chemistry, forensics, geology, and a host of other questions of physical science. Analysts can discern the origins of natural and man-made isotopes in the study of environmental radioactivity.

### Engineering

Analysts in the field of engineering look at structures, mechanisms, systems and dimensions. Electrical engineers analyze systems in electronics. Life cycles and system failures are broken down and studied by engineers.

### Intelligence

The field of intelligence employs analysts to break down and understand a wide array of questions. intelligence agencies may use heuristics, inductive and deductive reasoning, social network analysis, dynamic network analysis, link analysis, and brainstorming to sort through problems they face. Military intelligence may explore issues through the use of game theory, Red Teaming, and wargaming. Signals intelligence applies cryptanalysis and frequency analysis to break codes and ciphers. Business intelligence applies theories of competitive intelligence analysis and competitor analysis to resolve questions in the marketplace. Law enforcement intelligence applies a number of theories in crime analysis.

### Linguistics

Linguistics began with the analysis of Sanskrit; today it looks at individual languages and language in general. It breaks language down and analyzes its component parts: theory, sounds and their meaning, utterance usage, word origins, the history of words, the meaning of words and word combinations, sentence construction, basic construction beyond the sentence level, stylistics, and conversation. It examines the above using statistics and modeling, and semantics. It analyzes language in context of anthropology, biology, evolution, geography, history, neurology, psychology, and sociology. It also takes the applied approach, looking at individual language development and clinical issues.

### Literary criticism

• Analysis (Homer), an influential school of thought in Homeric scholarship in the 19th-20th centuries
• Psychocriticism, Charles Mauron's method based on Freud's own initial interpretations of literary works such as Hamlet

### Statistics

• Analysis of variance (ANOVA), a collection of statistical models and their associated procedures which compare means by splitting the overall observed variance into different parts
• Meta-analysis, combines the results of several studies that address a set of related research hypotheses
• Time-series analysis, methods that attempt to understand a sequence of data points spaced apart at uniform time intervals

### Other

• Aura analysis, a technique in which supporters of the method claim that the body's aura, or energy field is analysed
• Bowling analysis, a notation summarizing a cricket bowler's performance
• Lithic analysis, the analysis of stone tools using basic scientific techniques
• Protocol analysis, a means for extracting persons' thoughts while they are performing a task