Meaning of ANALYSIS in English

ANALYSIS

in physics and chemistry, determination of the physical properties or chemical composition of samples of matter or, particularly in modern physics, of the energy and other properties of subatomic particles produced in nuclear interactions. A large body of systematic procedures intended for these purposes has been continuously evolving in close association with the development of other branches of the physical sciences since their beginnings. Chemical analysis, which relies on the use of measurements, is divided into two categories depending on the manner in which the assays are performed. Classical analysis, also termed wet chemical analysis, consists of those analytical techniques that use no mechanical or electronic instruments other than a balance. The method usually relies on chemical reactions between the material being analyzed (the analyte) and a reagent that is added to the analyte. Wet techniques often depend on the formation of a product of the chemical reaction that is easily detected and measured. For example, the product could be coloured or could be a solid that precipitates from a solution. Most chemical analysis falls into the second category, which is instrumental analysis. It involves the use of an instrument, other than a balance, to perform the analysis. A wide assortment of instrumentation is available to the analyst. In some cases, the instrument is used to characterize a chemical reaction between the analyte and an added reagent; in others, it is used to measure a property of the analyte. Instrumental analysis is subdivided into categories on the basis of the type of instrumentation employed. Both classical and instrumental quantitative analyses can be divided into gravimetric and volumetric analyses. Gravimetric analysis relies on a critical mass measurement. As an example, solutions containing chloride ions can be assayed by adding an excess of silver nitrate. The reaction product, a silver chloride precipitate, is filtered from the solution, dried, and weighed. Because the product was formed by an exhaustive chemical reaction with the analyte (i.e., virtually all of the analyte was precipitated), the mass of the precipitate can be used to calculate the amount of analyte initially present. Volumetric analysis relies on a critical volume measurement. Usually a liquid solution of a chemical reagent (a titrant) of known concentration is placed in a buret, which is a glass tube with calibrated volume graduations. The titrant is added gradually, in a procedure termed a titration, to the analyte until the chemical reaction is completed. The added titrant volume that is just sufficient to react with all of the analyte is the equivalence point and can be used to calculate the amount or concentration of the analyte that was originally present. Since the advent of chemistry, investigators have needed to know the identity and quantity of the materials with which they are working. Consequently, the development of chemical analysis parallels the development of chemistry. The 18th-century Swedish scientist Torbern Bergman is usually regarded as the founder of inorganic qualitative and quantitative chemical analysis. Prior to the 20th century nearly all assays were performed by classical methods. Although simple instruments (such as photometers and electrogravimetric analysis apparatus) were available at the end of the 19th century, instrumental analysis did not flourish until well into the 20th century. The development of electronics during World War II and the subsequent widespread availability of digital computers have hastened the change from classical to instrumental analysis in most laboratories. Although most assays currently are performed instrumentally, there remains a need for some classical analyses. one of the main divisions of mathematics, the others being history and foundations, algebra, combinatorics and number theory, geometry, topology, and applied mathematics. In extent, analysis is the largest of these, comprising subdivisions that are nearly autonomous and easier to describe than the division as a whole. Analysis may be defined as that part of mathematics concerned with smooth abstract objects such as sets of numbers, sets of geometric points, or sets of functions that map numbers into numbers or points into points and with the processes, called limit processes, that depend on a measure of closeness between numbers, points, or functions. Analysis developed initially out of ad hoc arguments in which curves, geometric bodies, and physical motions were analyzed by an informal process of decomposition into infinitesimal parts. Limit and approximation processes, the usefulness of which was discovered during this early period of development, came to be used systematically in the 17th and 18th centuries in the differential and integral calculus of Sir Isaac Newton and Gottfried Wilhelm Leibniz. Limit processes were applied by their followers and by 19th-century French and German schools of analysts to sequences of values of a single function. Differential and integral calculus, ordinary differential equations, and the calculus of variations have thus arisen from mechanics; so-called Fourier series from acoustics and thermodynamics; complex analysis from optics, hydrodynamics, and electricity; and partial differential equations from elasticity, hydrodynamics, and electrodynamics. Even mathematical probability, although born from problems of gambling and human chance, drew much of its syllogistic strength in the 19th century from statistical theories of mechanics and thermodynamics. The Editors of the Encyclopdia Britannica in mathematics, the extended sequence of mathematical developments that flow out of the discovery, by Sir Isaac Newton and Gottfried Wilhelm Leibniz in the late 17th century, of the differential and integral calculus. Since that time, several more or less distinct fields of analysis have evolved, including infinite series, the calculus of variations, differential equations, Fourier analysis, complex analysis, vector and tensor analysis, and functional analysis. Other branches of mathematics have been profoundly affected by ideas originating in analysis, notably differential geometry, set theory, and topology. The development of the calculus introduced several ideas fateful for the subsequent history of mathematics. New techniques for calculating properties of functionsin particular their maxima and minima and the areas and volumes of plane and three-dimensional regionsmade it possible for many previously difficult results to be derived routinely. Important notions of indefinitely close approximation (limits) and related techniques for arbitrarily close approximation of solutions to general equations also were introduced. Another significant idea was that of the infinite seriesthe expression of a quantity or function as the sum of an unending series of smaller and smaller increments. Newton and his immediate followers realized that this procedure made it possible to adapt older techniques, developed for calculations with finite polynomials, to the investigation of more general mathematical relationships. The immediate successors of Newton and Leibniz, including the Bernoulli family and Leonhard Euler, extended the techniques of the calculus, defining the subject matter and approach that have subsequently remained standard. Euler, in particular, investigated the properties of the trigonometric, logarithmic, and exponential functions, establishing many relationships between them by manipulation of their representations as infinite series. He and the Bernoullis also extended Newton's techniques to quantities dependent upon the entire shape of a curve or surface, thereby inventing the so-called calculus of variations. These studies and Newton's use of the differential calculus to study physical motions led to an interest in differential equations, that is, relationships between functions of one or more variables and their derivatives. Early mathematical analysts sought explicit integrals or solutions of differential equations, in terms of finite combinations of known classical functions, or, at worst, by infinite series of such terms. By the latter part of the 18th century, however, it was realized that many important problems had no explicit solutions of this type. Investigations in analysis then became more qualitative, emphasizing the derivation of useful general properties of classes of functions, rather than requiring specific formulas for them. The calculus of functions of several variables proved useful in the study of curves and curved surfaces in space, an application pursued by Carl Friedrich Gauss, who, by applying the methods of the differential calculus to the study of geometric questions, founded differential geometry. Early in the 19th century, J.-B.-J. Fourier showed that arbitrary functions can be represented by infinite series of sines and cosines. These series are analogous to the representation of a vector in three-dimensional space. This analogy served in the 20th century to spur the rise of functional analysis. A.-L. Cauchy and others extended the calculus to complex functions. Cauchy showed that, if a function of a complex variable has a derivative, then its value at a given point can be expressed as an integral (Cauchy integral) of its values at other points. Cauchy's work eventually related analysis to topology, as emphasized in the work of Bernhard Riemann, which opened themes that were adequately explored only a century later. During the 19th century, increasing concern for logical rigour and a series of critical studies of the foundations of analysisamong which the works of Cauchy, Richard Dedekind, and Georg Cantor are particularly importantresulted in a stringent decomposition of the concept of continuity into more primitive set-theoretic notions and in the recognition that the whole structure of analysis built during the two preceding centuries could be based firmly on a handful of such principles. The exploitation of analogies between the geometry of vectors and the properties of collections of functions has been a major theme of 20th-century analysis. David Hilbert emphasized the close relationship between certain types of equations involving integrals and linear systems of algebraic equations in several variables. Hermann Weyl showed that Hilbert's methods allowed important classes of series expansions to be derived through the study of simple differential equations. John von Neumann and Stefan Banach systematically developed this new field, called functional analysis, giving polished abstract formulations to many of its principles and techniques. Functional analysis has led to deeper understanding of the solutions of partial differential equations important in physics and engineering. It also has revealed relationships between algebra, analysis, and topology. in physics and chemistry, determination of the physical properties or chemical composition of samples of matter or, particularly in modern physics, of the energy and other properties of subatomic particles produced in nuclear interactions. A large body of systematic procedures intended for these purposes has been continuously evolving in close association with the development of other branches of the physical sciences since their beginnings. Chemical analysis, in particular, has developed into a highly diversified discipline, with distinct branches oriented toward the solution of specific kinds of problems. A sample of a single compound may be analyzed to establish its elemental composition or its molecular structure; composition is most commonly found by procedures involving chemical reactions, which destroy the sample, but structure is more often studied by nondestructive measurements of physical properties. Many such measurements entail spectroscopic techniques that pinpoint the wavelengths at which electromagnetic radiation is absorbed or emitted by the substance. Mixtures of substances are ordinarily analyzed by separating, detecting, and identifying their components by methods that depend on differences in their physical properties, such as volatility, mobility in an electric or a gravitational field, or distribution between two immiscible substances. Additional reading Herbert A. Laitinen and Galen W. Ewing (eds.), A History of Analytical Chemistry (1977), provides a historical overview. General works on analytical chemistry include Larry G. Hargis, Analytical Chemistry: Principles and Techniques (1988); Douglas A. Skoog, Donald M. West, and F. James Holler, Fundamentals of Analytical Chemistry, 5th ed. (1988), also available in an abbreviated version, Analytical Chemistry: An Introduction, 5th ed. (1990); Kenneth A. Rubinson, Chemical Analysis (1987); Daniel C. Harris, Quantitative Chemical Analysis, 3rd ed. (1991); John H. Kennedy, Analytical Chemistry: Principles, 2nd ed. (1990); and Stanley E. Manahan, Quantitative Chemical Analysis (1986). Analytical Chemistry (semimonthly); and Analytical Biochemistry (16/yr.), are useful periodicals.The following are useful texts on qualitative analysis: Daniel J. Pasto and Carl R. Johnston, Organic Structure Determination (1969); Ralph L. Shriner et al., The Systematic Identification of Organic Compounds, 6th ed. (1980), a laboratory manual; John W. Lehman, Operational Organic Chemistry: A Laboratory Course, 2nd ed. (1988); and J.J. Lagowski and C.H. Sorum, Introduction to Semimicro Qualitative Analysis, 7th ed. (1991).Instrumental analysis is the focus of Robert D. Braun, Introduction to Instrumental Analysis (1987); Hobart H. Willard et al., Instrumental Methods of Analysis, 7th ed. (1988); Gary D. Christian and James E. O'Reilly (eds.), Instrumental Analysis, 2nd ed. (1986); J.D. Winefordner (ed.), Spectrochemical Methods of Analysis (1971); Joseph B. Lambert et al., Organic Structural Analysis (1976); James D. Ingle, Jr., and Stanley R. Crouch, Spectrochemical Analysis (1988); Allen J. Bard and Larry R. Faulkner, Electrochemical Methods (1980); E.P. Serjeant, Potentiometry and Potentiometric Titrations (1984); A.M. Bond, Modern Polarographic Methods in Analytical Chemistry (1980); and R. Belcher (ed.), Instrumental Organic Elemental Analysis (1977). Robert Denton Braun Additional reading Real analysis The history of the calculus is well covered in Otto Toeplitz, The Calculus: A Genetic Approach (1963; originally published in German, 1949); and Carl B. Boyer, The Concepts of the Calculus (1949, reprinted as The History of the Calculus and Its Conceptual Development (1959). Good introductions may be found in Richard Courant and Herbert Robbins, What Is Mathematics? (1941, reissued 1980); and Ian Stewart, The Problems of Mathematics (1987). Many excellent texts at all levels of difficulty are available. Two interesting examples are Sherman K. Stein, Calculus in the First Three Dimensions (1967); and Morris Kline, Calculus: An Intuitive and Physical Approach. 2nd ed. (1977). For modern theories of measure and integration, the following may be consulted: Stanislaw Saks, Theory of the Integral, trans. from Polish, 2nd rev. ed. (1937, reprinted 1964); Angus E. Taylor, General Theory of Functions and Integration (1965, reprinted 1985); and Edgar Asplund and Lutz Bungart, A First Course in Integration (1966). These all require a good foundation in calculus. Additional references include A.N. Kolmogorov and S.V. Fomin, Introductory Real Analysis, rev. ed. (1970; originally published in Russian, 2 vol., 195460); M.E. Munroe, Calculus (1970), and Introductory Real Analysis (1965); H.L. Royden, Real Analysis, 3rd ed. (1988); Walter Rudin, Principles of Mathematical Analysis, 3rd ed. (1976); G.H. Hardy, A Course of Pure Mathematics, 10th ed. (1952, reprinted 1975); and Angus E. Taylor and W. Robert Mann, Advanced Calculus, 3rd ed. (1983).General references to the elements of the theory of convergent series may be found in any book of calculus. More advanced texts include Tomlinson Fort, Infinite Series (1930); Thomas J. I'A. Bromwich, An Introduction to the Theory of Infinite Series, 2nd ed. rev. (1926, reprinted 1965); Konrad Knopp, Theory and Application of Infinite Series, 2nd ed. (1951, reissued 1971; originally published in German, 2nd enlarged ed., 1924), containing a wealth of material; and I.I. Hirschman, Infinite Series (1962, reprinted 1978). Also useful are Godfrey H. Hardy, Divergent Series (1949, reprinted 1973); H.R. Pitt, Tauberian Theorems (1958); and, for summability of integrals and related topics, K. Chandrasekharan and S. Minakshisundaram, Typical Means (1952).The following works concern vector and tensor analysis. Textbooks on the algebra of vectors include, in approximate order from elementary to advanced, Daniel Zelinsky, A First Course in Linear Algebra, 2nd ed. (1973); Daniel T. Finkbeiner II, Introduction to Matrices and Linear Transformations, 3rd ed. (1978); Kenneth Hoffman and Ray Kunze, Linear Algebra, 2nd ed. (1971); Paul R. Halmos, Finite-Dimensional Vector Spaces, 2nd ed. (1958, reprinted 1974); and Nathan Jacobson, Lectures in Abstract Algebra, vol. 2, Linear Algebra (1953, reprinted 1975). Vector analysis is examined by Louis Brand, Vector and Tensor Analysis (1947); Harry Lass, Vector and Tensor Analysis (1950); I.S. Sokolnikoff and R.M. Redheffer, Mathematics of Physics and Modern Engineering, 2nd ed. (1966); and S. Simons, Vector Analysis for Mathematicians, Scientists, and Engineers, 2nd ed. (1970). Claude Chevalley, The Construction and Study of Certain Important Algebras (1955); and A.I. Borisenko and I.E. Tarapov, Vector and Tensor Analysis with Applications, rev. ed. (1968, reissued 1979; originally published in Russian, 3rd ed., 1966), are works that may be consulted by the reader interested in tensor algebra. References on tensor analysis and its applications include Henry D. Block, Introduction to Tensor Analysis (1962); D.F. Lawden, An Introduction to Tensor Calculus,, Relativity, and Cosmology, 3rd ed. (1982); I.S. Sokolnikoff, Tensor Analysis: Theory and Applications to Geometry and Mechanics of Continua, 2nd ed. (1964); and Lon Brillouin, Tensors in Mechanics and Elasticity (1964; originally published in French, 1938). See also Jerrold E. Marsden and Anthony J. Tromba, Vector Calculus, 3rd ed. (1988). Hyman Kaufman Antoni Zygmund Charles L. Fefferman Daniel T. Finkbeiner II The Editors of the Encyclopdia Britannica Complex analysis The reader interested in the foundations of the subject and the Cauchy theory may consult Lars V. Ahlfors, Complex Analysis, 3rd ed. (1979); E.T. Copson, An Introduction to the Theory of Functions of a Complex Variable (1935, reprinted 1972); Einar Hille, Analytic Function Theory, 2 vol. (195962, reprinted 197782); Konrad Knopp, Theory of Functions, 2 vol. (194547; originally published in German, vol. 1, 5th ed., 1937, and vol. 2, 4th ed., 1931); Zeev Nehari, Conformal Mapping (1952, reissued 1975); E.C. Titchmarsh, The Theory of Functions, 2nd ed. (1939, reissued 1975); and E.T. Whittaker and G.N. Watson, A Course of Modern Analysis, 4th ed. (1927, reissued 1979).Elliptic functions and related topics are discussed in Frank Bowman, Introduction to Elliptic Functions, with Applications (1953, reissued 1961); Paul F. Byrd and Morris D. Friedman, Handbook of Elliptic Integrals for Engineers and Scientists, 2nd ed., rev. (1971); and A. Fletcher et al., An Index of Mathematical Tables, 2nd ed. (1962).Classic works dealing with the theory of functions of several complex variables and related subjects include Robert C. Gunning and Hugo Rossi, Analytic Functions of Several Complex Variables (1965), containing an extensive bibliography; Lars Hrmander, An Introduction to Complex Analysis in Several Variables, 2nd ed. (1979); Raghavan Narasimhan, Introduction to the Theory of Analytic Spaces (1966); and Kiyoshi Oka, Collected Papers, rev. ed. edited by R. Remmert (1984; originally published in French, 1961). Kenkichi Iwasawa Raghavan Narasimhan The Editors of the Encyclopdia Britannica Differential equations A good elementary introduction to the theory of ordinary differential equations is Walter Leighton, A First Course in Ordinary Differential Equations, 5th ed. (1981). The best accounts of the classical theory are contained in Earl A. Coddington and Norman Levinson, Theory of Ordinary Differential Equations (1955, reprinted 1984); Philip Hartman, Ordinary Differential Equations (1964, reprinted 1982); and Einar Hille, Lectures on Ordinary Differential Equations (1968). The qualitative theory is discussed in V.V. Nemytskii and V.V. Stepanov, Qualitative Theory of Differential Equations (1960, reprinted 1989; originally published in Russian, 1949). The theory of differential equations in Banach spaces is treated in Henri Cartan, Differential Calculus (1971; originally published in French, 1967). See also Roger C. McCann, Introduction to Ordinary Differential Equations (1982).Elementary accounts of partial differential equations, especially of the equations of mathematical physics, and of techniques of solving them may be found in Arnold Sommerfeld, Partial Differential Equations in Physics (1949, reissued 1967; originally published in German, 2nd ed., 1947); and Ian N. Sneddon, Elements of Partial Differential Equations (1957). The classical theory is treated fully in I.G. Petrovsky, Lectures on Partial Differential Equations (1954, reissued 1966; originally published in Russian, 1950); Bernard Epstein, Partial Differential Equations: An Introduction (1962, reprinted 1975); and Paul R. Garabedian, Partial Differential Equations, 2nd ed. (1986). Finite-difference methods of obtaining approximate solutions are contained in George E. Forsythe and Wolfgang R. Wasow, Finite-Difference Methods for Partial Differential Equations (1960). Dorothy L. Bernstein, Existence Theorems in Partial Differential Equations (1950, reissued 1965), gives a complete account of existence and uniqueness theorems. For an introduction to the abstract theory, see Avner Friedman, Partial Differential Equations (1969, reprinted 1976); and Robert W. Carroll, Abstract Methods in Partial Differential Equations (1969).The reader interested in dynamical systems on manifolds can obtain additional background on manifolds from Shlomo Sternberg, Lectures on Differential Geometry, 2nd ed. (1983); Serge Lang, Differential Manifolds (1972, reissued 1985), a very basic book; and Michael Spivak, Calculus on Manifolds: A Modern Approach to Classical Theorems of Advanced Calculus (1965), an especially readable elementary work. Works on the mathematics of chaos include James Gleick, Chaos: Making a New Science (1987); and Ian Stewart, Does God Play Dice? (1989). For more information on ordinary differential equations, in addition to the works mentioned above, see Witold Hurewicz, Lectures on Ordinary Differential Equations (1958, reissued 1970), a highly recommended and less-imposing book than most on this subject; and Solomon Lefshetz, Differential Equations: Geometric Theory, 2nd ed. (1963, reprinted 1977), which discusses structural stability and the van der Pol equation. For the mathematics of electrical circuits, see Charles A. Desoer and Ernest S. Kuh, Basic Circuit Theory (1969). Gerard Debreu, Theory of Value (1959, reissued 1979), gives a good background in mathematical economics. An excellent standard text with a traditional view is Herbert Goldstein, Classical Mechanics, 2nd ed. (1980). Other approaches are found in Lynn H. Loomis and Shlomo Sternberg, Advanced Calculus, rev. ed. (1990); Ralph Abraham and Jerrold E. Marsden, Foundations of Mechanics, 2nd ed. rev. and enlarged (1978, reprinted 1987); and V.I. Arnold and A. Avez, Ergodic Problems of Classical Mechanics (1968, reprinted 1988; originally published in French, 1967).Classical potential theory is examined in a number of works. Wolfgang J. Sternberg and Turner L. Smith, The Theory of Potential and Spherical Harmonics, 2nd ed. (1946), is an elementary work. Oliver Dimon Kellogg, Foundations of Potential Theory (1929, reprinted 1970), is more advanced. Griffith Conrad Evans, The Logarithmic Potential, Discontinuous Dirichlet and Neumann Problems (1927); and L.L. Helms, Introduction to Potential Theory (1969, reprinted 1975), use modern integration theory. Masatsugu Tsuji, Potential Theory in Modern Function Theory, 2nd ed. (1975), deals with applications. For modern developments in potential theory, see Marcel Brelot, Lectures on Potential Theory, expanded ed. (1967), and On Topologies and Boundaries in Potential Theory (1971); Paul A. Meyer, Probability and Potentials (1966; originally published in French, 1966); R.M. Blumenthal and R.K. Getoor, Markov Processes and Potential Theory (1968); and John Wermer, Potential Theory, 2nd ed. (1981). Ian Naismith Sneddon Stephen Smale The Editors of the Encyclopdia Britannica Special functions In addition to the many works on various aspects of analysis that contain discussions of special functions, there are also works devoted entirely to special functions. Milton Abramowitz and Irene A. Stegun (eds.), Handbook of Mathematical Functions, with Formulas, Graphs, and Mathematical Tables (1964, reissued 1972), is a large handbook covering many special functions, statements of mathematical properties, bibliographies, and other information. A very extensive treatise is Bateman Manuscript Project, Higher Transcendental Functions, ed. by Arthur Erdly, 3 vol. (195355, reprinted 1981). Ian N. Sneddon, Special Functions of Mathematical Physics and Chemistry, 3rd ed. (1980), contains a good treatment of special functions. For spherical harmonic functions, see T.M. MacRobert, Spherical Harmonics, 3rd ed. rev. (1967). There are also many works entirely devoted to one or another particular special function. The Editors of the Encyclopdia Britannica Fourier analysis Fourier series and related topics are discussed in G.H. Hardy and W.W. Rogosinski, Fourier Series, 3rd ed. (1956, reissued 1965); Antoni Zygmund, Trigonometric Series, 2nd ed., 2 vol. (1959, reissued 1988); Salomon Bochner, Harmonic Analysis and the Theory of Probability (1955); E.C. Titchmarsh, Introduction to the Theory of Fourier Integrals, 3rd ed. (1986); Norbert Wiener, The Fourier Integral and Certain of Its Applications (1933, reissued 1988); Gabor Szeg, Orthogonal Polynomials, 4th ed. (1975); M.J. Lighthill, Introduction to Fourier Analysis and Generalised Functions (1958, reissued 1970); Richard R. Goldberg, Fourier Transforms (1961); R.E. Edwards, Fourier Series, 2 vol. (1967); and Yitzhak Katznelson, An Introduction to Harmonic Analysis, 2nd corrected ed. (1976). H.S. Carslaw, Introduction to the Theory of Fourier's Series and Integrals, 3rd ed. rev. and enlarged (1930, reissued 1950); and Dunham Jackson, Fourier Series and Orthogonal Polynomials (1941, reissued 1961), contain elementary theory based on Riemann integration.In addition to the references on Fourier series, the reader interested in harmonic analysis and integral transforms may wish to consult Michael B. Marcus and Gilles Pisier, Random Fourier Series with Applications to Harmonic Analysis (1981); N.K. Bary (Bari), A Treatise on Trigonometric Series, 2 vol. (1964; originally published in Russian, 1961), which deals with the classical theory of Fourier series and integrals; Elias M. Stein and Guido Weiss, Introduction to Fourier Analysis on Euclidean Spaces (1971), which treats the n-dimensional theory extensively and systematically and also has a systematic treatment of convolutions; and I.I. Hirschman and D.V. Widder, The Convolution Transform, (1955) another good reference on convolutions. For variants of the Fourier transform (such as Hnkel or Mellin transforms), see George N. Watson, A Treatise on the Theory of Bessel Functions, 2nd ed. (1944, reprinted 1966); D.V. Widder, The Laplace Transform (1941); and Bateman Manuscript Project, Tables of Integral Transforms, ed. by Arthur Erdlyi, 2 vol. (1954). The topic of singular integrals is treated in Elias M. Stein, Singular Integrals and Differentiability Properties of Functions (1970, reissued 1986). An exposition of abstract harmonic analysis may be found in Walter Rudin, Fourier Analysis on Groups (1962). Lynn H. Loomis, An Introduction to Abstract Harmonic Analysis (1953); and Hermann Weyl, The Classical Groups: Their Invariants and Representations, 2nd ed. (1946, reissued 1966), deal with both abstract harmonic analysis and representations of groups and algebras. Some additional references on representations of groups and algebras are Claude Chevalley, Theory of Lie Groups (1946); Francis D. Murnaghan, The Theory of Group Representations (1938, reprinted 1963); L.S. Pontryagin, Topological Groups, 3rd ed. (1986; originally published in Russian, 1938); Hermann Weyl, The Theory of Groups and Quantum Mechanics (1931, reissued 1949); and G. de B. Robinson, Representation Theory of the Symmetric Group (1961). Antoni Zygmund Charles L. Fefferman The Editors of the Encyclopdia Britannica Calculus of variations There are many works available on the calculus of variations. For an introduction to the theory for problems involving arcs only, see Gilbert A. Bliss, Calculus of Variations (1925, reprinted 1962), and Lectures on the Calculus of Variations (1946, reprinted 1980). For problems involving multiple integrals, see Charles B. Morrey, Multiple Integrals in the Calculus of Variations (1966). For expositions of Morse's theory of critical points, see Marston Morse, The Calculus of Variations in the Large (1934). Charles B. Morrey, Jr. Functional analysis A comprehensive account of a large part of linear functional analysis is found in Nelson Dunford and Jacob T. Schwartz, Linear Operators, 3 vol. (195871). The first volume covers the general theory of linear operators and the second the theory of self-adjoint operators in Hilbert space. This work includes a bibliography of several thousand items and gives historical accounts of many of the topics that it discusses. A systematic study of semigroups of linear operators, together with more general background material on functional analysis, is contained in Einar Hille and Ralph S. Phillips, Functional Analysis and Semi-groups, rev. ed. (1957). For an account of the application of Hilbert space spectral theory to the analysis of differential equations, see M.A. Naimark, Linear Differential Operators (1967; originally published in Russian, 1954); and ch. 13 in vol. 2 of the work by Dunford and Schwartz cited above. For further developments in the application of functional analysis to differential equations, see Tosio Kato, Perturbation Theory for Linear Operators, 2nd ed. (1976, reprinted 1984); and Peter D. Lax and Ralph S. Phillips, Scattering Theory, rev. ed. (1989). The spectral analysis of non-self-adjoint operators is reviewed in vol. 3 of Dunford and Schwartz cited above. Ion Colojoara and Ciprian Foias, Theory of Generalized Spectral Operators (1968), contains additional material concerning this developing area. Patrick Billingsley, Ergodic Theory and Information (1965, reprinted 1978), gives an elegant short account of many of the main theorems of ergodic theory; see also William Parry, Entropy and Generators in Ergodic Theory (1969). Harmonic analysis is treated in many works; in particular, see Edwin Hewitt and Kenneth A. Ross, Abstract Harmonic Analysis, vol. 1, Structure of Topological Groups, Integration Theory, Group Representations, 2nd ed. (1979), which gives a useful picture of this area. Gerald M. Leibowitz, Lectures on Complex Function Algebras (1970), gives an account of some of the deeper issues in classical harmonic analysis. The generalized Fourier integral theory central to harmonic analysis is closely related to some of the principal theorems of group representation theory and of functional algebra. Martin Burrow, Representation Theory of Finite Groups (1965), contains an account of the representation theory of finite groups. The final chapter of Sigurdur Helgason, Differential Geometry and Symmetric Spaces (1962), contains the representation theory of noncompact continuous groups. M.A. Naimark, Normed Rings, rev. ed. (1970; originally published in Russian, 1956), covers many results of functional algebra. A more specialized discussion, devoted to the theory of von Neumann or W*-algebras, can be found in J.T. Schwartz, W*-Algebras (1967). For a general account of the use of global topological methods in functional analysis, see Schwartz's Nonlinear Functional Analysis (1965). The reader interested in the present state of this rapidly developing field should consult the current journal literature.The most used text on generalized functions is I.M. Gel'fand et al., Generalized Functions, 5 vol., trans. from Russian (196468), a very clearly written work, with numerous applications to partial differential equations, mathematical physics, and harmonic analysis. See also D.S. Jones, The Theory of Generalized Functions, 2nd ed. (1982).For a discussion on the general theory of locally convex spaces, see Alex P. Robertson and Wendy Robertson, Topological Vector Spaces, 2nd ed. (1973), which is very clear and very short. Franois Trves, Topological Vector Spaces, Distributions, and Kernels (1967), is another reference on distributions. Hans Bremermann, Distributions, Complex Variables, and Fourier Transforms (1965), is a short and well-written book, with a slant toward applications to physics and to electrical circuits, emphasizing the boundary values of analytic functions. Kosaku Yosida (Yoshida), Functional Analysis, 6th ed. (1980), is a wide-ranging presentation of functional analysis, from the theory of linear operators in Hilbert spaces to Banach algebras, ergodic theorems, and diffusion equations. Jacob T. Schwartz Franois Treves Calculus of variations The calculus of variations is an old subject; some of its problems were considered and partially solved by the ancient Greeks. Leonhard Euler in the mid-18th century deduced the first general rules for dealing with the subject, while much of the terminology was introduced soon after by Joseph-Louis Lagrange. Figure 21: Simple problem of arc length (see text). When two points, A and B, are given in a plane, as shown in Figure 21, there is an infinity of arcs joining them. A simple problem of the calculus of variations is that of finding in this group of arcs (such as E and E) one that has the shortest length, the solution of the problem being a straight line segment. It may also be desired, however, to find in the group of arcs joining A with B one down which a particle, started with a given initial velocity, will fall from A to B in the shortest time; or which one of these arcs, when rotated about the x-axis, will generate a surface of revolution of minimal area. These examples are typical problems of the calculus of variations of the so-called simplest type. These problems illustrate the usual situation in the calculus of variations, in which mathematicians seek to find that arc from some given class for which some quantity, whose value depends on the entire arc (its length, etc.), is a minimum or a maximum. The calculus of variations also deals with problems involving surfaces or functions of several variables. For example, if a circular wire is bent in any way, dipped in a soap solution, and then withdrawn, the soap film spanning the wire will assume the shape of the surface of least area bounded by the wire. The calculus of variations has been useful as a unifying principle in mechanics and as a guide for the determination of new laws of physics. Albert Einstein's theory of general relativity, for example, utilizes the calculus of variations extensively. One of the most widely applicable variational statements of classical mechanics is known as Hamilton's principle (after the 19th-century Irish mathematician William Rowan Hamilton); it states that the trajectories of many dynamical systems are the solutions of some variational problem involving an energy integral. The introduction of the calculus gave great impetus to the study and to the solution of variational problems. After a number of special problems had been solved, Euler in 1744 deduced the first general rule, now known as Euler's differential equation, for the characterization of the maximizing or minimizing arcs. Mathematical formulation of variational problems All the problems mentioned in the first paragraph of this section reduce to the determination of that differentiable arc y = y(x) joining the two points (x1, y1) and (x2, y2) that minimizes an integral extended over a segment of the x-axis and having its argument a function of x, of y, and of the derivative of y (see 603). The length of the arc and the area of the surface of revolution obtained by revolving it around the x-axis are found from elementary calculus to be given respectively by the integrals of the general type described above in which the function in the integrand involves the square root of one plus the square of the derivative of y (see 604). In the case of the curve of steepest descent, the time required for a bead to descend along a wire in the shape of an arc from the point (x1, y1) to the point (x2, y2) under the action of gravity is given by the integral of the above type with a special form in the integrand that accounts for the special type of problem involved (see 605) if friction is neglected; here the term a in the integrand is of the form v12 divided by twice g (see 606), in which v1 is the initial velocity of the bead and g is the acceleration of gravity. It is the case in the preceding examples that the function f (x, y, p) occurring in the minimizing integral (see 603) has one of several forms, each involving the square root of one plus p2 (see 607). The integrals arising from Hamilton's principle are of the type in formula (603), in which x is replaced by the time t (this usually does not appear in the function f in such applications) and the single variables y and p are replaced by several variables y1, . . . , yn, and p1, . . . , pn, the pi standing for the derivatives dyi/dt. Classical methods The majority of the classical analytical methods rely on chemical reactions to perform an analysis. In contrast, instrumental methods typically depend on the measurement of a physical property of the analyte. Classical qualitative analysis Classical qualitative analysis is performed by adding one or a series of chemical reagents to the analyte. By observing the chemical reactions and their products, one can deduce the identity of the analyte. The added reagents are chosen so that they selectively react with one or a single class of chemical compounds to form a distinctive reaction product. Normally the reaction product is a precipitate or a gas, or it is coloured. Take for example copper(II), which reacts with ammonia to form a copper-ammonia complex that is characteristically deep blue. Similarly, dissolved lead(II) reacts with solutions containing chromate to form a yellow lead chromate precipitate. Negative ions (anions) as well as positive ions (cations) can be qualitatively analyzed using the same approach. The reaction between carbonates and strong acids to form bubbles of carbon dioxide gas is a typical example. Prior to the qualitative analysis of any given compound, the analyte generally has been identified as either organic or inorganic. Consequently, qualitative analysis is divided into organic and inorganic categories. Organic compounds consist of carbon compounds, whereas inorganic compounds primarily contain elements other than carbon. Sugar (C12H22O11) is an example of an organic compound, while table salt (NaCl) is inorganic. Classical organic qualitative analysis usually involves chemical reactions between added chemical reagents and functional groups of the organic molecules. As a consequence, the result of the assay provides information about a portion of the organic molecule but usually does not yield sufficient information to identify it completely. Other measurements, including those of boiling points, melting points, and densities, are used in conjunction with a functional group analysis to identify the entire molecule. An example of a chemical reaction that can be used to identify organic functional groups is the reaction between bromine in a carbon tetrachloride solution and organic compounds containing carbon-carbon double bonds. The disappearance of the characteristic red-brown colour of bromine, due to the addition of bromine across the double bonds, is a positive test for the presence of a carbon-carbon double bond. Similarly, the reaction between silver nitrate and certain organic halides (those compounds con

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