meaning a bulging, convexity.
See also CUMULANT.
K-THEORY. The term came into use around 1960. It appears in a Mathematics Review article by R. Bott on M. F. Atiyah & F. Hirzebruch “Vector bundles and homogeneous spaces,” 1961 Proc. Sympos. Pure Math., Vol. III, pp. 7-38. Bott writes, “This paper summarizes some of the authors’ researches on the K-theory …” See the Encyclopedia of Mathematics entry K-theory.
KALMAN FILTER refers to the method introduced by R. E. Kalman in his A New Approach to Linear Filtering and Prediction Problems, Transactions of the ASME - Journal of Basic Engineering Vol. 82: pp. 35-45 (1960) See here for the paper. S. L. Lauritzen argues in his 2002 book Thiele: Pioneer in Statistics that Thiele anticipated Kalman by 80 years. In the 1820s Gauss had devised up-dating formulae for least squares estimates. [John Aldrich]
See FILTER.
KERNEL. David Hilbert used the German word kern in his first paper on integral equations, "Grundzüge einer allgemeinen Theorie der linearen Integralgleichungen", Nachrichten von d. Königl. Ges. d. Wissensch. zu Göttingen (Math.-physik. Kl.) (1904) p. 49. Kernel occurs in English in 1909 in M. Bôcher's Introduction to the Study of Integral Equations: "K is called the kernel of these equations." (OED2).
Kernel was an established term in Fourier analysis by the time of A. Zygmund's Trigonometrical Series (1935). A JSTOR search found the "Fejér kernel" and "Dirichlet kernel" in Charles N. Moore's "On the Application of Borel's Method to the Summation of Fourier's Series" (Proceedings of the National Academy, 11, (1925), 284-287) but it is unlikely that this was the first published use of these terms.
Kernel entered Statistics with the use of Fourier theory in describing estimates for spectral density and probability density functions. A JSTOR search found E. F. Schuster (Annals of Mathematical Statistics, 40, (1969), p. 1187) referring to "a so-called kernel class of estimates," introduced by M. Rosenblatt in "Remarks on Some Nonparametric Estimates of a Density Function," Annals of Mathematical Statistics, 27, (1956), 832-837. Earlier "Fejér kernel" was used in U. Grenander & M.Rosenblatt’s "Statistical Spectral Analysis of Time Series Arising from Stationary Stochastic Processes," Annals of Mathematical Statistics, 24, (1953), 537-558. An alternative term, especially popular in time series analysis, is WINDOW.
See also FEJÉR KERNEL.
The use of kernel in algebra appears to be unrelated to its use in integral equations and Fourier analysis. The OED gives the following quotation from Pontrjagin's Topological Groups i. 11 (translated by E. Lehmer 1946) "The set of all the elements of the group G which go into the identity of the group G* under the homomorphism g is called the kernel of this homomorphism."
G. D. Birkhoff & S. A. MacLane A Survey of Modern Algebra 3rd edition 1965, pp. 213-4 apply the concepts of homomorphism and kernel to a linear transformation T between vector spaces considered as an Abelian Group under addition. They remark, “Since O is the group identity, it follows that that the null-space of T is precisely the kernel of T regarded as a group-homomorphism.” See the entry NULL SPACE.
This entry was contributed by John Aldrich.
KITE. Deltoid appears in 1879 in Dictionary of Scientific Terms: "Deltoid, a four-sided figure formed of two unequal isosceles triangles on opposite sides of a common base" (OED2).
Kite appears as a geometric term in the 1893 Funk and Wagnalls Standard Dictionary.
KLEIN BOTTLE is found in 1940 in W. T. Puckett, "On 0-Regular Surface Transformations," Trans. American Mathematical Society [John M. Sullivan].
Klein bottle occurs in C. Tompkins, "A flat Klein bottle isometrically embedded euclidean 4-space," Bull. Am. Math. Soc. 47, 508 (1941).
Felix Klein describes--but does not illustrate--the construction in his short book Ueber Riemann's theorie der algebraischen functionen und ihrer integrale (1882) reprinted in Ges. Math. Abh, 3, 499-573.
KLEIN FOUR GROUP. Vierergruppe is found in 1884 in Vorlesungen uber das Ikosaeder und die Aufloesung der Gleichungen vom funften Grade by Felix Klein:
Offenbar umfasst unsere neue Gruppe von der Identitaet abgesehen nur Operationen von der Periode 2, und es ist zufaellig, das wir eine dieser Operationen an die Hauptaxe der Figur, die beiden anderen an die Nebenaxe geknupft haben. Dementsprechend will ich die Gruppe mit einem besonderen Namen belegen, der nicht mehr and die Dieder- configuration erinnert, und sie als Vierergruppe benennen.The above citation was provided by Gunnar Berg.
The term "(Kleinsche) Vierergruppe" was used by Bartel Leendert van der Waerden (1903-1996) in 1930 in his influential textbook Moderne Algebra. It denotes the permutation group generated by (12)(34) and (13)(24), rather than the abstract product of two 2-cyclic groups. The term does not occur in the older algebra books by Weber and by Perron [Peter Flor].
The term KLEINIAN GROUP was used by Henri Poincaré.
KNOT. The first mathematical paper which mentions knots is "Remarques sur les problemes de situation" (1771) by Alexandre-Theophile Vandermonde (1735-1796).
KNOT THEORY appears in 1932 in the title Knotentheorie by Kurt Werner Friedrick Reidemeister (1893-1971).
KOLMOGOROV EXTENSION THEOREM refers to the theorem in Section 4 of Chapter III of A. N. Kolmogorov's Grundbegriffe der Wahrscheinlichkeitsrechnung (1933). (English translation) It is also called the Daniell-Kolmogorov theorem in recognition of the work of P. J. Daniell, "Integrals in an Infinite Number of Dimensions," Annals of Mathematics, 20, (1919) 281-88.
The relationship between the work of Kolmogorov and Daniell is discussed by Glenn Shafer and Vladimir Vovk "The Sources of Kolmogorov’s Grundbegriffe," Statist. Sci. 21, (2006), 70-98 section 5.1.2.
KOLMOGOROV FORWARD AND BACKWARD EQUATIONS. See the entry FOKKER-PLANCK EQUATION.
KOLMOGOROV-SMIRNOV TEST appears in F. J. Massey Jr., "The Kolmogorov-Smirnov test of goodness of fit," J. Amer. Statist. Ass. 46 (1951).
See also W. Feller, "On the Kolmogorov-Smirnov limit theorems for empirical distributions," Ann. Math. Statist. 19 (1948) [James A. Landau].
The references are to A. N. Kolmogorov "Sulla determinazione empirica di una legge di distribuzione," Inst. Ital. Attuari, Gorn., 4, (1933), 1-11 and V. I. Smirnov (1900-66) "On the estimation of the discrepancy between empirical curves of distribution for two independent samples," Bulletin Mathématique de l’Université de Moscou, 2, (1939), fasc. 2.
KÖNIGSBERG BRIDGE PROBLEM. The problem was posed and solved in L. Euler "Solutio problematis ad geometriam situs pertinentis" (The solution of a problem relating to the geometry of position) Comment. Acad. Sci. U. Petrop. 8, 128-140, 1736. Reprinted in Opera Omnia Ser. I-7, pp. 1-10, 1766. There is an English translation in James Newman World of Mathematics vol. 1. 573-580. See also MacTutor History of Topology
See the entries GRAPH and TOPOLOGY.
The term KRONECKER DELTA is found in 1926 in Riemannian Geometry by Luther Pfahler Eisenhard: "These are called the Kronecker deltas and are used frequently throughout this work." The term is used because Kronecker used the construction in his lectures. [Joanne M. Despres of Merriam-Webster Inc.]
The KRONECKER PRODUCT or DIRECT PRODUCT of matrices appears in the English literature of the 1930s. F.D. Murnaghan Theory of Group Representations (1938, p. 68) refers to Kronecker products. C. C. MacDuffee, The Theory of Matrices, Springer (1933) refers to direct products. MacDuffee (p. 82) explains “The concept of direct product of matrices arises naturally from the concept of direct product in group theory.” For the latter he refers to O. Hölder “Die Gruppen der Ordnungen p3, pq2, pqr, p4,” Mathematische Annalen, 43, (1893), p. 305. [John Aldrich]
KUHN-TUCKER THEOREM, MULTIPLIER, etc. derive from H. W. Kuhn & A. W. Tucker’s "Nonlinear Programming", in J. Neyman, editor, Proceedings of the Second Berkeley Symposium (1951). T. H. Kjeldsen writes, "When Kuhn and Tucker proved the Kuhn-Tucker theorem ... they launched the theory of nonlinear programming. However, in a sense this theorem had been proven already: In 1939 by W. Karush in a master's thesis, which was unpublished; in 1948 by F. John in a paper that was at first rejected by the Duke Mathematical Journal; and possibly earlier by Ostrogradsky and Farkas." ("A Contextualized Historical Analysis of the Kuhn-Tucker Theorem in Nonlinear Programming: The Impact of World War II," Historia Mathematica, 27, (2000), 331-361)
A JSTOR search found the terms Kuhn-Tucker theorem and Kuhn-Tucker conditions in circulation by the end of the 1950s and Kuhn-Tucker multipliers by the end of the 60s.
KULLBACK-LEIBLER INFORMATION (NUMBER) appears in Herman Chernoff "Large-Sample Theory: Parametric Case," Annals of Mathematical Statistics, 27, (1956), 1-22. The reference is to S. Kullback and R. A. Leibler "On Information and Sufficiency," Annals of Mathematical Statistics, 22, (1951), pp. 79-86. (JSTOR)
KURTOSIS was used by Karl Pearson in
1905 in "Das Fehlergesetz und seine Verallgemeinerungen durch Fechner und
Pearson. A Rejoinder," Biometrika,
4, 169-212, in the phrase
"the degree of kurtosis." He states therein that he has used the term
previously (OED). According to the OED and to Schwartzman the
term is based on the Greek meaning a bulging, convexity.
He introduced the terms leptokurtic, platykurtic and mesokurtic, writing in Biometrika (1905), 5. 173: "Given two frequency distributions which have the same variability as measured by the standard deviation, they may be relatively more or less flat-topped than the normal curve. If more flat-topped I term them platykurtic, if less flat-topped leptokurtic, and if equally flat-topped mesokurtic" (OED2).
In his "Errors of Routine Analysis" Biometrika, 19, (1927), p. 160 Student provided a mnemonic:
