See also CUMULANT.

** K-THEORY.** The term came into use around 1960. It appears in a

**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.

The **KAPLAN-MEIER** estimator was proposed by E.
L. Kaplan and Paul Meier in “Nonparametric Estimation from Incomplete Observations,”
*Journal of the American Statistical Association*, **53**, (1958), 457-481.
Meier recalled the collaboration to Harry Marks in
“A Conversation with
Paul Meier,” *Clinical Trials*, **1**, (2004), 131-138. The Kaplan-Meier
paper is one of the most cited in biostatistics: an early reference to the
“Kaplan-Meier method” is in H. F. Stein “Disseminated Coccidioidomycosis,” *Diseases
of the Chest*, **36**, (1959), 136-145.

**KERNEL.** Ivar Fredholm used the French word, “noyau,” in his famous paper on INTEGRAL EQUATIONS,
“Sur une classe des équations fonctionnelles,” *Acta Math.*, **27**,
(1903), 365–390. David Hilbert put this into German as *Kern* in his “Grundzüge einer allgemeinen
Theorie der linearen Integralgleichungen,”
*Nachrichten
von d. Königl. Ges. d. Wissensch. zu Göttingen*
(Math.-physik.
Kl.) (1904) p. 49. The English word* kernel* appears in M. Bôcher’s *Introduction to the
Study of Integral Equations* (1909): “*K* is called the kernel of these equations.” (quoted in the *OED*).
See G. Birkhoff & E. Kreyszig (1984) “The Establishment of Functional Analysis,” *Historia Mathematica*, **11**, 258-321.

*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.
L. Pontrjagin used the term in 1931 on
page 102 of his paper
“Über den algebraischen Inhalt
topologischer Dualitätssätze” in *Math. Annalen* 105.

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* 3^{rd} 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.

[John Aldrich, Jan Peter Schäfermeyer]

**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 alsThe above citation was provided by Gunnar Berg.Vierergruppebenennen.

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].

In English,
“the Klein linear group of order 4 !” is found in 1898 in
*Science: A Weekly Journal devoted to the advancement of science.*
[Google print search by James A. Landau]

The term **KLEINIAN GROUP** was used by Henri Poincaré.

*Kleinian group* is found in 1891
in “On a Classs of Automorphic Functions” by W. Burnside
in *Proceedings of the London Mathematical Society*:
“In this paper I shall adhere as closely as possible to the notation and nomenclature used
by M. Poincaré. ... Hence this second proof certainly establishes the existence
of Kleinian groups of the kind considered ....”
[Google print search by James A. Landau]

**KNOT.** The first mathematical paper which mentions knots is "Remarques sur les problemes
de situation" (1771) by Alexandre-Theophile Vandermonde (1735-1796).

**KNOT THEORY.**
*Theory of knots* is found in 1911 in the *Encyclopaedia Britannica,* 11th ed.
“The founder of the theory of knots is undoubtedly Johann Benedict Listing (1808-1882).”
[Google print search by James A. Landau]

*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.

**KRIGING** refers to a family of least squares techniques for
spatial data first developed by Daniel
Gerhardus Krige in his Master’s thesis at the University of
Witwatersrand, *A Statistical Approach to Some Mine Valuations and
Allied Problems at the Witwatersrand* (1951). The eponym
*krigeage* was coined by Georges Matheron and appeared in his
*Krigeage d’un Panneau Rectangulaire par sa
Périphérie* (1960). The term went into English as
*kriging* and from the geological literature into the statistical
literature where it has been established since the 1970s.

It is found in 1970 in *Random kriging in Geostatistics — A
colloquium* by A. Marechal and J. Serra. [Google print search by James A.
Landau]

See the *Wikipedia* entry
Kriging.

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, ZEHFUSS** or **DIRECT PRODUCT** of matrices. In their
article “On the history of the kronecker product” *Linear and Multilinear Algebra*, **14**,
(1983), 113 – 120,
H. V. Jemderson, F. Pukelsheim & S. R. Searle find the origins of
such products in a paper on determinants by Johann Georg Zehfuss
“Ueber eine gewisse Determinante,” *Zeitschrift für Mathematik und Physik*, **3**,
(1858), 298-301.
The link to Kronecker was made by K. Hensel, “Ueber Gattungen,
welche durch Composititon aus zwei anderen Gattungen entstehen”
*Journal für die reine und angewandte Mathematik*, **105**, (1899),
329-344 who held that Kronecker discussed
them in his lectures in the 1880s.

When the products entered the English literature in the
1930s A. C. Aitken “The Normal Form of Compound and Induced Matrices,” *Proceedings of the London Mathematical Society*,
38, (1935) 354 – 376 called them **Zehfuss products**, F.D. Murnaghan *Theory of Group
Representations* (1938, p. 68) called them **Kronecker products** while C. C. MacDuffee, *The Theory of Matrices,* Springer (1933)
referred 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 p^{3}, pq^{2},
pqr, p^{4},” *Mathematische Annalen*, 43, (1893), p.
305. [John Aldrich]

**KRONECKER’S JUGENDTRAUM.** In a letter to Dedekind of 1880 Leopold Kronecker refers to *meinen
liebsten Jugendtraum* (the dearest dream of my youth): see
Kronecker Werke
V p. 455. The passage reads in translation:

It concerns the dearest dream of my youth, namely to prove that abelian equations with square-roots-rational numbers are reduced to transformation-equations of elliptic functions with singular moduli, just as integral abelian equations to cyclotomic equations.

from S. G. Vladut *Kronecker’s
Jugendtraum and Modular Functions* p. 79. The *Jugendtraum* is part of
the background to Hilbert’s 12^{th} problem. See N. Schappacher “On the
History of Hilbert’s Twelfth Problem: A Comedy of Errors,” *Matériaux pour
l'histoire des mathématiques au XXième siècle* (Nice, 1996), 243-273.
pdf

See HILBERT’S PROBLEMS.

**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: