# Earliest Known Uses of Some of the Words of Mathematics (U)

U-STATISTIC. The term was introduced by Wassily Hoeffding "A Class of Statistics with Asymptotically Normal Distribution," Annals of Mathematical Statistics, Vol. 19, No. 3 (Sep., 1948), pp. 293-325. The "U" stands for "unbiased" and Hoeffding noted that some properties of U-statistics had been obtained by P. R. Halmos "The Theory of Unbiased Estimation," Annals of Mathematical Statistics, 17, No. 1. (Mar., 1946), pp. 34-43.

ULAM-BORSUK THEOREM. See BORSUK-ULAM THEOREM.

The term ULTRAMETRIC was coined in 1944 by M. Krasner, according to an Internet web page.

The terms UMBRAL CALCULUS and UMBRAL NOTATION were coined by James Joseph Sylvester (1814-1897).

UNBIASED. See biased and estimation.

The term UNCONDITIONAL CONVERGENCE (of a continued fraction) was coined by Alfred Pringsheim (1850-1941).

UNDECAGON. Earlier terms for an 11-sided polygon, hendecagon and endecagon, are found in English in 1704 in Lexicon technicum, or an universal English dictionary of arts and sciences, by John Harris [OED].

Undecagon is found in English in 1728 in Chambers' Cyclopedia.

UNDECIDABLE was used by Kurt Gödel (1906-1978) in 1931 in the title Über formal unentscheidbare Sätze der Principia Mathematica und verwandter Systeme (On Formally Undecidable Propositions in Principia Mathematica and Related Systems).

UNGULA appears in 1710 in Lexicon technicum, or an universal English dictionary of arts and sciences, by John Harris: "Ungula, in Geometry, is the Section of a Cylinder cut off by a Plane, which passes obliquely thro' the Plane of the Basse, and part of the Cylindric Surface" [OED].

UNIFORM or RECTANGULAR DISTRIBUTION. Uniform distribution appears in 1937 in Introduction to Mathematical Probability by J. V. Uspensky. Page 237 reads, "A stochastic variable is said to have uniform distribution of probability if probabilities attached to two equal intervals are equal." This is a slight variant of the modern terminology, which would be "a variable is said to be uniformly distributed" or "a variable from the uniform distribution" [James A. Landau].

Uniformly distributed is found in H. Sakamoto, "On the distributions of the product and the quotient of the independent and uniformly distributed random variables," Tohoku Math. J. 49 (1943).

Rectangular distribution was once the more common term, favoured particularly by statisticians. A JSTOR search found Kirstine Smith using it in "On the Standard Deviations of Adjusted and Interpolated Values of an Observed Polynomial Function and its Constants and the Guidance they give Towards a Proper Choice of the Distribution of Observations" Biometrika, 12, (1918), 1-85.

The earliest writers on probability considered the behaviour of a fair die and thus the properties of the discrete uniform distribution but they did not present the distribution as it would be today. Nevertheless they did difficult things: thus finding the number of chances of throwing s points with n dice (treated by de Moivre in his Miscellanea Analytica (1730)) translates to finding the distribution of the sum of n independent discrete uniform random variables. Another important early use of the uniform was as a prior distribution. A uniform prior distribution is buried in Bayes's Essay towards solving a Problem in the Doctrine of Chances (1767) and in Gauss's Theoria Motus (Theory of the Motion of Heavenly Bodies moving around the Sun in Conic Sections) (1809). Gauss writes ( p. 243) that previous to the observations all systems of values of the unknowns are equally probable. The term uniform prior distribution now covers this application. The term is quite recent. A JSTOR search produced D. V. Lindley's "Fiducial Distributions and Bayes' Theorem," Journal of the Royal Statistical Society. Series B, 20, (1958), 102-107. In the role of a prior distribution, the uniform generated plenty of controversy.

See PROBABILITY GENERATING FUNCTION and PRINCIPLE OF INDIFFERENCE.

UNIFORM CONTINUITY. The concept was introduced by E. Heine "Die Elemente der Functionenlehre," Journal für die reine und angewandte Mathematik Journal für die reine und angewandte Mathematik, 74, (1872). 172-188. Heine writes on p. 184 that a function is called "gleichmäßig continuirlich when ..." See the Encyclopaedia of Mathematics entry.

UNIFORM CONVERGENCE. The concept of uniform convergence "first appeared in papers by Stokes in 1847, von Seidel in 1848, and Cauchy in 1853, but it was Weierstrass who discovered it first (in 1841; cf. Werke, Vol. 1, p. 67]) named it, and made its fundamental importance generally appreciated." From G. Birkhoff & E. Kreyszig (1984) "The Establishment of Functional Analysis," Historia Mathematica, 11, p. 261. The title of the 1841 paper was Zur Theorie der Potenzreihen, and the term Weierstrass used was gleichmäßig konvergent. See Wikipedia entry. [John Aldrich]

UNIFORMLY MOST POWERFUL. See POWER.

UNIMODAL is found in 1904 in F. de Helguero, "Sui massimi delle curve dimorfiche," Biometrika, 3, 84-98 (David, 1995).

UNION (of sets) is found in 1912 in James Pierpont's Lectures on the theory of functions of real variables (Vol. 2, p. 22). Sum was once the usual term, see e.g. die Summe in F. Hausdorff Grundzüge der Mengenlehre (1914, p. 5). Pierpont kept sum for the case of disjoint sets.

UNIT CIRCLE is found in 1852 in Cambridge and Dublin mathematical journal 7/134: "The imaginary unit in the solution of (II.) has the same reference to an unit circle, as that in the solution of (III.) has to an unit sphere." [OED]

UNITARY MATRIX was introduced by Alfred Loewy "Über Scharen reeller quadratischer und Hermitescher Formen" J. reine angew. Math. 122 (1900) 53-72.  This is according to C. C. MacDuffee, The Theory of Matrices, Springer (1933).

UNIVALENT. This French term appears in English in 1933 in Joseph L. Doob's "The Boundary Values of Analytic Functions. II," Transactions of the American Mathematical Society, 35, (2) p. 488. (JSTOR search.) In a footnote Doob explains, "A function is called univalent if f(z1)f(z2) unless z1 = z2.." At the time the German word SCHLICHT was the common term in the English literature on complex analysis.

See also SCHLICHT.

UNIVARIATE (in Statistics) is found in Karl Pearson's "The Fifteen Constant Bivariate Frequency Surface," Biometrika, 17, No. 3/4. (Dec., 1925), pp. 268-313.

OED2 has this quotation from John Wishart's "The generalized product moment distribution in samples from a normal multivariate population" Biometrika 20A, 32 (1928): "Various writers struggled with the problems that arise when samples are taken from uni-variate and bi-variate populations".

See BIVARIATE, MULTIVARIATE, N-VARIATE, and TRIVARIATE.

The term UNIVERSAL ALGEBRA was first used by James Joseph Sylvester (1814-1897) for the theory of matrix algebras in a paper, "Lectures on the Principles of Universal Algebra," published in the American Journal of Mathematics, vol. 6, 1884, according to Whitehead [Encyclopaedia Britannica, article: Algebraic Structures].

Charles S. Peirce had objected to the term in a letter to Sylvester of Jan. 5, 1882: "I confess I cannot see why the system should be called 'universal.' It is a favorite epithet among inventors of all kinds in this country; but it seems to me best to restrict it to a very exact signification. A dual 'relative' is a collection of pairs, and this is an algebra of collections of pairs."

UNIVERSE OF DISCOURSE (in logic) and UNIVERSAL CLASS or SET (in set theory).  The OED shows the development of these terms through quotations. For universe of discourse it gives: 1849 A. De Morgan in Trans. Cambridge Philos. Soc. VIII. 380 By not dwelling upon this power of making what we may properly (inventing a new technical name) call the universe of a proposition, or of a name, matter of express definition, all rules remaining the same, writers on logic deprive themselves of much useful illustrations. Ibid., Let the universe in question be 'man': then Briton and alien are simple contraries.

1881 J. Venn Symbolic Logic vi. 128 We must be supposed to know the nature and limits of the universe of discourse with which we are concerned...

For universal class the OED gives: 1910 Whitehead and Russell Principia Mathematica I. i. 30 The class determined by a function which is always true is called the universal class, and is represented by V.

UNKNOWN. See root.

UNLESSS. John H. Conway has suggested this term meaning "precisely unless," formed by analogy with IFF. However, the term appears not to have caught on. See Wolfram: precisely unless and A collection of word oddities: triple letters.

UPPER SEMICONTINUITY. See LOWER SEMICONTINUITY.

URELEMENT (Primitive element) appears in Ernst Zermelo "Uber Grenzzahlen und Mengenbereiche. Neue Untersuchungen über die Grundlagen der Mengenlehre, Fundamenta mathematicae, 16 (1930), 29-47.

In English a JSTOR search found the word (without quotes) in C. Ward Henson "Finite Sets in Quine's New Foundations," Journal of Symbolic Logic, 34, (1969), 589-596.

UTILITY. Although the modern axiomatic utility theory of decision-making in situations of risk dates from John von Neumann and Oskar Morgenstern's Theory of Games and Economic Behavior (1944), its antecedents go back to the work of Cramer and Daniel Bernoulli in the early 18th century. For "utility" Bernoulli used the term "emolumentum" and Cramer the term "valeur morale." Later Laplace used "fortune morale" to refer to the utility of wealth. When the economist W. Stanley Jevons (Theory of Political Economy (1871)) made utility the central concept in his mathematical theory of decision-making he identified Bernoulli's concept (see ch. IV, p. 125 in the 3rd edition) with the utility of Jeremy Bentham's An Introduction to the Principles of Morals and Legislation (1781). [John Aldrich]

For references and further information see MORAL EXPECTATION and ST. PETERSBURG PARADOX.

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