**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 (OED2).

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

**UNDECIDABLE** was used by Kurt Gödel (1906-1978) in 1931 in
the title *Uber 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" (OED2).

**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 Pierpoint’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). Pierpoint kept *sum* for the case of disjoint sets.

**UNIT CIRCLE** is found in 1890 in
*Five-place Logarithmic and Trigonometric Tables*
by George Albert Wentworth and George Anthony Hill:
"Find the length of an arc of 47° 32' 57" in a unit circle." [Google print search]

*Unit circle* is found in 1895 in
*Plane and spherical trigonometry, surveying and tables*
by George Albert Wentworth:
"The circle with radius equal to 1 is called a *unit circle,*
*AA'* the *horizontal,* and *BB'* the *vertical* diameter"
[University of Michigan Historic Math Collection].

**UNITARY MATRIX** was introduced by Alfred Loewy "Ueber 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*(*z*_{1})
≠ *f*(*z*_{2}) unless *z*_{1} = *z*_{2.}.”
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 & 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 18^{th} 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
3^{rd} 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.