BABBAGE , Charles (1792-1871). English analyst, statistician, and inventor. Prophet of the modern digital computing machine, envisaging a mechanical device using the basic principles of arithmetic operations and information storage and recall for astronomical and navigational computations.
BAC-TE'RI-AL , adj. law of bacterial growth. The rate of increase of bacteria growing freely in the presence of unlimited food is proportional to the number present. It is defined by the equation dN/dt = kN, where k is a constant, (the time, N the number of bacteria present, and kN the rate of in crease. The solution of this equation is N=ce kt , where c is the value of N when t= 0. This is also called the law of organic growth.
BAlRE, Louis Rene (1874-1932). French analyst.
Baire's category theorem. See category.
Baire function . A real-valued function f which has the property that for any real num- ber a the set of all x for which 
is a Bore/ set. Equivalent definitions result if the set of all x satisfying 
or the set of all x satisfying 
for arbitrary a and b, arc required to be Borel sets (and cither or both of the signs 
could be replaced by 
Any Bairc function is measurable. The Baire functions can be classified as follows. The set of continuous functions arc of the first Baire class . In general, a function is of Baire class 
if it is not of Baire class 
for any 
and is a point-wise limit of functions which belong to Baire classes corresponding to numbers preceding 
By transflnite induction, these classes are defined for all ordinal numbers corresponding to denumerable well-ordered sets. No additional functions are obtained by further extensions. Syn. Borel measurable functions. To every measurable function there corresponds a Borel measurable function which differs from /only on a set of measure zero.
property of Baire . ? set S contained in a set T has the property of Baire if each non empty open set U contains a point where either S or the complement of S is of first category. A set has the property of Baire if and only if it can be made into an open (or a closed) set by adjoining and taking away suitable sets of the first category, or if and only if it can be represented as a 
set plus a set of first category, or if and only if it can be represented as an 
set minus a set of first category. The class of all sets having the property of Baire is the ? -algebra generated by the open sets together with the sets of first category. Sec borll — Borel set, measurable — measurable set.
BAKER, Alan (1939- ). English math ematician and Fields Medal recipient ( 1 970). Extended [he Gel fond-Schneider theorem, showing that 
is transcendental if 
are algebraic numbers
are linearly independent, algebraic and rrational.
BALL , n . See sphere.
BANACH, Stefan (1892-1945). Polish algebraist, analyst, and topologist.
Banach algebr a. See algebra— Banach algebra.
Banach space . A vector space whose scalar multipliers are the real numbers (or the complex numbers) and which has associated with each clement ? a real number 
the norm of x , satisfying the postulates: (1) 
if 
for all real numbers a; (3) 
for all x and y; (4) the space is complete, a neighbor hood of an element ? being the set of all y satisfying 
for some fixed 
Without postulate (4), the space is a normed linear space or normed vector space . The Banach space is a real Banach space or a complex Banach space according as the scalar multipliers are real numbers or complex numbers. Examples of Banach spaces are Hilbert space, the spaces 
of all sequences 
for which 
is finite and and the space
C of all continuous functions f defined on the 
interval [0, 1] with 
for
Banach-Steinhaus theorem . Let X and Y be Banach spaces and let 
be a sequence of bounded linear transformations from X to Y. If the set 
is bounded for each x of X, then there is a number M such that 
for all x of X and each n .
Banach-Tarsk i paradox. The theorem of Banach and Tarski which states that if A and B arc bounded sets in a Euclidean space of dimension at least 3 and if both A and B have interior points, then A can be separated into a finite number of pieces and reassembled by moving the pieces by rigid motions (translations and rotations) to form a set congruent to B. In particular, it is possible to cut a solid sphere into a finite number of pieces and to reassemble these pieces to form two solid spheres the same size as the original sphere. No estimate of the number of pieces needed in this case was given by Banach and Tarski, but R. M. Robinson has proved that the smallest possible number of pieces is 5 and that one of these pieces can be a single point; he also prosed that the surface S of a sphere can be separated into two pieces each of which can be separated into two pieces congruent to itself (thus only four pieces are needed to make from S two identical copies of S). See HAUDORFF—Hausdorff paradox.
Hahn-Banach theorem . See HAHN.
Mazur-Banach game . See MAZUR.
BANK, n. bank discount . See DISCOUNT. bank note . A note given by a bank and used for currency. It usually has the shape and general appearance of government paper money.
mutual savings bank . See mutual,
BAR, n . See aggregation. bar graph. See graph.
BARROW , Isaac (1630-1677). English theologian, geometer, and analyst. Though highly talented and original, he is remembered primarily as Newton's teacher.
BAR'Y-CEN'TER, n. Same as center of mass. See center — center of mass.
BAR'Y-CEN'TRIC, adj. barycentric coordi nates. Let 
points of n- dimensional Euclidean (or vector) space 
that arc not in the same hyperplane of 
Then for each point ? of 
there is one and only one set 
of real numbers for which 
and 
The point ? is (by definition) the center of mass of point masses 
at the points 
respectively, and the numbers 
are said to be barycentric coordinates of the poini x. The motivation for this definition is that if three objects have weights 
with 
and their centers of mass are at the points 
then the center of mass of the three objects together is the point
BASE, n. A base of a geometric configura tion is usually a side (or face) upon which (perpendicular to which) an altitude is con structed, or is thought of as being constructed. See the particular geometric configuration. For an expression such as 
the quantity a is called the base and ? the exponent. Also see the various headings below.
base angles of a triangle. The two angles which have the base of the triangle for a common side.
base for a topology. A collection B of open sets is a base for the topology of a topolo- gica! space T if each open set is the union of some of the members of B. A subbase for a topology is a collection S of open sets such that the collection of all finite intersections of members of S is a base for the topology.
A collection N of open sets is a base for the neighborhood system of a point x (or a local base at X) if x belongs to each member of N and any open set which contains x also contains a member of N. A subbase for the neighborhood system of a point x (or a local subbase at x) is a collection S of sets such that the collection of all finite intersections of members of S is a base for the neighborhood system of x. A topological space is said to satisfy the first axiom of countability if each point has a countable base for its neighborhood system; it satisfies the second axiom of count- abilily if its topology has a countable base. A metric space satisfies the second axiom of countability if and only if it is separable. Syn. basis for a topology.
base in mathematics of finance . A number, usually a sum of money, of which some per cent is to be taken; a sum of money upon which interest is to be calculated.
base of a logarithmic system. See loga rithm.
base of a number system. The number of units, in a given digit's place or decimal place, which must be taken to denote 1 in the next higher place. E.g., if the base is ten, ten units in units place are denoted by 1 in the next higher place, which is ten's place; if the base is twelve, twelve units in units place are denoted by 1 in the next higher place, which is twelve's place. For example, when the base is twelve, 23 means 2 x twelve + 3. Tech., an integer to any base is of form 
where 
are each nonnegativc integers less than the base. A number between 0 and 1 can be written as
See binary, decimal -decimal number system, duodecimal,
BA'SIS, n. basis of a vector space. (1)A set of linearly independent vectors such that every vector of the space is equal to some finite linear combination of vectors of the basis. Syn. Hamel basis (see hamel). (2) For an infinite demensional (and separable) vector space with a vector length (or norm) defined, a basis usually means a sequence of elements 
such that every ? is uniquely ex pressible in the form 
(meaning that the limit as «becomes infinite of the length of 
is zero). If the vectors of the basis are mutually orthogonal, the basis is an orthogonal basis; if they are also all of unit length, the basis is a normal (or normalized ) orthogonal basis, or an orthonormal basis . If there is a finite number of vectors in the basis, t he space is said to be finite dimensional and ils dimension is equal to the number of vectors in its basis. Otherwise, it is infinite dimen- sional . The examples given of Banach spaces (see BANACH—Banach space) possess such a basis, but not all separable Banach spaces have bases. See INNER—inner- product space.
dual basis . (I) For a finite-dimensional linear space V with a basis 
the dual basis is the set of linear functionals 
defined by 
The dual basis is a basis for the first conjugate space V. ff V is taken as the dual of V" with x the linear functional on V defined by 
for all x in V and all f in V *,then the dual of 
(2) If a Banach space B has a basis 
then lhe sequence 
defined by 
is a sequence of continuous linear functionals and it is a basis (a dual basis) for the first conjugate space if and only if it is shrinking in the sense that 
for each continuous linear functional f where 
i.s the norm of/as a continuous linear functional with domain the linear span of 
This condition is satisfied by all bases in reflexive spaces. If 
is a complete orthonormal set for an inner product space T, then 
is a complete orthonormal set for the first conjugate space of T, where 
Analogously to (1), each of the orthonormal bases 
and 
is dual to the other. See inner— inner-product space,
BAYES, Thomas (1702-1761). English theologian and probabilist.
Bayes' theorem . Suppose A and 
are events for which the probability I'(A) of A is not 0, 
and 
hen the conditional probability 
of given that A has occurred is given by 
Sometimes 
is called the inverse probability of the event 
E.g., suppose 4 urns are equally likely to be sampled. Number 1 contains 1 white and 2 red balls. number 2 has 1 white and 3 red, number 3 has 1 white and 4 red, and number 4 has 1 while and 5 red. The probability of an urn being sampled is 
equals 
respectively, for 
where A is the draw of a white ball. Appli cation of Bayes' formula yields
See probability -conditional probability.
BEAR ING, n. bearing of a line (Surveying). The angle which the line makes with the north and south line; its direction relative to the north-south line.
bearing of a point, with reference to an- other point. The angle the line through lhe points makes with the north-south line.
BEHRENS, Walter Ulrich . German agricul tural statistician.
Behrens-Fisher problem . The problem of determining confidence intervals for the difference of the means of two normal populations when the variances of the populations are unknown, and the means of two random samples are known.
BEI, adj. bei function . See bhr ber function.
BEND , adj. bend point . A point on a plane curve where the ordinate is a maximum or minimum.
BENDING MOMENT . See moment.
BEN'E-FI'CI-ARY, n. {Insurance) The one to whom lhe amount guaranteed by the policy is to be paid.
BEN'E-FIT , n. benefits of an insurance policy. The sum or sums which the company promises to pay provided a specified event occurs, such as the dealh of the insured or his attainment of a certain age.
BER , adj. ber function . The ber, bei, her, hei, ker, and kei functions are defined by the relations:
where 
is a Besset function, 
and 
are Hankel /unctions, and 
is a modified Bessel function of the second kind. The following conventions arc also used: 
etc. It follows that:
These sis functions are real when n is real and ? is real and positive. In particular,
these still being valid if ber is replaced by ker and bei by kei.
BERNOULLI, Daniel (1700-1782). Swiss anatomist, botanist, hydrodynaniicist, ana lyst, and probabilist. The most celebrated of his generation of Bernoullis. See PETERS- BERG paradox, RICCATI-Riccati equa tion.
Bernoulli's polynomials. (1) The poly nomials 
defined by
The first four Bernoulli polynomials arc
and
(2) The polynomials 
defined by
It follows that 
and that 
Trivial variations of these defini tions are sometimes given.
BERNOULLI, James (or Jacques or Jakob } (1654-1705). Swiss physicist, analyst, combinatorist, probabilist, and statistician. First and perhaps most famous of the Bernoulli family of mathematicians. See BRACHlSTOCHRONE .
Bernoulli distribution. ? random variable X has a Bernoulli distribution, or is a Bernoulli random variable, if there is a number ? such that X is the number of successes in a single Bernoulli experiment with probability of success p. The range of X is the set 
and the probability of k successes is
where 
The mean is ? and the variance is pi/. See BINOMIAL— binomial distribution.
Bernoulli equation. A differential equa tion of the form
Bernoulli experiment (Statistics). An ex periment or trial for which there are two possible outcomes, such as "heads" or "tails" when tossing a coin, or "A" or "B" when asking if candidate A or candidate B is favored. See above, Bernoulli distribution. Syn. Bernoulli trial.
Bernoulli inequality. The inequality
and n an integer greater than 1.
Bernoulli numbers. (1) The numerical values of the coeliicients of 
in the expansion of 
or 
Substituting the exponential series for 
and starting the division by the e xpansion of 
one obtains, for the first four terms 
of this quotient.
The odd terms all drop out after the term 
Some authors denote the Bernoulli numbers by B 1 , B 2 , etc. Others use B 2 , B 4 . etc. With the 
first notation ; 
In 
general, (2) The numbers defined by the relation 
It follows that 
except possibly for sign, that 
for all 
and that 
where 
is the n th Bernoulli polynomial.
See BERNOULLI, Daniel-Bernoulli's poly nomials. Various trivial variations of these definitions arc sometimes given.
Bernoulli's theorem (Statistics). See large— law of large numbers.
lemniscate of Bernoulli. See LEMNISCATE,
BERNOULLI, John (or Jean or Johann ) (1667-1748). Swiss mathematician. Stu dent, rival, and perhaps equal of his older brother James. Gave impetus to the study of the calculus of variations through posing his famous hmchistochrone problem. See brachistochronu, EULFR-Euler's equation.
BERNOULLI, Nikolaus (or Nicolaus ) (1623 -1708). Progenitor of celebrated Swiss family of mathematicians, the best known of whom are listed in the following chart. The family came originally from Antwerp , but because of religious persecution fled to Basel in 1583.
BESSEL, Friedrich Wilhelm (1784-1846). German astronomer and mathematician.
Bessel functions. For n a positive or negative integer, the 
Bessel function, 
is the coefficient of 
BERNOULLI, Nikolaus (or Nicolaus ), II 11 687-1759). Swiss mathematician. Edu cated by his uncles, James and John, he solved many of their problems.
BERNSTEIN, Sergei Natanovich (1880- 1968). Russian analyst and approximation theorist.
Bernstein polynomials. If f is a real-valued function with domain the closed interval [0, l],then
are Bernslein polynomials. If f is continu- ous, then 
converges uniformly to f on [0, 1 ].
BERTRAND, Joseph Louis Francois (1822- 1903). French analyst, differential geometer, and probabilist.
BE'TA , n. The second letter in the Greek alphabet: lower case, 
capital, B.
beta coefficient . See CORRELATION — multiple correlation.
beta distribution . A random variable X has a beta distribution or is a beta random variable if X has the interval (0, 1) as range and there are positive numbers a and ? for which the probability density function f satisfies
where 
is the gamma function and B is the beta function. The mean is 
and the variance is 
The k th moment about zero is 
If F is an F random variable with 
degrees of freedom, then 
is a beta random variable with 
beta function . The function 
defined by
form and ? positive. In terms of the ? function, 
See ????? gamma function. The in- complete beta function is defined by
which is equal to 
where F is the hypcrgcomctric function.
beta weight . See correlation normal correlation.
BETTI, Enrico (1823-1892). Italian alge braist, analyst, topologjst, and politician. See HOMOLOGY-homology group.
Betti number. Let 
be the r-dimensional homoiogy group (of a simplicial complex K) formed by using the group G. If G is lhe group of integers modulo 
where 
is a prime, then G is a field, 
is a linear (vector) space, and the dimension of 
is the r- dimcnsional Belli number (modulo 
) of K. If G is the group of integers, then 
is a commutative group with a Unite number of generators and is the Cartesian product of infinite cyclic groups 
and cyclic groups 
of finite orders 
(sec torsion torsion coefficients of a group). The number m is the r-dimensional Betti number and 
are the r -dimensional torsion coefficients of K. The Betti numbers (.especially the 1 -dimensional Betti number modulo 2, or 1 plus this number) arc sometimes called connectivity numbers (see conni ctivity). For an ordinary closed surface, 
where 
is the Euler characteristic and 
is the 1 -dimensional Betti number modulo 2. If the surface is not closed (has boundary curves), then 
If the surface is orientable, then the genus of the surface is equal to 