BE.-TWEEN , prep. Used in mathematics in many senses that arc roughly consistent with English usage. E.g., if a and c are real numbers and a < c, then b being between a and c means lhat a < b < c; for a point B on a line to be between distinct points A and C on the line means that B is on the same side of ? as C and on the same side of C as A; for a set S to be between sets R and T means that 
(usually it is not required lhat S not be equal to cither R or T).
BEZOUT, Etienne (1730-1783). French analyst and geometer.
Bezout's theorem , if two algebraic plane curves of degrees m and n do not have a common component, then they have ex- actly m n points of inlersection [points of intersection are counted to the degree of their multiplicity and include points of intersection at infinity (see coordinate — homogeneous coordinates, and projectiVF. -projective plane)]. For n -dimensional Euclidean space, if ? algebraic hypersurfaces have degrees 
and have only
a finite number of common points, then there are at most 
common
points (exactly 
if points at
infinity are counted and if multiplier les are defined suitably and points of intersection are counted to the degrees of their multi plicities).
BI-AN'NU-AL , adj. Twice a year. Syn. semiannual.
Bl'ASED, or BI'ASSED , adj. biased esti- mator. See UNBIASED—unbiased estimator. biased test . See hypothesis -test of a hypothesis.
BI-COM-PACT ', adj. See COMPACT.
BI-COM-PAC'TUM , i. Same as COMPAC- TUM.
BI'CON-DI'TION-AL , adj. See EQUIVA LENCE- equivalence of propositions.
BIENAYME, Irenee Jules (1796-1878). French probabilist.
Bienayme-Chebyshev inequality {Sta- tistics). See CHEBYSHEV -Chebyshev's inequality.
BI-EN 'NI-AL , adj. Once in two years ; every two years.
BI'HAR-MON'IC, adj. bi harmonic boundary value problem. See boundary.
biharmonic function . A solution of the fourth order partial differential equation 
where 
is the Laplace operator 
thus, a solution u ( x,y, z) of the equation
The definition applies equally well to functions of two, four, or any other number of indepen dent variables. Biharmonic functions occur in the study of electrostatic boundary value problems and elsewhere in mathematical physics.
Bl-JEC'TION , n. A bijection from a set ? to a set B is a one-to-one correspondence between A and B, i.e., a function from A into B that is both an injection and a sur- jection. Syn, bijective function. See in jection , SORJECTION .
BI-JEC'TIVE, adj. See BIJE.CTION.
BI-LIN'E.-AR , adj. A mathematical ex pression is bilinear if it is linear with respect to each of two variables or positions. E.g. : The function /(*, y) = 3*>' is linear in jc and >', since 
and 
The scalar pro duel of vectors 
which is bilinear since 
and 
The scalar product and the function 3xy are bilinear forms (see form). The function F for which F( u, v) is a function whose value at ? is 
is a bilinear function of u and v, where u and v are functions of two variables.
bilinear concomitant . See adjoint — adjoint of a differential equation.
BILL , n. A statement of money due, usually containing an itemized statement of the goods or services For which payment is asked.
BIL'LION , n. (1) In the U.S. and France, a thousand millions (1,000,000,000). (2) In England and Germany, a million miltions (1,000,000,000,000).
BI-MO'DAL , adj. bimodal distribution . A distribution with two modes; i.e., there are two different values which are conspicuously more frequent than neighboring values.
Bl'NA-RY , adj. binary number system . A system of numerals for representing real numbers that uses the base 2 instead of the base 10. Only the digits 0 and 1 are needed. For example, 101110 in binary notation is equal to 
or 46 in decimal notation. This ex ample illustrates the process of changing from binary notation to decimal notation. The reverse process is accomplished by suc cessive division by 2. E.g. , 29 in decimal notation is equal to
which is 11101 in binary notation. As with decimal notation, a real number has a re peating infinite sequence representation in binary notation if and only if it is rational. E.g., 0.1100000·- and 0.10,10,10,·.. in binary notation are equal to 
and 
, respectively, in decimal notation. Binary arilhmetic is useful in connection with electronic computers, since the digits 0 and 1 can be described electrically as "off" and "on." Syn. dyadic number system. See base — base of a number system.
binary operation . An operation which is applied to two objects. For example, any two numbers can be added, two numbers can be multiplied, the intersection of two sets is a set, the product of a matrix with n columns and a matrix with n rows is a matrix, and the composition of two functions produces a function. A set is closed with respect to a particular binary operation if, whenever the operation is applied to a pair of members of the set, it gives a member of the set. The set of positive integers is closed under addition and multiplication. It is not closed under division or subtraction, since 2/3 is not an integer and 2 — 3 is not a positive integer. The set 
is nol closed under addition, since 2 + 3 is not a member of the set. Tech. A binary operation is a function f whose domain is a set of ordered pairs of members of a set S; the set S is closed with respect to the binary operation f if and only if f(x,y) belongs to S whenever x and y belong to S . See ternary — ternary operation.
BI-NO'MI-AL , n . A polynomial of two terms, such as 2x + 5y or 2 — (a + b). See TRINOMIAL.
binomial coclficients . The coefficients in the expansion of 
For example, 
so that the binomial coefficients of order 2 are 1, 2, and 1. The (r+ l)st binomial coefficient of order n ( n a positive integer) is 
the number of combinalions of ings r at a time, and is denoted by 
The sum of the binomial coefficients is equal to 
, shown by putting 1 for each of ? and y in 
See pascal — Pascal's triangle, and below, binomial theorem.
binomial differential. A differential of the form 
where a and b are any constants and the exponents m, n, and ? are rational numbers.
binomial distribution. A random variable X is binomially distributed or is a binomial random variable if there is an integer n and a number p such that X is the number of successes in n independent Bernoulli experi-ments, where the probability of success in a single experiment is p. The range of X is the set 
and the probability of k successes is
where q - 1 - p. E.g., if three coins are thrown, then 
and the probabilities of 0, 1, 2 or 3 heads are 
; these are the terms in the expansion ot 
by the binomial theorem. In general,
The mean of the binomial distribution is np, the variance is npq, and the moment generating function is 
When n is large, the binomial distribution can be approximated by a normal distribu- tion with mean np and variance npq. The binomial distribution can be approximated by a Poisson distribution mean np if n is large. See BERNOULLI-Bernoulli distribu- tion, Bernoulli experiment, CENTRAL— central limit theorem, MOMENT—moment generating function, MULTINOMlAL-multi- nomial distribution, normal— normal distribution, POI SSon -Poisson distribution.
binomial equation . An equation of the form 
binomial expansion . The expansion given by the binomial theorem.
binomial formula . The formula given by the binomial theorem,
binomial series . A binomial expansion which contains infinitely many terms. That is, the expansion of 
where n is not a
positive integer or zero. Such an expansion converges and its sum is 
or if 
and 
, or if 
and 
E.g.,
binomial surd . See surd.
binomial theorem , A theorem (or rule) for the expansion of a power of a binomial The theorem can be stated thus: The first term in the expansion of 
; the
second term has n for its coefficient, and the other factors are 
and y; in subsequent
terms the powers of x decrease by 1 for each term and those of y increase by 1, while the next coefficient can be obtained from a given coefficient by multiplying by the exponent of y and dividing by one more than the exponent of y. E.g., 
In general, if n is a positive integer,
The coefficient of 
See above, binomial coefficients, binomial series.
negative binomial distribution. A random variable S has a negative binomial distribu- tion or is a negative binomial random variable if there are numbers r and ? such that X is the number of repeated indepen- dent Bernoulli trials with probability of
success ? that are performed to obtain r successes. The range of X is the infinite set 
and the probability of n trials is
where q = 1 - p. The mean is r/p, the variance is 
, and the moment gen-
erating function is
Syn. Pascal distribution. If r = 1, X has a geometric distribution or is a geometric random variable . Then P(X — n) is 
if 
ihe mean is 1/p, and the variance
is 
. Sometimes V = X ~ 1, the number
ot trials before the first success, is called a geometric random variable; then P(X - n) is 
and the mean is qjp.
BI-NOR'MAL, n. See normal — normal to a curve or surface.
BI-PAR'TITE , n. bipartite cubic . The locus of the equation
The curve is symmetric about the x -axis and intersects the x -axis at the origin and at the points (a, O) and ( b , 0). It is said to be bi- partite because it has two entirely separate branches.
BI-QUAD-RAT'IC , adj. biquadratic equation .
An algebraic equation of the fourth degree. Syn. quadratic.
Bl-REC-TANG'U-LAR , adj. Having two right angles. E.g., a birectangular spherical triangle is a spherical triangle with two right angles.
BIRKHOFF, George David (1884-1944). Leading American topologist, analyst, and applied mathematician of his time. Worked on coloring of maps, calculus of variations, and dynamical systems. Proved Poincare's last theorem, concerning fixed points in a ring. See ERGODIC— ergodic theory,
POINCARE—Poincare-Birkhoff fixed point theorem.
BI-SECT ', v. To divide in half.
bisect an angle . To draw a line through the vertex dividing the angle into two equal angles.
bisect a line segment . To find the point on the line segment and equally distant from the ends. Analytically, the Cartesian coordinates of the midpoint can be found as the arithmetic means or averages of the corresponding coordinates of the two end points. See point — point of division. If 
and 
are the end points of a line segment, the coordinates of the midpoint are 
BI-SECT'ING , adj. bisecting point of a line segment. Same as midpoint of a line segment.
Bl-SEC'TOR , n. bisector of an angle . The straight line which divides the angle into two equal angles. The equation of an angle bi- sector can be obtained by equating the dis- tances of a variable point from the two lines. See DISTANCE—distance from a line to a point.
bisector of the angle between two inter- secting planes. ? plane containing all the points equidistant from the two planes. There are two such bisectors for any two such planes. Their equations are obtained by equating the distances of a variable point from the two planes- first giving these distances like signs, and then unlike signs See DISTANCE—distance from a plane to a point.
BI-VAR'I-ATE , adj. Involving two vari ables.
bivariate distribution . See DISTRIBU TION—distribution function.
bivariate normal distribution . The vector random variable (X, Y) has a bivariate normal distribution if its probability density function is given by
where 
ind 
are the means of X and Y 1 and 
and 
are the vari- ances of X and Y. The conditional distribu- tion of X given Y (or of Y given X) is nor- mal. The conditional mean of X given that Y = y is 
The parameter r is the correlation parameter and is equal to the correlation coefficient between the random variables X and Y.
BLASCHKE, Wilhelm (1885-1962). Austrian-German analyst and geometer. Blaschke product . A product of type
where 
for each 
converges, and k is a nonnegative integer. The function B is bounded and analytic on the set of all complex numbers ? such that 
The zeros of B are the numers 
and O (if k > O).
Blaschke's theorem . Each bounded closed convex plane set of width 1 contains a circle of radius 
. See JUNG—Jung's theorem.
BLISS, Gilbert Ames (1876-1951). Ameri can analyst. Best known for his work in the calculus of variations, he also contributed to the study of algebraic functions of a com plex variable and to the study of exterior ballistics.
BLOCK , n . randomized blocks . An experi- mental design for which an experiment is repeated for each of several situations, called blocks . E.g. , the yields of three types of corn might be tested in several fields, the blocks, by planting each type of corn in a plot in each field, assuming all plots in a given field have equal fertility. When study ing the quality of a product, the machines might be grouped into several types, the blocks, and the operators chosen randomly. See VARIANCE - analysis of variance,
BOARD MEASURE . The system of measur ing used for measuring lumber. See MEASURE — board measure.
BOD'Y , n. convex body . See convex- convex set.
BOHR, Harold (1887-1951). Danish ana lyst and number theorist. Brother of physi cist Niels Bohr. Worked on theory ofsum- mability, Dirichlet series, and zeta function. Founded theory of almost periodic func tions. See PERIODIC-almost periodic function.
BOLYAI, John (1802-1860). Hungarian geometer. Invented non-Euclidean geom etry independently of Lohachevski. See GEOMETRY —non-Euclidean geometry, LOBACHEVSKl.
BOLYAl, Wolfgang (1775-1856). Hun garian mathematician; friend of Gauss and father of John Bolyai.
BOLZA, Oskar (1857-1942). German analyst who spent many years in the U.S. Best known for his work in the calculus of variations, he also contributed to the study of elliptic and hyperelliptic functions.
problem of Bolza . In the calculus of variations, the general problem of deter mining, in a class of curves subject to con straints of the form 
and
an arc thai minimizes a function of the form
BOLZANO, Bernhard (1781-1848). Czech- oslovakian analyst.
Bolzano's theorem . A real-valued func- tion f of a real variable x is zero for at least one value of x between a and b if it is continuous on the closed interval [a, b] and f(a) and f(b) have opposite signs.
Bolzano-Weierstrass theorem , if E is a bounded set containing infinitely many points, there is a point x which is an accumulation point of E. The set E may be a set of real numbers, a set in a plane, or a set in n- dimensional Euclidean space. An equivalent statement of the theorem is tbat for any (finite dimensional) Euclidean space the concepts of bounded closed sets and sets with the Bolzano-Wcicrstrass property arc equivalent (see COMPACT). This theorem is frequently credited to Wcicrstrass, but was proved by Bolzano in 1817 and seems to have been known to Cauchy.
BOMBIERI, Enrico (1940- ). Italian mathematician awarded a Fields Medal |1974) for his contributions to number theory and the theory of minimal surfaces.
BOND , n. A written agreement to pay interest (dividends) on a certain sum of money and to pay the sum in some specified manner, unless it be a pcrpcluai bond (which draws interest, but whose principal need never be paid). Callable (or optional ) bonds are redeemable prior to maturity at the option of the issuing corporation, usually under certain specified conditions and at certain specified times. An annuity bond is re- deemed in equal installments which include the interest on the unpaid balance and suffi- cient payment of the face of the bond to redeem it by the end of a specified time. Coupon bonds are bonds for which the interest is paid by means of coupons (in effect, the coupons are post-dated checks, attached to the bond, which may be detached and used at the specified dale); registered bonds are bonds whose ownership is registered with the debtor, the interest being paid by check directly to the registered owner. If an issue of bonds is such that part of the bonds mature on a certain date and part of the bonds mature at each of certain dates thereafter (usually each year), the bonds arc said to be serial bonds . Collateral trust bonds are bonds issued by corporations whose assets consist primarily of securities of subsidiaries and of other corpor- ations (the securities are deposited with a trust company as trustee); guaranteed bonds are bonds for which some corporation (in addition to the one which issues the bonds) guarantees payment of principal or interest or both; debenture bonds are unsecured and usually protected only by the credit and earning power of the issuer; mortgage bonds have the highest priority in case of liquidation of the corporation (they arc called first mortgage bonds, second mortgage bonds, etc.).
"and interest price," purehase price , and redemption price of a bond. See price.
bond rate . Sec dividend — dividend on a bond.
bond table . A table showing the values of a bond at a given bond, rate for various investment rates, and for various periods. Most tables are based on interest computed semiannually (the usual practice) and on the assumption that the bonds will be redeemed at par.
par value of a bond . The principal named in the bond. Syn, face value.
premium bonds . Sec premium.
valuation of bonds . Computing the present value, at the investor's rale of interest, of the face value of the bond and of the interest pay ments (an annuity whose rental is equal to the dividend payments on the bond).
where P denotes the value of the bond, C its redemption value, R the interest payments (coupon value of a coupon bond), ? the number of periods before redemption, and ? the investor's (purchaser's) rale per period. yield of a bond . See vteld.
BONNET, Pierre Ossian (1819-1892). French analyst arid differential geometer.
Bonnet's mean-value theorem. See MEAN —mean-value theorems (or laws of the mean) for integrals.
BO'NUS , M. A sum paid in addilion to a sum that is paid periodically, as bonuses added to dividends, wages, etc. See insur ance — participating insurance policy.
BOOK , n. book value . See value.
BOOLE, George (1815-1864). Pioneering British logician. He also worked in algebra, analysis, calculus of variations, and probability theory.
Boolean algebra . A ring which has the properties that 
for each x, and there
is an element / such that 
for each x.
If the members <>f the ring are sets, then addition and multiplication for the ring correspond to symmetric difference and intersection of sets and / is a set which con tains all sets belonging to the ring. If a collection of subsets of a set S contains the complement of each of its members and the union of any two of its members, then it is a Boolean algebra if the ring operations of addi tion and multiplication are taken to be sym metric difference and intersection. Conversely, any Boolean algebra is an algebra of subsets of a collection of subsets of some set. If, for any Boolean algebra, the operations 
and 
, and the concept of inclusion, are defined by
if and only if 
then these correspond to the union, inter section, and inclusion concepts for sets and the following statements can easily be proved. {A +A can be proved to have the same value for all elements A of the Boolean algebra and lhis common value is denoted by 
):
If the complement of A' is defined to be A+1, then 
The simplest Boolean algebra is the one whose elements are the empty set and the set of one point, 
and /. Then 
if and only if one (or both) of A and B is I , and 
if and only if one (or both) of A and B is 
As well as being interpreted as an algebra of sets, a Boolean algebra can also be interpreted as an algebra of elementary logical properties of statements (propositions). The statement P = q means that the statements denoted by "p" and "q" are logically equivalent; 
denotes the statement "p or q"\ 
denotes the statement "p and 17"; and p' denotes the statement "not p." If ? is the statement "triangle ? is isosceles", and "q" is the state ment "triangle ? is equilateral", then 
is the statement "triangle ? is isosceles or triangle ? is equilateral"; 
is the statement "triangle ? is isosceles and triangle ? is equilateral", and 
is the statement 
(i.e., "for any triangle ?, ? is isosceles if ? is equilateral"). See atom, LATTICE.
BOR'DER-ING , v. bordering a determinant .
Annexing a column and a row. Usually refers Io annexing a column and a row which have 1 as a common element all the other elements of cither the column or the row being zero. This increases the order of the determinant by I but does not change its value.
BOREL, Felix Edouard Justin Emile (1 871-1956) French math, and politician. Founded modern theory of measure, divergent series; contributed to probability, game theory, etc. Borel covering theorem . Same as the HElNE-BOREL THEOREM.
BOREL first definition of the sum of a di-vergent series. If 
is the scries to be summed, 
the sum by this definition is where This definition is regular.
See summation 
— summation of divergent
series.
Borel's integral definition of the sum of a divergent series. The sum of 
is defined as where .? is real, if this limit 
exists. This definition is regular. See summation summation of divergent series.
Borel measurable function . See baire — Baire function.
Borel set . Any set which can be obtained from lhe closed and open sets on the real line by repeated applications of operations of union and intersection to denumerable num bers of sets. The class of all Borel sets is the 
generated by the class of all open sets (or by the class of all closed sets, or by the class of all intervals). Examples of Borel sets arc 
sets, which are countable unions of closed sets, and 
sets, which are countable intersections of open sets. Any Borel set is a measurable set. A Borcl set is sometimes called a Borel measurable set.
BOUND , n . A lower bound of a set of numbers is a number which is less than or equal to every number in the set; an upper bound is a number which is greater than or equal to every number in the set. The greatest lower bound or infimum (abbreviated g.l.b. or inf ) of a set of numbers is the largest of its lower bounds. This is either the smallest number in the set or the largest number that is less than all the numbers in the set. The least upper bound or supremum (abbreviated l.u.b . or sup ) of a set of numbers is the smallest of its upper bounds. This is either the largest number in the set or the smallest number that is greater than all the numbers in the set. If a g.l.b. or a l.u.b. of a set does not belong to the set (and sometimes if it does), then it is an accumulation point of the set. E.g., the set 
has
the l.u.b. 1/3, which also is an accumulation point. These concepts can be extended to any partially ordered set. E.g., an upper bound of a collection of sets is a set U that contains each of lhe given sets. See lattice.
bound of a function . A bound of a function f o n a set S is a bound for the set of numbers f(x) for which x is in S.
least upper bound axiom . The statement: "A set of real numbers that has an upper bound has a least upper bound." This often is taken as one of the axioms for the real number system, but otherwise is proved. It is equiv- alent to the greatest lower bound axiom; "A set of real numbers that has a lower bound has a greatest lower bound."