BOUND'A-RY , n. biharmonic boundary value problem . For a region R with boundary surface S 7 the biharmonic boundary value problem is the problem of determining a function U(x, y, z) that is biharmonic in R and is such that its first-order partial deriv atives coincide with prescribed boundary value functions on S. This problem, along with the Dirichlet problem, arises in particular problems concerning clastic bodies.
boundary of a set . See INTERIOR—interior of a set.
boundary of a simplex and a chain and boundary operator . See chain chain of simplexes.
boundary of half-line, half-plane, and half- space. See half —half-plane, half-space, and RAV.
boundary-value problem . {Differential Equations) The problem of finding a solution to a given differential equation or set of equations which will meet certain specified requirements for a given set of values of the independent variables -the boundary points. Many of the problems of mathematical physics arc of this type.
first boundary-value problem of potential theory (the Dirichlet prohlem). Given a region R, its boundary surface S, and a function /defined and continuous over 5, to determine a solution U of Laplace's equation 
which is regular in R , is continuous in R + S, and satisfies the equation U=f on the boundary. This problem occurs in electrostatics and heat flow. It has at most one solution. See green —Green's function.
second boundary-value problem of poten-tial theory (the Neumann problem). Given a region R, its boundary surface 5, and a function f defined and continuous over 5 and such that 
over S vanishes, to find a solution of Laplace 's equation 
which is regular in R, which together with its normal derivative is continuous in R\ S, and which has a normal derivative equal to / on the boundary S. This problem occurs in fluid dynamics. Any two of its solutions differ at most by a constant. See Neumann — Neumann's function.
third boundary-value problem of potential theory. As in the two above problems, except the function XJ is required to satisfy the equation 
on the boundary, where k, h, and f are prescribed functions that are continuous on S. This problem includes the other two and is of importance in heat flow and fluid mechanics. If h/k > 0, it has at most one solution. See robin — Robin's function.
BOUND'ED, adj. bounded convergence theorem. Suppose m is a countably additive measure on a 
of subsets of a set T for which 
and thai 
is a sequence of measurable functions for which there is a number M such that 
for all n and all x in T. Then each 
is integrable and, if there is a function S for which 
a.e. on T, then S is integrable and
For Riemann integration, the theorem can be stated as follows: Suppose that for a sequence of functions 
and an interval / there is a number M such that 
for all n and all x in I . Also suppose that each 
is Riemann integrable on I and that there is a function S which is a Riemann integrable on /and for which 
a.e. on I .
Then the integral of S over I is equal to the limit of the integral of 
over I as 
See LEBESGUE — Lebesgue convergence theorem, monotone — monotone convergence theorem, and series — integration of an infinite series,
bounded linear transformation. See LINEAR —linear transformation.
bounded quantity, or function . A quantity whose numerical value is always less than or equal to some properly chosen constant. The ratio of a leg of a right triangle to tbe hypotenuse is a bounded quantity since it is always less than or equal to 1; that is, the functions sin x and cos x are bounded functions since they are always less than or equal to 1. The function tan x in the interval 
is not bounded.
bounded sequenc e. See SEQUENCE — bound to a sequence.
bounded set of numbers . A set of numbers which has both a lower bound and an upper bound ; a set of numbers For which there are numbers A and B such that for each number ? of the set. 
bounded set of points . A set of points for which the set of distances between pairs of points is a bounded set. The least upper bound of such distances is called the diameter of the set. A set T is totally bounded if, for any 
there is a finite set of points in T such that each point of T is at distance less than 
from at least one of these points. A metric space is compact if and only if it is complete and totally bounded.
bounded variation . See variation total variation of a function.
essentially bounded function . A function / for which there is a number K such that the set of all ? for which 
is of measure zero. The greatest lower bound of such numbers K is the essential supremum of 
BOX , n . three boxes same . A game in which there are three boxes marked 1, 2, and 3. For a given play of the game, player A re- moves the bottom of one of the boxes, but player B docs not know which one it is. Player B then puts an amount of money equal to the number marked on the box in each of two of the three boxes. He loses the money put in the box with no bottom and wins the money put in the others. This is a zero-sum game with imperfect information. The payoff matrix docs not have a saddle point and the solutions are mixed strategies. The solutions arc 
for A and 
for B,
meaning that A removes the bottoms ot boxes 2 and 3, each with probability 
, player B puts money into boxes 1 and 2, or 1 and 3, with respective probabilities 
and 
(never in 2 and 3). The value of this game is 1 (with B the maximizing player).
BOYLE, Robert (1627-1691). British chemist and natural philosopher.
Boyle's law . At a given temperature, the product of the volume of a gas and the pressure (pv) is constant. Also called Boyle and Mariott's law. Approximately true for moderate pressures.
BRAC E, n. See aggregation.
BRA-CHIS'TO-CHRONE, adj. brachisto- chrone problem . The calculus of variations problem of finding the equation of the path down which a particle will fall from one point to another in the shortest time. Proposed by John Bernoulli in 1696 as a challenge to the mathematicians of Europe. The time re quired for a particle with the initial velocity 
to fall along a path y = f(x) from a point 
to a point 
is
where 
The solution of the problem then requires the determination of y that minimizes the integral. Sec calculus — calculus of variations. Newton, Leibniz, l 'Hopital, and James and John Bernoulli all found the correct solution, which is a cycloid through the two points. See cycloid.
BRACK'ET , n. See aggregation.
BRANCH , n. branch of a curve . Any section of a curve separated from the other sections of the curve by discontinuities or special points such as vertices, maximum or minimum points, cusps, nodes, etc. One would speak of the two branches of an hyperbola, or even of four branches of an hyperbola; or of two branches of the semi-cubical parabola, or of the branch of a curve above (or below) the x -axis.
branch cut of a Riemann surface . A line on curve C on a Riemann surface such that on crossing C a variable point is considered as passing from one sheet to another.
branch of a multiple-valued analytic function. The single-valued analytic function w=f(z) corresponding to values of z on a single sheet of the Riemann surface of definition.
branch-point of a Riemann surface . A point of the Riemann surface al which two or more sheets of the surface hang together.
infinite branch . See INFINITE.
BREADTH , n . Same as width.
BRIANCHON, Charles Julien (1783-1864). French geometer.
Brianchon's theorem . If a hexagon is circumscrihed about a conic section, the three diagonals (lines through opposite vertices) arc concurrent. This is the dual of Pascal's theorem. See duality —principle of duality of projective geometry, PASCAL — Pascal's theorem.
BRIDG'ING , v. bridging in addition . 1 ?
adding a one-place number to a second number, bridging is said to occur if the sum is in a decade different from that in which the second number lies. Thus bridging occurs in 14 + 9 = 23 but not in 14 + 3-17. See DECADE.
bridging in subtraction . If the difference obtained by subtracting a number from a second number (the minuend) is in a decade different from that in which the minuend lies, bridging is said to have occurred. Thus bridging occurs in the examples 64 — 9 = 55, 34-27 = 7, but not in 64-3 = 61.
BRIGGS, Henry (1561-1630). English astronomer, geometer, and numerical-table maker.
Briggs logarithms . Logarithms using 10 as a base. Syn. common logarithms. See LOGARITHM .
BRITISH , adj. British thermal unit or B.T.U . The amount of heat required to raise the tem perature of 1 Ib. of water 
when the water is at its maximum density, which is at 
or 
F.
BRO'KEN , adj. broken line . A curve con sisting of segments of lines joined end to end and not forming a single straight line segment. When defining the length of a curve, it is customary to approximate the curve by a broken line inscribed in the curve (i.e., having its vertices on the curve).
BRO'KER , n. One who buys and sells stocks and bonds on commission, that is, for pay equal to a given percentage of the value of the paper. Broker is sometimes applied to those who sell any kind of goods on commission, but commission merchant, or commission man, is more commonly applied to those who deal in staple goods.
BRO'KER-AGE , n. A commission charged fo r selling or buying stocks, bonds, notes, mortgages, and other financial contracts. See BROKER.
BROUWER, Luitzen Egbertus Jan (1881-1966). Dutch topologist and logician. Founder of modern intuitionism, wherein the positive integers furnish the prototype for the intellectual construction of all mathematical objects. In accordance with this philosophy, he objected to the un restricted use of Aristotelian logic (includ ing the law of the excluded middle), par ticularly in dealing with infinite sets.
Brouwer's fixed-point theorem . Let C be a circular disk consisting of a circle and the region within the circle. Then, for any continuous transformation which transforms each point of C into a point of C, there is some point which the transformation leaves fixed. The transformation is not assumed to be one-to-one. This theorem is also true for cl osed n-cells 
, e.g. , for a closed interval or for a sphere with its interior.
BROWN, Robert (1773-1858). Scottish botanist.
Brownian motion process . Same as WIENER PROCESS.
BUDAN DE BOIS LAURENT, Ferdinand Francois Desire (c. 1800-1853 or later). French physician and amateur mathe matician.
Budan's theorem. The number of real roots of f(x) = 0 between a and 
, where f(x) is a polynomial of degree n, is V(a) - V(b), or less by an even number, V(a) and V(b) being the numbers of variations in sign of the sequence 
when x — a and x = b, respectively. (Vanishing terms on the sequence are not counted and m -tuple roots are counted as m roots.) E.g., to find the number of roots of 
between 0 and 1, we form the sequence 
then substitute C and 1 for x, successively. This gives the sequences 1, -5, 0, 6 and —3, — 2, 6, 6, whence V (O)- V (1) = 2- 1 - 1. Thus there is one root between 0 and 1. Similarly the other roots can be located between 2 and 3 and between — 3 and —2.
BUFF'ER , n. In a computing machine, a switch that transmits a signal if any one of several signals is received by the switch; thus a buffer is the machine equivalent of the logical "or". See DISJUNCTION, GATE. Syn. inverse gate.
BUFFON, Georges Louis Leclerc, Comte de (1707-1788). French naturalist and proba- bilist.
Buffon's needle problem . Suppose a board is ruled with equidistant parallel lines and that a needle fine enough to be con sidered a segment of length 
less than the distance d between consecutive lines is thrown on the board. What is the prob ability it will hit one of the lines? The answer is 
It is possible to approximate the value off by throwing such a needle a large number of times.
BLIlLD'ING , n. building and loan association . A financial organization whose objective is to loan money for building homes. One plan, called the individual account plan, is essentially as follows: Members may buy shares purely as an investment, usually paying lor them in monthly installments at an annual nominal rate; or they may borrow money (shares) from the company with which to build, secur ing (guaranteeing) this money with mortgages on their homes. In both cases the monthly payments arc called dues. Failure to meet monthly payments on time is sometimes subjected to a fine which goes into the profits of the company. The profits of the company are distributed to the share purchasers, thus helping to mature (complete the payments on) their shares. In practice, the interest rale is usually figured so that it returns all profits automatically. A serial plan is a plan under which shares are issued at different times to accommodate new members. Monthly dues arc paid and profits distributed to all share holders. This plan naturally resolves into the individual account plan. A guaranteed stock plan is a plan in which certain investors provide certain funds and guarantee the payment of certain dividends on all shares, any surplus over this guarantee being divided among these basic stockholders. A terminat ing plan is a plan under which the members pay dues for a certain number of years to facilitate their building homes, the highest bidder getting the use of the money, since there is not enough to go around. New members coming in have to pay back-dues and back-earnings. This is the earliest plan of building and loan association and is not usually practiced now.
BULK , n. bulk modulus . See modulus. BUN'DLE , n. bundle of planes . See sheaf.
BUNIAK0VSKI (or BOUNIAKOWSKY), Victor Jakowlewitsech (1804-1899). Rus sian probabilist.
Buniakovski's inequality. See SCHWARZ — Schwarz's inequality.
BUR ALI-FORTI, Cesare (1861-1931). Ital ian mathematician.
Burali-Forti paradox. The "set of all ordinal numbers"' (each of which is an order type of a well-ordered set) is a well-ordered set. However, the order type Y of this set is then a largest ordinal number. This is impossible, since Y + 1 is a larger ordinal number (Y is the older type of a certain well-ordered set and Y + 1 is the order type of the well-ordered set obtained by intro ducing a single new element to follow every member of this set).
BUSH, Vannevar (1890- ). American electrical engineer. Starting about 1925, he built the first large-scale mechanical but electrically powered analog computer.