A.E . An abbreviation for almost everywhere. Sec measure —measure zero.
AB'A-CUS , n. [pi- abaci]. A counting frame to aid in arithmetic computation; an instruc tive plaything for children, used as an aid in teaching place value; a primitive predecessor of the modern computing machine. One form consists of a rectangular frame carrying as many parallel wires as there are digits in the largest number to be dealt with. Each wire contains nine beads free to slide on it. A bead on the lowest wire counts unity, on the next higher wire 10, on the next higher 100. etc. Two beads slid fo the right on the lowest wire, three on the next higher, five on the next and four on the next denote 4532.
Abel's method of summation. The method of summation for which a series 
is summable and has sum S if 
exists. A convergent series is summable by this method [sec below, Abel's theorem on power series (2)]. Also called Euler's method of summation. See summation —summation of a divergent series.
Abel's problem. Suppose a particle is con strained (without friction) to move along a certain path in a vertical plane under the force of gravity. Abel's problem is to find the path for which the time of descent is a given function / of x, where the x-axis is the hori zontal axis and the particle starts from rest. This reduces to the problem of finding a solution s(x) of the Volterta integral equation 
ABEL, Niels Henrik (1802-1829). Norwe gian algebraist and analyst. When about nineteen, he proved the genera! quintic equation in one variable cannot be solved by a finite number of algebraic operations (see BUFF1N1). Made fundamental contri butions to the theories of infinite series, transcendental functions, groups, and elliptic functions. This is easily obtained from Ihe evident identity:
Abel's identity. The identity. Abel's inequality. If 
for all positive integers n, then where L is the largest of the
quantities: jai , 
This inequality can be easily deduced from Abel's identity.
Abel's tests for convergence. (1) If the series 
converges and 
is a bounded mono- ic sequence, then 
converges. (2) If
is equal to or less than a properly chosen constant for all k and 
is a positive, monotonic decreasing sequence which ap proaches zero as a limit, then 
con verges. (3) If a series of complex numbers 
is convergent, and the series 
is absolutely convergent, then 
is convergent. (4) If the series 
is uniformly convergent in an interval 
is positive and monotonic decreasing for any value of x in the interval, and there is a number k such that 
for all x in the interval, then 
is uniformly convergent (this is Abel's test for uniform convergence).
Abel's theorem on power series. (1) If a power series, 
..., converges for 
it converges absolutely for 
(2) If 
is convergent, then 
where the limit is the limit on the left at +1. An equivalent statement is that if 
converges when x = R, then S is continuous if $(x) is defined as 
when x is in the closed interval
with end points 0 and R. This theorem is designated in various ways, most explicitly by "Abel's theorem on continuity up to the circle of convergence."
A-BEL'IAN , adj. Abelian group. See group.
A-BRIDGED ', adj. abridged multiplication.
See multiplication.
PI ticker's abridged nutation. A notation used for studying curves. Consists of the use of a single symbol lo designate the expression (function) which, equated to zero, has a given curve for its locus; hence reduces Ihe study of curves to the study of polynomials of the first degree, 
and 
denotes the family of lines passing through their common point (1, 1). See pencil- pencil of lines through a point.
AB-SCIS'SA , n. [pi. abscissas or abscissae].
The horizontal coordinates in a two-dimen sional system of rectangular coordinates; usually denoted by x. Also used in a similar sense in systems of oblique coordinates. See c ar Ttsi an — Cartesian coordinates.
AB'SO-LUTE , adj. absolute constant, con tinuity, convergence, inequality, maximum (minimum), symmetry. See constant, con tinuous, CONVERGENCF, INEQUALITY, MAXI MUM, SYMMETRIC —symmetric function,
absolute moment (Statistics). For a ran dom variable X or the associated distribu tion function, the kth absolute moment about a is the expected value of 
whenever this exists. See moment— mo ment of a distribution.
absolute number . A number represented by figures such as 2, 3, or 
rather than by fetterS as in algebra.
absolute property of a surface . Same as INTRINSIC PROPERTY OF A SURFACE.
absolute term in an expression . A term which does not contain a variable. Syn. constant term. In the expression ax 2 + bx + c, c is the only absolute term.
absolute value of a complex number . See modulus —modulus of a complex number.
absolute value of a real number . The absolute value of a, written \a\, is the non- negative number which is equal to a if a is nonnegative and equal to —a if a is negative; e.g., 3 = [3|,0 = |0|, and 3 = |-3|. Useful properties of the absolute value are that 
for all real numbers x and y. Syn. numerical value.
absolute value of a vector. See vector — absolute value of a vector.
AB'STRACT , adj. abstract mathematics. See mathematics —pure mathematics.
abstract number . Any number as such, simply as a number, without reference to any particular objects whatever except in so far as these objects possess Ihe number property. Used to emphasize the disiinclion between a number, as such, and concrete numbers. See CONCRFTL, NUMBER, DENOMINATE.
abstract space . A formal mathematical system consisting of undefined objects and axioms of a geometric nature. Examples are Euclidean spaces, metric spaces, topological spaces, and vector spaces.
abstract word or symbol. (1) A word or symbol that is nut concrete; a word or symbol denoting a concept built up from consideration of many special cases; a word or symbol denoting a property common to many individ uals or individual sets, as yellow, hard, two, three, etc. (2) A word or symbol which has no specific reference in the sense that the concept it represents exists quite independently of any specific cases whatever and may or may not have specific reference.
AC-CEL'ER-A'TION , n. The time rate of change of velocity. Since velocity is a directed quantity, the acceleration a is a vector equal
to 
where 
is the increment
in the velocity v which the moving object acquires in t units of time. Thus, if an air plane moving in a straight line with the speed of 2 miles per minute increases its speed until it is flying at the rate of 5 miles per minule at the end of the next minute, its average acceleration during that minute is 3 miles per minute per minule. If the increase in speed during this one minute interval of time is uniform, the average acceleration is equal to the actual acceleration. If the in crease in speed in this example is not uniform, the instantaneous acceleration at the time Ii is determined by evaluating the limit of the
quotient 
as the time interval 
where 
is the derivative of speed v along the is made to approach zero by making r approach 
For a particle moving along a curved path, the velocity v is directed along the tangent to the path and the acceleration a can be shown to be given by the formula path, c is the curvature of the path at the point, and t and v are vectors of unit lengths directed along the tangent and normal to the path. The first of these terms, 
is called the tangential component, and the second, 
the normal (or centripetal) component of acceleration. If the path is a
straight line, the curvature c is zero, and hence the acceleration vector will be directed along the path of motion. If the path is not recti linear, the direction of the acceleration vector is determined by its tangential and normal components as shown in the figure.
acceleration of Coriolis. If S' is a reference frame rotating with the angular velocity <o about a fixed point in another reference frame S, the acceleration a of a particle, as measured by the observer fixed in the reference frame S, is given by the sum of three terms: 
where 
is the acceleration of the particle relative to 
is the acceleration of the moving space, and 
is the acceleration of Coriolis. The symbol 
denotes the vector product of the angular velocity u, and the velocity 
relative to 
, so that the acceleration of Coriolis is normal to the plane determined by the vectors m and 
and has the magnitude 
sin 
The acceleration of Coriolis is also called the complementary acceleration.
acceleration of a falling body. The accelera tion with which a body falls in vacua at a given point on or near a given point on the earth's surface. This acceleration, frequently denoted by #, varies by less than one percent over the entire surface of the earth. Its "average value" has been defined by the International Commission of Weights and Measures as 9.80665 meters (or 32.174 feet) per second per second. Its value at the poles is 9.8321 and at the equator 9.7799. Syn. acceleration of gravity.
angular acceleration. The time rate of change of angular velocity. If the angular velocity is represented by a vector 
directed along the axis of rotation, then the angular acceleration 
in the symbolism of calculus, is given by See velocity —angular velocity. 
centripetal, normal, and tangential com ponents of acceleration. See above, acce-.le-ration .
uniform acceleration . Acceleration in which there are equal changes in the velocity in equal intervals of time. Syn, constant acceleration.
ACCENT , n. A mark above and to the right of a quantity (or letter), as in a' or x'; the mark used in denoting that a letter is primed. See prime —prime as a symbol.
AC-CEPT'ANCE, adj. acceptance region. See HYPOTHESIS-test of a hypothesis.
AC-CU'MU-LAT'ED , adj. accumulated value.
Same as amount at simple or compound interest. The accumulated value (or amount) of an annuity at a given date is the sum of the compound amounts of the annuity payments to that date.
AC-CU'MU-LA'TION, adj., n. Same as ACCUMULATED VALUE.
accumulation of discount on a bond . Writing up the book value of a bond on each dividend dale by an amount equal to the interest on the investment (interest on book value at yield rale) minus the dividend. See value —book value.
accumulation factor . The name sometimes given to the binomial (1- r ), or (1 + i ), where r, or /, is the rate of interest. The formula for compound interest is 
where A
is the amount accumulated at the end of n periods from an original principal P at a rate r per period. See compound —compound amount.
accumulation point . An accumulation point of a set of points is a point P such that there is at least one point of the set distinct from P in any neighborhood of the given point; a point which is the limit of a sequence of points of the set (for spaces which satisfy the first axiom of countability). An accumulation point of a sequence is a point P such that, for any integer n, each neighborhood of P contains at least one term of the sequence after the nth term; e.g., the sequence 
has two accumulation points, the numbers 0 and 1 (also see skqukno —accumulation point of a sequence). Syn, cluster point, limit point. See BOLZANO-Bolzano-Weierstrass theorem, CONDENSATION—condensation point.
accumulation problem. The determination of the amount when the principal, or princi pals, interest rate, and time for which each principal is invested are given.
accumulation schedule of bond discount. A table showing the accumulation of bond discounts on successive dates. Interest and book values are usually listed also.
AC-CU'MU-LA'TOR, n. In a computing machine, an adder or counter that augments its stored number by each successive number is receives.
AC'CU-RA-CY, n. Correctness, usually re- ferring to numerical computations. The accuracy of a table may mean cither: (1) The number of significant digits appearing in the numbers in the table {e.g., in the mantissas of a logarithm table); (2) the number of correct places in computations made with the table. (This number of places varies with the form of computation, since errors may repeatedly combine so as to become of any size whatever.)
AC'CU-RATE, adj. Exact, precise, without error. One speaks of an accurate statement in the sense that it is correct or true and of an accurate computation in the sense that it contains no numerical error. Accurate to a certain decimal place means that all digits preceding and including the given place are correct and the next place has been made zero if less than 5 and 10 if greater than 5 (if it is equal to 5, the most usual convention is to call it zero or 10 as is necessary to leave the last digit even). E.g., 1,26 is accurate to two places if obtained from 1.264 or 1.256 or 1.255. See rounding —rounding off.
AC'NODE , ii. See point- isolated point,
A-COUS'TI-CAL, adj. acoustical property of conies . See ELLIPSE—focal property of ellipse, hyperbola- focal property of hyperbola, PAHABOLA -focal property of parabola.
A'CRE , n. The unit commonly used in the United States in measuring land; contains 43,560 square feet, 4,840 square yards, or 160 square rods.
AC'TION , n. A concept in advanced dynam ics defined by the line integral 
called the action integral , where m is the mass of the particle, 
is its velocity, and 
is the vector element of the arc of the trajectory joining the points 
and 
The dot in the integrand denotes the scalar product of the momentum vector 
and 
The acti on 
plays an important part in the development of dynamics from variational rinciples. See below, principle of least action.
law of action and reaction . The basic law of mechanics asserting that two particles interact so that the forces exerted by one on another are equal in magnitude, act along the line joining the particles, and arc opposite in direction. See newton —Newton's laws of motion,
principle of least action . Of all curves passing through two fixed points in the neigh borhood of the natural trajectory, and which are traversed by the particle at a rate such that for each (at every instant of time) the sum of the kinetic and potential energies is a constant, that one for which the action integral has an extremal value is the natural trajectory of the particle. See action.
A-CUTE' , adj. acute angle . An angle nu merically smaller than a right angle (usually a positive angle less than a right angle). acute triangle . See triangle,
AD'DEND , n. One of a set of numbers to be added, as 2 or 3 in the sum 2 + 3.
AD'DER , n. In a computing machine, any arithmetic component that performs the operation of addition of positive numbers. An arithmetic component that performs the operations of addition and subtraction is said to be an algebraic adder . See accumula tor, counter,
AD-DI'TION , n. addition of angles, directed line segments, integers, fractions, irrational numbers, mixed numbers, matrices, and vectors .
See various headings under sum.
addition of complex numbers . Sec com- plex —complex numbers,
addition of decimals . The usual procedure for adding decimals is to place digits with like place value under one another, i.e., place decimal points under decimal points, and add as with integers, putting the decimal point of the sum directly below those of the addends. See sum- sum of real numbers.
addition formulas of trigonometry . See TRIGONOMETRY.
addition of series. Sec series.
addition of similar terms in algebra. The process of adding the coefficients of terms which are alike as regards their other factors. See dissimilar —dissimilar terms. addition of tensors. See tensor. algebraic addition . See sum- algebraic sum, sum of real numbers.
arithmetic addition. See sum arithmetic
sum.
proportion by addition (and addition and
subtraction). See proportion.
ADD'I-TIVE, adj. additive function. A function 
which has the property thai 
is defined and equals 
whenever 
and 
are defined. A continuous additive function is necessarily homogeneous. A function 
is subaddirive or superadditivc according as 
or 
for all 
and 
in the domain of (this domain is usually taken to be an interval of the form 
additive inverse . See inverse —inverse of an clement.
additive set function . A function which assigns a number 
t o each set 
of a 
for all disjoint members X and Y of F. The function 
is countably additive (or completely additive) if the union of any finite or countable set of members of F is a member of F and 
family F of sets is additive (or finitely additive] if the union of any two members of F is a member of F and for each finite or countable collection of sets 
which are pairwise disjoint and belong to F. If 
then 
is said to be subadditive (it is then not necessary to assume the sets arc pairwise disjoint). See MEASURE— measure of a set.
AD'I-A-BAT'IC , adj. adiabatic curves. Curves showing the relation between pressure and volume of substances which are assumed to have adiabatic expansion and contraction. adiabatic expansion (or contraction). {Thermodynamics) A change in volume without loss or gain of heat.
AD IN'FI-NI'TUM . Continuing without end (according to some law); denoted by three dots, as ...; used, principally, in writing infinite series, infinite sequences, and infinite products.
AD-JA'CENT , adj. adjacent angles. Two angles having a common side and common vertex and lying on opposite sides of their common side. In the figure, AOB and BOC arc adjacent angles.
AD-JOINED' , adj. adjoined number. See held —number field.
AD'JOINT , adj., n. adjoint of a differential
the adjoint is the differential equation
equation. For a homogeneous differential equation.This relation is symmetric, 
being the adjoint of 
A function is a solution of one of these equations if and only if it is an integrating factor of the other. There is an expression 
for which
is linear and homogeneous in 
and in 
It is known as the bilinear concomitant. An equation is self- adjoint if 
E.g., Sturm-Liouville differentia! equations and Lcgendrc differential equations are self-adjoint.
adjoint of a matrix . The transpose of the matrix obtained by replacing each element by its cofactor; the matrix obtained by replacing each element 
(in row r and column s) by the cofactor of the element 
(in row . s and column r). The adjoint is defined only for square matrices. Sometimes (rarely) the adjoint is called the adjugatc, although adjugatc has also been used for the square matrix of order 
formed from a square matrix of order n by arranging all r th-order minors in some specified order. The Ifermi- tian conjugate matrix is frequently called the adjoint matrix ' by writers on quantum mechanics.
adjoint of a transformation. For a bounded linear transformalion T which maps a Hilbert space H into H (with domain of T equal to H), there is a unique bounded linear transforma tion 
the adjoin! of T. such that the inner products 
and 
are equal for all .v and y of H. It follows lhat 
Two linear transformations 
and 
are said to be adjoint if 
for each x in the domain of 
and y in the domain of 
If 7" is a linear transformation whose domain is dense in H, there is a unique trans formation 
(called the adjoint of T) such that 7"and 
are adjoint and, if S is any other transformalion adjoint to T, then the domain of S is contained in the domain of 
and S and 
coincide on the domain of S. For a finite dimensional space and a transformation T which maps vectors 
into 
with 
(for each ('), the adjoin! of T is the transformation for which 
with 
and the matrices of the coefficients of T and of 
are Hermitian conjugates of each other. If T is a bounded linear transformation which maps a Banacli space X into a Banaeh space Y, and 
and 
are the first conjugate spaces of X and Y, then the adjoint of T is the linear transformation 
which maps 
into 
and is such that 
members of 
and 
respectively) if / is the continuous linear functional defined by 
For two bounded linear transformations 
and 
the adjoints of 
and 
are 
and 
respectively. If 7* has an inverse whose domain is all of H (or Y), then 
For Banaeh spaces, the adjoint 
of 
is a mapping of 
into 
which is a norm-preserving extension of T (T maps a subset of 
, which is isometric with X, into Y**). For Hilbert space, T** is identical with Tif T is bounded with domain H; T** is a linear extension of T otherwise. See shlf — self-adjoint transformation.
adjoint space . See conjucjatl —conjugate space.