CEN-TRIP'E-TAL, adj. centripetal accelera tion. See ACCELERATION.
centripetal force. The force which restrains a body, in motion, from going in a straight line. H is directed toward the center of curvature. A force equal, but opposite in sign, Io the centrifugal force .
CEN'TROID, n. centroid of a set . The point whose coordinates are the mean values of the coordinates of the points in the set. The center is the centroid of a circle; the centroid of a triangle is the point of intersection of its medians. For sets in space over which integration can be performed, the coordinates of the centroid, 
and 
are given by
and 
where denotes the integral over the set, 
denotes 
an clement of area, arc length, or volume, and s denotes the area, arc length, or volume of the set. The centroid is the same as the center of mass if the set is regarded as having constant density (constant mass per unit area, length, or volume). See CENTER— center of mass, INTEGRAL—definite inte gral, mean mean value of a function.
centroid of a triangle . See median — me dian of a triangle.
CESARO, Ernesto (1859-1906). Italian geometer and analyst.
Cesaro's summation formula. A specific method of attributing a sum to certain di vergent series. A sequence of partial sums
is replaced by the sequence 
where
and
being the rth binomial coeflicient of order
n. 
If thesequence has 
a limit,
Ihe series 
is summable 
or 
to
this limit, in terms of the a, of the original
series,
Cesaro's summation formula is regular. See Summation summation of divergent series.
CEULEN, Ludolph van (1540-1610). Dutch mathematician who worked most of his life calculating 
to 35 digits; 
was the entire epitaph on his tombstone.
CHAIN , adj., n. (1) A linearly ordered set. See ORDERKD-ordered set, NESTED—nested sets. (2) See below, chain of simplexes, etc. chain conditions on rings. A ring R satisfies t!he descending chain condition on right ideals (or is Artinian on right ideals) if every non empty set of right ideals has a minimal member, or, equivalenliy, if no sequence of right ideals 
for which 
for each 
has more than a finite number of different members; R satisfies the ascending chain con- • dilion on right ideals (or is Noetherian on right ideals) if every nonempty set of right ideals has a maximal member, or, equivalently, if no sequence of right ideals 
for which
for each k has more than a finite number of different members. Similar defini tions can be given for left ideals. See wedder-
BURN.
chain discounts . See discount — discount scries.
chain of simplexes . Let G be a com- mutalive group with the group operation indi cated as addition. Let 
be oriented r-dimnensional simplexes of a simplicial complex K. Then
is an r-dimensional chain, or an ( r-chain . It
is understood that if 
is the simplex 
with its orientation changed, then 
for any g of G. The set of all r- chains is a group if chains are added in the natural way, i.e., by adding coefficients of each oriented simplex. The group 
is usually taken as either the group 
of integers or one of the finite groups 
of integers modulo an integer 
Of the latter, the group 
of integers modulo 2 is especially useful. If G is one of these groups of integers, then the boundary of an r-simplex 
is defined to be the 
-chain
where 
is the set of all 
dimensional faces of 
and 
is +1 or — 1 according as 
and 
are coherently oriented or noneoheremly oriented. If 
the boundary 
is defined to be 
The
boundary of the chain x : is defined to be
It follows that the boundary of a boundary is 
i.e., 
for ? any chain. A
chain whose boundary is 
is called a cycle (any boundary is a cycle). E.g., a chain of " 'edges" 
is a cycle if the
"edges" are joined so as to form a closed oriented path. See homolog ? — homology group.
chain rule . For ordinary differentiation , the rule of differentiation which states that, if F is the composite function of f and u defined by 
for all ? in the domain of u for which u{x) is in the domain of f , then
For example, the derivative of 
with 
is 
-du}dx, or 
Suffi-
cient conditions for the chain rule to be \alid at ? are that u be differentiate at x, / be differentiable at 
and each neighborhood
of ? contain points in the domain of F other than x. This rule can be used repeatedly, e.g., as 
or along with
other differentiation formulas in an explicit differentiation, such as
The chain rule can also be used to change
variables; e.g., if y is replaced by 
in
the differential equation 
one uses the formula
to obtain 
or 
Now let F be a function of one or more vari ables 
and each of these variables be a function of one or more variables 
The chain rule for partial differentiation is
This formula can be applied at a point 
if P 0 is an interior point of the domains of each of 
each of these
functions is differentiable with respect to 
at 
and F is differentiabie at the point 
obtained by evaluating each 
at 
If each of the variables 
is a
function of one variable x, then the formula becomes 
This is the total derivative of F with respect to x. E.g., if 
and 
the total derivative of z with respect to / is given by
See EPSILON — epsilon chain. surveyor's chain . A chain 66 feet long containing 100 links, each link 7.92 inches long. Ten square chains equal one acre. See denominate mjmbers in the appendix.
CHANCE , adj., n. Same as probability, Has considerable popular, but little technical, usage.
chance variable . Same as random vari able. See random.
CHANGE , n. change of base in logarithms.
See base change of base in logarithms.
change of coordinates . See transformation — transformation of coordinates.
change of variable in integration . See INTEGRATION —change of variables in inte gration.
cyclic change of variables . Same as cyclic
PERMUTATION. See PERMUTATION.
CHAR'AC-TER, n. finite character.
collection of sets is of finite character if A contains any set all of whose finite subsets belong to A and each finite subset of a member of A belongs to A. A property of subsets of a set is of finite character if a subset S has the property if and only if each nonempty finite subset of S has the property. E.g., the property of being simply ordered is of finite character, while the property of being well- ordered is not. If a property is of finite character, then the collection of all sets with this property is of finite character. If a collection A of sets is of finite character, then the property of belonging to A is of finite character. See zorn — Zorn's lemma.
group character. ? character of a group G is a homomorphism of G into the group of complex numbers of absolute value 1 ; i.e., it is a continuous function f defined on G for which 
is a complex number with
for all ? and y of G (the group operation of G is indicated here by multiplication). The set of all characters of G is called a character group. the "product" of characters f and g being defined to be the character h defined by 
for each ? of G. If G is commutative and locally compact, then G is algebraically isomorphic with the characler group of its character group. The character group can be given a topology by defining neighborhoods of a point so that U is a neigh borhood of a character f if there are elements 
of G and a positive number 
such that U is the set of all characters C for which
It then follows that the character group is a topological group and is locally compact if G is locally compact; it is discrete if G is compact. If G is the group of translations of the real line, then the character group of G is isomorphic with G.
CHAR'AC-TER-IS'TIC, adj., n. characteristic curves of a surface. That conjugate system of curves on a surface S such that the directions of the tangents of lhe two curves of the system through any point P of S are the characteristic directions at P on S. See conjugate — con jugate system of curves on a surface, and characteristic— characteristic directions on a surface. The characteristic curves arc parametric if and. only if 
and 
See FUNDAMENTAL — fundamental coefficients of a surface.
characteristic directions on a surface. The pair of conjugate directions on a surface S at a point P of S which are symmetric with respect to the directions of the lines of curva ture on S through P. The characteristic directions on S at P are unique except at umbilical points, and are the directions which minimize the angle between pairs of conjugate directions on S at P.
characteristic equation of li matrix. Let I be the unit matrix of the same order as the square matrix A. If 
is the deter-minant of the matrix xl— A, then 
is lhe characteristic equation of A and 
is the characteristic function of A. Thus the characteristic equation of the matrix 
or
It is known that every matrix satisfies its characteristic equation (the Hamiltonian-Cayley Theorem). E.g., the matrix 
is 7cro. The re-
duced characteristic equation of a matrix is the equation of lowest degree which is satisfied by the matrix. If A is a matrix of order n and / is the unit matrix of order n, then the reduced characteristic equation is 
where 
is the determinant of the matrix
arid 
is the greatest common
divisor of the 
—rowed minor deter-
minants of 
The reduced characteristic
function is 
The reduced characteristic equation is also called the minimal (or minimum ) equation , and the matrix derogatory of its order is greater than that of its reduced characteristic equation. See eigenvalue.
characteristic function {Statistics). Fora random variable X or the associated distri bution function, the characteristic function
is the function for which 
is the expected value of 
for real numbers t.
For a discrete random variable with values 
and probability function 
for a continuous random variable with probability density function f ,
is the n th derivative of 
then 
is the 
moment if the 
moment exists. The characteristic function 
of a vector random variable 
is defined by letting 
be the expected value of 
Then the random variables 
are independent if and only if 
where 
is the characteristic function of 
are independent, then the chifunction for the random variable See CUMULANTS, FOURIER-Fourier 
transform, INVARIANT—semi-invariant, MOMENT—mo ment generating function.
characteristic function of a matrix . See above, characteristic equation of a matrix.
characteristic function of a set. The function f which is defined by 
for each point ? in the set, and 
is not in the set.
characteristic of the logarithm of a number. See logarithm characteristic and mantissa of a logarithm.
characteristic number of a matrix . Same as CHARACTERISTIC root. See below.
characteristic number, functions, and vector in the study of symmetric operators. See EIGENVALUE.
characteristic of a one-para meter family of surfaces. The limiting curve of intersection of two neighboring members of the family as they approach coincidence—i.e., as the two values of the parameter determining the two members of the family of surfaces approach a common value. The equations of a given characteristic curve are the equation of the family taken with the partial derivative of this equation with respect to the parameter, each equation being evaluated for a particular value of the parameter. The locus of the characteristic curves, as the parameter varies, is the envelope of the family of surfaces. E.g., if the family of surfaces consists of all spheres of a given, fixed radius with their centers on a given line, the characteristic curves are circles having their centers on the line, and the envelope is the cylinder generated by these circles.
characteristic of a ring or field . If there is a smallest positive integer n such that 
for all x in the ring, then n is the characteristic of the ring; otherwise, the characteristic is zero. If the ring is an integral domain (e.g., a field), then the characteristic is a prime if it is not zero. Sometimes "characteristic 
" is used instead of "characteristic zero."
characteristic root of a matrix . A root of t he characteristic equation of the matrix. Syn. characteristic number, latent root. See EIGENVALUE.
Euler characteristic . See EULER—Euler characteristic.
Segre characteristic of a matrix. See canonical —canonical form of a matrix.
CHARGE , n. Coulomb's law for point-charges . See coulomb.
density of charge. See various headings under density.
electrostatic unit of charge. See ELECTROSTATIC
point-charge . A point endowed with elec trical charge. The electrical counterpart of point-mass or particle, i.e., electrical charge considered as concentrated at a point.
set (or complex) of point-charges . A collec tion of charges located at definite points of space. Sometimes the term complex carries the connotation that the maximum distance between the various pairs of charges is small in comparison to the distance to the field- points at which the electrical elfects are to be determined.
surrender charge. See surrender.
CHARLIER, Carl Vilhelm Ludvig (1862-1934). Swedish astronomer.
Gram-Charlier series. See GRAM, J. P.
CHART , n. flow chart . In machine computation, a diagram with labeled boxes,arrows, etc., showing the logical pattern of a prob lem, but not ordinarily including machine-language instructions and commands. See CODING, PROGRAMMING-programming for a computing machine.
CHEBYSHEV, Pafnuti Lvovich (1821-1894). Russian mathematician who worked in alge bra, analysis, geometry, number theory, and probability theory. Numerous other trans literations are used, e.g., Tchebycheff, Tche- bychev. See bertrand— Bertrand's postu late, MOMENT—moment problem.
Chebyshev differential equation . The dif ferential equation
Chebyshev inequality. If X is a random variable, f is a nonnegative real-valued func tion, and 
then
where 
is the probability of
and 
is the mean or ex-
pected value of f(X). The special case for which 
is the
Bienayme-Chebyshev inequality (often cal led simply the Chebyshev inequality) which was discovered by Bienayme in 1853 and rediscovered by Chebyshev in 1867:
where 
and 
are the variance and mean of X, and 
is finite and 
this can be written as
Chebyshev net of parametric curves on a surface . See parametric— equidistant sys tem of parametric curves on a surface.
Chebyshev polynomials . The polynomi als defined by 
and
cos (if arc cos x) 
or by
For 
while 
is a solution of Chebyshev's differen tial 
equation. Also,
is sometimes defined as 
times the
above value. See JACOBI—Jacobi polynomials.
CHECK, n.,v. (1) A draft upon a bank, usu ally drawn by an individual. (2) To verify by repetition or some other device. Any process used to increase the probability of correctness of a solution.
CHI, adj., n. The Greek letter 
chi-square distribution . A random vari able has a chi-square (or 
) distribution or
is a chi-square (or 
) random variable with n degrees of freedom if it has a probability density function f for which 
and
where 
is the gamma function. The mean is n, the variance is 
, and the moment gen erating function is 
If 
are independent normal ran dom variables with means 
and variances 
,
then is a chi-square random
variable 
with n degrees of freedom. The 
distribution is the same as the gamma dis tribution with its parameters given The values 
For large n, 
has approxn ' a normal distribution with mean 
and unit variance. Chi-square random variables have the additive property that 
random variable with degrees of freedom if 
are 
independent 
random variables with 
degrees of freedom. The preceding are sometimes called central chi- square distributions ; if 
are in dependent normal random variables with means 
and variances 1, then as a non cenlral chi-square dis tribution 
which has mean 
and moment generating function 
, where 
is the noncentrality parameter . Chi-square distributions are used widely for testing hypotheses; e.g., concerning contingency tables, goodness of fit, estimates of variances, and parameters of random samples. See COCHRAN—Coch- ran's theorem, F - F distribution, gamma— gamma distribution, and below, chi-square test.
chi-square test . Suppose one wishes to test the hypothesis that a certain random variable X with a finite number of values 
has a specified probability function p. For a random sample of size n , let
where 
is the number of times 
occurs in the sample. If n is sufficiently large (one empirical rule is that 
for all 0, then Y has approximately the chi-square dis tribution with 
degrees of freedom; us ing a significance level 
one accepts or rejects the hypothesis "p is the probability function of X" according as 
or 
where 
for the distri bution function F of this chi-square dis tribution. This test can be used for random variables with infinitely many values by grouping the values, e.g., by using intervals. The chi-square test can be adapted to many other situations, e.g., when the distribution is only partially specified or when there are unknown parameters.
CHI'NESE' , adj. Chinese remainder theo rem. If the integers 
are relatively prime in pairs and 
are any n integers, then there is an integer x which satisfies all the congruences 
and any two solutions of these congruences are congruent
CHOICE , n. An alternative elected by one of the players, or determined by a random device, for a move in the play of a game. See GAME , MOVE , PLAY .
axiom of choice . Given any collection of sets, there exists a "method" of designating a particular element of each set as a "special" element of thai set; for any collection A of sets there exists a function f such that 
is anelement of S for each set 5* of the collection A. See ordered -well-ordered set, zorn —Zorn's lemma. Syn. Zermelo's axiom.
finite axiom of choice . The axiom of choice for the special case that the collection of sets is a finite collection.