198 fixed point
statistical theory of natural selection. Fisher died in
Adelaide, Australia, on July 29, 1962.
See also
HISTORY OF PROBABILITY AND STATISTICS
(essay); K
ARL
P
EARSON
.
fixed point Any point that is mapped to itself by a
given
TRANSFORMATION
is called a fixed point. For
example, the points x= 0 and x= 1 are fixed points
for the function f(x) = x2. Any point on the line of
REFLECTION
for a reflection in a plane is a fixed point
for that reflection.
Any continuous map fthat maps points in the unit
interval [0,1] to points in the same interval must pos-
sess a fixed point. (By the
INTERMEDIATE
-
VALUE THEO
-
REM
, the graphs of y= f(x) and y= xfor 0 ≤x≤1 must
intersect.) This is a special case of a more general result
proven by Luitzen Egbertus Jan Brouwer in 1915 stat-
ing that, for all values n, any continuous map f:[0, 1]n
→[0,1]nmust possess at least one fixed point. (For
n= 2, [0,1]2= [0,1] × [0,1] is the unit square in the
plane, and for n= 3, [0,1]3= [0,1] × [0,1] × [0,1] is a
unit cube in three-dimensional space.) This theorem has
the following amusing consequences:
Consider two square sheets of paper, one lying
directly on top of the other. Initially each point
of the upper sheet lies directly above its corre-
sponding point on the lower sheet. Now crum-
ple the top sheet and rest the crumpled ball
anywhere on the lower sheet. By the Brouwer
fixed-point theorem there is still at least one
point of the crumpled sheet lying directly above
its corresponding point on the bottom sheet.
As a thought experiment, imagine the
molecules of the liquid in a cup of coffee as the
points in a three-dimensional cube. After the
coffee is stirred, the Brouwer fixed point theo-
rem assures that at least one molecule will
return to its original location.
See also
ITERATION
.
floor/ceiling/fractional part functions The floor
function, also known as the greatest-integer function,
takes a real number xand returns the greatest integer
not exceeding x. This quantity is denoted: x. For
example, 7.2= 7, 7.9998= 7 and 7= 7. Also,
–6.34= –7.
The ceiling function, also known as the least-
integer function, takes a real number xand returns the
least integer not smaller than x. This quantity is
denoted: x. For example, 7.2= 8, 7.998= 8
and 7= 7. Also, –6.34= –6. The fractional part of
a real number x, denoted {x}, is given by: {x} = x– x.
For example, {7.2} = 0.2, {7.998} = 0.998 and {7} = 0.
Also, {–6.34} = 0.66.
The names “floor” and “ceiling,” as well as the
notation for these functions, were introduced by Ken-
neth Iverson in his 1962 computer science text A Pro-
gramming Language. These functions often arise in
applications of counting. For example, there are
multiples of 4 less than, or equal, to N, and, for any
two real numbers xand ywith x< y, the closed interval
[x, y] contains y– x+ 1 integers.
See also
DAYS
-
OF
-
THE
-
WEEK FORMULA
.
floor function See
FLOOR
/
CEILING
/
FRACTIONAL PART
FUNCTIONS
.
fluxion In his version of
CALCULUS
, S
IR
I
SAAC
N
EW
-
TON
thought of variable xas a flowing quantity, or a
fluent, and called the rate of change of xthe “fluxion”
of x. He used the notation .
xfor the fluxion of x, ¨xfor
the fluxion of the fluxion of x, and so forth. Thus if
x=f(t),where xis the distance and tthe time for a
moving object, then .
xis the instantaneous
VELOCITY
of
the object, and ¨xits instantaneous
ACCELERATION
.
Today the term fluxion is considered obsolete, and we
use the word
DERIVATIVE
in its stead. The raised-dot
notation for derivative, however, is still used by physi-
cists for denoting derivatives with respect to time.
See also
CALCULUS
;
HISTORY OF CALCULUS
(essay).
focal chord Any
CHORD
of a conic curve—a
PARABOLA
, an
ELLIPSE
, or a
HYPERBOLA
—that passes
through a
FOCUS
of the conic is called a focal chord.
See also
CONIC SECTIONS
;
FOCAL RADIUS
.
focal radius Any line from the
FOCUS
of a conic
curve—a
PARABOLA
, an
ELLIPSE
, or a
HYPERBOLA
—to a
point on the conic is called a focal radius.
See also
CONIC SECTIONS
;
FOCAL CHORD
.
N
4