statistical theory of natural selection. Fisher died in
Adelaide, Australia, on July 29, 1962.
See also HISTORY OF PROBABILITY AND STATISTICS
(essay); KARL PEARSON.
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) = x
2 . Any point on the line of
REFLECTION for a reflection in a plane is a fixed point
for that reflection.
Any continuous map f that maps points in the unit
interval [0,1] to points in the same interval must possess a fixed point. (By the INTERMEDIATE-VALUE THEOREM, the graphs of y = f(x) and y = x for 0 ≤ x ≤ 1 must
intersect.) This is a special case of a more general result
proven by Luitzen Egbertus Jan Brouwer in 1915 stating that, for all values n, any continuous map f:[0, 1]
n
→ [0,1]
n must 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 corresponding point on the lower sheet. Now crumple 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 theorem 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 x and 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 leastinteger function, takes a real number x and 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 Kenneth Iverson in his 1962 computer science text A Programming 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 x and y with 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, SIR ISAAC NEWTON thought of variable x as a flowing quantity, or a
fluent, and called the rate of change of x the “fluxion”
of x. He used the notation
.
x for the fluxion of x, ¨
x for
the fluxion of the fluxion of x, and so forth. Thus if
x = f(t), where x is the distance and t the time for a
moving object, then
.
x is the instantaneous VELOCITY of
the object, and ¨
x its 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 physicists 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
198 fixed point
Adelaide, Australia, on July 29, 1962.
See also HISTORY OF PROBABILITY AND STATISTICS
(essay); KARL PEARSON.
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) = x
2 . Any point on the line of
REFLECTION for a reflection in a plane is a fixed point
for that reflection.
Any continuous map f that maps points in the unit
interval [0,1] to points in the same interval must possess a fixed point. (By the INTERMEDIATE-VALUE THEOREM, the graphs of y = f(x) and y = x for 0 ≤ x ≤ 1 must
intersect.) This is a special case of a more general result
proven by Luitzen Egbertus Jan Brouwer in 1915 stating that, for all values n, any continuous map f:[0, 1]
n
→ [0,1]
n must 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 corresponding point on the lower sheet. Now crumple 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 theorem 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 x and 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 leastinteger function, takes a real number x and 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 Kenneth Iverson in his 1962 computer science text A Programming 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 x and y with 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, SIR ISAAC NEWTON thought of variable x as a flowing quantity, or a
fluent, and called the rate of change of x the “fluxion”
of x. He used the notation
.
x for the fluxion of x, ¨
x for
the fluxion of the fluxion of x, and so forth. Thus if
x = f(t), where x is the distance and t the time for a
moving object, then
.
x is the instantaneous VELOCITY of
the object, and ¨
x its 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 physicists 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
198 fixed point