This numerical value represents the proportion in
total variation in the y variable that can be accounted
for by the line of best fit. This proportion has values
between 0 and 1, with a value of 1 indicating that all
variation is due to a linear fit, that is, the data values
lie perfectly on the regression line, and an R
2 value of
zero indicates that none of the variation in the y-values is due to a linear correlation. If R
2 = 0.84, for
example, then we can say that 84 percent of the variation in the y-values is accounted for by a linear relationship with the values of x.
See also REGRESSION.
Lebesgue, Henri-Léon (1875–1941) French Analysis Born on June 28, 1875, French mathematician
Henri Lebesgue is remembered for his revolutionary
ideas in CALCULUS and in the theory of integration. By
generalizing the notion of AREA to one of an abstract
“measure theory,” Lebesgue transformed the object of
an integral into a tool applicable to an extraordinarily
large class of settings. He published work on this topic
at the young age of 27.
Lebesgue studied at the École Normale Supérieure,
France, and taught at the University of Nancy for 3
years. He presented his famous work to his university
colleagues during his final year there in 1902. The
idea behind his approach is relatively simple. One typically computes an integral by subdividing the range
of inputs, the x-axis, into small intervals and then
adding the areas of rectangles above these intervals of
heights given by the function. This is akin to counting
the value of a pocket full of coins by taking one coin
out at a time and adding the outcomes as one goes
along. Lebesgue’s approach, however, is to subdivide
the range of outputs, the y-axis, into small intervals
and to measure the size of the sets on the x-axis for
which the function gives the desired output on the yaxis. This is akin to counting coins by first collecting
all the pennies and determining their number, then all
the nickels and ascertaining the size of that collection,
and so forth. Of course the shape of the sets one
encounters along the x-axis can be complicated and
difficult to measure in size. The work of French mathematicians Émile Borel (1871–1956) and MARIE
ENNEMOND CAMILLE JORDAN (1838–1922) in developing so-called measure theory provided Lebesgue the
means to do this.
Lebesgue wrote over 50 papers and two books,
including his 1902 paper “Intégrale, longeur, aire”
(Integrals, lengths, area), his 1910 article “Sur l’intégration des functions discontinues” (On the integration of
discontinuous functions), and his 1906 monograph
Leçons sur les séries trigonométriques (Lectures on
trigonometric series). He also made important contributions to the fields of TOPOLOGY and FOURIER SERIES,
and was appointed professor at the Sorbonne, University of Paris, in 1910.
At one point in his life, Lebesgue expressed serious
unease about continuing the work in integration theory
he himself had founded. He feared that by making
mathematics abstract, topics in the subject would begin
to lose meaningful context. He died in Paris, France, on
July 26, 1941.
Lebesgue’s revolutionary approach to integration
theory is taught to all upper-level college students and
graduate students in mathematics today. It is considered a core component of any serious study of analysis.
left derivative/right derivative The DERIVATIVE of a
function f(x) at position x is defined as the LIMIT f′(x) =
. The value of the limit, if it exists,
represents the slope of the tangent line to the graph of
the function at position x. If the quantity h is restricted
to run only through negative values as it approaches
the value zero, that is, if the limit above is replaced by
a limit from the left, we obtain the left derivative of the
function at x:
Restricting h to run only through positive values produces the right derivative of the function at x:
(See LIMIT.) The general derivative f′(x) exists if, and
only if, the left and right derivatives both exist and agree
in value. This need not always be the case. Consider the
lim
(
) ( )
h
f x h f x
h
→ +
+ −
0
lim
(
) ( )
h
f x h f x
h
→ −
+ −
0
lim
(
) ( )
h
f x h f x
h
→
+ −
0
R
S
S S
xy
xx yy
2
2
=
( )
left derivative/right derivative 307
total variation in the y variable that can be accounted
for by the line of best fit. This proportion has values
between 0 and 1, with a value of 1 indicating that all
variation is due to a linear fit, that is, the data values
lie perfectly on the regression line, and an R
2 value of
zero indicates that none of the variation in the y-values is due to a linear correlation. If R
2 = 0.84, for
example, then we can say that 84 percent of the variation in the y-values is accounted for by a linear relationship with the values of x.
See also REGRESSION.
Lebesgue, Henri-Léon (1875–1941) French Analysis Born on June 28, 1875, French mathematician
Henri Lebesgue is remembered for his revolutionary
ideas in CALCULUS and in the theory of integration. By
generalizing the notion of AREA to one of an abstract
“measure theory,” Lebesgue transformed the object of
an integral into a tool applicable to an extraordinarily
large class of settings. He published work on this topic
at the young age of 27.
Lebesgue studied at the École Normale Supérieure,
France, and taught at the University of Nancy for 3
years. He presented his famous work to his university
colleagues during his final year there in 1902. The
idea behind his approach is relatively simple. One typically computes an integral by subdividing the range
of inputs, the x-axis, into small intervals and then
adding the areas of rectangles above these intervals of
heights given by the function. This is akin to counting
the value of a pocket full of coins by taking one coin
out at a time and adding the outcomes as one goes
along. Lebesgue’s approach, however, is to subdivide
the range of outputs, the y-axis, into small intervals
and to measure the size of the sets on the x-axis for
which the function gives the desired output on the yaxis. This is akin to counting coins by first collecting
all the pennies and determining their number, then all
the nickels and ascertaining the size of that collection,
and so forth. Of course the shape of the sets one
encounters along the x-axis can be complicated and
difficult to measure in size. The work of French mathematicians Émile Borel (1871–1956) and MARIE
ENNEMOND CAMILLE JORDAN (1838–1922) in developing so-called measure theory provided Lebesgue the
means to do this.
Lebesgue wrote over 50 papers and two books,
including his 1902 paper “Intégrale, longeur, aire”
(Integrals, lengths, area), his 1910 article “Sur l’intégration des functions discontinues” (On the integration of
discontinuous functions), and his 1906 monograph
Leçons sur les séries trigonométriques (Lectures on
trigonometric series). He also made important contributions to the fields of TOPOLOGY and FOURIER SERIES,
and was appointed professor at the Sorbonne, University of Paris, in 1910.
At one point in his life, Lebesgue expressed serious
unease about continuing the work in integration theory
he himself had founded. He feared that by making
mathematics abstract, topics in the subject would begin
to lose meaningful context. He died in Paris, France, on
July 26, 1941.
Lebesgue’s revolutionary approach to integration
theory is taught to all upper-level college students and
graduate students in mathematics today. It is considered a core component of any serious study of analysis.
left derivative/right derivative The DERIVATIVE of a
function f(x) at position x is defined as the LIMIT f′(x) =
. The value of the limit, if it exists,
represents the slope of the tangent line to the graph of
the function at position x. If the quantity h is restricted
to run only through negative values as it approaches
the value zero, that is, if the limit above is replaced by
a limit from the left, we obtain the left derivative of the
function at x:
Restricting h to run only through positive values produces the right derivative of the function at x:
(See LIMIT.) The general derivative f′(x) exists if, and
only if, the left and right derivatives both exist and agree
in value. This need not always be the case. Consider the
lim
(
) ( )
h
f x h f x
h
→ +
+ −
0
lim
(
) ( )
h
f x h f x
h
→ −
+ −
0
lim
(
) ( )
h
f x h f x
h
→
+ −
0
R
S
S S
xy
xx yy
2
2
=
( )
left derivative/right derivative 307