Chebyshev, Pafnuty Lvovich 71

y

Intuitively, the concept is easy to grasp if we think of

derivatives as rates of change. For example, if ychanges

atimes as fast as u, and uchanges btimes as fast as x,

then we expect yto change ab times faster than x.

The chain rule extends to functions of more than

one variable. For example, if z= f(x, y) is a function of

two variables with each of xand ya function of t, then

one can show that the total derivative of zwith respect

to tis given by:

See also

PARTIAL DERIVATIVE

.

change of variable See

INTEGRATION BY SUBSTITUTION

.

chaos A situation in which a

DYNAMICAL SYSTEM

can

appear to be random and unpredictable is called chaos.

More precisely, mathematicians define a dynamical sys-

tem to be chaotic if the set of all equilibrium points for

the system form a

FRACTAL

.

The term chaos was introduced by American math-

ematician James Yorke and Chinese mathematician

Tien-Yien Li in their 1975 seminal paper on iterations

of functions on the real number line.

characteristic polynomial See C

AYLEY

-H

AMILTON

THEOREM

.

Chebyshev,Pafnuty Lvovich (Tchebyshev) (1821–

1894) Russian Number theory, Analysis, Statistics

Born on May 16, 1821, in Okatova, Russia, Pafnuty

Chebyshev is remembered for his significant contribu-

tions to

NUMBER THEORY

,

ANALYSIS

,

PROBABILITY

the-

ory, and the development of inferential statistics. In

1850 he proved a conjecture posed by French mathe-

matician Joseph-Louis François Bertrand (1822–1900)

stating that for any value n> 3, there is at least one

PRIME

between nand 2n– 2. Chebyshev is also noted

for founding an influential school of mathematics in

St. Petersburg.

Chebyshev entered Moscow University in 1837

and graduated four years later with an undergraduate

degree in mathematics. Driven by an unabashed

desire to achieve international recognition, Cheby-

shev immersed himself in mathematical work. He

earned a master’s degree in 1846 at the same institu-

tion, all the while publishing results on integration

theory and methods, the convergence of T

AYLOR

SERIES

, and the development of analysis. Chebyshev

also examined the principles of probability theory

and developed new insights that prove the main

results of the theory in an elementary, but rigorous,

way. In particular, Chebyshev was able to offer an

elegant proof of S

IMÉON

-D

ENIS

P

OISSON

’s weak

LAW

OF LARGE NUMBERS

.

In 1849 Chebyshev wrote a thesis on the theory

of

MODULAR ARITHMETIC

, earning him a doctorate in

mathematics from Moscow University, as well as a

prize from the Russian Academy of Science in recog-

nition of its originality and its significance. In his

study of prime numbers, Chebyshev not only estab-

lished Bertrand’s conjecture, but also made signifi-

cant steps toward proving the famous

PRIME

-

NUMBER

THEOREM

.

Chebyshev was elected as a full professor in math-

ematics at the University of St. Petersburg in 1850,

and, by this time, had indeed achieved international

fame. He traveled extensively throughout Europe and

collaborated with many scholars on research projects

on topics as diverse as mechanics, physics, mechanical

inventions, and the construction of calculating

machines, as well as continued work in mathematics.

He was awarded many honors throughout his life,

including membership to the Berlin Academy of Sci-

ences in 1871, the Bologna Academy in 1873, the

R

OYAL

S

OCIETY

of London in 1877, the Italian Royal

Academy in 1880, and the Swedish Academy of Sci-

ences in 1893. Every Russian university elected him to

an honorary faculty position, and Chebyshev was even

awarded honorary membership to the St. Petersburg

Artillery Academy, as well as to the French Légion

d’Honneur. He died on December 8, 1894, in St.

Petersburg, Russia.

A number of results and concepts are today named

in Chebyshev’s honor. For example, in analysis, the

“Chebyshev polynomials” provide a basis for the

VEC

-

TOR SPACE

of

CONTINUOUS FUNCTION

s and have impor-

tant applications to approximation theory. In statistics

and probability theory, C

HEBYSHEV

’

S THEOREM

pro-

vides a “weak law of large numbers.”

See also

STATISTICS

:

INFERENTIAL

.