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The encycpedia of Mathematics, "Sugaku-jiten", 3rd edition, edited by Japanese
Mathematics Society, has the topic for Hlibert's problems. However it's written
in Japanese :-) The English translation of the encyclopedia is published by MIT
press.
kaz
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This is a short description of the 23 problems, i found in small book
"A concise History of mathematics" by Dirl J. Struik, Dover.
Iam also trying to make a list of who solved what problems, i'll post these
results once i complete the list.
/nasser
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1. Cantor's problem of the cardinality of the continuum: Is the continuum
the cardinality next to the denumerable set- and can the continuum
be considered as well ordered ?
2. the consistency of (lack of contradictions among) the arithmetical
axioms: IIf the consistency exists, then the geometrical axioms can be
established.
3. the equality of two tetrahedra in volumne, if base, area, and altitude
are equal. : prove this with the aid of division and combination alone
(hence without infinitesimals).
4. the problem of the straight line as the shortest connection of two
points: This question is raised, for example, by the geometry of
Minkowski and by certain problems in the calculus of variations.
5. Lie's concept of the continuous transformation group without postulating
the differentiability of the functions defining the group: this question
may lead to functional equations.
6. mathematical treatment of the axioms of physics: from the axioms of
geometry we can pass to those of rational mechanics (as, e.g. Blotzmann
did in 1897) and to such fields as statistical mechanics, probability, etc.
7. Irrationality and transcendence of certin numbers: Examples are numbers
of the form a^b for algebraic a /= 0 , and algebraic irrational b, such
as 2^SQRT(2) or EXP(Pi)= i^(-2i), are they transcendental or irrational?
Hilbert thought of Hermite and Lindemann on PI.
8. problems in Prime number theory: Here we think of Riemman's Zeta function
and Goldbach's conjecture that every even number is at least in one way the
sum of two primes (1742, letter to Euler).
9. proof of the most general law of reciprocity in arbitrary number fields:
this referred immediatly to some recent work by Hilbert on relative
quadratic number fields.
10. Decision whether a diophantine equation with integral rational numbers
is solvable in such numbers. this was an ancient problem already tackled
for some equations of higher degree than the second and realted to Fermat's
"great problem".
11. The theory of quadratic forms with algebraic coefficients: this again had
a direct bearing on Hilbert's work on number fields.
12. Generalizations of Kronecker's theorm on Abelian fields to an arbitrary
field of rationality: this brings us to a domain where algebraic functions
number theory, and abstract algebra meet.
13. the impossibility of solving the general equation of degree seven by
means of functions of only two variables: this was a problem suggested by
nomography, as Maurice d'Ocagne had explained it.
14. the proof of the finite character of certin systems of "relative integral"
functions: extending here the notion of integral functions to relativganz,
this problem asks for the generalization of the theorm on the finiteness
in the classical theory of invariants due to hilbert and Gordan.
15. Rigorous foundations of schubert's enumerative geometry: for this a
firm foundation in algebra will be necessary.
16. The problem of topology of algebraic curves and surfaces: solution of
this problem is only in its infancy, though we have some knowledge,
especially in the case of curves.
17. the representation of difinite functions (functions never negative for
real value of the variables) by quotients of sum of squares of functions.
18. construction (filling) of space by congruent polyhedra: this relates to a
question of group theory and crystallography, and the work of
E.S.Von Fedorov and A. Schoenfliesz.
19. Are the solutions of reqular variational problems always analytic?:
here "reqular" is speciffically defined. Hilbert remarks that surface of
positive constant curvature has to be analytic, but this does not
hold for surfaces of negative constant curvature.
20. the general boundary problem: especially demonstrating the existence of
solutions of partial differential equations with given boundary values, and
generalizations of reqular variational problems.
21. proof of the existence of linear differential equations with presecribed
monodromy group: this problem was suggested by poicare's theory of
fuchsian functions.
22. Uniformization of analytic relations by means oautomorphic functions:
this was also suggested by Hilbert's proof that the uniformization of any
algebraic relation between two variables can be accomplished by means
of automorphic functions of one variable.
23. Extenstion of the methods of the calculus of variations: Hilbert added
This "propaganda" suggestion because he found the despite Weierstrss'
contributions in this field it still contained many angles that had
been poorly investigated and were potentially useful in serveral fields
of mathematics and mechanics (such as the three-body problem).
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| i dont know Dan, you might be right, this is how the author named them
i assume, the original edition of this book is 1948, so ..
by the way, i find that going over history of mathematics seems to help
to appreciate the theory one is studying, and it might help also in
understanding it if one knows the historical stages it went through,
example is how fourier series came about and used and many other examples.
at school, i find that teachers do not give background of how a certin
theory or approach to solving a problem came about, historically
speaking... may be we dont have time for that..
/nasser
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