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Smale's problems are a list of 18 challenging problems for the twenty-first century proposed by Field medalist Steven Smale. These problems were inspired in part by Hilbert's famous list of problems presented in 1900 (Hilbert's problems), and in part in response to a suggestion by V. I. Arnold on behalf of the International Mathematical Union that mathematicians describe a number of outstanding problems for the 21st century.1. The Riemann hypothesis. 2. The Poincaré conjecture. 3. Does (i.e., are P-problems equivalent to NP-problems)? 4. Integer zeros of a polynomial. 5. Height bounds for Diophantine curves. 6. Finiteness of the number of relative equilibria in celestial mechanics. 7. Distribution of points on the 2-sphere. 8. Introduction of dynamics into economic theory. 9. The linear programming problem. 10. The closing lemma. 11. Is 1-dimensional dynamics generally hyperbolic? 12. Centralizers of diffeomorphisms...

A set of 15 open problems on Schrödinger operators proposed by mathematical physicist Barry Simon (2000). This set of problems follows up a 1984 list of open problems in mathematical physics also proposed by Simon, of which thirteen involved Schrödinger operators.1. Extended states. Prove for and suitable values of that the Anderson model has purely absolutely continuous spectrum in some energy range. 2. Localization in two dimensions. Prove that for , the spectrum of the Anderson model is dense pure point for all values of . 3. Quantum diffusion. Prove that for and values of where there is a.c. spectrum that grows as as . 4. Ten Martini problem. Prove for all and all irrational that (which is independent) is a Cantor set, that is, that it is nowhere dense. 5. Prove for all irrational and that has measure zero. 6. Prove for all irrational and that the spectrum is purely absolutely continuous. 7. Do there exist potentials on so that..

Landau's problems are the four "unattackable" problems mentioned by Landau in the 1912 Fifth Congress of Mathematicians in Cambridge, namely: 1. The Goldbach conjecture, 2. Twin prime conjecture, 3. Legendre's conjecture that for every there exists a prime between and (Hardy and Wright 1979, p. 415; Ribenboim 1996, pp. 397-398), and 4. The conjecture that there are infinitely many primes of the form (Euler 1760; Mirsky 1949; Hardy and Wright 1979, p. 19; Ribenboim 1996, pp. 206-208). The first few such primes are 2, 5, 17, 37, 101, 197, 257, 401, ... (OEIS A002496). Although it is not known if there always exists a prime between and , Chen (1975) has shown that a number which is either a prime or semiprime does always satisfy this inequality. Moreover, there is always a prime between and where (Iwaniec and Pintz 1984; Hardy and Wright 1979, p. 415). The smallest primes between and for , 2, ..., are 2, 5, 11,..

Hilbert's problems are a set of (originally) unsolved problems in mathematics proposed by Hilbert. Of the 23 total appearing in the printed address, ten were actually presented at the Second International Congress in Paris on August 8, 1900. In particular, the problems presented by Hilbert were 1, 2, 6, 7, 8, 13, 16, 19, 21, and 22 (Derbyshire 2004, p. 377). Furthermore, the final list of 23 problems omitted one additional problem on proof theory (Thiele 2001).Hilbert's problems were designed to serve as examples for the kinds of problems whose solutions would lead to the furthering of disciplines in mathematics. As such, some were areas for investigation and therefore not strictly "problems."1. "Cantor's problem of the cardinal number of the continuum." The question of if there is a transfinite number between that of a denumerable set and the numbers of the continuum was answered by Gödel and Cohen in their..

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