Avraham Trahtman

From Wikipedia, the free encyclopedia

Avraham Trahtman

Residence Jerusalem, Israel
Fields Mathematics
Institutions Bar-Ilan University
Known for solving the road coloring problem

Avraham Trahtman (Trakhtman) (b. 1944, Ph.D. (1973) and former degree (1967) at Ural University, Sverdlovsk) is a mathematician at Bar-Ilan University (Israel). In 2007, Trahtman solved a problem in combinatorics that had been open for 37 years, the Road Coloring Conjecture[1] posed in 1970.

Contents

[edit] Road coloring problem posed and solved

Trahtman's road coloring problem solution was accepted and published in 2007 by the Israel Journal of Mathematics. The problem arose in the subfield of symbolic dynamics, an abstract part of the field of dynamical systems. The road coloring problem was raised by R. L. Adler and L. W. Goodwyn from the United States and Israeli mathematician B. Weiss[2] [3]. The proof used results from earlier work [4] [5] [6].

[edit] Other work

The problem of the finite basis question for semigroups of order less than six in the theory of semigroups was posed by Alfred Tarski [7] in 1966 and repeated immediately by Anatoly Maltsev and Shevrin. The problem was solved by Trahtman 17 years later in 1983 [8] [9].

In the theory of varieties of semigroups and universal algebras the problem of existence of covering elements in the lattice of varieties was posed by Evans[10] in 1971. The positive solution of the problem was found by Trakhtman[11] . He also found a six-element semigroup that generates a variety with a continuum of subvarieties [12] and varieties of semigroups having no irreducible base of identities [13].

The theory of locally testable automata can be based on the theory of varieties of locally testable semigroups [14]. Trahtman found the precise estimation on the order of local testability of finite automata.[15]

There are results in theoretical mechanics [16] and in the promising area of extracting moisture from the air [17] mentioned in "New Scientist" [18].

[edit] External links

[edit] References

  1. ^ J.E. Pin. On two combinatorial problems arising from automata theory. Annals of Discrete Math., 17, 535-548, 1983.
  2. ^ R.L. Adler, B. Weiss. Similarity of automorphisms of the torus, Memoirs of the Amer. Math. Soc. 98, Providence, RI, 1970.
  3. ^ R.L. Adler, L.W. Goodwyn, B. Weiss. Equivalence of topological Markov shifts, Israel J. of Math. 27, 49-63, 1977.
  4. ^ K. Culik II, J. Karhumaki, J. Kari. A note on synchronized automata and Road Coloring Problem. Developments in Language Theory (5th Int. Conf., Vienna, 2001), Lecture Notes in Computer Science, 2295, 175-185, 2002.
  5. ^ J. Friedman. On the road coloring problem. Proc. of the Amer. Math. Soc. 110, 1133-1135, 1990.
  6. ^ J. Kari. Synchronizing finite automata on Eulerian digraphs. Springer, Lect. Notes in Comp. Sci., 2136, 432-438, 2001.
  7. ^ A. Tarski. Equational logic and equational theories of algebras. Contrib. to math. Logic. Hannover, 1966, (Amst. 1968), 275-288.
  8. ^ A. N. Trahtman. The finite basis question for semigroups of order less than six. Semigroup Forum, NY, 27(1983), 387-389.
  9. ^ A.N. Trahtman. Finiteness of a basis of identities of 5-element semigroups. Polugruppy i ih gomomorphismy, Ross. Gos. ped. Univ., Leningrad, 1991, 76-98.
  10. ^ T. Evans. The lattice of semigroup varieties. Semigroup Forum. 2, 1(1971), 1-43.
  11. ^ A.N. Trahtman. Covering elements in the lattice of varieties of universal algebras. Mat. Zametky, Moscow, 15(1974), 307-312.
  12. ^ A.N. Trahtman. A sixelement semigroup that generates a variety with a continuum of subvarieties. Ural Gos. Univ. Mat. zap., Alg. syst. i ih mnogoobr., Sverdlovsk, 14(1988), no. 3, 138-143.
  13. ^ A. N. Trahtman. A variety of semigroups without an irreducible basis of identities. Math. Zametky, Moscow, 21(1977), 865-871.
  14. ^ A. N. Trahtman. Identities of locally testable semigroups. Comm. Algebra, 27(1999), no. 11, 5405-5412.
  15. ^ A. N. Trahtman. Optimal estimation on the order of local testability of finite automata. Theoret. Comput. Sci., 231(2000), 59-74.
  16. ^ S.A. Kazak, G.G. Kozhushko, A.N. Trahtman. Calculation of load in discrete chains. Teorija mashin i met. gorn. ob. Sverdlovsk, rel. 1, 1978, 39-51.
  17. ^ B Kogan., A.N. Trahtman. The Moisture from the Air as Water Resource in Arid Region: Hopes, Doubts and Facts. J of Arid Env., London, 2, 53(2003), 231-240.
  18. ^ F. Pearce. Pyramids of dew. "New Scientist". 16 April 2005. 52-53.
  • Jerusalem Post 9-2-2008 Health and Sci Tech