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{{Short description|Soviet-born Israeli mathematician (1944–2024)}}
{{Infobox scientist
{{Infobox scientist
|name = Avraham Trahtman
|name = Avraham Naumovich Trahtman
|image = Abram 008.jpg |
|image = Abram 008.jpg |
|image_size = 250px
|image_size = 250px
|caption =
|caption =
|birth_date = 10 February 1944
|birth_date = {{birth date|df=y|1944|02|10}}
|birth_place = Kalinovo, [[Nevyansky District]], [[Sverdlovsk Oblast]]
|birth_place = Kalinovo, [[Nevyansky District]], [[Sverdlovsk Oblast]], [[Russian SFSR]], [[USSR]]
|death_date =
|death_date = {{death date and age|df=y|2024|07|17|1944|02|10}}
|death_place =
|death_place = [[Jerusalem]]
|residence = [[Jerusalem]], [[Israel]]
|nationality =
|nationality =
|ethnicity =
|ethnicity =
Line 14: Line 14:
|work_institutions = [[Bar-Ilan University]]
|work_institutions = [[Bar-Ilan University]]
|alma_mater = [[Ural State University]]
|alma_mater = [[Ural State University]]
|doctoral_advisor = [[Lev Shevrin]]
|doctoral_advisor = [[Lev N. Shevrin]]
|doctoral_students =
|doctoral_students =
|known_for = solving the [[road coloring problem]]
|known_for = solving the [[road coloring problem]]
|prizes =
|prizes =
}}
}}

'''Avraham Trahtman''' (Trakhtman) ({{lang-ru|Абрам Наумович Трахтман}}; b. 1944, [[USSR]]) 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]] posed in 1970.<ref>J.E. Pin. On two combinatorial problems arising from automata theory. Annals of Discrete Math., 17, 535-548, 1983.</ref>
'''Avraham Naumovich Trahtman''' (Trakhtman) ({{langx|ru|Абрам Наумович Трахтман}}; 10 February 1944 – 17 July 2024) was a Soviet-born Israeli mathematician and academic at [[Bar-Ilan University]] ([[Israel]]). In 2007, Trahtman solved a problem in [[combinatorics]] that had been open for 37 years, the [[Road Coloring Conjecture]] posed in 1970.<ref>J. E. Pin. On two combinatorial problems arising from automata theory. Annals of Discrete Math., 17, 535-548, 1983.</ref> Trahtman died in [[Jerusalem]] on 17 July 2024, at the age of 80.<ref>{{cite web |title=Avraham Trakhtman 1944 – 2024 |url=https://www.forevermissed.com/avraham-trakhtman/about |website=Forever Missed |access-date=5 August 2024}}</ref>


==Road coloring problem posed and solved==
==Road coloring problem posed and solved==
Trahtman's solution to the [[road coloring problem]] was accepted in 2007 and published in 2009 by the ''[[Israel Journal of Mathematics]]''.<ref>Avraham N. Trahtman: The Road Coloring Problem. ''[[Israel Journal of Mathematics]]'', Vol. 172, 51&ndash;60, 2009</ref> 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 the Israeli mathematician B. Weiss.<ref>R.L. Adler, B. Weiss. Similarity of automorphisms of the torus, Memoirs of the Amer. Math. Soc. 98, Providence, RI, 1970</ref><ref>R.L. Adler, L.W. Goodwyn, B. Weiss. Equivalence of topological Markov shifts, Israel J. of Math. 27, 49-63, 1977</ref> The proof used results from earlier work.<ref>K. Culik II, J. Karhumaki, J. Kari. A note on synchronized automata and Road Coloring Problem. [[International Conference on Developments in Language Theory|Developments in Language Theory]] (5th Int. Conf., Vienna, 2001), Lecture Notes in Computer Science, 2295, 175-185, 2002</ref><ref>J. Friedman. On the road coloring problem. Proc. of the Amer. Math. Soc. 110, 1133-1135, 1990</ref><ref>A.N. Trahtman. An Algorithm for Road Coloring. Lect. Notes in Comp. Sci, 7056 (2011), Springer, 349--360</ref>
Trahtman's solution to the [[road coloring problem]] was accepted in 2007 and published in 2009 by the ''[[Israel Journal of Mathematics]]''.<ref>Avraham N. Trahtman: The Road Coloring Problem. ''[[Israel Journal of Mathematics]]'', Vol. 172, 51&ndash;60, 2009</ref> 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 [[Roy Adler|R. L. Adler]] and L. W. Goodwyn from the United States, and the Israeli mathematician [[Benjamin Weiss|B. Weiss]].<ref>R.L. Adler, B. Weiss. Similarity of automorphisms of the torus, Memoirs of the Amer. Math. Soc. 98, Providence, RI, 1970</ref><ref>R.L. Adler, L.W. Goodwyn, B. Weiss. Equivalence of topological Markov shifts, [[Israel Journal of Mathematics]] 27, 49-63, 1977</ref> The proof used results from earlier work.<ref>K. Culik II, J. Karhumaki, J. Kari. A note on synchronized automata and Road Coloring Problem. [[International Conference on Developments in Language Theory|Developments in Language Theory]] (5th Int. Conf., Vienna, 2001), Lecture Notes in Computer Science, 2295, 175-185, 2002</ref><ref>J. Friedman. On the road coloring problem. [[Proceedings of the American Mathematical Society]] 110, 1133-1135, 1990</ref><ref>A.N. Trahtman. An Algorithm for Road Coloring. Lect. Notes in Comp. Sci, 7056 (2011), Springer, 349--360</ref>


==Cerny conjecture==
==Černý conjecture==
The problem of estimating the length of synchronizing word has a long history and was posed independently by several authors, but it is commonly known as the Černý conjecture. In 1964 Jan Černý conjectured that (n-1)<sup>2</sup> is the upper bound for the length of the shortest synchronizing word for any n-state complete DFA (a DFA with complete state transition graph).<ref>J. Černý, Poznamka k homogenym eksperimentom s konechnymi automatami, Math.-Fyz. Čas., 14(1964) 208--215.
The problem of estimating the length of synchronizing word has a long history and was posed independently by several authors, but it is commonly known as the [[Černý conjecture]]. In 1964 Jan Černý conjectured that <math>(n-1)^2</math> is the upper bound for the length of the shortest synchronizing word for any n-state complete DFA (a DFA with complete state transition graph).<ref name="cerny">{{citation |first=Ján |last=Černý |title=Poznámka k homogénnym experimentom s konečnými automatmi |journal=Matematicko-fyzikálny časopis Slovenskej Akadémie Vied |volume=14 |year=1964 |pages=208–216 |url=http://dml.cz/bitstream/handle/10338.dmlcz/126647/MathSlov_14-1964-3_2.pdf}} (in Slovak). English translation: [https://www.jalc.de/issues/2019/issue_24_2-4/jalc-2019-123-132.php A Note on Homogeneous Experiments with Finite Automata]. J. Autom. Lang. Comb. 24(2019), 123-132</ref> If this is true, it would be tight: in his 1964 paper, Černý exhibited a class of automata (indexed by the number n of states) for which the shortest reset words have this length. In 2011 Trahtman published a proof<ref>A.N. Trahtman. Modifying the Upper Bound on the Length of Minimal Synchronizing Word. Lect. Notes in Comp. Sci, 6914(2011) Springer, 173-180</ref> of upper bound <math>n (7n^2+6n-16)/48</math>, but then he found an error in it.<ref>{{Cite arXiv |eprint = 1104.2409v6|last1 = Trahtman|first1 = A. N|title = Modifying the upper bound on the length of minimal synchronizing word|class = cs.DM|year = 2011}}</ref> The conjecture holds in many partial cases, see for instance, Kari<ref>J. Kari. Synchronizing finite automata on Eulerian digraphs. Springer, Lect. Notes in Comp. Sci., 2136, 432-438, 2001.</ref> and Trahtman.<ref>A.N. Trahtman. The Černý Conjecture for Aperiodic Automata. Discrete Math. Theor. Comput. Sci. v. 9, 2(2007), 3-10</ref>
</ref> If this is true, it would be tight: in his 1964 paper, Černý exhibited a class of automata (indexed by the number n of states) for which the shortest reset words have this length. The best upper bound known is n(7n<sup>2</sup>+6n-16)/48, far from the lower bound.<ref>A.N. Trahtman. Modifying the Upper Bound on the Length of Minimal Synchronizing Word. Lect. Notes in Comp. Sci, 6914(2011) Springer, 173-180</ref> The conjecture holds in many partial cases, see for instance<ref>J. Kari. Synchronizing finite automata on Eulerian digraphs. Springer, Lect. Notes in Comp. Sci., 2136, 432-438, 2001.</ref><ref>A.N. Trahtman. The Černý Conjecture for Aperiodic Automata. Discr. Math. & Theor. Comput. Sci. v. 9, 2(2007), 3-10</ref>


==Other work==
==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]] in 1966,<ref>A. Tarski. Equational logic and equational theories of algebras. Contrib. to math. Logic. Hannover, 1966, (Amst. 1968), 275-288.</ref> and repeated immediately by [[Anatoly Maltsev]] and Shevrin. The problem was solved by Trahtman 17 years later in 1983.<ref>A. N. Trahtman. The finite basis question for semigroups of order less than six. Semigroup Forum, NY, 27(1983), 387-389.</ref><ref>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.</ref>
The finite basis problem for [[semigroups]] of order less than six in the theory of semigroups was posed by [[Alfred Tarski]] in 1966,<ref>A. Tarski. Equational logic and equational theories of algebras. Contrib. to math. Logic. Hannover, 1966, (Amst. 1968), 275-288.</ref> and repeated by [[Anatoly Maltsev]] and L. N. Shevrin. In 1983, Trahtman solved this problem by proving that all semigroups of order less than six are finitely based.<ref>A. N. Trahtman. The finite basis question for semigroups of order less than six. [[Semigroup Forum]], 27(1983), 387-389.</ref><ref>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.</ref>


In the theory of [[Variety (universal algebra)|varieties]] of semigroups and [[universal algebra]]s the problem of existence of covering elements in the [[lattice (order)|lattice]] of varieties was posed by Evans in 1971.<ref>T. Evans. The lattice of semigroup varieties. Semigroup Forum. 2, 1(1971), 1-43.</ref> The positive solution of the problem was found by Trahtman.<ref>A.N. Trahtman. Covering elements in the lattice of varieties of universal algebras. Mat. Zametky, Moscow, 15(1974), 307-312.</ref> He also found a six-element semigroup that generates a variety with a continuum of subvarieties,<ref>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.</ref> and varieties of semigroups having no irreducible base of identities.<ref>A. N. Trahtman. A variety of semigroups without an irreducible basis of identities. Math. Zametky, Moscow, 21(1977), 865-871.</ref>
In the theory of [[Variety (universal algebra)|varieties]] of semigroups and [[universal algebra]]s the problem of existence of covering elements in the [[lattice (order)|lattice]] of varieties was posed by Evans in 1971.<ref>T. Evans. The lattice of semigroup varieties. [[Semigroup Forum]]. 2, 1(1971), 1-43.</ref> The positive solution of the problem was found by Trahtman.<ref>A.N. Trahtman. Covering elements in the lattice of varieties of universal algebras. Mat. Zametky, Moscow, 15(1974), 307-312.</ref> He also found a six-element semigroup that generates a variety with a continuum of subvarieties,<ref>A.N. Trahtman. A six-element 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.</ref> and varieties of semigroups having no irreducible base of identities.<ref>A. N. Trahtman. A variety of semigroups without an irreducible basis of identities. Math. Zametky, Moscow, 21(1977), 865-871.</ref>


The theory of [[locally testable]] [[automata theory|automata]] can be based on the theory of varieties of locally testable semigroups.<ref>A. N. Trahtman. Identities of locally testable semigroups. Comm. Algebra, 27(1999), no. 11, 5405-5412.</ref> Trahtman found the precise estimation on the order of local testability of finite automata.<ref>A. N. Trahtman. Optimal estimation on the order of local testability of finite automata. Theoret. Comput. Sci., 231(2000), 59-74.</ref>
The theory of [[locally testable]] [[automata theory|automata]] can be based on the theory of varieties of locally testable semigroups.<ref>A. N. Trahtman. Identities of locally testable semigroups. Comm. Algebra, 27(1999), no. 11, 5405-5412.</ref> Trahtman found the precise estimation on the order of local testability of finite automata.<ref>A. N. Trahtman. Optimal estimation on the order of local testability of finite automata. Theoret. Comput. Sci., 231(2000), 59-74.</ref>


There are results in theoretical mechanics<ref>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.</ref> and in the promising area of extracting moisture from the air<ref>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.</ref> mentioned in "''[[New Scientist]]''".<ref>F. Pearce. Pyramids of dew. "New Scientist". 16 April 2005. 52-53.</ref>
There are results in theoretical mechanics<ref>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.</ref> and in the promising area of extracting moisture from the air<ref>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.</ref> mentioned in "''[[New Scientist]]''".<ref>F. Pearce. Pyramids of dew. "New Scientist". 16 April 2005. 52-53.</ref>
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==External links==
==External links==
*{{MathGenealogy |id=59507 }}
*[http://www.cs.biu.ac.il/~trakht/ Trahtman's page at Bar-Ilan University's Website]
*[http://www.cs.biu.ac.il/~trakht/ Trahtman's page at Bar-Ilan University's Website]
*[http://www.cs.biu.ac.il/~trakht/cv.html Trahtman's Curriculum Vitae]
*[http://www.cs.biu.ac.il/~trakht/cv.html Trahtman's Curriculum Vitae]
*[http://arxiv.org/pdf/0709.0099v4 Trahtman's paper (in PDF format)]
*[https://arxiv.org/abs/0709.0099 Trahtman's paper (in PDF format)]
*[http://www.msnbc.msn.com/id/23729600/ "63-year-old solves riddle from 1970" on MSNBC]
*[https://web.archive.org/web/20140306205750/http://www.nbcnews.com/id/23729600/ "63-year-old solves riddle from 1970" on MSNBC]
*[http://www.britannica.com "Encyclopedia - Britannica Online Encyclopedia", article: Avraham Trahtman]
*[https://www.britannica.com/biography/Avraham-Trahtman "Encyclopedia - Britannica Online Encyclopedia", article: Avraham Trahtman]
*[http://www-history.mcs.st-andrews.ac.uk/ "MacTutor History of Mathematics. Trahtman biography"]
*[http://www-history.mcs.st-andrews.ac.uk/ "MacTutor History of Mathematics. Trahtman biography"]
⚫
*[https://books.google.com/books?id=WUrVAwAAQBAJ ''A Mathematical Medley Fifty Easy Pieces on Mathematics''] by [[George Szpiro|George G. Szpiro]]
*[http://www.uprack.com/israeli-mathematicians-adi-shamir-giulio-racah-sahron-shelah-zlil-sela=robert-aumann-michael-0-rabin-oded-schramm-avraham-trahtman "Israeli Mathematicians, Adi Shamir, Giulio Racah, Saharon Shelah, Zlil Sela, Robert Aumann, Michael O. Rabin, Oded Schramm, Avraham Trahtman, Llc Books, 2010"]

⚫
*[http://books.google.com/books/ "A Mathematical Medley Fifty Easy Pieces on Mathematics. George G. Szpiro"]
{{Authority control}}


{{Persondata <!-- Metadata: see [[Wikipedia:Persondata]]. -->
| NAME = Trahtman, Avraham
| ALTERNATIVE NAMES =
| SHORT DESCRIPTION = Israeli mathematicians
| DATE OF BIRTH = 1944
| PLACE OF BIRTH = [[USSR]]
| DATE OF DEATH =
| PLACE OF DEATH =
}}
{{DEFAULTSORT:Trahtman, Avraham}}
{{DEFAULTSORT:Trahtman, Avraham}}
[[Category:1944 births]]
[[Category:1944 births]]
[[Category:Living people]]
[[Category:2024 deaths]]
[[Category:Israeli academics]]
[[Category:Academic staff of Bar-Ilan University]]
[[Category:Israeli Jews]]
[[Category:Israeli Jews]]
[[Category:Israeli mathematicians]]
[[Category:21st-century Israeli mathematicians]]
[[Category:Russian Jews]]
[[Category:Russian Jews]]
[[Category:21st-century mathematicians]]
[[Category:20th-century Israeli mathematicians]]
[[Category:Mathematics and culture]]
[[Category:Ural State University alumni]]

[[de:Avraham Trakhtman]]
[[fr:Avraham Trahtman]]
[[he:אברהם טרכטמן]]
[[ru:Трахтман, Абрам Наумович]]

Latest revision as of 13:15, 6 November 2024

Avraham Naumovich Trahtman
Born(1944-02-10)10 February 1944
Kalinovo, Nevyansky District, Sverdlovsk Oblast, Russian SFSR, USSR
Died17 July 2024(2024-07-17) (aged 80)
Jerusalem
Alma materUral State University
Known forsolving the road coloring problem
Scientific career
FieldsMathematics
InstitutionsBar-Ilan University
Doctoral advisorLev N. Shevrin

Avraham Naumovich Trahtman (Trakhtman) (Russian: Абрам Наумович Трахтман; 10 February 1944 – 17 July 2024) was a Soviet-born Israeli mathematician and academic at Bar-Ilan University (Israel). In 2007, Trahtman solved a problem in combinatorics that had been open for 37 years, the Road Coloring Conjecture posed in 1970.[1] Trahtman died in Jerusalem on 17 July 2024, at the age of 80.[2]

Road coloring problem posed and solved

[edit]

Trahtman's solution to the road coloring problem was accepted in 2007 and published in 2009 by the Israel Journal of Mathematics.[3] 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 the Israeli mathematician B. Weiss.[4][5] The proof used results from earlier work.[6][7][8]

Černý conjecture

[edit]

The problem of estimating the length of synchronizing word has a long history and was posed independently by several authors, but it is commonly known as the Černý conjecture. In 1964 Jan Černý conjectured that is the upper bound for the length of the shortest synchronizing word for any n-state complete DFA (a DFA with complete state transition graph).[9] If this is true, it would be tight: in his 1964 paper, Černý exhibited a class of automata (indexed by the number n of states) for which the shortest reset words have this length. In 2011 Trahtman published a proof[10] of upper bound , but then he found an error in it.[11] The conjecture holds in many partial cases, see for instance, Kari[12] and Trahtman.[13]

Other work

[edit]

The finite basis problem for semigroups of order less than six in the theory of semigroups was posed by Alfred Tarski in 1966,[14] and repeated by Anatoly Maltsev and L. N. Shevrin. In 1983, Trahtman solved this problem by proving that all semigroups of order less than six are finitely based.[15][16]

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 in 1971.[17] The positive solution of the problem was found by Trahtman.[18] He also found a six-element semigroup that generates a variety with a continuum of subvarieties,[19] and varieties of semigroups having no irreducible base of identities.[20]

The theory of locally testable automata can be based on the theory of varieties of locally testable semigroups.[21] Trahtman found the precise estimation on the order of local testability of finite automata.[22]

There are results in theoretical mechanics[23] and in the promising area of extracting moisture from the air[24] mentioned in "New Scientist".[25]

References

[edit]
  1. ^ J. E. Pin. On two combinatorial problems arising from automata theory. Annals of Discrete Math., 17, 535-548, 1983.
  2. ^ "Avraham Trakhtman 1944 – 2024". Forever Missed. Retrieved 5 August 2024.
  3. ^ Avraham N. Trahtman: The Road Coloring Problem. Israel Journal of Mathematics, Vol. 172, 51–60, 2009
  4. ^ R.L. Adler, B. Weiss. Similarity of automorphisms of the torus, Memoirs of the Amer. Math. Soc. 98, Providence, RI, 1970
  5. ^ R.L. Adler, L.W. Goodwyn, B. Weiss. Equivalence of topological Markov shifts, Israel Journal of Mathematics 27, 49-63, 1977
  6. ^ 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
  7. ^ J. Friedman. On the road coloring problem. Proceedings of the American Mathematical Society 110, 1133-1135, 1990
  8. ^ A.N. Trahtman. An Algorithm for Road Coloring. Lect. Notes in Comp. Sci, 7056 (2011), Springer, 349--360
  9. ^ Černý, Ján (1964), "Poznámka k homogénnym experimentom s konečnými automatmi" (PDF), Matematicko-fyzikálny časopis Slovenskej Akadémie Vied, 14: 208–216 (in Slovak). English translation: A Note on Homogeneous Experiments with Finite Automata. J. Autom. Lang. Comb. 24(2019), 123-132
  10. ^ A.N. Trahtman. Modifying the Upper Bound on the Length of Minimal Synchronizing Word. Lect. Notes in Comp. Sci, 6914(2011) Springer, 173-180
  11. ^ Trahtman, A. N (2011). "Modifying the upper bound on the length of minimal synchronizing word". arXiv:1104.2409v6 [cs.DM].
  12. ^ J. Kari. Synchronizing finite automata on Eulerian digraphs. Springer, Lect. Notes in Comp. Sci., 2136, 432-438, 2001.
  13. ^ A.N. Trahtman. The Černý Conjecture for Aperiodic Automata. Discrete Math. Theor. Comput. Sci. v. 9, 2(2007), 3-10
  14. ^ A. Tarski. Equational logic and equational theories of algebras. Contrib. to math. Logic. Hannover, 1966, (Amst. 1968), 275-288.
  15. ^ A. N. Trahtman. The finite basis question for semigroups of order less than six. Semigroup Forum, 27(1983), 387-389.
  16. ^ 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.
  17. ^ T. Evans. The lattice of semigroup varieties. Semigroup Forum. 2, 1(1971), 1-43.
  18. ^ A.N. Trahtman. Covering elements in the lattice of varieties of universal algebras. Mat. Zametky, Moscow, 15(1974), 307-312.
  19. ^ A.N. Trahtman. A six-element 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.
  20. ^ A. N. Trahtman. A variety of semigroups without an irreducible basis of identities. Math. Zametky, Moscow, 21(1977), 865-871.
  21. ^ A. N. Trahtman. Identities of locally testable semigroups. Comm. Algebra, 27(1999), no. 11, 5405-5412.
  22. ^ A. N. Trahtman. Optimal estimation on the order of local testability of finite automata. Theoret. Comput. Sci., 231(2000), 59-74.
  23. ^ 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.
  24. ^ 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.
  25. ^ F. Pearce. Pyramids of dew. "New Scientist". 16 April 2005. 52-53.
[edit]
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