Arie Bialostocki is an Israeli American mathematician with expertise and contributions in discrete mathematics and finite groups.[2][1]
Education and career
Arie received his BSc, MSc, and PhD (1984) degrees from Tel-Aviv University in Israel.[1] His dissertation was done under the supervision of Marcel Herzog.[3] After a year of postdoc at University of Calgary, Canada, he took a faculty position at the University of Idaho, became a professor in 1992, and continued to work there until he retired at the end of 2011.[2] At Idaho, Arie maintained correspondence and collaborations with researchers from around the world who would share similar interests in mathematics.[2] His Erdős number is 1.[4] He has supervised seven PhD students and numerous undergraduate students who enjoyed his colorful anecdotes and advice.[2] He organized the Research Experience for Undergraduates (REU) program at the University of Idaho from 1999 to 2003 attracting many promising undergraduates who themselves have gone on to their outstanding research careers.[2]
Mathematics research
Arie has published more than 50 publications in reputed mathematics journals.[5][6] The following are some of Arie's most important contributions:
Bialostocki[7] redefined[8] a B-injector in a finite group G to be any maximal nilpotent subgroup B of G satisfying \( {\displaystyle d_{2}(B)=d_{2}(G)} \), where \( {\displaystyle d_{2}(X)} \) is the largest cardinality of a subgroup of G which is nilpotent of class at most 2. Using his definition, it was proved by several authors[9][10][11][12] that in many non-solvable groups the nilpotent injectors form a unique conjugacy class.
Bialostocki contributed to the generalization of the Erdős-Ginzburg-Ziv theorem (also known as the EGZ theorem).[13][14] He conjectured: \( {\displaystyle A=(a_{1},a_{2},\ldots ,a_{n})} \) is a sequence of elements of \( {\displaystyle {\mathbb {Z} }_{m}} \), then A contains at least \( {\displaystyle {\lfloor {n/2}\rfloor \choose {m}}+{\lceil {n/2}\rceil \choose {m}}} \) zero sums of length m m. The EGZ theorem is a special case where \( {\displaystyle n=2m-1} \). The conjecture was partially confirmed by Kisin,[15] Füredi and Kleitman,[16] and Grynkiewicz.[17]
Bialostocki introduced the EGZ polynomials and contributed to generalize the EGZ theorem for higher degree polynomials.[18][19] The EGZ theorem is associated with the first degree elementary polynomial.
Bialostocki and Dierker[20][21] introduced[22] the relationship of EGZ theorem to Ramsey Theory on graphs.
Bialostocki, Erdős, and Lefmann[4] introduced[23] the relationship of EGZ theorem to Ramsey Theory on the positive integers.
In Jakobs and Jungnickel's book "Einführung in die Kombinatorik",[24] Bialostocki and Dierker are attributed for introducing Zero-sum Ramsey theory. In Landman and Robertson's book "Ramsey Theory on the Integers",[25] the number \( {\displaystyle b(m,k;r)} \) is defined in honor of Bialostocki's contributions to the Zero-sum Ramsey theory.
Bialostocki, Dierker, and Voxman[26] suggested[27] a conjecture offering a modular strengthening of the Erdős–Szekeres theorem proving that the number of points in the interior of the polygon is divisible by k k, provided that total number of points \( {\displaystyle n\geqslant k+2} \). Károlyi, Pach and Tóth[28] made further progress toward the proof of the conjecture.
In Recreational Mathematics, Arie's paper[1] on application of elementary group theory to Peg Solitaire is a suggested reading in Joseph Gallian's book[29] on Abstract Algebra.
References
Bialostocki, Arie (1998). "An Application of Elementary Group Theory to Central Solitaire". The College Mathematics Journal. 29 (3): 208–212. doi:10.1080/07468342.1998.11973941.
"Professor Arie Bialostocki retires". 2023-05-22.
Arie Bialostocki at the Mathematics Genealogy Project
Bialostocki, Arie; Erdős, Paul; Lefmann, Hanno (1995). "Monochromatic and zero-sum sets of nondecreasing diameter". Discrete Mathematics. 137 (1–3): 19–34. doi:10.1016/0012-365X(93)E0148-W.
Arie Bialostocki at zbMATH Open
Arie Bialostocki at Google scholar
Bialostocki, Arie (1982). "Nilpotent injectors in symmetric groups". Israel Journal of Mathematics. 41 (3): 261–273. doi:10.1007/BF02771725. S2CID 122321992.
Review by A. R. Camina at zbMATH Open
Sheu, Tsung-Luen (1993). "Nilpotent injectors in general linear groups". Journal of Algebra. 160 (2): 380–418. doi:10.1006/jabr.1993.1192.
Mohammed, Mashhour Ibrahim (2009). "On nilpotent injectors of Fischer group M ( 22 ) {\displaystyle M(22)}". Hokkaido Mathematical Journal. 38 (4): 627–633. doi:10.14492/hokmj/1258554237.
Flavell, Paul (1992). "Nilpotent injectors in finite groups all of whose local subgroups are N-constrained". Journal of Algebra. 149 (2): 405–418. doi:10.1016/0021-8693(92)90024-G.
Alali, M. I. M.; Hering, Ch.; Neumann, A. (2000). "More on B-injectors of sporadic groups". Communications in Algebra. 28 (4): 2185–2190. doi:10.1080/00927870008826951. S2CID 120962734.
Bialostocki, A.; Lotspeich, M. (1992). "Some developments of the Erdős-Ginzburg-Ziv theorem I". Sets, graphs, and numbers: a birthday salute to Vera T. Sós and András Hajnal. Colloquia mathematica Societatis János Bolyai. pp. 97–117.
Bialostocki, Arie; Dierker, Paul; Grynkiewicz, David; Lotspeich, Mark (2003). "On some developments of the Erdős-Ginzburg-Ziv theorem II". Acta Arithmetica. 110 (2): 173–184. Bibcode:2003AcAri.110..173B. doi:10.4064/aa110-2-7.
Kisin, M. (1994). "The number of zero sums modulo m in a sequence of length n". Mathematika. 41 (1): 149–163. doi:10.1112/S0025579300007257.
Füredi, Z.; Kleitman, D. J. (1993). "The minimal number of zero sums". Combinatorics, Paul Erdős is eighty (volume 1). Bolyai Society Mathematical Studies. János Bolyai Mathematical Society. pp. 159–172.
Grynkiewicz, David J. (2006). "On the number of m m-term zero-sum subsequences". Acta Arithmetica. 121 (3): 275–298. Bibcode:2006AcAri.121..275G. doi:10.4064/aa121-3-5.
Bialostocki, Arie; Luong, Tran Dinh (2014). "Cubic symmetric polynomials yielding variations of the Erdős-Ginzburg-Ziv theorem". Acta Mathematica Hungarica. 142: 152–166. doi:10.1007/s10474-013-0346-4. S2CID 254240326.
Ahmed, Tanbir; Bialostocki, Arie; Pham, Thang; Vinh, Le Anh (2019). "Power sum polynomials as relaxed EGZ polynomials" (PDF). Integers. 19: A49.
Bialostocki, A.; Dierker, P. (1990). "Zero sum Ramsey theorems". Congressus Numerantium. 70: 119–130.
Bialostocki, A.; Dierker, P. (1992). "On the Erdős-Ginzburg-Ziv theorem and the Ramsey numbers for stars and matchings". Discrete Mathematics. 110 (1–3): 1–8. doi:10.1016/0012-365X(92)90695-C.
Review by R. L. Graham at MathSciNet
Review by Ralph Faudree at zbMATH Open
Jakobs, Conrad; Jungnickel, Dieter (2004). Einführung in die Kombinatorik. de Gruyter Lehrbuch. doi:10.1515/9783110197990. ISBN 3-11-016727-1.
Landman, Bruce; Robertson, Aaron (2015). Ramsey Theory on the Integers. Student Mathematical Library. Vol. 73 (Second ed.). American Mathematical Society. ISBN 978-0-8218-9867-3.
Bialostocki, Arie; Dierker, P.; Voxman, B. (1991). "Some notes on the Erdős-Szekeres theorem". Discrete Mathematics. 91 (3): 231–238. doi:10.1016/0012-365X(90)90232-7.
Review by Yair Caro at MathSciNet
Károlyi, Gy.; J., Pach; Tóth, G. (2001). "A modular version of the Erdős-Szekeres theorem". Studia Scientiarum Mathematicarum Hungarica. 38 (1–4): 245–259. doi:10.1556/sscmath.38.2001.1-4.17.
Gallian, Joseph A. (2015). Contemporary Abstract Algebra. Cengage Learning. ISBN 978-1-305-65796-0.
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