Biography:Ilona Palásti
Ilona Palásti (1924–1991) was a Hungarian mathematician who worked at the Alfréd Rényi Institute of Mathematics. She is known for her research in discrete geometry, geometric probability, and the theory of random graphs.[1] With Alfréd Rényi and others, she was considered to be one of the members of the Hungarian School of Probability.[2]
Contributions
In connection to the Erdős distinct distances problem, Palásti studied the existence of point sets for which the [math]\displaystyle{ i }[/math]th least frequent distance occurs [math]\displaystyle{ i }[/math] times. That is, in such points there is one distance that occurs only once, another distance that occurs exactly two times, a third distance that occurs exactly three times, etc. For instance, three points with this structure must form an isosceles triangle. Any [math]\displaystyle{ n }[/math] evenly-spaced points on a line or circular arc also have the same property, but Paul Erdős asked whether this is possible for points in general position (no three on a line, and no four on a circle). Palásti found an eight-point set with this property, and showed that for any number of points between three and eight (inclusive) there is a subset of the hexagonal lattice with this property. Palásti's eight-point example remains the largest known.[3][4][E]
Another of Palásti's results in discrete geometry concerns the number of triangular faces in an arrangement of lines. When no three lines may cross at a single point, she and Zoltán Füredi found sets of [math]\displaystyle{ n }[/math] lines, subsets of the diagonals of a regular [math]\displaystyle{ 2n }[/math]-gon, having [math]\displaystyle{ n(n-3)/3 }[/math] triangles. This remains the best lower bound known for this problem, and differs from the upper bound by only [math]\displaystyle{ O(n) }[/math] triangles.[3][D]
In geometric probability, Palásti is known for her conjecture on random sequential adsorption, also known in the one-dimensional case as "the parking problem". In this problem, one places non-overlapping balls within a given region, one at a time with random locations, until no more can be placed. Palásti conjectured that the average packing density in [math]\displaystyle{ d }[/math]-dimensional space could be computed as the [math]\displaystyle{ d }[/math]th power of the one-dimensional density.[5] Although her conjecture led to subsequent research in the same area, it has been shown to be inconsistent with the actual average packing density in dimensions two through four.[6][A]
Palásti's results in the theory of random graphs include bounds on the probability that a random graph has a Hamiltonian circuit, and on the probability that a random directed graph is strongly connected.[7][B][C]
Selected publications
| A. | Palásti, Ilona (1960), "On some random space filling problems", Magyar Tud. Akad. Mat. Kutató Int. Közl. 5: 353–360 |
| B. | Palásti, I. (1966), "On the strong connectedness of directed random graphs", Studia Scientiarum Mathematicarum Hungarica 1: 205–214 |
| C. | Palásti, I. (1971), "On Hamilton-cycles of random graphs", Period. Math. Hungar. 1 (2): 107–112, doi:10.1007/BF02029168 |
| D. | "Arrangements of lines with a large number of triangles", Proceedings of the American Mathematical Society 92 (4): 561–566, 1984, doi:10.2307/2045427 |
| E. | Palásti, I. (1989), "Lattice-point examples for a question of Erdős", Period. Math. Hungar. 20 (3): 231–235, doi:10.1007/BF01848126 |
References
- ↑ Former Members of the Institute, Alfréd Rényi Institute of Mathematics, https://old.renyi.hu/former_new.html, retrieved 2018-09-13.
- ↑ "Rényi, Alfréd", Leading personalities in statistical sciences: From the seventeenth century to the present, Wiley Series in Probability and Statistics: Probability and Statistics, New York: John Wiley & Sons, 1997, pp. 205–207, doi:10.1002/9781118150719.ch62, ISBN 0-471-16381-3. See in particular p. 205.
- ↑ 3.0 3.1 Horváth, János, ed. (2006), "Discrete and convex geometry", A panorama of Hungarian mathematics in the twentieth century. I, Bolyai Soc. Math. Stud., 14, Springer, Berlin, pp. 427–454, doi:10.1007/978-3-540-30721-1_14 See in particular p. 444 and p. 449.
- ↑ Which Way Did the Bicycle Go?: And Other Intriguing Mathematical Mysteries, Dolciani Mathematical Expositions, 18, Cambridge University Press, 1996, Plate 3, ISBN 9780883853252.
- ↑ Gani, J. M., ed. (1986), "Looking at life quantitatively", The craft of probabilistic modelling: A collection of personal accounts, Applied Probability, New York: Springer-Verlag, pp. 10–30, doi:10.1007/978-1-4613-8631-5_2, ISBN 0-387-96277-8. See in particular p. 23.
- ↑ Blaisdell, B. Edwin (1982), "Random sequential packing in Euclidean spaces of dimensions three and four and a conjecture of Palásti", Journal of Applied Probability 19 (2): 382–390, doi:10.2307/3213489
- ↑ Random graphs, Cambridge Studies in Advanced Mathematics, 73 (2nd ed.), Cambridge, UK: Cambridge University Press, 2001, doi:10.1017/CBO9780511814068, ISBN 0-521-80920-7. See in particular p. 198 and p. 201.
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