Joos Ulrich Heintz | |
|---|---|
 | |
| Born | 27 October 1945 |
| Died | 03 October 2024 Buenos Aires |
| Nationality | Swiss |
| Alma mater | University of Zurich |
| Occupation(s) | Mathematician, Philosopher and Anthropologist |
Joos Ulrich Heintz (27 October 1945 - 3 October 2024) was an Argentinean-Swiss mathematician. He was a professor emeritus at the University of Buenos Aires. [1]
Heintz was born on 27 October 1945 in Zürich, Switzerland. After studying Mathematics and Cultural Anthropology at the University of Zurich to undergraduate level, he went on to receive a PhD in mathematics in 1982 under the supervision of Volker Strassen. [2] He performed his habilitation in 1986 at the J.W.von Goethe University in Frankfurt am Main [3] where he also studied Turcology and Sephardic history and culture. He was appointed Privatdozent at the J.W. Goethe university Frankfurt am Main. Until his retirement in 2017, he worked as a Full Professor at the University of Buenos Aires and University of Cantabria/Spain and as a Senior Researcher at the National Council for Scientific and Technological Development (CONICET).
Heintz worked mainly in algebraic complexity theory, computational algebraic geometry, and semi algebraic geometry. For this purpose, he developed with his collaborators, different mathematical tools, e.g. the Bezout Inequality [4] or the first effective Nullstellensatz in arbitrary characteristic. [5] This allowed him and his collaborators to adapt Kronecker's elimination theory [6] to the complexity requirements of modern computer algebra and to prove that all reasonable geometric (not algebraic) computation problems are solvable in PSPACE. Later, he extended these complexity results to polynomial input systems given by arithmetic circuits. The outcome was a worst case optimal probabilistic elimination algorithm with the ability to recognize “easily solvable” input systems which was later implemented by Grégoire Lecerf. [7] Finally, Heintz and his collaborators demonstrated that under fragile and natural assumptions, the worst case complexity of elimination algorithms is unavoidably exponential, independent of the chosen data structure. [8] He applied his results and methods also to mixed integer optimization [9] and foundations of software engineering. [10]
Furthermore, in the field of linguistics, he identified the morphology and phonology of the Turkish languages as a regular language. [11]
In 1987, Heintz founded the Argentinean research group Noaï Fitchas in Buenos Aires. This group was transformed into the international working group TERA (Turbo Evaluation and Rapid Algorithms) with collaborators from several Argentinean, French, Spanish and German universities and research institutions, such as the University of Buenos Aires, CONICET, University of Nice, École Polytechnique at Paris, Universidad de Cantabria (Spain), and Humboldt University at Berlin. Noaï Fitchas was used as a pseudonym for the Argentinean group and numerous influential papers in Computer Algebra were published in the nineties under this name. [12] [13]
Heintz was a member of the editorial boards of several international journals, including the Foundations of Computational Mathematics, Computational Complexity and Applicable Algebra in Engineering, and Communication and Computing, from which he received three best paper awards.
In 2003, Heintz was awarded the Argentinean Konex Medal of Merit. [14]
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Smale's problems is a list of eighteen unsolved problems in mathematics proposed by Steve Smale in 1998 and republished in 1999. Smale composed this list in reply to a request from Vladimir Arnold, then vice-president of the International Mathematical Union, who asked several mathematicians to propose a list of problems for the 21st century. Arnold's inspiration came from the list of Hilbert's problems that had been published at the beginning of the 20th century.
In mathematics, cylindrical algebraic decomposition (CAD) is a notion, along with an algorithm to compute it, that is fundamental for computer algebra and real algebraic geometry. Given a set S of polynomials in Rn, a cylindrical algebraic decomposition is a decomposition of Rn into connected semialgebraic sets called cells, on which each polynomial has constant sign, either +, − or 0. To be cylindrical, this decomposition must satisfy the following condition: If 1 ≤ k < n and π is the projection from Rn onto Rn−k consisting in removing the last k coordinates, then for every pair of cells c and d, one has either π(c) = π(d) or π(c) ∩ π(d) = ∅. This implies that the images by π of the cells define a cylindrical decomposition of Rn−k.
In computational complexity theory, and more specifically in the analysis of algorithms with integer data, the transdichotomous model is a variation of the random-access machine in which the machine word size is assumed to match the problem size. The model was proposed by Michael Fredman and Dan Willard, who chose its name "because the dichotomy between the machine model and the problem size is crossed in a reasonable manner."
In mathematical logic, computational complexity theory, and computer science, the existential theory of the reals is the set of all true sentences of the form where the variables are interpreted as having real number values, and where is a quantifier-free formula involving equalities and inequalities of real polynomials. A sentence of this form is true if it is possible to find values for all of the variables that, when substituted into formula , make it become true.
In mathematical optimization, the criss-cross algorithm is any of a family of algorithms for linear programming. Variants of the criss-cross algorithm also solve more general problems with linear inequality constraints and nonlinear objective functions; there are criss-cross algorithms for linear-fractional programming problems, quadratic-programming problems, and linear complementarity problems.
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Deepak Kapur is a Distinguished Professor in the Department of Computer Science at the University of New Mexico.
