János Pach

Bio: János Pach is an academic researcher from Moscow Institute of Physics and Technology. The author has contributed to research in topics: Planar graph & Vertex (geometry). The author has an hindex of 60, co-authored 485 publications receiving 14319 citations. Previous affiliations of János Pach include Institute of Science and Technology Austria & University College London.

Research Problems in Discrete Geometry

Abstract: Note: Professor Pach's number: [045]; Also in: Facsimile edition: World Classics in Mathematics Series, vol. 28, China Science Press, Beijing, 2006. Mimeographed from 100 Research Problems in Discrete Geometry (with W. O. J. Moser). Reference DCG-BOOK-2008-001 URL: http://www.math.nyu.edu/~pach/research_problems.html Record created on 2008-11-18, modified on 2017-05-12

How to draw a planar graph on a grid

Abstract: Answering a question of Rosenstiehl and Tarjan, we show that every plane graph withn vertices has a Fary embedding (i.e., straight-line embedding) on the 2n−4 byn−2 grid and provide anO(n) space,O(n logn) time algorithm to effect this embedding. The grid size is asymptotically optimal and it had been previously unknown whether one can always find a polynomial sized grid to support such an embedding. On the other hand we show that any setF, which can support a Fary embedding of every planar graph of sizen, has cardinality at leastn+(1−o(1))√n which settles a problem of Mohar.

Combinatorial Geometry

Abstract: \indent This beautiful discipline emerged from number theory after the fruitful observation made by Minkowski (1896) that many important results in diophantine approximation (and in some other central fields of number theory) can be established by easy geometric arguments.

On the union of Jordan regions and collision-free translational motion amidst polygonal obstacles

Abstract: Let ?1,..., ?m bem simple Jordan curves in the plane, and letK1,...,Km be their respective interior regions. It is shown that if each pair of curves ?i, ?j,i ?j, intersect one another in at most two points, then the boundary ofK=?i=1mKi contains at most max(2,6m ? 12) intersection points of the curves?1, and this bound cannot be improved. As a corollary, we obtain a similar upper bound for the number of points of local nonconvexity in the union ofm Minkowski sums of planar convex sets. Following a basic approach suggested by Lozano Perez and Wesley, this can be applied to planning a collision-free translational motion of a convex polygonB amidst several (convex) polygonal obstaclesA1,...,Am. Assuming that the number of corners ofB is fixed, the algorithm presented here runs in timeO (n log2n), wheren is the total number of corners of theAi's.

Graphs drawn with few crossings per edge

Abstract: Keywords: graphs drawn on plane ; crossing number Note: Professor Pach's number: [119]. Also in: Lecture Notes in Computer Science 1190, Springer-Verlag, Berlin, 1997, 345-354. Reference DCG-ARTICLE-1997-008doi:10.1007/BF01215922 Record created on 2008-11-14, modified on 2017-05-12

I and i

Abstract: There is, I think, something ethereal about i —the square root of minus one. I remember first hearing about it at school. It seemed an odd beast at that time—an intruder hovering on the edge of reality. Usually familiarity dulls this sense of the bizarre, but in the case of i it was the reverse: over the years the sense of its surreal nature intensified. It seemed that it was impossible to write mathematics that described the real world in …

Random graphs

Abstract: We will review some of the major results in random graphs and some of the more challenging open problems. We will cover algorithmic and structural questions. We will touch on newer models, including those related to the WWW.

The Probabilistic Method

Abstract: The use of randomness is now an accepted tool in Theoretical Computer Science but not everyone is aware of the underpinnings of this methodology in Combinatorics - particularly, in what is now called the probabilistic Method as developed primarily by Paul Erdoős over the past half century. Here I will explore a particular set of problems - all dealing with “good” colorings of an underlying set of points relative to a given family of sets. A central point will be the evolution of these problems from the purely existential proofs of Erdős to the algorithmic aspects of much interest to this audience.

Planning Algorithms: Introductory Material

Abstract: Planning algorithms are impacting technical disciplines and industries around the world, including robotics, computer-aided design, manufacturing, computer graphics, aerospace applications, drug design, and protein folding. This coherent and comprehensive book unifies material from several sources, including robotics, control theory, artificial intelligence, and algorithms. The treatment is centered on robot motion planning but integrates material on planning in discrete spaces. A major part of the book is devoted to planning under uncertainty, including decision theory, Markov decision processes, and information spaces, which are the “configuration spaces” of all sensor-based planning problems. The last part of the book delves into planning under differential constraints that arise when automating the motions of virtually any mechanical system. Developed from courses taught by the author, the book is intended for students, engineers, and researchers in robotics, artificial intelligence, and control theory as well as computer graphics, algorithms, and computational biology.

Rhythms of the brain

Abstract: Prelude. Cycle 1. Introduction. Cycle 2. Structure defines function. Cycle 3. Diversity of cortical functions is provided by inhibition. Cycle 4. Windows on the brain. Cycle 5. A system of rhythms: from simple to complex dynamics. Cycle 6. Synchronization by oscillation. Cycle 7. The brain's default state: self-organized oscillations in rest and sleep. Cycle 8. Perturbation of the default patterns by experience. Cycle 9. The gamma buzz: gluing by oscillations in the waking brain. Cycle 10. Perceptions and actions are brain state-dependent. Cycle 11. Oscillations in the "other cortex:" navigation in real and memory space. Cycle 12. Coupling of systems by oscillations. Cycle 13. The tough problem. References.