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Polygons in billiard orbits
[article]

2011
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arXiv
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pre-print

We study the geometry of billiard orbits on rectangular billiards. A truncated billiard orbit induces a partition of the rectangle into polygons. We prove that thirteen is a sharp upper bound for the number of different areas of these polygons.

arXiv:1106.2030v1
fatcat:qhnmpb4sijdoraj5wugidva6qe
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Crushing candies on the line
[article]

2015
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arXiv
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pre-print

We investigate stability properties of a probabilistic cellular automaton based on the candy crush game.

arXiv:1501.03351v1
fatcat:67mx3gz6png7vhgvydpycs5xqa
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Synchronizing non-deterministic finite automata
[article]

2017
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arXiv
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pre-print

In this paper, we show that every D3-directing CNFA can be mapped uniquely to a DFA with the same synchronizing word length. This implies that Černý's conjecture generalizes to CNFAs and that the general upper bound for the length of a shortest D3-directing word is equal to the Pin-Frankl bound for DFAs. As a second consequence, for several classes of CNFAs sharper bounds are established. Finally, our results allow us to detect all critical CNFAs on at most 6 states. It turns out that only very few critical CNFAs exist.

arXiv:1703.07995v1
fatcat:5bd3roxlkfgdjgk3gib5gu5kxu
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Counting symbol switches in synchronizing automata
[article]

2018
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arXiv
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pre-print

Instead of looking at the lengths of synchronizing words as in Černý's conjecture, we look at the switch count of such words, that is, we only count the switches from one letter to another. Where the synchronizing words of the Černý automata C_n have switch count linear in n, we wonder whether synchronizing automata exist for which every synchronizing word has quadratic switch count. The answer is positive: we prove that switch count has the same complexity as synchronizing word length. We give

arXiv:1812.04050v1
fatcat:3n3vcp2xyvhizd2pbzud4tqyga
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... some series of synchronizing automata yielding quadratic switch count, the best one reaching 2/3 n^2 + O(n) as switch count. We investigate all binary automata on at most 9 states and determine the maximal possible switch count. For all 3≤ n≤ 9, a strictly higher switch count can be reached by allowing more symbols. This behaviour differs from length, where for every n, no automata are known with higher synchronization length than C_n, which has only two symbols. It is not clear if this pattern extends to larger n. For n≥ 12, our best construction only has two symbols.##
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Constructing and searching conditioned Galton-Watson trees
[article]

2014
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arXiv
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pre-print

We investigate conditioning Galton-Watson trees on general recursive-type events, such as the event that the tree survives until a specific level. It turns out that the conditioned tree is again a type of Galton-Watson tree, with different types of offspring and a level-dependent offspring distribution, which will all be given explicitly. As an interesting application of these results, we will calculate the expected cost of searching a tree until reaching a given level.

arXiv:1412.5890v1
fatcat:lywmy64eobfxta2ujywspnx5zu
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Agenda for the Housing Market

2009
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De Economist
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INTRODUCTION This communication gives an overview of the Reports to the 2008 Annual Meeting of the Royal Netherlands Economic Association (

doi:10.1007/s10645-009-9114-9
fatcat:pfiknxxq5vcd5nzsexrslzhbvq
*Don*(2008) ). ... CONCLUSION Three years ago, I called the housing market the missing item on the reform agenda (*Don*(2005) ). Previous research in this field had not gained much attention. ...##
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Polygons in billiard orbits

2012
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Journal of Number Theory
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We study the geometry of billiard orbits on rectangular billiards. A truncated billiard orbit induces a partition of the rectangle into polygons. We prove that thirteen is a sharp upper bound for the number of different areas of these polygons.

doi:10.1016/j.jnt.2011.12.012
fatcat:s47kgksknnfezgoovmwupaspzy
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The Cerny conjecture and 1-contracting automata
[article]

2015
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arXiv
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pre-print

A deterministic finite automaton is synchronizing if there exists a word that sends all states of the automaton to the same state. Černý conjectured in 1964 that a synchronizing automaton with n states has a synchronizing word of length at most (n-1)^2. We introduce the notion of aperiodically 1-contracting automata and prove that in these automata all subsets of the state set are reachable, so that in particular they are synchronizing. Furthermore, we give a sufficient condition under which

arXiv:1507.06070v2
fatcat:wqkuqo6lgzbbjpdz64yf4jlida
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... Černý conjecture holds for aperiodically 1-contracting automata. As a special case, we prove some results for circular automata.##
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Self-averaging sequences which fail to converge
[article]

2016
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arXiv
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pre-print

We consider self-averaging sequences in which each term is a weighted average over previous terms. For several sequences of this kind it is known that they do not converge to a limit. These sequences share the property that nth term is mainly based on terms around a fixed fraction of n. We give a probabilistic interpretation to such sequences and give weak conditions under which it is natural to expect non-convergence. Our methods are illustrated by application to the group Russian roulette problem.

arXiv:1609.07971v2
fatcat:35fu6cz7encxld265wkfzoard4
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Self-averaging sequences which fail to converge

2017
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Electronic Communications in Probability
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We consider self-averaging sequences in which each term is a weighted average over previous terms. For several sequences of this kind it is known that they do not converge to a limit. These sequences share the property that nth term is mainly based on terms around a fixed fraction of n. We give a probabilistic interpretation to such sequences and give weak conditions under which it is natural to expect non-convergence. Our methods are illustrated by application to the group Russian roulette problem.

doi:10.1214/17-ecp48
fatcat:ia6dujqzofhezj3ng7u52phhj4
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Finding DFAs with Maximal Shortest Synchronizing Word Length
[chapter]

2017
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Lecture Notes in Computer Science
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It was conjectured byČerný in 1964 that a synchronizing DFA on n states always has a shortest synchronizing word of length at most (n − 1) 2 , and he gave a sequence of DFAs for which this bound is reached. In 2006 Trahtman conjectured that apart fromČerný's sequence only 8 DFAs exist attaining the bound. He gave an investigation of all DFAs up to certain size for which the bound is reached, and which do not contain other synchronizing DFAs. Here we extend this analysis in two ways: we drop

doi:10.1007/978-3-319-53733-7_18
fatcat:xohbwpyqz5gbxov6dr6hzuhsoy
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... latter condition, and we drop limits on alphabet size. For n ≤ 4 we do the full analysis yielding 19 new DFAs with smallest synchronizing word length (n − 1) 2 , refuting Trahtman's conjecture. Several of these new DFAs admit more than one synchronizing word of length (n − 1) 2 , and even the synchronizing state is not unique. All these new DFAs are extensions of DFAs that were known before. For n ≥ 5 we prove that none of the DFAs in Trahtman's analysis can be extended similarly. In particular, as a main result we prove that theČerný examples Cn do not admit non-trivial extensions keeping the same smallest synchronizing word length (n − 1) 2 .##
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Conditioned multi-type Galton-Watson trees
[article]

2015
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arXiv
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pre-print

We consider multi-type Galton Watson trees, and find the distribution of these trees when conditioning on very general types of recursive events. It turns out that the conditioned tree is again a multi-type Galton Watson tree, possibly with more types and with offspring distributions, depending on the type of the father node and on the height of the father node. These distributions are given explicitly. We give some interesting examples for the kind of conditioning we can handle, showing that our methods have a wide range of applications.

arXiv:1506.02610v3
fatcat:ib466vvlqvglxiq5efvh654fqe
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Measuring the Economic Effects of Competition Law Enforcement

2008
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De Economist
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Together with a move from a rules-based legal approach to a more economic approach in competition cases, the economic effects of competition law enforcement have received increasing attention. Measuring these effects is important for external accountability of the Competition Authority, for quality control of its decisions and for evaluating the effectiveness of the competition law. This raises many issues in measurement, including the choice of counterfactual, the choice of effects to be

doi:10.1007/s10645-008-9107-0
fatcat:7beuagjkmvbdvcqnaxzpt6jvgy
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... ed, and the proper use of available data. The papers in this Special Issue of De Economist discuss these and related issues, based on a broad range of experience in competition law enforcement.##
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Extremal Binary PFAs with Small Number of States
[article]

2022
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arXiv
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pre-print

The largest known reset thresholds for DFAs are equal to (n-1)^2, where n is the number of states. This is conjectured to be the maximum possible. PFAs (with partial transition function) can have exponentially large reset thresholds. This is still true if we restrict to binary PFAs. However, asymptotics do not give conclusions for fixed n. We prove that the maximal reset threshold for binary PFAs is strictly greater than (n-1)^2 if and only if n≥ 6. These results are mostly based on the

arXiv:2108.13927v2
fatcat:62xes4irinc7blfwlxzpdvu2am
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... of synchronizing word lengths for a certain family of binary PFAs. This family has the following properties: it contains the well-known Černý automata; for n≤ 10 it contains a binary PFA with maximal possible reset threshold; for all n≥ 6 it contains a PFA with reset threshold larger than the maximum known for DFAs. Analysis of this family reveals remarkable patterns involving the Fibonacci numbers and related sequences such as the Padovan sequence. We derive explicit formulas for the reset thresholds in terms of these recurrent sequences. Asymptotically the Černý family gives reset thresholds of polynomial order. We prove that PFAs in the family are not extremal for n≥ 41. For that purpose, we present an improvement of Martyugin's prime number construction of binary PFAs.##
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New methods to bound the critical probability in fractal percolation
[article]

2013
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arXiv
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pre-print

Fractal percolation has been introduced by Mandelbrot in 1974. We study the two-dimensional case, with integer subdivision index M and survival probability p. It is well known that there exists a non-trivial critical value p_c(M) such that a.s. the largest connected component in the limiting set K is a point for p 0.881 and p_c(3)>0.784. For the upper bounds, we introduce the idea of classifications. The fractal percolation iteration process now induces an iterative random process on a finite

arXiv:1210.4150v2
fatcat:h3wpacngqbht7htywlfvmop2ya
## more »

... phabet, which is easier to analyze than the original process. This theoretical framework is the basis of computer aided proofs for the following upper bounds: p_c(2)<0.993, p_c(3)<0.940 and p_c(4)<0.972.
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