Papers by Michiel de Bondt
arXiv (Cornell University), Apr 3, 2024
A camel can carry B bananas on its back. It can have 2 bananas at a time in its stomach. For each... more A camel can carry B bananas on its back. It can have 2 bananas at a time in its stomach. For each mile the camel walks, the amount of bananas in its stomach decreases 1. As soon as the amount of bananas in the camel's stomach is at most 1, it can eat a new banana. When the camel's stomach is empty, the camel must eat a new banana (in order to be able to continue its itinerary). Let there be a stock of N bananas at the border of the desert. How far can the camel penetrate into the desert, starting at this point? (Of course it can form new stocks with transported bananas.) The case B = 1 is solved completely. The round trip variant is solved for B = 1 as well. For B = 2, the round trip variant is solved for N which are a power of 2 and N ≤ 8, and estimated up to 1/(N − 1) miles for general N .
arXiv (Cornell University), Feb 10, 2024
A camel can carry one banana at a time on its back. It is on a diet and therefore can only have o... more A camel can carry one banana at a time on its back. It is on a diet and therefore can only have one banana at a time in its stomach. As soon as it has eaten a banana it walks a mile and then it needs a new banana (in order to be able to continue its itinerary). Let there be a stock of N bananas at the border of the desert. How far can the camel penetrate into the desert, starting at this point? (Of course it can form new stocks with transported bananas.) 1 * The proof of Lemma B has been replaced by a shorter proof on February 10, 2024.
arXiv (Cornell University), Apr 2, 2013
J ' edrzejewicz showed that a polynomial map over a field of characteristic zero is invertible, i... more J ' edrzejewicz showed that a polynomial map over a field of characteristic zero is invertible, if and only if the corresponding endomorphism maps irreducible polynomials to irreducible polynomials. Furthermore, he showed that a polynomial map over a field of characteristic zero is a Keller map, if and only if the corresponding endomorphism maps irreducible polynomials to square-free polynomials. We show that the latter endomorphism maps other square-free polynomials to square-free polynomials as well. In connection with the above classification of invertible polynomial maps and the Jacobian Conjecture, we study irreducible properties of several types of Keller maps, to each of which the Jacobian Conjecture can be reduced. Herewith, we generalize the result of Bakalarski, that the components of cubic homogeneous Keller maps with a symmetric Jacobian matrix (over C and hence any field of characteristic zero) are irreducible. Furthermore, we show that the Jacobian Conjecture can even be reduced to any of these types with the extra condition that each affinely linear combination of the components of the polynomial map is irreducible. This is somewhat similar to reducing the planar Jacobian Conjecture to the so-called (planar) weak Jacobian Conjecture by Kaliman.
arXiv (Cornell University), Mar 14, 2018
Let K be any field with charK = 2, 3. We classify all cubic homogeneous polynomial maps H over K ... more Let K be any field with charK = 2, 3. We classify all cubic homogeneous polynomial maps H over K with rkJH ≤ 2. In particular, we show that, for such an H, if F = x + H is a Keller map then F is invertible, and furthermore F is tame if the dimension n = 4.
International Journal of Foundations of Computer Science, Nov 16, 2022
The largest known reset thresholds for DFAs are equal to (n − 1) 2 , where n is the number of sta... more 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 analysis of synchronizing word lengths for a certain family of binary PFAs. This family has the following properties: it contains the well-knownČerný 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 theČerný 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.
arXiv (Cornell University), Sep 30, 2016
We classify Keller maps x + H in dimension n over fields with 1 6 , for which H is homogeneous, a... more We classify Keller maps x + H in dimension n over fields with 1 6 , for which H is homogeneous, and (1) deg H = 3 and rk J H ≤ 2; (2) deg H = 3 and n ≤ 4; (3) deg H = 4 and n ≤ 3; (4) deg H = 4 = n and H1, H2, H3, H4 are linearly dependent over K. In our proof of these classifications, we formulate (and prove) several results which are more general than needed for these classifications. One of these results is the classification of all homogeneous polynomial maps H as in (1) over fields with 1 6 .
arXiv (Cornell University), Nov 28, 2018
We shorten the proof of a theorem of J.-E. Pin (theorem 1.1 below), which can be found in his the... more We shorten the proof of a theorem of J.-E. Pin (theorem 1.1 below), which can be found in his thesis. The part of the proof which is my own (not Pin's) is a complete replacement of the same part in an earlier version of this paper.
arXiv (Cornell University), Jul 7, 2018
This paper contains results which arose from the research which led to
arXiv (Cornell University), Aug 25, 2021
The largest known reset thresholds for DFAs are equal to (n − 1) 2 , where n is the number of sta... more 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 analysis of synchronizing word lengths for a certain family of binary PFAs. This family has the following properties: it contains the well-knownČerný 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 theČerný 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.
arXiv (Cornell University), Jan 31, 2018
It was conjectured byČerný in 1964, that a synchronizing DFA on n states always has a synchronizi... more It was conjectured byČerný in 1964, that a synchronizing DFA on n states always has a synchronizing word of length at most (n − 1) 2 , and he gave a sequence of DFAs for which this bound is reached. Until now a full analysis of all DFAs reaching this bound was only given for n ≤ 5, and with bounds on the number of symbols for n ≤ 12. Here we give the full analysis for n ≤ 7, without bounds on the number of symbols. For PFAs (partial automata) on ≤ 7 states we do a similar analysis as for DFAs and find the maximal shortest synchronizing word lengths, exceeding (n − 1) 2 for n ≥ 4. Where DFAs with long synchronization typically have very few symbols, for PFAs we observe that more symbols may increase the synchronizing word length. For PFAs on ≤ 10 states and two symbols we investigate all occurring synchronizing word lengths. We give series of PFAs on two and three symbols, reaching the maximal possible length for some small values of n. For n = 6, 7, 8, 9, the construction on two symbols is the unique one reaching the maximal length. For both series the growth is faster than (n − 1) 2 , although still quadratic. Based on string rewriting, for arbitrary size we construct a PFA on three symbols with exponential shortest synchronizing word length, giving significantly better bounds than earlier exponential constructions. We give a transformation of this PFA to a PFA on two symbols keeping exponential shortest synchronizing word length, yielding a better bound than applying a similar known transformation. Both PFAs are transitive. Finally, we show that exponential lengths are even possible with just one single undefined transition, again with transitive constructions.
arXiv (Cornell University), Sep 22, 2016
It was conjectured byČerný in 1964 that a synchronizing DFA on n states always has a shortest syn... more 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 this paper, we investigate the role of the alphabet size. For each possible alphabet size, we count DFAs on n ≤ 6 states which synchronize in (n − 1) 2 − e steps, for all e < 2 n/2. Furthermore, we give constructions of automata with any number of states, and 3, 4, or 5 symbols, which synchronize slowly, namely in n 2 − 3n + O(1) steps. In addition, our results proveČerný's conjecture for n ≤ 6. Our computation has led to 27 DFAs on 3, 4, 5 or 6 states, which synchronize in (n − 1) 2 steps, but do not belong toČerný's sequence. Of these 27 DFA's, 19 are new, and the remaining 8 which were already known are exactly the minimal ones: they will not synchronize any more after removing a symbol. So the 19 new DFAs are extensions of automata which were already known, including theČerný automaton on 3 states. But for n > 3, we prove that thě Cerný automaton on n states does not admit non-trivial extensions with the same smallest synchronizing word length (n − 1) 2 .
arXiv (Cornell University), Oct 18, 2020
We give a simple proof of that determining solvability of Shisen-Sho boards is NP-complete. Furth... more We give a simple proof of that determining solvability of Shisen-Sho boards is NP-complete. Furthermore, we show that under realistic assumptions, one can compute in logarithmic time if two tiles form a playable pair. We combine an implementation of the algoritm to test playability of pairs with my earlier algorithm to solve Mahjong Solitaire boards with peeking, to obtain an algorithm to solve Shisen-Sho boards. We sample several Shisen-Sho and Mahjong Solitaire layouts for solvability for Shisen-Sho and Mahjong Solitaire.
ArXiv, 2017
We show that Boolean matrix multiplication, computed as a sum of products of column vectors with ... more We show that Boolean matrix multiplication, computed as a sum of products of column vectors with row vectors, is essentially the same as Warshall's algorithm for computing the transitive closure matrix of a graph from its adjacency matrix. Warshall's algorithm can be generalized to Floyd's algorithm for computing the distance matrix of a graph with weighted edges. We will generalize Boolean matrices in the same way, keeping matrix multiplication essentially equivalent to the Floyd-Warshall algorithm. This way, we get matrices over a semiring, which are similar to the so-called "funny matrices". We discuss our implementation of operations on Boolean matrices and on their generalization, which make use of vector instructions.
We give a short proof of a theorem of J.-E. Pin (theorem 1.1 below), which can be found in his th... more We give a short proof of a theorem of J.-E. Pin (theorem 1.1 below), which can be found in his thesis.
In this paper, we show that the Jacobian conjecture holds for gradient maps in dimension n <= ... more In this paper, we show that the Jacobian conjecture holds for gradient maps in dimension n <= 3 over a field K of characteristic zero. We do this by extending the following result for n <= 2 by F. Dillen to n <= 3: if f is a polynomial of degree larger than two in n <= 3 variables such that the Hessian determinant of f is constant, then after a suitable linear transformation (replacing f by f(Tx) for some T in GL_n(K)), the Hessian matrix of f becomes zero below the anti-diagonal. The result does not hold for larger n. The proof of the case det Hf in K* is based on the following result, which in turn is based on the already known case det Hf = 0: if f is a polynomial in n <= 3 variables such that det Hf <> 0, then after a suitable linear transformation, there exists a positive weight function w on the variables such that the Hessian determinant of the w-leading part of f is nonzero. This result does not hold for larger n either (even if we replace `positive'...
Let $K$ be any field and $x = (x_1,x_2,\ldots,x_n)$. We classify all matrices $M \in {\rm Mat}_{m... more Let $K$ be any field and $x = (x_1,x_2,\ldots,x_n)$. We classify all matrices $M \in {\rm Mat}_{m,n}(K[x])$ whose entries are polynomials of degree at most 1, for which ${\rm rk} M \le 2$. As a special case, we describe all such matrices $M$, which are the Jacobian matrix $J H$ (the matrix of partial derivatives) of a polynomial map $H$ from $K^n$ to $K^m$. Among other things, we show that up to composition with linear maps over $K$, $M = J H$ has only two nonzero columns or only three nonzero rows in this case. In addition, we show that ${\rm trdeg}_K K(H) = {\rm rk} J H$ for quadratic polynomial maps $H$ over $K$ such that $\frac12 \in K$ and ${\rm rk} J H \le 2$. Furthermore, we prove that up to conjugation with linear maps over $K$, nilpotent Jacobian matrices $N$ of quadratic polynomial maps, for which ${\rm rk} N \le 2$, are triangular (with zeroes on the diagonal), regardless of the characteristic of $K$. This generalizes several results by others. In addition, we prove the s...
We show that the Generalized Vanishing Conjecture ∀_m > 1 [^m f^m = 0] ∀_m ≫ 0 [^m (g f^m) = 0... more We show that the Generalized Vanishing Conjecture ∀_m > 1 [^m f^m = 0] ∀_m ≫ 0 [^m (g f^m) = 0] for a fixed differential operator ∈ k[∂] follows from a special case of it, namely that the additional factor g is a power of the radical polynomial f. Next we show that in order to prove the Generalized Vanishing Conjecture (up to some bound on the degree of ), we may assume that is a linear combination of powers of distinct partial derivatives. At last, we show that the Generalized Vanishing Conjecture holds for products of linear forms in ∂, in particular homogeneous differential operators Λ∈ k[∂_1,∂_2].
We compute all synchronizing DFAs with 7 states and synchronization length >= 29. Furthermore,... more We compute all synchronizing DFAs with 7 states and synchronization length >= 29. Furthermore, we compute alphabet size ranges for maximal, minimal and semi-minimal synchronizing DFAs with up to 7 states.
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Papers by Michiel de Bondt