Exploring Mount Neverest
The problem `Exploring Mount Neverest' by Henry Ernest Dudeney is solved. Dudeney formulated the problem in the beginning of the 20th century and gave a non-optimal solution (without estimating the non-optimality).
math.HO↗
arXiv subjects
Michiel de Bondt
Publications and source records attributed to Michiel de Bondt.
The Camel-Banana Problem
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.)
math.HO↗
A camel with a less strict diet
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 $N \le 8$, and estimated up to $1/(N-1)$ miles for general $N$.
math.HO↗
Another generalization of Mason's ABC-theorem
We show a generalization of Mason's ABC-theorem, with the only conditions that the greatest common divisor has been divided out and no proper subsum of the (possibly multivariate) polynomial sum f_1 + f_2 + ... + f_n = 0 vanishes. As a result, we show that the generalized Fermat-Catalan equation for polynomials: g_1^{d_1} + g_1^{d_2} + ... + g_n^{d_n} = 0 has no non-constant solutions if the greatest common divisor of the terms equals one, no proper subsum vanishes and the hyperbolic sum 1/d_1 + 1/d_2 + ... + 1/d_n is at most 1/(n-2). Furthermore, we show that the generalized Fermat-equation for polynomials g_1^d + g_1^d + ... + g_n^d = 0 has no 'interesting' solutions if d >= n(n-2).
math.NT↗
Extremal Binary PFAs with Small Number of States
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 $n\geq 6$. 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-known Černý automata; for $n\leq 10$ it contains a binary PFA with maximal possible reset threshold; for all $n\geq 6$ 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 $n\geq 41$. For that purpose, we present an improvement of Martyugin's prime number construction of binary PFAs.
cs.FL↗
A short and elegant proof of a theorem of J.-E. Pin
We give a short 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.
cs.FL↗
Polynomial Hessians with small rank
In this paper, the results in [Singular Hessians, J. Algebra 282 (2004), no. 1, 195--204], for polynomial Hessians with determinant zero in small dimensions $r+1$, are generalized to similar results in arbitrary dimension, for polynomial Hessians with rank $r$. All of this is over a field $K$ of characteristic zero. The results in [Singular Hessians, J. Algebra 282 (2004), no. 1, 195--204] are also reproved in a different perspective. One of these results is the classification by Gordan and Noether of homogeneous polynomials in $5$ variables, for which the Hessians determinant is zero. This result is generalized to homogeneous polynomials in general, for which the Hessian rank is 4. Up to a linear transformation, such a polynomial is either contained in $K[x_1,x_2,x_3,x_4]$, or contained in $$ K[x_1,x_2,p_3(x_1,x_2)x_3+p_4(x_1,x_2)x_4+\cdots+p_n(x_1,x_2)x_n] $$ for certain $p_3,p_4,\ldots,p_n \in K[x_1,x_2]$ which are homogeneous of the same degree. Furthermore, a new result which is similar to those in [Singular Hessians, J. Algebra 282 (2004), no. 1, 195--204], is added, namely about polynomials $h \in K[x_1,x_2,x_3,x_4,x_5]$, for which the last four rows of the Hessian matrix of $t h$ are dependent. Here, $t$ is a variable, which is not one of those with respect to which the Hessian is taken. This result is generalized to arbitrary dimension as well: the Hessian rank of $t h$ is $4$ and the first row of the Hessian matrix of $t h$ is independent of the other rows.
math.AG↗
Solving Shisen-Sho boards
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.
cs.DS↗
On the theory of Gordan-Noether on homogeneous forms with zero Hessian (Improved version)
We give a detailed proof for Gordan-Noether's results in "Ueber die algebraischen Formen, deren Hesse'sche Determinante identisch verschwindet" published in 1876 in Mathematische Annahlen. C. Lossen has written a paper in a similar direction as the present paper, but did not provide a proof for every result. In our paper, every result is proved. Furthermore, our paper is independent of Lossen's paper and includes a considerable number of new observations. An earlier version of this paper has been printed in Proceedings of the School of Science of Tokai University, Vol.49, Mar. 2014. In this version, a serious error has been corrected and some new results have been added.
math.AC↗
Slowly synchronizing DFAs of 7 states and maximal slowly synchronizing DFAs
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.
cs.FL↗
The classification of some polynomial maps with nilpotent Jacobians
In the paper, we first classify all polynomial maps $H$ of the following form: $H=\big(H_1(x_1,x_2,\ldots,x_n),H_2(x_1,x_2),H_3(x_1,x_2),\ldots,H_n(x_1,x_2)\big)$ with $JH$ nilpotent. After that, we generalize the structure of $H$ to $H=\big(H_1(x_1,x_2,\ldots,x_n),H_2(x_1,x_2),H_3(x_1,x_2,H_1),\ldots,H_n(x_1,x_2,H_1)\big)$.
math.AG↗
Lower Bounds for Synchronizing Word Lengths in Partial Automata
It was conjectured by Černý 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 $n \leq 5$, and with bounds on the number of symbols for $n \leq 12$. Here we give the full analysis for $n \leq 7$, without bounds on the number of symbols. For PFAs (partial automata) on $\leq 7$ states we do a similar analysis as for DFAs and find the maximal shortest synchronizing word lengths, exceeding $(n-1)^2$ for $n \geq 4$. 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 $\leq 10$ 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.
cs.FL↗
Subset synchronization of DFAs and PFAs, and some other results
This paper contains results which arose from the research which led to arXiv:1801.10436, but which did not fit in arXiv:1801.10436. So arXiv:1801.10436 contains the highlight results, but there are more results which are interesting enough to be shared.
cs.FL↗
Quadratic homogeneous polynomial maps $H$ and Keller maps $x+H$ with $rk JH=3$
We classify all quadratic homogeneous polynomial maps $H$ and Keller maps of the form $x + H$, for which $rk J H = 3$, over a field $K$ of arbitrary characteristic. In particular, we show that such a Keller map (up to a square part if $char K=2$) is a tame automorphism.
math.AG↗
Classification of cubic homogeneous polynomial maps with Jacobian matrices of rank two
Let $K$ be any field with $\textup{char}K\neq 2,3$. We classify all cubic homogeneous polynomial maps $H$ over $K$ with $\textup{rk} JH\leq 2$. 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 $n\neq 4$.
math.AG↗
Slowly synchronizing automata with fixed alphabet size
It was conjectured by Černý 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 $n \le 6$ states which synchronize in $(n-1)^2 - e$ steps, for all $e < 2\lceil n/2 \rceil$. 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 prove Černý's conjecture for $n \le 6$. 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 to Černý's sequence. Of these $27$ DFA's, $19$ are new, and the remaining $8$ which were already known are exactly the \emph{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 the Černý automaton on $3$ states. But for $n > 3$, we prove that the Černý automaton on $n$ states does not admit non-trivial extensions with the same smallest synchronizing word length $(n-1)^2$.
cs.FL↗
Fast algorithms for anti-distance matrices as a generalization of Boolean matrices
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.
cs.DM↗
Rational maps $H$ for which $K(tH)$ has transcendence degree 2 over $K$
We classify all rational maps $H \in K(x)^n$ for which ${\rm trdeg}_K K(tH_1,tH_2,\ldots,tH_n) \le 2$, where $K$ is any field and $t$ is another indeterminate. Furthermore, we classify all such maps for which additionally $JH \cdot H = {\rm tr} JH \cdot H$ (where $JH$ is the Jacobian matrix of $H$), i.e. $$ \sum_{i=1}^n H_i \frac{\partial}{\partial x_i} H_k = \sum_{i=1}^n H_k \frac{\partial}{\partial x_i} H_i $$ for all $k \le n$. This generalizes a theorem of Paul Gordan and Max Nöther, in which both sides and the characteristic of $K$ are assumed to be zero. Besides this, we use some of our tools to obtain several results about $K$-subalgebras $R$ of $K(x)$ for which ${\rm trdeg}_K L = 1$, where $L$ is the fraction field of $R$. We start with some observations about to what extent, Lüroth's theorem can be generalized.
math.AC↗